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tesseract. The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results. Of the 24 center-to-vertex radii of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction, but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.
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plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell. However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons. The ring has two sets of three great hexagons, each on three
Clifford parallel great circles. The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.
25376:. Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles. Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a
27294:, unless they are separated by two angles of 90Β° (completely orthogonal planes) or 0Β° (coincident planes). Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180Β° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths) of 90Β° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each
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great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are
Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.
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26828:. Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.
18708:-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.
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24137:-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a
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32258:, Β§7. Conclusions; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."
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plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration). We can pick out 6 mutually isoclinic (Clifford parallel) great squares (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration). Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.
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isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The convex regular 4-polytopes nest inside each other, and hide next to each other in the
Clifford parallel spaces that comprise the 3-sphere. For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation.
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8948:-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are
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also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).
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may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.
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4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.
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square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.
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2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.
18662:. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.
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18237:. The left displacements of this class are not congruent with the right displacements, but enantiomorphous like a pair of shoes. Each left (or right) isoclinic rotation takes left planes to right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.
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8606:, so that the two strands of the double helix form a continuous single strand in a closed loop. In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic MΓΆbius
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26205:, which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.
27344:, where they both follow chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart. Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the
7555:. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell. They overlap with each other, but all of their element sets are disjoint: they do not share any vertex count, edge length, or face area, but they do share cell volume. They also share 4-content, their common core.
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hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram contains one
7517:. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell. They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume. They do share 4-content, their common core.
8277:, depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell) was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other. The distance from one of these orientations to another is an
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32406:. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a
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24945:), the way the vertices of a cube surround its center. The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are
26605:
four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do
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32672:, p. 138; "We allow the SchlΓ€fli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."
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volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers. Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).
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28968:
Let Q denote a rotation, R a reflection, T a translation, and let Q R T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q is a double rotation (in
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the vertex belongs to. Plane (b) contains the 120Β° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the
Clifford parallel (isoclinic) great hexagon planes
27043:
is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of
26256:
We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a
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points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by
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bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do,
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to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's
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than a simple 60Β° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60Β° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds
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by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30Β° isoclinic rotation takes the vertex 60Β° to its nearest-neighbor vertex by a
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equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away. For example, when the unit-radius 24-cell rotates isoclinically 60Β° in a hexagon invariant plane and 60Β° in its completely orthogonal invariant plane, each vertex
23215:
as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of
10724:
The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell). If the regular 24-cell has radius and edge length π
8626:
Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the
Clifford sense. Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells
8622:
if an isoclinic rotation will bring them together. The isoclinic planes are precisely those central planes with
Clifford parallel geodesic great circles. Clifford parallel great circles do not intersect, so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central
8228:
A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a
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away, without passing through any intervening vertices. Each left square rotates 45Β° (like a wheel) at the same time that it tilts sideways by 45Β° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720Β°
30200:
The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell. Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one
29940:
away), without passing through any intervening vertices. Each left hexagon rotates 180Β° (like a wheel) at the same time that it tilts sideways by 180Β° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through
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in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great
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two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation
26842:
In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The
25751:
the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is
Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal
24908:
In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy,
9032:
The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90Β° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90Β° around
8916:
of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation. When it has traversed one chord from each of the six great hexagons, after 720
8654:
Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An
8650:
All
Clifford parallel 4-polytopes are related by an isoclinic rotation, but not all isoclinic polytopes are Clifford parallels (completely disjoint). The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to
8369:
to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great
7705:
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core. The tesseracts and the 16-cells are rotated 60Β° isoclinically with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are
29875:
away), without passing through any intervening vertices. Each left hexagon rotates 30Β° (like a wheel) at the same time that it tilts sideways by 30Β° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through
29016:
The left planes are
Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation, but the two sets of planes are not all mutually Clifford parallel; they are different
28740:
edge of the great triangle inscribed in the great hexagon misses the vertex, because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three
28533:
The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2. Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by
26379:
Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract,
23348:
Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space;
18665:
The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two interlocking great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf
8755:
Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60Β° in the fourth dimension at each of the six octahedron centers, in a
8402:
in two perpendicular non-intersecting planes of rotation at once. In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their
8232:
The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are
7763:
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell
7696:
The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells). Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices
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are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices, and the 32 remaining
7464:
We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (β) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three
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away, without passing through any intervening vertices. Each left digon rotates 45Β° (like a wheel) at the same time that it tilts sideways by 45Β° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720Β°
31136:
away), without passing through any intervening vertices. Each left square rotates 30Β° (like a wheel) at the same time that it tilts sideways by 30Β° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through
31071:
away), without passing through any intervening vertices. Each left square rotates 180Β° (like a wheel) at the same time that it tilts sideways by 180Β° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through
30925:
90Β° orthogonally like coins flipping, displacing each vertex by 180Β°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares. This
30419:
away), without passing through any intervening vertices. Each left square rotates 180Β° (like a wheel) at the same time that it tilts sideways by 180Β° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through
30190:
away), without passing through any intervening vertices. Each left hexagon rotates 30Β° (like a wheel) at the same time that it tilts sideways by 30Β° (in an orthogonal central plane) into its corresponding right square plane. Repeated 12 times, this rotational displacement turns the 24-cell through
30008:
away), without passing through any intervening vertices. Each left hexagon rotates 90Β° (like a wheel) at the same time that it tilts sideways by 90Β° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through
29794:
30Β° orthogonally like coins flipping, displacing each vertex by 60Β°, as their vertices move along parallel helical isocline paths through successive
Clifford parallel hexagon planes. Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a
29717:
away), without passing through any intervening vertices. Each left hexagon rotates 60Β° (like a wheel) at the same time that it tilts sideways by 60Β° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through
28612:
An isoclinic rotation by a multiple of 60Β° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra. It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic
28410:
the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration means choosing a cell-disjoint set of four 6-cell rings that is the
28196:
there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between
26633:
Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius
26604:
This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that
24379:
be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60Β° away (a nearest vertex) by rotating isoclinically by 30Β° in two completely orthogonal
23704:
for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has
23329:
To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w=0, z=0) shares no axis with the wz central plane (where
23316:
to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite
11287:, each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its great hexagon planes. The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete
11258:
chord of the rotation. The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between. Although the isocline does not follow any one great circle, it is contained within a ring of another
8902:
chords bends 60 degrees in two central planes at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to. Thus the path follows one great hexagon from each
7742:
edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells. This is reminiscent of the
30354:
away), without passing through any intervening vertices. Each left hexagon rotates 60Β° (like a wheel) at the same time that it tilts sideways by 60Β° (in an orthogonal central plane) into its corresponding right square plane. Repeated 6 times, this rotational displacement turns the 24-cell through
29758:
Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they
28942:
The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the
28323:
angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a
25371:
are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great
23311:
Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the
18791:
Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells.
8485:
of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or
7263:
The sum of the squared lengths of all these distinct chords of the 24-cell is 576 = 24. These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices
32724:
forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind. Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell
27281:
to only one of them. Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal
26085:
an infinite hyperplane (from the inside). These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex
8999:
axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them. The vertex-bonded octahedra of the 4-cell ring also lie in
8437:
In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great
7767:
Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the
7716:
The tesseracts are inscribed in the 24-cell such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell such that only their vertices are
24352:
reach their orthogonally nearest neighbor vertices by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of
18505:
Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere. The cells labeled equatorial in the following table are interstitial to the meridian great circle
8840:
Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively. Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation
8832:
chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation, a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell
7803:
cubic cells: they are bisected by a square face into two square pyramids, the apexes of which also lie at a vertex of a cube. The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.
26266:
Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.
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in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space. Although he described a
8236:
Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not
30095:
Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices. Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies
28594:. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.
26783:, the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves.
8794:
fibrations. The 24-cell contains 16 great hexagons, divided among four fibrations, each of which is a set of four 6-cell rings, but the 24-cell has only four distinct 6-cell rings. Each 6-cell ring contains 3 of the great hexagons in each of three fibrations: only 3 of the 4 Clifford parallel
8341:
of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed
7750:
The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its
26572:
chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections). The third
26396:
The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2. Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius
24129:
edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are
8920:
At each vertex, there are four great hexagons and four hexagram isoclines (all black or all white) that cross at the vertex. Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right)
28635:
tilts 60Β° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60Β° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an
26414:
chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90Β° away outside the vanished cube, at the new nearest vertices; before
26913:
longer, its circumference is 4π
instead of 2π
, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent. Nevertheless, to avoid confusion we always refer to it as an
23689:
Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60Β° isoclinically with respect to each other (so their corresponding vertices are 120Β° =
28753:
Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of
28000:
doubles back across itself in each revolution, reversing its chirality but without ever changing its even/odd parity of rotation (black or white). The 6-vertex isoclinic path forms a MΓΆbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices
27697:
vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle
26970:. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination
26881:
An isoclinic rotation by 60Β° is two simple rotations by 60Β° at the same time. It moves all the vertices 120Β° at the same time, in various different directions. Six successive diagonal rotational increments, of 60Β°x60Β° each, move each vertex through 720Β° on a MΓΆbius double loop called an
28820:
away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel; they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by
27505:
Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way. The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates.
25844:
The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices. Each set constitutes a discrete
8969:
Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60Β° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on
8383:
25670:
chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight
8561:
edges. Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has
29640:
60Β° orthogonally like coins flipping, displacing each vertex by 120Β°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes. Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel
28160:
called the {1,1} torus knot or Villarceau circle in which each of two "circles" linked in a MΓΆbius "figure eight" loop traverses through all four dimensions. The double loop is a true circle in four dimensions. Even and odd isoclines are also linked, not in a MΓΆbius loop but as a
29081:
sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel
27204:β 0.866 displacements summing to a 120Β° degree displacement in the 24-cell's characteristic isoclinic rotation are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a
26365:). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).
28411:
unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container, but the fibration and the octahedral cell rings themselves are not chiral objects.
25460:
is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a
29577:, with all the left (or right) displacements taking place concurrently. Each left plane is separated from the corresponding right plane by two equal angles, each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the
28484:
There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are
8730:
great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane, and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these
8177:
lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The
23188:
The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration, the order in which they nest inside each other as compounds. Each greater polytope in the sequence is
23312:
tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time. Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is
8285:
of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point). This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.
32175:
was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and
30737:
is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group
26110:
Two angles are required to fix the relative positions of two planes in 4-space. Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in
8318:
29785:
with a circumference of 4π
. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right)
28741:
pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside. The three black and three white isoclines belong to the same 6-cell ring of the same fibration.
7816:
represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.
26772:
may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection). But in four dimensions there is yet another way in which rotations can occur, called a
25685:
chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90Β° away in a second concentric shell of six
23197:
or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point 4-polytope to 600-point
18506:
cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a
25849:
of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do
18639:
Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.
27534:
Isoclinic rotations partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the
24940:
to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's
24165:
around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical
18833:
31620:
11275:
is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two
8578:
direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic
8225:, whose vertices can be described by 4-integer Cartesian coordinates. The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.
8921:
isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines. There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white). Each isocline is a skew
18830:
19533:. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an
644:
or any other number of dimensions, either below or above. It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the
8457:
in many directions at once. Each vertex moves an equal distance in four orthogonal directions at the same time. In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates
7220:
edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.
6716:
The 24 vertices and 96 edges form 16 non-orthogonal great hexagons, four of which intersect at each vertex. By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are
3913:
28141:
of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4π
circumference. The 600-cell edge-rotates on isoclines of 5π
26513:
The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the
25149:). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell. The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its
27102:
right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There
18635:
of four interlocking rings. One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.
8774:
Each isoclinically displaced octahedron is also rotated itself. After a 360Β° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720Β° isoclinic rotation, its vertices are returned to their original
7822:
26573:
perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.
680:
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their SchlΣfli symbol, and the regular polygons with 7 or more sides. In other words, the 24-cell contains
26267:
Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.
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to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the
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through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do
28517:β 0.816. When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still
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lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices
888:; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.
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The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length
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The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and
8430:(like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the
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60 degrees apart. Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons). Each set of Clifford parallel great circles is a parallel
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edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each
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Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing
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plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90 distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell
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disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract. This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.
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boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are
29007:= 4 in particular, every displacement is either a double rotation Q, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.
18693:. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.
27348:, exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of
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edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12
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to only one of them. Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually
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are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a
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which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is
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The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.
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away, without passing through any intervening vertices. Each left square rotates 90Β° (like a wheel) at the same time that it tilts sideways by 90Β° (in an orthogonal central plane) into its corresponding right square plane,
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great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices. Each set constitutes a discrete
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is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is
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isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.
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away, without passing through any intervening vertices. Each left digon rotates 90Β° (like a wheel) at the same time that it tilts sideways by 90Β° (in an orthogonal central plane) into its corresponding right digon plane,
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path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a
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Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel
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may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when
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are Clifford parallels (completely disjoint). Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and
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x=0, y=0). The xy plane exists at only a single instant in time (w=0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).
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away, without passing through any intervening vertices. Each left square rotates 90Β° (like a wheel) at the same time that it tilts sideways by 90Β° (in an orthogonal central plane) into its corresponding right square,
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as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white
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It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the
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pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side
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from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell
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sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:
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Triangles and squares come together uniquely in the 24-cell to generate, as interior features, all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the
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The 24 vertices form 18 great squares (3 sets of 6 orthogonal central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of
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25424:, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90 bend in it as the path through the center). If their Pythagorean distance is
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31453:. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.
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Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.
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Excessive explanatory footnotes, some of which include other explanatory footnotes, which include other explanatory footnotes, and so on. Linearize by trimming for brevity, inserting into main text, or spawning
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pairs of completely orthogonal invariant planes. Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple
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is thus representative of the 4 disjoint great hexagons pictured, a quaternion group which comprise one distinct fibration of the great hexagons (four fibrations of great hexagons) that occur in the 24-cell.
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In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
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chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices, and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a
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every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a
24161:. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the
8790:, and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles). Each great hexagon belongs to just one fibration, but each 6-cell ring belongs to
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which, however, is not regular. The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction, equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described
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is thus representative of the 6 disjoint great squares pictured, a quaternion group which comprise one distinct fibration of the great squares (three fibrations of great squares) that occur in the 24-cell.
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Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in
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The 24-cell's cubical vertex figure has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6
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The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.
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with a circumference of 4π
, visible in the {24/9}=3{8/3} orthogonal projection. The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint
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chords run vertex-to-every-other-vertex in the same planes as the hexagonal great circles. They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2
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just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.
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octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.
26760:. Coxeter cites this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.
25431:, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60 bend, or as a straight line with one 60 bend in it through the center). Finally, if their Pythagorean distance is
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sequentially in 3 steps of a single 360Β° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.
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27197:-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal
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as they move: they are invariant planes. Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.
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in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.
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of 60Β° will not take one whole 16-cell to another 16-cell, because their vertices are 60Β° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell
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chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube, which is another vertex of the 24-cell: one located 120Β° away in a third concentric shell of eight
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8694:(chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.
27298:), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.
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fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a
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chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than
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The three great hexagons are Clifford parallel, which is different than ordinary parallelism. Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a
11184:, first from a 24-cell vertex to a 24-cell edge center, then turning 90Β° to a 24-cell face center, then turning 90Β° to a 24-cell octahedral cell center, then turning 90Β° to the 24-cell center.
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Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by
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angles of separation. Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V
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708:. The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.
32509:
32374:, pp. 2β3, Motivation; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the
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than its predecessor, enclosing more content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing
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A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.
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each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).
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Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders in order to move the short distance between Clifford parallel subspaces.
8663:
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are
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envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable 4-dimensional interstices between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are
31629:, p. 298, Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (Β§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column
27230:β 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is
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Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.
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in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the
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which are 120Β° away, in an isoclinic rotation. But in a rigid rotation of this kind, all hexagon planes move in congruent rotational displacements, so this rotation class also includes
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26779:. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of
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24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.
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It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the
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The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature. The regular 5-cell is not found in the interior of any convex regular 4-polytope except the
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In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in
31569:} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.
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do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.
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Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red
27340:. The isocline is a helical MΓΆbius double loop which reverses its chirality twice in the course of a full double circuit. The two loops are both entirely contained within the same
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Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right
25417:, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is
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in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes (
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In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.
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The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished,
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edges that zig-zag 90Β° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360Β° rotation. In contrast, the isoclinic hexagram
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figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The
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4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These
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chords converge from the corners of the 24-cell's cubical vertex figure and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight
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polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
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occurs: all the great circle planes Clifford parallel to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates
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match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: Β±i, Β±j, Β±k * 8 order-6: (+1Β±iΒ±jΒ±k)/2 * 8 order-3: (-1Β±iΒ±jΒ±k)/2.
29649:, so a fibration of 4 Clifford parallel great hexagon planes is represented. This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the
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edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a
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apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an
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31510:, p. 289, Epilogue; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."
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invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a
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The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The
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degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.
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tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.
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31708:, p. 7, Β§6 Angles between two Planes in 4-Space; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally,
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such 16-cell vertex during the 360Β° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations. The vertex follows an
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fibration of 4 Clifford parallel great hexagon planes is represented. This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the
29790:. The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30Β° like wheels
29636:. The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60Β° like wheels
26816:, all the Clifford parallel invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted
26692:; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).
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chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a
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long diameters of the octahedral cell. Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a
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vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex". That is what serves the illustrative purpose here.
23098:. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.
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are the diagonals of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract) cells are not cells of the unit radius coordinate lattice.
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great circle polygon (square, hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a
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Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by
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25295:) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.
18767:
In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
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of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A
32203:, pp. 1438β1439, Β§4.5 Regular Convex 4-Polytopes; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group πΉ
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realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by
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and cannot reach squares of the opposite color, even those immediately adjacent. Things moving diagonally move farther than 1 unit of distance in each movement step (
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in any of the great hexagon planes by a multiple of 60Β° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An
8108:). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D
7653:
7643:
7633:
7033:
edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting
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chord of each color, and visits 6 of the 9 different color-pairs of vertex. Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.
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chord of the helical geodesic crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both. The
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25723:
One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the
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to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a
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29569:
is the path followed by a vertex, which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant
24306:
24257:
7937:{\displaystyle {\begin{bmatrix}{\begin{matrix}24&8&12&6\\2&96&3&3\\3&3&96&2\\6&12&8&24\end{matrix}}\end{bmatrix}}}
7206:
chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers. The 72
30782:
30759:
30592:, intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete
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tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The
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apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.
28283:
The length of a strip can be measured at its centerline, or by cutting the resulting MΓΆbius strip perpendicularly to its boundary so that it forms a rectangle.
26152:
Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.
25089:
16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only
32533:
29628:
with a circumference of 4π
. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew
25438:, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).
6990:
of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.
31553:
33139:
30201:
other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).
25678:
edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six
7743:
way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler. In fact it is the exact dimensional analogy (the
26821:
26813:
25389:
A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does
24937:
23727:
The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only
13286:
8959:
rotation by 90Β° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and
8454:
26438:
chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).
24420:
The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60Β° isoclinically with respect to each other. The corresponding vertices of two 8-cells are
11941:
11506:
3392:
32298:; Β§3. The Dodecagonal Aspect; Coxeter considers the 150Β°/30Β° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the
31617:, pp. 4β5, Β§3.4 The 24-cell: points, lines and Reye's configuration; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.
28733:
Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the
18410:, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.
8767:
by 60Β° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a
32390:
memorial lecture 1986 to show that a single rotation (2π
) is not equivalent in all respects to no rotation at all, whereas a double rotation (4π
) is."
27247:
25977:
17876:
may be representative not only of its own fibration of Clifford parallel planes but also of the other congruent fibrations. For example, rotation class
12402:
8399:
8357:
8282:
2979:
29449:
29350:
27026:
rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a
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14017:
13604:
13160:
12848:
12725:
12276:
11818:
11426:
3473:
31558:
31556:
26380:
and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.
26182:
to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great
25816:
apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement
7717:
exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior 16-cell edges have length
6312:
5987:
28052:, which are the 4-dimensional analogues of great circles (great 1-spheres). Discrete isoclines are polygons; discrete great 2-spheres are polyhedra.
25604:
24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the
18244:. The left (or right) rotations carry the left planes to the right planes simultaneously, through a characteristic rotation angle. For example, the
8937:
isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the
28429:
disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).
7324:. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees
7037:
geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.
3669:
3619:
25887:
The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.
25731:
of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).
24909:
xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is
24890:
are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.
11263:
of spherical octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.
7461:(dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.
6666:
6616:
31728:, p. 153, 8.5. Gosset's construction for {3,3,5}: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."
25639:
of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do
25018:
rotate together, but in four different rotational directions, taking each 16-cell to another 16-cell. But since an isoclinic 60Β° rotation is a
26990:, by rotating on a single helical geodesic (so it is the shortest path). Conversely, any simple rotation can be seen as the composition of two
25207:
The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.
6755:
in the unit-radius 24-cell, inscribed in the 16 great hexagons. Each great triangle is a ring linking three completely disjoint great squares.
29446:
that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate
28388:, because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the
10721:. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.
8574:
in the 4-space in which it is embedded is now different. Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the
33460:
15365:
14949:
5804:
3331:
3188:
3130:
2803:
32285:
26721:+ 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the
23139:. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.)
20228:
8504:
and 120Β° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between. Each
6131:
734:
30674:
30599:
30514:
30448:
27172:
chiralities (left and right) but only one color (black or white), visiting one vertex of each of those same 6 octahedra in a 720Β° rotation.
15772:
15663:
15253:
14837:
14539:
14430:
8610:
is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.
5936:
28645:
Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.
28085:
All 3-sphere isoclines of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference
6261:
5862:
2928:
30287:
has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.
19065:
contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is
11251:(a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each
8566:
arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees
8297:
can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. Thus the general rotation in 4-space is a
689:
of the pentagonal polytopes. The geometric relationships among all of these regular polytopes can be observed in a single 24-cell or the
182:
31827:
29777:, each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12}
29749:, so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).
25654:. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.
8426:
rotations. In a double rotation each vertex moves in a spiral along two orthogonal great circles at once. Either the path is right-hand
6887:
away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.
132:
32967:
31513:
27040:
26769:
26676:
25358:
8925:
of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.
8583:
arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each
8294:
7958:
The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the
32696:, p. 18, Β§8. The simplex, cube, cross-polytope and 24-cell; Coxeter studied cell rings in the general case of their geometry and
31856:
subspaces of the tangent hyperplane to the sphere , so their only common point is the point of contact itself.... In fact, the radii
28872:
constant β 1.618, for which Coxeter uses π (tau), we reverse Coxeter's conventions, and use π to represent the characteristic angle.
24141:. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.
23731:
of the 4 coordinate system axes. The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of
621:
cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The
32675:
28657:
Each vertex of a 6-cell ring is missed by the two halves of the same MΓΆbius double loop hexagram, which curve past it on either side.
28442:
we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.
27123:
9119:
5510:
28952:
The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional)
28835:
edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.
25069:
it (120Β° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60Β° isoclinic rotation
24390:
of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).
30596:
that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate
28898:
28156:
Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity. A single black or white isocline forms a
8837:
hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided MΓΆbius loop.
8570:
been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's
6767:
Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.
5752:
5705:
2751:
2704:
34632:
33184:
Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem".
28187:
is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an
8716:. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the
6189:
32482:, pp. 14β16, Β§8.3 Properties of the Hopf Fibration; Corollary 9. Every great circle belongs to a unique right Hopf bundle.
28671:
At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is
27332:
The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's
27146:' paths along the diagonals of either the black or white squares of the chessboard. The Petrie dodecagon is a circular helix of
26475:
10661:
8862:
edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The
30784:
also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).
27967:
The composition of two simple 60Β° rotations in a pair of completely orthogonal invariant planes is a 60Β° isoclinic rotation in
24375:
half the destinations. However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination
30877:, each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.
28780:
The 12 black-white pairs of hexagram isoclines in each fibration and the 16 distinct hexagram isoclines in the 24-cell form a
26466:
faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own
25122:). The three pairs of 16-cells form three tesseracts. The tesseracts share vertices, but the 16-cells are completely disjoint.
9033:
the great square, and 90Β° orthogonally into a different 4-cell ring entirely. The 180Β° four-edge arc joining two ends of each
3763:
2861:
2481:
33411:
33083:
33048:
32977:
32847:
22825:
19542:
8411:
completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.
32402:, p. 44, Β§1. Villarceau Circles; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a
25539:
is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex
18308:= 120Β° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720Β° and back to itself in the same
31420:. Repeated 4 times, this rotational displacement turns the 24-cell through 720Β° and returns it to its original orientation.
31209:. Repeated 4 times, this rotational displacement turns the 24-cell through 720Β° and returns it to its original orientation.
30856:. Repeated 4 times, this rotational displacement turns the 24-cell through 720Β° and returns it to its original orientation.
28631:(a) and (c) is also 60Β°. In this 60Β° interval of the isoclinic rotation, great hexagon plane (a) rotates 60Β° within itself
28111:; simple rotations of unit-radius polytopes take place on 2π
isoclines. Double rotations may have isoclines of other than
27336:, as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic
25026:
orthogonal great circles at once, the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120Β° apart:
9662:
7536:
whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the
29933:
isoclinic rotation in great hexagon invariant planes takes each vertex through a 360Β° rotation and back to itself (360Β° =
10612:
10515:
10418:
10163:
9837:
9424:
7260:
chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.
5665:
5353:
2641:
2601:
2561:
34067:
31608:
31064:
isoclinic rotation in great square invariant planes takes each vertex through a 360Β° rotation and back to itself (360Β° =
30921:, in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90Β° like wheels
30412:
isoclinic rotation in great square invariant planes takes each vertex through a 360Β° rotation and back to itself (360Β° =
28376:
polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline polygrams, but the two
11204:
mirrors is a rotation. Consequently, regular polytopes can be generated by reflections or by rotations. For example, any
33433:
33424:
32627:
27443:
of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings found in the
27356:. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).
11052:(edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is
8602:
The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a
28577:
through all four dimensions instead of lying in a single plane. These helical great circles occur in Clifford parallel
26283:. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18
18358:
right set once each. The picture in the isocline column represents this union of the left and right plane sets. In the
3282:
32227:, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression , 2
28293:
27030:
rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.
18631:
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete
5472:
33453:
33238:
31596:
28205:
for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference
27142:
isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the
26672:
75:
27127:
25993:
8548:
8273:
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell
685:
of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but
32302:, a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.
30870:
Icositetragon {24/9}=3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.
29781:, a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a
28006:
26702:
26352:
in the projections and rotating animations illustrating this article; the others contain invisible interior chords.
22961:
19476:
has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
32615:
31840:; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of ".
25968:
are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere they always occur in pairs as
24210:
There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of distinct great hexagons), designated
18678:
Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)
6532:
3535:
533:
31843:
31648:
30210:
Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups
27286:, or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a
10000:
9543:
8631:
polytopes. A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all
7489:) are only some of those planes. Here we shall expose some of the others: the face planes of interior polytopes.
7320:
The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a
20:
33058:
Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.).
31441:
by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are
23383:
There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of distinct great squares), designated
3246:
33725:
33311:
33115:
32998:
32986:
32963:
32927:
32897:
32885:
32831:
32819:
32805:
30261:(for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares
22819:
22814:
8739:
in the great square plane by a multiple of 90Β° takes each octahedron in the ring to an octahedron in the ring.
7448:, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the
700:(in order of size and complexity). It can be deconstructed into 3 overlapping instances of its predecessor the
31784:
11089:
10795:
8941:. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of
8092:(which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a
8015:
The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the
5393:
2448:
2415:
2359:
31408:
of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180Β° =
29710:
isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120Β° =
28332:
28076:
are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.
26938:
Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations
26898:
around the 24-cell on an ordinary great circle. The helical double loop 4π
isocline is just another kind of
23514:, the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.
18716:
11155:
11122:
11056:
11023:
10990:
10957:
10898:
10865:
10832:
10762:
10729:
8481:
rotation animation appears to turn itself inside out. It appears to, because it actually does, reversing the
7444:(8-cell). The tesseract gives Gosset's construction of the 24-cell, equivalent to cutting a tesseract into 8
31852:, p. 147, Β§8.1 The simple truncations of the general regular polytope; "At a point of contact, lie in
31740:, p. 304, Table VI(iv) II={5,3,3}: Faceting {5,3,3}{3,3,5} of the 120-cell reveals 120 regular 5-cells.
28622:
Two central planes in which the path bends 60Β° at the vertex are (a) the great hexagon plane that the chord
25956:, and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in
23645:, each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a
11196:
in its own facets (its tetrahedral mirror walls). Reflections and rotations are related: a reflection in an
3908:{\displaystyle \left(\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}\right)}
34649:
33446:
32449:
31719:
31129:
isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60Β° =
30183:
isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60Β° =
30056:
lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the
29868:
isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60Β° =
26591:
a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and
23531:. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the
8903:
octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram
8365:
to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex
2521:
32135:; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.
30001:
isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180Β° =
29561:
Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right)
26902:
full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is
23482:
23132:
22809:
10927:(the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the
9370:
8097:
7813:
31218:
At the mid-point of the isocline arc (45Β° away) it passes directly over the mid-point of a 24-cell edge.
30939:
At the mid-point of the isocline arc (45Β° away) it passes directly over the mid-point of a 24-cell edge.
29808:
At the mid-point of the isocline arc (30Β° away) it passes directly over the mid-point of a 24-cell edge.
28489:
helices: they lie on ordinary flat great circles. Three of them are Clifford parallel and belong to one
25077:
isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the
24855:
triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the
20746:
10245:
10208:
10082:
10045:
9919:
9882:
9744:
9707:
9625:
9588:
9506:
9469:
9333:
8104:(i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the
7477:
the 24-cell by cutting through interior cells bounded by vertex chords to remove vertices, exposing the
33709:
33230:
33109:
32687:
32410:.... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."
32219:, p. 119, Β§7.1. Dimensional Analogy: "For instance, seeing that the circumference of a circle is 2
32194:
30347:
isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120Β° =
30126:. This is possible because some great hexagon planes lie Clifford parallel to some great square planes.
26595:
two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).
23045:
21971:
21245:
20325:
9189:
The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with
32339:
31796:
31541:
27584:
apart the vertex moves along a helical arc called an isocline (not a planar great circle), which does
26022:
chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel
25578:
25549:
25509:
25477:
25134:
The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares, each forming a
24448:
chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.
24108:
squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.
19122:
8844:
Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of
7336:
Each great circle intersects with the other great circles to which it is not Clifford parallel at one
3930:
The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as
27365:
Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are
26906:
true circle, as perfectly round and geodesic as the simple great circle, even through its chords are
26785:
In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves
24932:
inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other.
24434:
chords (its long diameters). The 8-cells are not completely disjoint (they share vertices), but each
23064:
22950:
22939:
18786:
Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
18469:". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "
8782:
Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a
8533:
chords and 360Β° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of
8135:
7347:
or otherwise Clifford parallel do not intersect at all: they pass through disjoint sets of vertices.
5633:
5601:
5569:
697:
115:
47:
31465:
27605:
lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.
27134:{12/5} which zig-zags 90Β° left and right like the edges dividing the black and white squares on the
26279:
The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the
23155:
The convex regular polytopes in the first four dimensions with a 5 in their SchlΣfli symbol are the
8493:
completely orthogonal planes one of which is a great hexagon, each vertex rotates first to a vertex
8202:
34090:
33693:
33684:
33661:
32809:
32497:
29632:
are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right)
26807:
26587:
Because there are three overlapping tesseracts inscribed in the 24-cell, each octahedral cell lies
23091:
21229:
21222:
19561:
18856:
18309:
10713:, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular
8776:
8571:
8471:
8450:
8105:
8074:
8050:
7026:(in planes inclined at 60 degrees to each other), 4 of which cross at each vertex. The 96 distinct
6987:
6729:
840:
57:
29548:
move together, remain Clifford parallel while moving, and carry all their points with them to the
25503:, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the
25164:
of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices
18627:
An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator
10718:
8786:. Four cell-disjoint 6-cell rings comprise the same discrete fibration. The 24-cell has four such
34060:
31992:
31989:, p. 150: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the
31922:
29109:
28510:
When unit-edge octahedra are placed face-to-face the distance between their centers of volume is
28299:
25605:
24966:
distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The
23128:
21962:
21236:
19019:
18904:
18075:
17973:
17847:
17797:
10565:
10468:
10371:
10291:
10116:
9953:
9790:
9040:
diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded
8702:
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4
7041:
2327:
2300:
2273:
2246:
2219:
2192:
704:(8-cell), as the 8-cell can be deconstructed into 2 overlapping instances of its predecessor the
32317:
32019:
31356:
31149:
30951:
30796:
30299:
29953:
29820:
28048:
that curve in 4-space in two orthogonal great circles at once. They should not be confused with
27448:
26495:
The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.
17664:
17480:
16567:
16383:
15159:
14743:
14143:
13923:
13066:
12182:
7570:
which meet at the center of the polytope, each contributing two radii and an edge. They form 96
5077:
5047:
2059:
2029:
33067:
32603:
32485:
31295:
31230:
31084:
31019:
30926:
illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the
30367:
30138:
29888:
29665:
29306:
28886:
26479:
23494:
23490:
21955:
21948:
20213:
20201:
19442:
has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
18746:
envelope. The layout of cells in this image is similar to the image under parallel projection.
18739:
18453:). One can stack octahedrons face to face in a straight line bent in the 4th direction into a
18361:
18247:
18180:
18129:
18027:
17879:
17574:
17436:
17298:
17211:
17120:
16927:
16840:
16749:
16477:
16339:
16064:
15973:
15572:
15115:
14699:
14339:
13879:
13727:
13513:
13022:
12634:
12138:
11727:
11313:
10673:
6445:{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}
3736:, having the same number of vertices (24) as cells and the same number of edges (96) as faces.
555:
32473:
29250:
29191:
26994:
double rotations (a left isoclinic rotation and a right isoclinic rotation), as discovered by
24848:
axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the
18813:
17617:
17530:
17424:
17254:
17167:
17079:
17067:
16883:
16796:
16708:
16696:
16520:
16433:
16327:
16107:
16020:
15932:
15920:
15728:
15619:
15531:
15519:
15318:
15209:
15103:
14902:
14793:
14687:
14495:
14386:
14298:
14286:
14096:
13973:
13867:
13683:
13560:
13472:
13460:
13239:
13116:
13010:
12804:
12681:
12593:
12581:
12355:
12232:
12126:
11897:
11774:
11686:
11674:
8374:, and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.
34604:
34597:
34590:
32425:
31853:
31572:
30023:
28845:
27337:
27278:
26338:
26179:
26120:
25765:
25535:. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the
25362:
25354:
24985:. The left and right 16-cells form a tesseract. Two 16-cells have vertex-pairs which are one
24977:
hexagons cross at each vertex (and its antipodal vertex), inclined at 60Β° to each other. The
24479:
These triangles lie in the same planes containing the hexagons; two triangles of edge length
23354:
23313:
22804:
22656:
22649:
21941:
21215:
21201:
20220:
20117:
19492:
18959:
18952:
9237:
9094:
8841:
traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.
8807:, can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every
8362:
8222:
7344:
6519:{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}
6113:{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}
3972:
3721:
3112:{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{4\phi ^{4}}}\right)\approx 180.73}
954:
34129:
34107:
34095:
33059:
30264:
29058:
29032:
28767:
The isoclines themselves are not left or right, only the bundles are. Each isocline is left
28234:
28208:
28114:
28088:
26504:
The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.
26333:. There are an infinite number of central planes through exactly two vertices (great circle
25650:
that is the rotational circle traversed by those 6 vertices in one particular left or right
23438:
23412:
23386:
22663:
21814:
21807:
20774:
17629:
16532:
15330:
14914:
14108:
13251:
12367:
34261:
34208:
33702:
33654:
33384:
33284:
32709:
32579:
31699:
30236:
30060:
30026:
29223:
29164:
28274:
geodesic loops (six vertices circling in each loop) and return to their original positions.
27491:
are black or white because they connect vertices which are all of the same color, and they
27060:
is a different special case, similar but not identical to two simple rotations through the
26825:
26531:) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.
26478:
surrounding its center. The characteristic orthoscheme has the shape described by the same
26361:
The 24-cell's cubical vertex figure has been truncated to a tetrahedral vertex figure (see
26243:
26236:
25865:
that is the rotational cirle traversed by those 8 vertices in one particular left or right
23650:
23646:
23082:
The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a
23059:
21934:
21821:
21208:
18687:
18647:
route, through the octahedrons' opposing vertices, that is four cells long. These are the
18422:
17542:
17266:
17179:
16895:
16808:
16445:
16119:
16032:
15740:
15631:
15221:
14805:
14507:
14398:
13985:
13695:
13572:
13128:
12816:
12693:
12244:
11909:
11786:
8977:
In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal
8173:
lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F
7449:
4073:
3964:
1055:
946:
650:
630:
595:
32834:(1995), Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic (eds.),
32651:
31437:
is performed by rotating left and right planes in "opposite" directions, according to the
29741:
chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel
28774:
22642:
22628:
21779:
20753:
18674:
18441:. For visualization purposes, it is convenient that the octahedron has opposing parallel
9008:
8825:
edges. The hexagram does not lie in a single central plane, but is composed of six linked
7213:
chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2
8:
34616:
34515:
34265:
33207:
33093:
Copher, Jessica (2019). "Sums and Products of Regular Polytopes' Squared Chord Lengths".
33031:
32305:
31529:
31197:
isoclinic rotation in great square invariant planes takes each vertex to a vertex 180Β° =
30844:
isoclinic rotation in great square invariant planes takes each vertex to a vertex 180Β° =
27277:
Each great square plane is isoclinic (Clifford parallel) to five other square planes but
24973:
edges form great hexagons of 6 vertices which run around the 24-cell in a central plane.
24285:
24236:
23502:
22635:
21927:
21800:
21786:
21194:
21187:
20767:
20195:
20190:
20095:
19538:
19514:
18743:
18724:
18704:
envelope. Two of the octahedral cells, the nearest and farther from the viewer along the
18690:
18623:
8254:
8201:
in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the
8138:
also form multiplicative groups of quaternions, but few of them generate a root lattice.
8131:
of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.
7533:
4059:
3951:
3924:
1041:
927:
33388:
33288:
33252:
32413:
32353:
32273:
31660:
30999:
isoclinic rotation in great square invariant planes takes each vertex to a vertex 90Β° =
30764:
30741:
30213:
30022:
lie in the great square central planes. Rotations of this type are an expression of the
29141:
27807:
27805:
27803:
27273:
27271:
27269:
25828:
can be represented equivalently as a linear chordal distance, or as an angular distance.
25799:(a great 1-sphere) is a great 2-sphere, which is an ordinary sphere that constitutes an
24262:
24213:
22621:
22614:
21793:
20781:
20739:
19548:
The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a
18338:
18315:
17950:
17927:
11650:
11625:
11400:
11377:
7747:), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.
5913:
2905:
34485:
34435:
34385:
34342:
34312:
34272:
34235:
34053:
33469:
33359:
33308:
Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics
33224:
33211:
33193:
33172:
33124:
33094:
32944:
32932:
Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics
32836:
32639:
32383:
31731:
31275:
isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90Β° =
30873:
This orthogonal projection of a 24-cell to a 24-gram {24/9}=3{8/3} exhibits 3 disjoint
28985:, the number of dimensions. Transformations involving a translation are expressible as
28781:
28498:
28494:
27797:
wide; it actually has only one edge, which is a single continuous circle with 6 chords.
27353:
27246:
That a double rotation can turn a 4-polytope inside out is even more noticeable in the
25743:
25741:
25739:
25737:
25407:
21772:
20207:
19553:
19488:
11272:
10935:
10338:
9305:
8732:
8190:
8124:
8093:
7667:
6725:
6593:
6570:
6238:
5449:
5426:
5325:
5302:
5279:
5256:
5233:
5210:
3596:
3573:
2681:
2392:
563:
504:
32741:, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the
32591:
30914:
30874:
28809:
27444:
27295:
26748:
dimensions, the most well-known examples being that the circumference of a circle is 2
25142:): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length
23498:
22878:
22871:
22864:
22505:
21663:
9111:
9104:
8971:
8938:
8679:
8427:
8387:
8356:
When a 4-polytope is rotating with only one invariant central plane, the same kind of
8322:
8179:
6983:
5166:
5159:
5083:
5059:
5053:
4083:
2148:
2141:
2065:
2041:
2035:
1065:
571:
122:
34624:
33372:
33363:
33234:
33215:
33079:
33060:
33044:
33011:
32973:
32918:
32901:
32843:
32726:
32569:
32403:
30887:
29774:
29731:
29603:
28370:
27800:
27266:
26132:
25969:
25631:
25368:
25343:
23023:
22996:
22928:
21920:
21913:
20185:
20175:
20142:
20135:
20106:
19565:
19507:
19500:
18909:
13368:{\displaystyle (-{\tfrac {1}{2}},-{\tfrac {1}{2}},-{\tfrac {1}{2}},-{\tfrac {1}{2}})}
8218:
8214:
8182:
for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of
8166:
7563:
The 24-cell can be constructed radially from 96 equilateral triangles of edge length
7074:
7034:
7002:
6718:
690:
661:
437:
42:
33176:
32684:, pp. 1438β1439, Β§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations.
28024:, as the 720Β° isoclinic rotation takes each hexagon through all six hexagons in the
25734:
25085:
isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the
12020:{\displaystyle ({\tfrac {1}{2}},-{\tfrac {1}{2}},-{\tfrac {1}{2}},-{\tfrac {1}{2}})}
11585:{\displaystyle ({\tfrac {1}{2}},-{\tfrac {1}{2}},-{\tfrac {1}{2}},-{\tfrac {1}{2}})}
11358:
rotation class consists of distinct rotational displacements by an arc-distance of
8599:
around the 24-cell twice, returning the 24-cell to its original chiral orientation.
8434:
by which we conventionally say which way is "up" on each of the 4 coordinate axes).
8157:
root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the
8069:
7946:
Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.
6740:
to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.
5180:
3455:{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}}}\right)\approx 51.246}
2162:
637:
are the only convex regular 4-polytopes in which the edge length equals the radius.
34628:
34193:
34182:
34171:
34160:
34151:
34142:
34081:
34077:
33392:
33349:
33315:
33292:
33203:
33162:
33154:
33071:
32913:
32795:
32237:
31868:πΉ, ... determine the polytopes ... whose vertices are the centers of elements ππ
30057:
28328: – the ratio of the strip's length to its width – is
26722:
26467:
26415:
truncation those were 24-cell vertices in the second shell of surrounding vertices.
25764:
Each square plane is isoclinic (Clifford parallel) to five other square planes but
23034:
23012:
22763:
22599:
21757:
20724:
18781:
18771:
18762:
11603:
11397:
and a corresponding set of 16 great hexagon planes represented by quaternion group
8407:
in the completely orthogonal rotation. A rotation in 4-space always has (at least)
8337:
because each point in the plane moves in a circle but stays within the plane. Only
8262:
8151:
8020:
7982:
7974:
7959:
7482:
7478:
6772:
3980:
962:
641:
548:
526:
476:
102:
32636:, pp. 17β20, Β§10 The Coxeter Classification of Four-Dimensional Point Groups.
31433:
is performed by rotating the left and right planes in the "same" direction, and a
29017:
fibrations, except in table rows where the left and right planes are the same set.
28402:
28400:
28398:
28044:
are 4-dimensional great circles in the sense that they are 1-dimensional geodesic
25921:
A point under isoclinic rotation traverses the diagonal straight line of a single
12475:{\displaystyle (-{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
7364:). Consequently, there are numerous ways to construct or deconstruct the 24-cell.
7199:(3 sets of 6 orthogonal planes), 3 of which cross at each vertex. The 72 distinct
6890:
To visualize how the interior polytopes of the 24-cell fit together (as described
3036:{\displaystyle 1200\left({\tfrac {2{\sqrt {3}}}{4\phi ^{2}}}\right)\approx 396.95}
884:
with 8 vertices permutations of (Β±2,0,0,0). The vertex figure of a 16-cell is the
34218:
34203:
33428:
33416:
33123:
Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions".
33075:
32791:
32461:
32083:.... Thus the general simplex may alternatively be defined as a finite region of
31584:
31438:
30204:
29519:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
29420:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
28165:
of two non-intersecting circles, as are all the Clifford parallel isoclines of a
27693:
away. More generally, isoclines are geodesics because the distance between their
27588:
pass through an intervening vertex: it misses the vertex nearest to its midpoint.
27282:
planes are in a distinguished category of Clifford parallel planes: they are not
25952:
either, because they form a closed loop like any circle. Isoclinic geodesics are
23735:
coordinate vertices (one of the four coordinate axes), and two opposite pairs of
23136:
19518:
18442:
18438:
16221:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
14087:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
13674:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
13230:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
12918:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
12795:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
12346:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
11888:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
11496:{\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})}
11306:
pictured. The vertices of the moving planes move in parallel along the polygonal
11212:
of the 24 vertices to and through 5 other vertices and back to itself, on a skew
10665:
9243:
Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with
9222:
8627:
inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are
8431:
8194:
8162:
7727:
7663:
7509:
4108:
3522:{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\sqrt {5}}\right)^{4}\approx 2.329}
1090:
665:
611:
522:
518:
514:
427:
233:
32573:
32210:
31605:, pp. 145β146, Β§8.1 The simple truncations of the general regular polytope.
30909:
29782:
29625:
27993:
27954:
is a 360Β° isoclinic rotation, and one half of the 24-cell's double-loop hexagram
27787:
27345:
27215:β 0.866 steps of each 120Β° displacement are concurrent, not successive, so they
26541:
25985:
25925:, reaching its destination directly, instead of the bent line of two successive
25812:
in the fourth dimension. Their corresponding points (on their two surfaces) are
24427:(120Β°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four
22843:
19408:
24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
18984:(Each edge corresponds to one triangular face, colored by symmetry arrangement)
18225:
which has distinct rotational displacements rather than because there are two
9143:
9127:
9077:
8895:
chords belong to different great hexagons. At each vertex the isoclinic path of
8603:
8446:
When the angles of rotation in the two invariant planes are exactly the same, a
34568:
33718:
33677:
33670:
32746:
32742:
32713:
32407:
32329:
32172:
31743:
30593:
29443:
29138:
corresponds to a distinct set of Clifford parallel great circle polygons, e.g.
28868:π, π, π of a regular polytope. Because π is commonly used to represent the
28578:
28574:
28395:
28385:
28363:
28166:
27536:
27473:
27143:
26894:(six rotational units) that it would take a simple rotation to take the vertex
26463:
25989:
25973:
25961:
25846:
25636:
25109:
Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional
24904:
24902:
24900:
24898:
24896:
24341:
23176:
23095:
22836:
19557:
18990:
18880:
18632:
11288:
9089:
8909:
8783:
8596:
8518:
chords meet at a 60Β° angle, but since they lie in different planes they form a
8101:
8058:
8038:
7330:
6362:{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}
6037:{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}
5194:
5187:
5153:
2176:
2169:
2135:
567:
459:
33354:
33337:
33319:
33158:
32249:
28157:
26681:
25948:
rather than simple 2-dimensional circles. But they are not like 3-dimensional
25760:
25758:
24207:
The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.
23207:
The edge length will always be different unless predecessor and successor are
19446:
17622:
17535:
17429:
17259:
17172:
16888:
16801:
16701:
16332:
14692:
13015:
9181:
9160:
8833:
before completing its six-vertex loop. Rather than a flat hexagon, it forms a
8193:
inscribed in the 24-cells of this tessellation give rise to the densest known
5173:
2155:
34643:
34585:
34473:
34466:
34459:
34423:
34416:
34409:
34373:
34366:
33421:
33014:
32437:
30866:
30119:
29536:
29534:
29532:
28930:
28389:
28377:
28270:
24-cell the 24 vertices rotate along four separate Clifford parallel hexagram
27219:
actually symmetrical radii in 4 dimensions. In fact they are four orthogonal
26995:
25949:
25783:
In 4-space, two great circles can be perpendicular and share a common center
25724:
25536:
25457:
25195:
24203:
24167:
24158:
24138:
23698:
23376:
22607:
20732:
18979:
18701:
18499:
18458:
16525:
16438:
16112:
16025:
15925:
15733:
15624:
15323:
15214:
15108:
14907:
14798:
14500:
14391:
14101:
13978:
13872:
13688:
13565:
13465:
13244:
13121:
12809:
12686:
12360:
12237:
12131:
11902:
11779:
11679:
9174:
9153:
8912:), but it is not: any isoclinic path we can pick out always zig-zags between
8334:
8061:
of order 1152. The rotational symmetry group of the 24-cell is of order 576.
8000:
on a central hyperplane. These vertices, combined with the 8 vertices of the
7997:
7744:
7671:
7505:
7458:
7445:
7321:
7049:
5138:
5121:
3959:
3706:{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 16.770}
3656:{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 15.451}
2120:
2103:
941:
914:
664:. As a polytope that can tile by translation, the 24-cell is an example of a
646:
622:
499:
471:
447:
97:
32624:, pp. 130β133, Β§7.6 The symmetry group of the general regular polytope.
28497:
of three ordinary circles, but they are not twisted: the 6-cell ring has no
27205:
25158:
each great triangle is a ring linking the three completely disjoint 16-cells
24959:
away from its 8 surrounding vertices in other 16-cells. Now visualize those
24893:
24353:
vertices, the way bishops are confined to the white or black squares of the
23380:
The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.
22934:
21765:
9218:
9167:
8988:
chord (the diameter of the great square and of the isocline) connects them.
8747:
7954:
7580:
The 24-cell can be constructed from 96 equilateral triangles of edge length
6703:{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}
6653:{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}
5065:
2047:
34525:
32697:
30584:
chord of 90Β° arc-length. Each such distinct chord is an edge of a distinct
30019:
29434:
chord of 60Β° arc-length. Each such distinct chord is an edge of a distinct
28869:
28653:
28651:
28325:
28002:
27997:
27679:, the isoclinic rotation has gone the long way around the 24-cell over two
27439:
27118:
27116:
27114:
26614:
lie at the corner of a cube: but a cube in another (overlapping) tesseract.
26135:
great circles. A great square and a great hexagon in different hyperplanes
25866:
25796:
25755:
25748:
25346:
23317:
each other, as two edges of the tetrahedron are perpendicular and opposite.
23164:
23001:
20180:
20149:
19530:
19526:
18943:
18454:
18174:
is the conventional representation for all congruent plane displacements.
11291:
of 4 non-intersecting great circles which visit all 24 vertices just once.
10714:
9200:
9084:
9074:
9059:
the 24-cell's characteristic rotation, and it does not take whole 16-cells
8834:
8346:
perpendicular to the invariant plane), but the circles do not lie within a
8250:
8158:
8147:
8120:
and is given by the subring of Hurwitz quaternions with even norm squared.
8113:
8089:
7752:
6998:
5113:
2095:
32393:
32046:
29529:
26474:). Every regular polytope can be dissected radially into instances of its
25182:
edge in each tesseract, so it is also a ring linking the three tesseracts.
24934:
The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes
23305:
18838:
Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
8795:
hexagons of each of the three fibrations, and only 18 of the 24 vertices.
7264:
that are helical rather than simply circular; they correspond to diagonal
6763:
5145:
4976:
2127:
1958:
34534:
34495:
34445:
34395:
34352:
34322:
34254:
34240:
32343:
28713:
triangles which meet at its center, this is a mid-edge of one of the six
26451:
24486:
are inscribed in each hexagon. For example, in unit radius coordinates:
23325:
23323:
23160:
22979:
22917:
22895:
20788:
18997:
18819:
9260:
chords apart on the circular isocline are antipodal vertices joined by a
8198:
8016:
7681:
7674:
6806:
Each vertex is joined to 8 others by an edge of length 1, spanning 60Β° =
4007:
3988:
989:
970:
726:
722:
191:
141:
30890:, each point represents two vertices, and each line represents multiple
29734:, each point represents two vertices, and each line represents multiple
29606:, each point represents two vertices, and each line represents multiple
28648:
28639:
27111:
26174:
26172:
26170:
26168:
25992:
around the 3-sphere through the non-adjacent vertices of a 4-polytope's
25854:
lie in a square central plane, but comprise a 16-cell and lie on a skew
25394:
but some pairs of great circles are non-intersecting Clifford parallels.
23182:
20795:
19412:
18510:(8-cell), although they touch at their vertices instead of their faces.
11420:. One of the distinct rotations of this class moves the representative
8851:
chords of an isocline from octahedron to octahedron. In the 24-cell the
8723:
axes of the four octahedra occupy the same plane, forming one of the 18
34520:
34504:
34454:
34404:
34361:
34331:
34245:
33297:
33272:
33167:
32704:
in its own right which fills a three-dimensional manifold (such as the
29624:
edges: two open skew triangles with their opposite ends connected in a
28953:
28706:
radii. Since the 24-cell can be constructed, with its long radii, from
28469:
28465:
28461:
27540:
27372:
27135:
27131:
27098:
the mirror image of the hand on your right side: a hand is either left
26115:
angles. Great squares in different hyperplanes are 90 degrees apart in
26057:
25960:
orthogonal great circles at once. They are true circles, and even form
25728:
25105:
25103:
25101:
25099:
24981:
are not perpendicular to each other, or to the 16-cells' perpendicular
24354:
23506:
23486:
23225:
The edges of six of the squares are aligned with the grid lines of the
23175:{5,3,3}. The 5-cell {3, 3, 3} is also pentagonal in the sense that its
23007:
22923:
22906:
19534:
18970:
18963:
18866:
18470:
18466:
18435:
15427:{\displaystyle (0,0,-{\tfrac {\sqrt {2}}{2}},-{\tfrac {\sqrt {2}}{2}})}
15011:{\displaystyle (-{\tfrac {\sqrt {2}}{2}},-{\tfrac {\sqrt {2}}{2}},0,0)}
11374:= 120Β° between 16 great hexagon planes represented by quaternion group
8869:
chords are great hexagon diagonals, joining great hexagon vertices two
8082:
8042:
7993:
7989:
5849:{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}
4021:
3379:{\displaystyle 600\left({\tfrac {4}{12\phi ^{3}}}\right)\approx 47.214}
3233:{\displaystyle 16\left({\tfrac {2{\sqrt {2}}}{3}}\right)\approx 15.085}
3175:{\displaystyle 5\left({\tfrac {5{\sqrt {10}}}{12}}\right)\approx 6.588}
2848:{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{4}}\right)\approx 21.651}
1003:
885:
618:
608:
402:
33438:
33396:
28626:
the vertex belongs to, and (b) the great hexagon plane that the chord
27901:
vertices in common; they do not intersect.) The third vertex reached V
27208:. In an actual left (or right) isoclinic rotation the four orthogonal
25338:
23320:
19556:. Relaxing the condition of convexity admits two further figures: the
18802:
18240:
Each rotation class (table row) describes a distinct left (and right)
8026:. The 24 vertices of the original 24-cell form a root system of type D
6176:{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}
34576:
34490:
34440:
34390:
34347:
34317:
34286:
33502:
33248:
33019:
30908:
edges: two open skew squares with their opposite ends connected in a
30730:{\displaystyle ({\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}},0,0)}
30655:{\displaystyle ({\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}},0,0)}
30570:{\displaystyle ({\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}},0,0)}
30504:{\displaystyle (0,0,{\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}})}
29778:
28591:
28162:
28069:
27406:
27405:
which are black at one end and white at the other. Things which have
26165:
25377:
25119:
25110:
24933:
22857:
21182:
18507:
18473:" cell. This skeleton accounts for 18 of the 24 cells (2 +
18446:
17072:
15828:{\displaystyle (0,0,{\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}})}
15719:{\displaystyle (0,0,{\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}})}
15524:
15309:{\displaystyle ({\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}},0,0)}
14893:{\displaystyle ({\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}},0,0)}
14595:{\displaystyle ({\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}},0,0)}
14486:{\displaystyle ({\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {2}}{2}},0,0)}
14291:
12586:
9211:
8482:
8329:
In 3 dimensions a spinning polyhedron has a single invariant central
7514:
7441:
6923:
5974:{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}
3733:
701:
657:
634:
464:
19:"Octaplex" redirects here. For the clotting factors concentrate, see
33434:
Petrie dodecagons in the 24-cell: mathematics and animation software
33273:"Orientational Sampling Schemes Based on Four Dimensional Polytopes"
28969:
four dimensions). Every orthogonal transformation is expressible as
27992:
geodesic is bent into a twisted ring in the fourth dimension like a
27539:' diagonal moves restrict them to the black or white squares of the
26106:
26104:
26102:
26100:
26098:
26096:
26094:
26092:
25175:
of 16-cells forms a tesseract (8-cell). Each great triangle has one
25096:
23541:
of unit edge length squares of the coordinate lattice. For example:
23018:
20045:
11294:
Each row of the table describes a class of distinct rotations. Each
8037::1. This is likewise true for the 24 vertices of its dual. The full
7012:
edges is always 1, 2, or 3, and it is 3 only for opposite vertices.
6748:
The 24 vertices form 32 equilateral great triangles, of edge length
6299:{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}
5900:{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}
4945:
4938:
2966:{\displaystyle 96\left({\sqrt {\tfrac {3}{4}}}\right)\approx 83.138}
1927:
1920:
34550:
34305:
34301:
34228:
33522:
33517:
33129:
33099:
32705:
32701:
32561:, pp. 292β293, Table I(ii); 24-cell Petrie polygon orthogonal
32299:
31687:
30898:
29614:
28828:= 60Β° chords. The triple helix of 3 isoclines contains 24 disjoint
28566:
28073:
26215:
25931:
25792:
25788:
25504:
25373:
25350:
25011:
24830: (
24488: (
24083: (
23741: (
23637:
23635:
23633:
23631:
23629:
23627:
23625:
23623:
23595: (
23543: (
23172:
23168:
23156:
22850:
22758:
21159:
21030:
20951:
20873:
20059:
20031:
18498:. One can easily follow this path in a rendering of the equatorial
18462:
18450:
18407:
9048:
8934:
8812:
8551:
8265:
of rotations about a fixed point in 4-dimensional Euclidean space.
7697:
is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).
7513:
edges divide the surface into 24 square faces and 8 cubic cells: a
7474:
7372:
The 8 integer vertices (Β±1, 0, 0, 0) are the vertices of a regular
7367:
7361:
6995:
4067:
4053:
1049:
1035:
899:
669:
420:
222:{\displaystyle \left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}}
33198:
32191:, p. 156: "...the chess-board has an n-dimensional analogue."
31657:, pp. 5β6, Β§3. Clifford's original definition of parallelism.
30429:
A quaternion Cartesian coordinate designates a vertex joined to a
29291:
A quaternion Cartesian coordinate designates a vertex joined to a
29287:
29285:
27223:
centered at the rotating vertex. Finally, in 2 dimensional units,
26583:
26581:
26579:
26127:
angles. Planes which are separated by two equal angles are called
22956:
21152:
21138:
21081:
21074:
21023:
21009:
20944:
20930:
20866:
20852:
18700:
parallel projection of the 24-cell into 3-dimensional space has a
18686:
parallel projection of the 24-cell into 3-dimensional space has a
9210:
edges each. The 24-cell can be decomposed into 2 disjoint zig-zag
8221:. The third regular tessellation of four dimensional space is the
171:{\displaystyle \left\{{\begin{array}{l}3\\3,4\end{array}}\right\}}
34559:
34529:
34291:
34282:
34223:
33507:
33111:
Symmetry groups of regular polytopes in three and four dimensions
32102:
31898:
28749:
28747:
28064:
28062:
28060:
28058:
26459:
26155:
26089:
26070:
25135:
24929:
23087:
23083:
21908:
21145:
21131:
21124:
21110:
21103:
21067:
21053:
21016:
21002:
20995:
20981:
20974:
20937:
20923:
20916:
20902:
20859:
20845:
20838:
20824:
20817:
20760:
20088:
20080:
20073:
19549:
8001:
7552:
7373:
6771:
The 24 vertices of the 24-cell are distributed at four different
4015:
997:
910:
909:. Remarkably, the edge length equals the circumradius, as in the
844:
705:
33271:
Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010).
32365:
31565:, pp. 292β293, Table I(ii): The sixteen regular polytopes {
31550:, p. 136, Β§7.8 The enumeration of possible regular figures.
29161:
corresponds to a set of four disjoint great hexagons. Note that
28464:
occupies a different (2-dimensional) face plane, each cell of a
26713:+ 1, and the much rarer and less obvious kind between dimension
26655:. The three 16-cells inscribed in each 24-cell have edge length
26224:
26139:
be isoclinic, but often they are separated by a 90 degree angle
25065:
edge away) by rotating toward it; it can only reach the 16-cell
23620:
22945:
22699:
22685:
21117:
21096:
21060:
21046:
20988:
20909:
20888:
20831:
20810:
20066:
20052:
19569:
19513:
and truncating at half the depth to the dual 24-cell yields the
19403:
18854:
There are two lower symmetry forms of the 24-cell, derived as a
10824:
around its exterior right-triangle face (the edges opposite the
7170:
7163:
7121:
7066:
6954:
like the square; and the long diameter of the 24-cell itself is
4931:
1913:
34499:
34449:
34399:
34356:
34326:
34277:
34213:
33497:
33371:
Koca, Mehmet; Al-Ajmi, Mudhahir; Koc, Ramazan (November 2007).
32725:
tiles 4-dimensional Euclidean space by translation to form the
32721:
32458:, pp. 20β33, Clifford Parallel Spaces and Clifford Reguli.
32261:
31774:
31772:
31770:
31581:, p. 302, Table VI (ii): ππ = {3,4,3}: see Result column
31474:, p. 118, Chapter VII: Ordinary Polytopes in Higher Space.
31442:
30423:
29282:
29220:
generally are distinct sets. The corresponding vertices of the
28864:) uses the greek letter π (phi) to represent one of the three
28686:
28667:
28665:
28663:
28036:
28034:
27283:
27090:
27018:
27016:
27014:
27012:
27010:
27008:
26576:
26455:
25051:
24336:
24334:
24332:
24330:
24328:
24326:
24324:
24322:
24320:
24318:
22985:
22794:
22734:
21892:
20967:
20038:
19148:
has a real representation as a 24-cell in 4-dimensional space.
19004:
18226:
8636:
8415:
8382:
7587:, where the three vertices of each triangle are located 90Β° =
7357:
7235:
in 16 planes, 4 of which cross at each vertex. The 96 distinct
7114:
5551:{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}
4910:
4029:
4001:
1892:
1011:
983:
414:
396:
28744:
28231:, and the former is an isocline of circumference greater than
28055:
27359:
23617:
coordinates comprise the vertices of the 6 orthogonal squares.
23040:
23029:
22974:
22912:
22890:
22727:
22720:
21885:
21878:
20895:
8394:
The points in the completely orthogonal central plane are not
7988:
form the vertices of a 24-cell. The vertices can be seen in 3
7949:
6926:
of dimensions 1 through 4: the long diameter of the square is
544:
32387:
31501:
30106:
30104:
30102:
28848:
takes whole 16-cells to other 16-cells in different 24-cells.
28548:
28546:
28544:
28542:
28540:
27387:
which connect them by isoclinic rotation. Everything else is
27002:
commutative, and is possible for any double rotation as well.
26847:
because the points in each stay in their places in the plane
26446:
26444:
26334:
26315:
rectangles. The planes through exactly 3 vertices are the 96
26183:
25981:
25945:
24441:
chord occurs as a cube long diameter in just one 8-cell. The
24345:
24308:. Each named set of 4 Clifford parallel hexagons comprises a
24198:
24196:
22901:
22713:
22706:
22692:
21871:
21864:
21857:
21850:
11599:
8955:= 90Β° apart, so the circumference of the isocline is 4π
. An
8818:, which in the unit-radius, unit-edge-length 24-cell has six
8519:
8371:
8258:
7985:
406:
33140:"On Cayley's Factorization of 4D Rotations and Applications"
32756:
32058:
31910:
31767:
28660:
28031:
27005:
26843:
two completely orthogonal planes of rotation are said to be
26820:
by that same angle in the completely orthogonal rotation. A
24340:
In an isoclinic rotation vertices move diagonally, like the
24315:
24194:
24192:
24190:
24188:
24186:
24184:
24182:
24180:
24178:
24176:
23461:. Each named set of 6 Clifford parallel squares comprises a
22678:
21843:
21836:
18418:
18406:
example it can be seen as a set of 4 Clifford parallel skew
5786:{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}
5739:{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}
4924:
4917:
2785:{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309}
2738:{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309}
1906:
1899:
725:
of its vertices which can be described as the 24 coordinate
34249:
32716:
and some (but not all) cell rings and their honeycombs are
32150:
32114:
32016:. (Their centres are the mid-points of the 24 edges of the
31968:
31423:
30886:
In this orthogonal projection of the 24-point 24-cell to a
29773:
In this orthogonal projection of the 24-point 24-cell to a
29730:
In this orthogonal projection of the 24-point 24-cell to a
29613:
chords. Each disjoint triangle can be seen as a skew {6/2}
29602:
In this orthogonal projection of the 24-point 24-cell to a
27815:
in the original great hexagon plane of isoclinic rotation P
27555:
Although adjacent vertices on the isoclinic geodesic are a
26890:
around the 24-cell and back to its point of origin, in the
26362:
25114:
24924:
24922:
24920:
22990:
11229:
11194:
constructed by the reflections of its characteristic 5-cell
8664:
7996:
cell on each of the outer hyperplanes and 12 vertices of a
7772:
6225:{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}
4035:
1017:
891:
In this frame of reference the 24-cell has edges of length
626:
452:
32326:, pp. 7β10, Β§6. Angles between two Planes in 4-Space.
32184:
32182:
31793:, p. 148, Β§8.2. Cesaro's construction for {3, 4, 3}..
31677:
31675:
30918:
30099:
29629:
28856:
28854:
28699:
chord passes through the mid-edge of one of the 24-cell's
28606:
28537:
27772:
Each half of a skew hexagram is an open triangle of three
27551:
27549:
26974:
without passing through the intermediate point touched by
26441:
26375:
26373:
26371:
26275:
26273:
26231:
25693:-distant vertices that surrounds the first shell of eight
25644:
25474:
The cube is not radially equilateral in Euclidean 3-space
25452:
25450:
25448:
25446:
25444:
25251:-distant vertices that surrounds the first shell of eight
25130:
25128:
23723:
23721:
23719:
23717:
23715:
23371:
23369:
23367:
23365:
23363:
23090:. The other two are also 4-polytopes, but not convex: the
18873:
or symmetry, and drawn tricolored with 8 octahedra each.
11248:
11213:
9064:
8584:
8317:
32938:, Providence, Rhode Island: American Mathematical Society
31477:
31418:
which in this rotation is the completely orthogonal plane
31207:
which in this rotation is the completely orthogonal plane
30897:
chords. Each disjoint square can be seen as a skew {8/3}
30854:
which in this rotation is the completely orthogonal plane
27056:
is a special case in which one rotational angle is 0. An
26801:
26799:
26797:
26795:
26793:
26752:
times 1, and the surface area of the ordinary sphere is 2
26544:, with their apexes filling the corners of the tesseract.
26260:
25855:
25201:
24173:
24153:
24151:
24149:
24147:
11225:
8884:
chord is a chord of just one great hexagon (an edge of a
8709:. The axis can then be bent into a square of edge length
7735:
The 16-cells are also inscribed in the tesseracts: their
7700:
33373:"Polyhedra obtained from Coxeter groups and quaternions"
32542:, pp. 292β293, Table I(ii); 24-cell Petrie polygon
32494:, p. 12, Β§8 The Construction of Hopf Fibrations; 3.
32138:
31888:
31886:
30089:
26695:
26554:
26552:
26550:
24917:
24118:
24116:
24114:
23739:
coordinate vertices (not coordinate axes). For example:
21168:
The 24-cell can also be derived as a rectified 16-cell:
19525:
The 96 edges of the 24-cell can be partitioned into the
10725:= 1, its characteristic 5-cell's ten edges have lengths
8928:
8798:
6724:
The 12 axes and 16 hexagons of the 24-cell constitute a
2892:{\displaystyle 32\left({\sqrt {3}}\right)\approx 55.425}
2508:{\displaystyle {\tfrac {\sqrt {2}}{\phi }}\approx 0.874}
32663:
32506:, pp. 34β57, Linear Systems of Clifford Parallels.
32179:
32162:
31980:
31672:
31636:
28987: Q R T
28851:
28079:
27546:
26684:). Folding around a square face is just folding around
26368:
26270:
25808:, displaced relative to each other by a fixed distance
25626:
25624:
25622:
25620:
25618:
25616:
25614:
25441:
25125:
23712:
23685:
23683:
23681:
23360:
23344:
23342:
23340:
23338:
23336:
18229:
ways to perform any class of rotations, designated its
8972:
8-chord helical isoclines characteristic of the 16-cell
8268:
32434:, p. 8, Left and Right Pairs of Isoclinic Planes.
31805:, p. 302, Table VI(ii) II={3,4,3}, Result column.
30699:
30682:
30624:
30607:
30539:
30522:
30485:
30468:
29502:
29487:
29472:
29457:
29403:
29388:
29373:
29358:
29025:
29023:
28804:
28802:
28727:
28445:
28152:
28150:
28148:
28001:
cross-connected to each other. This 60Β° isocline is a
26790:
25917:
25915:
25913:
25333:
25331:
25329:
25327:
25325:
25323:
25321:
25244:-distant vertices surrounding the second shell of six
24878:
24876:
24874:
24872:
24870:
24868:
24144:
23151:
23149:
23122:
23120:
23118:
23116:
23114:
23112:
23110:
23108:
23106:
23104:
18465:
description. Pick an arbitrary cell and label it the "
16204:
16189:
16174:
16159:
15809:
15792:
15700:
15683:
15408:
15388:
15278:
15261:
14980:
14960:
14862:
14845:
14564:
14547:
14455:
14438:
14070:
14055:
14040:
14025:
13657:
13642:
13627:
13612:
13351:
13333:
13315:
13297:
13213:
13198:
13183:
13168:
12901:
12886:
12871:
12856:
12778:
12763:
12748:
12733:
12458:
12443:
12428:
12413:
12329:
12314:
12299:
12284:
12003:
11985:
11967:
11949:
11871:
11856:
11841:
11826:
11568:
11550:
11532:
11514:
11479:
11464:
11449:
11434:
11236:
such an orbit visits 3 * 8 = 24 distinct vertices and
11161:
11128:
11095:
11062:
11029:
10996:
10963:
10904:
10871:
10838:
10801:
10768:
10735:
10618:
10521:
10424:
10250:
10213:
10169:
10087:
10050:
10006:
9924:
9887:
9843:
9749:
9712:
9668:
9630:
9593:
9549:
9511:
9474:
9430:
9375:
9338:
8547:
chords: a 720Β° rotation twice around the 24-cell on a
7835:
7831:
6671:
6621:
6537:
6487:
6468:
6387:
6324:
6273:
6201:
6143:
6062:
5999:
5949:
5875:
5816:
5758:
5711:
5671:
5638:
5606:
5574:
5515:
5477:
5359:
3889:
3871:
3853:
3835:
3674:
3624:
3540:
3478:
3404:
3343:
3294:
3200:
3142:
3061:
2991:
2941:
2815:
2757:
2710:
2646:
2606:
2566:
2486:
792:{\displaystyle (\pm 1,\pm 1,0,0)\in \mathbb {R} ^{4}.}
32857:
Two aspects of the regular 24-cell in four dimensions
32826:(2nd ed.), Cambridge: Cambridge University Press
32732:
32296:
Two aspects of the regular 24-cell in four dimensions
32022:
31995:
31925:
31919:, p. 269, Β§14.32. "For instance, in the case of
31883:
31838:
Two aspects of the regular 24-cell in four dimensions
31817:, pp. 149β150, Β§8.22. see illustrations Fig. 8.2
31524:
Two aspects of the regular 24-cell in four dimensions
31359:
31298:
31233:
31152:
31087:
31022:
30954:
30799:
30767:
30744:
30677:
30602:
30517:
30451:
30370:
30302:
30267:
30239:
30216:
30141:
30063:
30029:
30012:
29956:
29891:
29823:
29745:, where two opposing great triangles lie in the same
29668:
29645:, where two opposing great triangles lie in the same
29452:
29353:
29309:
29253:
29226:
29194:
29167:
29144:
29112:
29061:
29035:
28971: Q R
28936:
28335:
28302:
28237:
28211:
28117:
28091:
27499:
27328:
27326:
27324:
26547:
26081:
a finite simplex figure (from the outside), and they
25911:
25909:
25907:
25905:
25903:
25901:
25899:
25897:
25895:
25893:
25700:-distant vertices. The face-center through which the
25581:
25552:
25512:
25480:
25319:
25317:
25315:
25313:
25311:
25309:
25307:
25305:
25303:
25301:
25190:
25188:
25171:
apart: there are 32 distinct linking triangles. Each
24288:
24265:
24239:
24216:
24111:
23441:
23415:
23389:
19125:
18364:
18341:
18318:
18250:
18183:
18132:
18078:
18030:
17976:
17953:
17930:
17882:
17850:
17800:
17667:
17632:
17577:
17545:
17483:
17439:
17301:
17269:
17214:
17182:
17123:
17082:
16930:
16898:
16843:
16811:
16752:
16711:
16570:
16535:
16480:
16448:
16386:
16342:
16154:
16122:
16067:
16035:
15976:
15935:
15775:
15743:
15666:
15634:
15575:
15534:
15368:
15333:
15256:
15224:
15162:
15118:
14952:
14917:
14840:
14808:
14746:
14702:
14542:
14510:
14433:
14401:
14342:
14301:
14146:
14111:
14020:
13988:
13926:
13882:
13730:
13698:
13607:
13575:
13516:
13475:
13289:
13254:
13163:
13131:
13069:
13025:
12851:
12819:
12728:
12696:
12637:
12596:
12405:
12370:
12279:
12247:
12185:
12141:
11944:
11912:
11821:
11789:
11730:
11689:
11653:
11628:
11509:
11429:
11403:
11380:
11316:
11158:
11125:
11092:
11059:
11026:
10993:
10960:
10938:
10901:
10868:
10835:
10798:
10765:
10732:
10615:
10568:
10518:
10471:
10421:
10374:
10341:
10294:
10248:
10211:
10166:
10119:
10085:
10048:
10003:
9956:
9922:
9885:
9840:
9793:
9747:
9710:
9689:{\displaystyle {\sqrt {\tfrac {1}{12}}}\approx 0.289}
9665:
9628:
9591:
9546:
9509:
9472:
9427:
9373:
9336:
9308:
8237:
require (or permit) any of the pentagonal polytopes.
7825:
7618:), centered at the 24 mid-edge-radii of the 24-cell.
6669:
6619:
6596:
6573:
6535:
6466:
6378:
6315:
6264:
6241:
6192:
6134:
6053:
5990:
5939:
5916:
5865:
5807:
5755:
5708:
5668:
5636:
5604:
5572:
5513:
5475:
5452:
5429:
5396:
5356:
5328:
5305:
5282:
5259:
5236:
5213:
3825:
3766:
3672:
3622:
3599:
3576:
3538:
3476:
3395:
3334:
3285:
3249:
3191:
3133:
3052:
2982:
2931:
2908:
2864:
2806:
2754:
2707:
2684:
2644:
2604:
2564:
2524:
2484:
2451:
2418:
2395:
2362:
2330:
2303:
2276:
2249:
2222:
2195:
737:
185:
135:
32768:
32660:, pp. 217β218, Β§12.2 Congruent transformations.
31538:, p. 70, Β§4.12 The Classification of Zonohedra.
29102:
29100:
29098:
29096:
29094:
29092:
29090:
29088:
28846:
600-cell's isoclinic rotation in great square planes
27322:
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27316:
27314:
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between a pair of their corresponding points is the
25611:
25402:
25400:
25054:
diagonal nature of isoclinic rotations, the 16-cell
24475:
24473:
23678:
23333:
23211:
radially equilateral, i.e. their edge length is the
18849:
18461:
of 6 cells. The cell locations lend themselves to a
11280:, in a distinct pair of non-parallel mirror planes.
11208:
of the 24-cell in a hexagonal invariant plane takes
10719:
characteristic tetrahedron of the regular octahedron
10639:{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}
10542:{\displaystyle {\sqrt {\tfrac {2}{3}}}\approx 0.816}
10445:{\displaystyle {\sqrt {\tfrac {3}{4}}}\approx 0.866}
10190:{\displaystyle {\sqrt {\tfrac {1}{6}}}\approx 0.408}
9864:{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}
9451:{\displaystyle {\sqrt {\tfrac {1}{3}}}\approx 0.577}
9055:
This is the characteristic rotation of the 16-cell,
8966:= 180Β° distant across the diameter of the isocline.
5692:{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}
5380:{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}
2668:{\displaystyle {\tfrac {\sqrt {2}}{2}}\approx 0.707}
2628:{\displaystyle {\tfrac {\sqrt {2}}{2}}\approx 0.707}
2588:{\displaystyle {\tfrac {\sqrt {2}}{4}}\approx 0.354}
31669:, pp. 8β10, Relations to Clifford Parallelism.
30917:are the circular vertex paths characteristic of an
30194:
29726:
29724:
29555:
29020:
28964:
28962:
28799:
28324:hexagonal loop over six equilateral triangles. Its
28145:
27984:
27982:
27980:
27978:
27530:
27528:
27526:
27524:
27522:
27520:
27518:
27516:
27164:edges which all bend either left or right at every
26934:
26932:
26930:
26928:
26877:
26875:
26873:
26871:
26869:
26867:
26865:
25058:reach the adjacent 16-cell (whose vertices are one
24952:, each vertex of the compound of three 16-cells is
24865:
23146:
23101:
8274:
7607:away from each other on the 3-sphere. They form 48
7457:). The analogous construction in 3-space gives the
6947:. Moreover, the long diameter of the octahedron is
3917:all 24 of which lie at distance 1 from the origin.
33270:
33186:Journal of Physics A: Mathematical and Theoretical
33137:
32838:Kaleidoscopes: Selected Writings of H.S.M. Coxeter
32835:
32681:
32422:, pp. 8β9, Relations to Clifford parallelism.
32311:
32279:
32267:
32255:
32200:
32035:
32008:
31957:
31489:
31396:
31332:
31267:
31189:
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30776:
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30339:
30279:
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29180:
29153:
29130:
29073:
29047:
28351:
28315:
28249:
28223:
28129:
28103:
28028:, and each 8-cell through all three 8-cells twice.
27477:, and non-chiral cell rings such as the 24-cell's
25890:
25779:
25777:
25775:
25596:
25567:
25527:
25495:
25298:
25185:
25138:. The three 16-cells are completely disjoint (and
24300:
24274:
24251:
24225:
23517:
23453:
23427:
23401:
19140:
18723:parallel projection has a nonuniform hexagonal bi-
18398:
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16010:
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9863:
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9762:
9725:
9688:
9643:
9606:
9569:
9524:
9487:
9450:
9393:
9351:
9314:
8301:. There are two important special cases, called a
7936:
6826:of arc. Next nearest are 6 vertices located 90Β° =
6702:
6652:
6602:
6579:
6556:
6518:
6444:
6361:
6298:
6247:
6224:
6175:
6112:
6036:
5973:
5922:
5899:
5848:
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5412:
5379:
5334:
5311:
5288:
5265:
5242:
5219:
3907:
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2340:
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2286:
2259:
2232:
2205:
791:
221:
170:
32648:, pp. 33β38, Β§3.1 Congruent transformations.
32126:
29085:
27768:
27766:
27764:
27762:
27301:
27076:
27074:
27072:
27070:
26831:
26763:
26682:folding a flat net of 8 cubes up into a tesseract
26665:
26189:
25976:, the geodesic paths traversed by vertices in an
25964:like ordinary 2-dimensional great circles. These
25831:
25397:
25091:sense in which the two 16-cells are left or right
24470:
24456:
24454:
24416:
19156:has 24 vertices, and 24 3-edges. Its symmetry is
8888:inscribed in that great hexagon), but successive
8403:circularly moving points remain within the plane
8208:
7621:The 24-cell can be constructed directly from its
7343:diameter of the 24-cell. Great circles which are
6994:The vertex chords of the 24-cell are arranged in
6968:chords are the edges of central squares, and the
3318:{\displaystyle 24\left({\tfrac {4}{3}}\right)=32}
696:The 24-cell is the fourth in the sequence of six
660:four-dimensional space face-to-face, forming the
34641:
33032:"4D uniform polytopes (polychora) x3o4o3o - ico"
33009:
32842:(2nd ed.), Wiley-Interscience Publication,
31449:(like a pair of shoes), not opposite rotational
31137:720Β° and returns it to its original orientation.
31072:720Β° and returns it to its original orientation.
30420:720Β° and returns it to its original orientation.
30355:720Β° and returns it to its original orientation.
30191:720Β° and returns it to its original orientation.
30009:720Β° and returns it to its original orientation.
29941:720Β° and returns it to its original orientation.
29876:720Β° and returns it to its original orientation.
29721:
29718:720Β° and returns it to its original orientation.
28959:
27975:
27513:
26925:
26862:
26427:
26425:
26423:
26421:
25805:
25630:The 24-cell has four sets of 4 non-intersecting
25160:. There are four different ways (four different
25139:
25015:
24414:
24412:
24410:
24408:
24406:
24404:
24402:
24400:
24398:
24396:
19506:truncating at 1/2 of the edge length yields the
19499:truncating at 1/3 of the edge length yields the
18822:3D projection of an icositetrachoron (24-cell).
18292:rotation moves all hexagonal planes at once by
11237:
7368:Reciprocal constructions from 8-cell and 16-cell
5497:{\displaystyle {\tfrac {1}{\phi }}\approx 0.618}
3920:
3739:If the dual of the above 24-cell of edge length
640:The 24-cell does not have a regular analogue in
33326:
32600:, p. 139, Β§7.9 The characteristic simplex.
32359:
29565:in multiple invariant planes concurrently. The
28480:
28478:
28392:of the same woven fabric that is the fibration.
27457:an even/odd parity nor a chirality include all
27433:
27427:
27220:
26219:
25772:
24309:
23489:is simply a (w, x, y, z) Cartesian coordinate.
23462:
8203:highest possible kissing number in 4 dimensions
7622:
33370:
33183:
33003:The Theory of Uniform Polytopes and Honeycombs
32612:, p. 290, Table I(ii); "dihedral angles".
32156:
31614:
27759:
27434:sets of Clifford parallel great polygon planes
27253:
27067:
26392:
26390:
26388:
26386:
26322:equilateral triangle (16-cell) faces, and the
26048:
26046:
26044:
26042:
25944:lie in a single plane; they are 4-dimensional
25156:chords) have one vertex in each 16-cell. Thus
24451:
23523:The edges of the orthogonal great squares are
23477:
23475:
23473:
23471:
18742:of the 24-cell into 3-dimensional space has a
11220:that winds twice around the 3-sphere on every
11193:
8613:
8064:
7789:cubic cells, sharing the same volume. 24-cell
3724:great squares which intersect at no vertices.
617:The boundary of the 24-cell is composed of 24
34061:
33454:
33222:
33138:Perez-Gracia, Alba; Thomas, Federico (2017).
32503:
32455:
32443:
31654:
29769:
29767:
29765:
28917:-orthoscheme that surround its center, where
28720:triangles in a great hexagon, as seen in the
28569:lies on a different kind of circle called an
27664:chord apart on some other isocline, and just
26982:, or the other intermediate point touched by
26418:
26036:chords to form one of the 18 central squares.
25795:. The dimensionally analogous structure to a
25073:take every 16-cell to another 16-cell: a 60Β°
24393:
23512:ordered four-element multiple of real numbers
18177:These rotation classes are all subclasses of
11266:
9273:
7782:tetrahedral cells are inscribed in tesseract
7684:in its own cells (which are 3-orthoschemes).
7005:between two 24-cell vertices along a path of
6975:chords are the diagonals of central squares.
6894:), keep in mind that the four chord lengths (
33329:Elementary particles and the laws of physics
33107:
32246:, p. 6, Β§5. Four-Dimensional Rotations.
32108:
31974:
31904:
29598:
29596:
28475:
28179:
28177:
28175:
27830:chord lying in a different hexagonal plane P
27180:
27178:
25878:The sum of 1γ»96 + 2γ»72 + 3γ»96 + 4γ»12 is 576.
25869:as they rotate positions within the 16-cell.
25643:lie in a hexagonal central plane, each skew
23675:chord length) in four orthogonal directions.
19575:
10670:characteristic 5-cell of the regular 24-cell
8674:(polygonal rings running through vertices):
7558:
6940:; and the long diameter of the tesseract is
6557:{\displaystyle {\tfrac {2}{3}}\approx 0.667}
3560:{\displaystyle {\tfrac {8}{3}}\approx 2.666}
33327:Feynman, Richard; Weinberg, Steven (1987).
32797:Harmonices Mundi (The Harmony of the World)
32588:, pp. 292β293, Table I(ii); "24-cell".
32576:as each Petrie polygon is half the 24-cell.
32470:, pp. 292β293, Table I(i): Octahedron.
31593:, p. 156, Β§8.7. Cartesian Coordinates.
31283:and returns it to its original orientation.
31007:and returns it to its original orientation.
28810:in the 16-cell, the isocline is an octagram
27842:at a 60Β° angle. The second vertex reached V
27779:chords, the two open ends of which are one
26383:
26242:This animation shows the construction of a
26039:
25988:in the fourth dimension, taking a diagonal
25543:radii, so the cube is radially equilateral
23468:
19013:
18865:or symmetry drawn bicolored with 8 and 16
11226:four orthogonal pairs of antipodal vertices
10027:{\displaystyle {\sqrt {\tfrac {1}{4}}}=0.5}
9570:{\displaystyle {\sqrt {\tfrac {1}{4}}}=0.5}
8053:through the hyperplanes orthogonal to the F
7950:Symmetries, root systems, and tessellations
7796:octahedral cells overlap their volume with
7731:Kepler's drawing of tetrahedra in the cube.
3934:to distinguish it from others, such as the
16:Regular object in four dimensional geometry
34068:
34054:
33461:
33447:
32745:, showed how the honeycombs correspond to
32338:, p. 141, Β§7.x. Historical remarks; "
30122:and great hexagon (great triangle) planes
29762:
28468:occupies a different (3-dimensional) cell
27639:chord, unless they are antipodal vertices
26922:for an ordinary great circle in the plane.
26770:Rotations in 4-dimensional Euclidean space
26701:There are (at least) two kinds of correct
26341:to one of the 16 hexagonal great circles.
26178:In the 24-cell each great square plane is
25645:hexagram forming an isoclinic geodesic or
24844:are two opposing central triangles on the
24380:invariant planes (one of them a hexagon),
23505:, publishing his discovery of the regular
19167:Related figures in orthogonal projections
18869:cells. Lastly it can be constructed from D
18730:
9065:24-cell's rotation in great hexagon planes
9000:different tesseracts. The isocline's four
8639:forms: there is a disjoint 16-cell to the
8386:A 3D projection of a 24-cell performing a
8321:A 3D projection of a 24-cell performing a
8295:Rotations in 4-dimensional Euclidean space
3269:{\displaystyle 8{\sqrt {8}}\approx 22.627}
33353:
33296:
33197:
33166:
33128:
33098:
32972:, Cambridge: Cambridge University Press,
32942:
32917:
32518:, pp. 292β293, Table I(ii); 24-cell
32144:
32120:
31681:
31642:
30919:isoclinic rotation in great square planes
30880:
29593:
28792:, just the way the 24-cell's 12 axes and
28172:
27175:
26998:; perhaps surprisingly, this composition
25785:which is their only point of intersection
25584:
25555:
25515:
25483:
19128:
18445:(a trait it shares with the cells of the
8540:chords closes into a loop only after six
776:
76:Learn how and when to remove this message
33057:
32762:
32738:
32314:, pp. 2β3, Β§2. Isoclinic rotations.
31347:
31344:of the 24-cell, in which no points move.
31140:
30942:
30865:
30787:
30761:corresponds to a set of those. However,
30290:
29944:
29811:
29787:
29633:
29562:
29541:
29347:, the Cartesian "north pole". Thus e.g.
28905:polytope is subdivided by its symmetry (
28558:
28188:
27916:chord lying in another hexagonal plane P
27857:chord lying in another hexagonal plane P
27417:
26230:
26218:, though every convex 4-polytope can be
25651:
25337:
25004:
24202:
23642:
23375:
18977:
18941:
18673:
18622:
18417:
18241:
11205:
11110:{\displaystyle {\sqrt {\tfrac {1}{12}}}}
10816:{\displaystyle {\sqrt {\tfrac {1}{12}}}}
9293:
9290:
8804:
8764:
8746:
8736:
8687:
8381:
8316:
8278:
8068:
8004:, represent the 32 root vectors of the B
7953:
7726:
7333:which visits all 24 vertices just once.
7265:
6982:
6961:like the tesseract. In the 24-cell, the
6874:away, along an interior chord of length
6846:away, along an interior chord of length
6762:
5413:{\displaystyle {\sqrt {2}}\approx 1.414}
3804:{\displaystyle \left(\pm 1,0,0,0\right)}
2468:{\displaystyle {\sqrt {2}}\approx 1.414}
2435:{\displaystyle {\sqrt {2}}\approx 1.414}
2379:{\displaystyle {\sqrt {5}}\approx 2.236}
801:Those coordinates can be constructed as
543:
34633:List of regular polytopes and compounds
33468:
33122:
32997:
32985:
32962:
32926:
32896:
32884:
32830:
32818:
32804:
32774:
32751:24-cell's 4 rings of 6 octahedral cells
32693:
32669:
32657:
32645:
32633:
32621:
32609:
32597:
32585:
32558:
32539:
32515:
32491:
32479:
32467:
32431:
32419:
32335:
32323:
32291:
32282:, pp. 12β13, Β§5. A useful mapping.
32243:
32216:
32188:
32168:
32064:
32052:
31986:
31916:
31849:
31833:
31814:
31802:
31790:
31778:
31737:
31725:
31705:
31666:
31626:
31602:
31590:
31578:
31562:
31547:
31535:
31519:
31507:
31483:
31471:
31286:
31221:
31075:
31010:
30358:
30129:
29879:
29656:
29303:in standard (vertex-up) orientation is
28861:
28501:, either clockwise or counterclockwise.
28352:{\displaystyle {\sqrt {3}}\approx 1.73}
27686:chords to reach a vertex that was only
27221:mid-edge radii of a unit-radius 24-cell
26855:tilting sideways by the angle that the
25806:Clifford parallel 3-dimensional objects
25787:, because there is more than one great
23709:axis which is a coordinate system axis.
23194:
18669:
11259:kind: in the 24-cell it stays within a
11176:{\displaystyle {\sqrt {\tfrac {1}{2}}}}
11143:{\displaystyle {\sqrt {\tfrac {1}{6}}}}
11077:{\displaystyle {\sqrt {\tfrac {1}{4}}}}
11044:{\displaystyle {\sqrt {\tfrac {1}{2}}}}
11011:{\displaystyle {\sqrt {\tfrac {2}{3}}}}
10978:{\displaystyle {\sqrt {\tfrac {3}{4}}}}
10919:{\displaystyle {\sqrt {\tfrac {1}{6}}}}
10886:{\displaystyle {\sqrt {\tfrac {1}{4}}}}
10853:{\displaystyle {\sqrt {\tfrac {1}{2}}}}
10783:{\displaystyle {\sqrt {\tfrac {1}{4}}}}
10750:{\displaystyle {\sqrt {\tfrac {1}{3}}}}
8100:. The vertices of the 24-cell form the
3746:is taken by reciprocating it about its
34642:
33305:
33092:
32877:Regular and Semi-Regular Polytopes III
32790:
32720:, occurring in left- and right-handed
32371:
31892:
31693:
30434:
29296:
28721:
27988:Because the 24-cell's helical hexagram
27671:apart on some great hexagon. Between V
26343:Only the polygons composed of 24-cell
26029:chords joins another pair of parallel
19482:
18811:
18312:, passing through all 4 planes of the
8441:
7701:Relationships among interior polytopes
7687:
668:, the simplest one that is not also a
33637:
33480:
33442:
33342:Advances in Applied Clifford Algebras
33335:
33223:Tyrrell, J. A.; Semple, J.G. (1971).
33038:
33010:
32890:The Beauty of Geometry: Twelve Essays
32870:Regular and Semi-Regular Polytopes II
32749:, and made a particular study of the
32399:
31495:
30875:octagram {8/3} isoclines of a 16-cell
28201:for the former, and reserve the term
28193:
26220:deconstructed into irregular 5-cells.
25714:chord, so it lies inside the 24-cell.
25707:chord passes is the mid-point of the
25357:. They have a common center point in
25090:
25003:be taken to another 16-cell by a 60Β°
24993:
24992:edge (one hexagon edge) apart. But a
8929:Helical octagrams and their isoclines
8799:Helical hexagrams and their isoclines
8760:
8632:
8289:
7807:
7376:, and the 16 half-integer vertices (Β±
7269:
3754:8 vertices obtained by permuting the
656:Translated copies of the 24-cell can
33417:24-cell in stereographic projections
33247:
33029:
32863:Regular and Semi Regular Polytopes I
32223:, while the surface of a sphere is 4
32132:
31749:
27487:an even/odd parity and a chirality:
27130:{12} and also (orthogonally) a skew
26482:as the regular polytope without the
26337:); 16 are distinguished, as each is
26331:equilateral triangle (24-cell) faces
25824:which intersects both 2-spheres, so
25120:Gosset's construction of the 24-cell
24371:in the 24-cell), but at the cost of
24348:. Vertices in an isoclinic rotation
23351:they can intersect in a single point
21028:
21014:
21000:
20979:
19491:can be derived from the 24-cell via
18844:
8803:Another kind of geodesic fiber, the
8643:of each 16-cell, and another to its
8591:to complete a circuit through every
8269:The 3 Cartesian bases of the 24-cell
8169:. The 24-cells are centered at the D
7440:) are the vertices of its dual, the
6758:
2543:{\displaystyle 2-\phi \approx 0.382}
25:
32055:, p. 12, Β§1.8. Configurations.
31756:half-integer vertices (tesseract),
31752:, animation of a rotating 24-cell:
30437:. The conventional top vertex of a
30118:to each other: great square planes
30052:. The edges and 4π
characteristic
29299:. The conventional top vertex of a
28197:them we generally reserve the term
27465:(shared by black and white cells),
26744:enclosed by the sphere embedded in
26705:: the usual kind between dimension
25010:, because an isoclinic rotation is
23527:aligned with the grid lines of the
23483:four-dimensional Euclidean geometry
11224:vertex of the hexagram. Any set of
9394:{\displaystyle {\tfrac {2\pi }{3}}}
8989:
8939:helical construction of the 16-cell
8670:The 24-cell contains four kinds of
8377:
8312:
7614:-edge tetrahedra (the cells of the
7310:chords occur in 48 parallel pairs,
7296:chords occur in 36 parallel pairs,
6933:; the long diameter of the cube is
13:
32712:. He found that cell rings follow
32700:, identifying each cell ring as a
32171:, p. 163: Coxeter notes that
32071:+1 points which do not lie in an (
28266:In a 720Β° isoclinic rotation of a
27946:chords lie in different 8-cells. V
27869:lies in both intersecting planes P
27190:β 0.866 is the long radius of the
27168:vertex along a geodesic spiral of
26787:(as in a 2-dimensional rotation!).
26202:
26131:. Planes which are isoclinic have
25016:Clifford parallel hexagonal planes
23065:Uniform 4-polytope Β§ The F4 family
19529:to produce the 96 vertices of the
18807:Animated cross-section of 24-cell
18756:Cell-first perspective projection
10264:{\displaystyle {\tfrac {\pi }{2}}}
10227:{\displaystyle {\tfrac {\pi }{6}}}
10101:{\displaystyle {\tfrac {\pi }{2}}}
10064:{\displaystyle {\tfrac {\pi }{6}}}
9938:{\displaystyle {\tfrac {\pi }{2}}}
9901:{\displaystyle {\tfrac {\pi }{4}}}
9763:{\displaystyle {\tfrac {\pi }{3}}}
9726:{\displaystyle {\tfrac {\pi }{6}}}
9644:{\displaystyle {\tfrac {\pi }{3}}}
9607:{\displaystyle {\tfrac {\pi }{6}}}
9525:{\displaystyle {\tfrac {\pi }{4}}}
9488:{\displaystyle {\tfrac {\pi }{4}}}
9352:{\displaystyle {\tfrac {\pi }{3}}}
8635:, isoclinic rotations come in two
8462:hexagons by 60 degrees, and takes
8073:The 24 quaternion elements of the
7282:edges occur in 48 parallel pairs,
6891:
589:(short for "octahedral complex"),
14:
34661:
33405:
32750:
32446:, pp. 1β9, Β§1. Introduction.
32075:-1)-space are the vertices of an
30018:The edges and 4π
characteristic
28562:
28025:
26558:Consider the three perpendicular
25860:forming an isoclinic geodesic or
25111:hypercube (a tesseract or 8-cell)
24460:These triangles' edges of length
23649:. The point is displaced a total
23493:did not see them as such when he
18850:Three Coxeter group constructions
18335:left set and all 4 planes of the
11310:paths pictured. For example, the
11260:
10660:Every regular 4-polytope has its
8978:
8607:
8405:as the whole plane tilts sideways
7758:
34024:great grand stellated dodecaplex
33422:24-cell description and diagrams
33226:Generalized Clifford parallelism
33043:. New York: Dover Publications.
32943:Stillwell, John (January 2001).
32892:(2nd ed.). New York: Dover.
32814:(3rd ed.). New York: Dover.
32682:Mamone, Pileio & Levitt 2010
32201:Mamone, Pileio & Levitt 2010
31880:, ... of the original polytope."
31212:
30933:
30859:
29802:
29752:
29584:
29010:
28956:consists of 3-dimensional cells.
28946:
28913:instances of its characteristic
28897:-polytope's diagram without the
28875:
28838:
28761:
28616:
28597:
28584:
28527:
28504:
28454:
28432:
28414:
28286:
28277:
28260:
27961:
27701:
27608:
27591:
27240:
27138:. In contrast, the skew hexagram
26627:
26617:
26598:
26534:
26064:-spaces which can be defined by
25597:{\displaystyle \mathbb {R} ^{4}}
25568:{\displaystyle \mathbb {S} ^{3}}
25528:{\displaystyle \mathbb {S} ^{3}}
25496:{\displaystyle \mathbb {R} ^{3}}
24936:; the other two are rotated 60Β°
23503:four-dimensional Euclidean space
23039:
23028:
23017:
23006:
22995:
22984:
22973:
22955:
22944:
22933:
22922:
22911:
22900:
22889:
22877:
22870:
22863:
22856:
22849:
22842:
22835:
22733:
22726:
22719:
22712:
22705:
22698:
22691:
22684:
22677:
22662:
22655:
22648:
22641:
22634:
22627:
22620:
22613:
22606:
22495:
22490:
22485:
22480:
22475:
22470:
22465:
22456:
22451:
22446:
22441:
22436:
22431:
22426:
22417:
22412:
22407:
22402:
22397:
22388:
22383:
22378:
22373:
22368:
22363:
22358:
22349:
22344:
22339:
22334:
22329:
22320:
22315:
22310:
22305:
22300:
22295:
22290:
22281:
22276:
22271:
22266:
22261:
22256:
22251:
22242:
22237:
22232:
22227:
22222:
22213:
22208:
22203:
22198:
22193:
22188:
22183:
22174:
22169:
22164:
22159:
22154:
22145:
22140:
22135:
22130:
22125:
22120:
22115:
22106:
22101:
22096:
22091:
22086:
22077:
22072:
22067:
22062:
22057:
22052:
22047:
22038:
22033:
22028:
22023:
22018:
22009:
22004:
21999:
21994:
21989:
21984:
21979:
21891:
21884:
21877:
21870:
21863:
21856:
21849:
21842:
21835:
21820:
21813:
21806:
21799:
21792:
21785:
21778:
21771:
21764:
21653:
21648:
21643:
21638:
21633:
21628:
21623:
21614:
21609:
21604:
21599:
21594:
21589:
21584:
21575:
21570:
21565:
21560:
21555:
21550:
21545:
21536:
21531:
21526:
21521:
21516:
21507:
21502:
21497:
21492:
21487:
21482:
21477:
21468:
21463:
21458:
21453:
21448:
21443:
21438:
21429:
21424:
21419:
21414:
21409:
21404:
21399:
21390:
21385:
21380:
21375:
21370:
21365:
21360:
21351:
21346:
21341:
21336:
21331:
21322:
21317:
21312:
21307:
21302:
21297:
21292:
21283:
21278:
21273:
21268:
21263:
21258:
21253:
21158:
21151:
21144:
21137:
21130:
21123:
21116:
21109:
21102:
21095:
21080:
21073:
21066:
21059:
21052:
21045:
21029:
21022:
21015:
21008:
21001:
20994:
20987:
20980:
20973:
20966:
20950:
20943:
20936:
20929:
20922:
20915:
20908:
20901:
20894:
20887:
20872:
20865:
20858:
20851:
20844:
20837:
20830:
20823:
20816:
20809:
20794:
20787:
20780:
20773:
20766:
20759:
20752:
20745:
20738:
20731:
20714:
20709:
20704:
20699:
20694:
20689:
20684:
20675:
20670:
20665:
20660:
20655:
20650:
20645:
20636:
20631:
20626:
20621:
20616:
20611:
20606:
20597:
20592:
20587:
20582:
20577:
20572:
20567:
20558:
20553:
20548:
20543:
20538:
20533:
20528:
20519:
20514:
20509:
20504:
20499:
20494:
20489:
20480:
20475:
20470:
20465:
20460:
20455:
20450:
20441:
20436:
20431:
20426:
20421:
20416:
20411:
20402:
20397:
20392:
20387:
20382:
20377:
20372:
20363:
20358:
20353:
20348:
20343:
20338:
20333:
20079:
20072:
20065:
20058:
20051:
20044:
20037:
20030:
20020:
20015:
20010:
20005:
19997:
19992:
19987:
19982:
19977:
19968:
19963:
19958:
19953:
19945:
19940:
19935:
19930:
19925:
19916:
19911:
19906:
19901:
19893:
19888:
19883:
19878:
19873:
19864:
19859:
19854:
19849:
19841:
19836:
19831:
19826:
19821:
19812:
19807:
19802:
19797:
19792:
19784:
19779:
19774:
19769:
19764:
19755:
19750:
19745:
19740:
19735:
19727:
19722:
19717:
19712:
19707:
19698:
19693:
19688:
19683:
19678:
19670:
19665:
19660:
19655:
19650:
19641:
19636:
19631:
19626:
19621:
19613:
19608:
19603:
19598:
19593:
19470:
19465:
19460:
19445:
19436:
19431:
19426:
19411:
19402:
19360:
19355:
19350:
19333:
19328:
19323:
19306:
19301:
19296:
19291:
19286:
19281:
19276:
19261:
19256:
19251:
19234:
19229:
19224:
19207:
19202:
19197:
19192:
19187:
19182:
19177:
19141:{\displaystyle \mathbb {C} ^{2}}
19113:
19108:
19103:
19095:
19090:
19085:
19059:
19054:
19049:
19041:
19036:
19031:
19003:
18996:
18989:
18942:
18827:
18812:
18801:
18780:
18770:
18761:
18425:at Pennsylvania State University
18413:
17621:
17534:
17428:
17258:
17171:
17071:
16887:
16800:
16700:
16524:
16437:
16331:
16111:
16024:
15924:
15732:
15623:
15523:
15322:
15213:
15107:
14906:
14797:
14691:
14499:
14390:
14290:
14100:
13977:
13871:
13687:
13564:
13464:
13243:
13120:
13014:
12808:
12685:
12585:
12359:
12236:
12130:
11901:
11778:
11678:
11277:
10707:
10702:
10697:
10692:
10687:
10682:
10677:
9180:
9173:
9166:
9159:
9152:
8933:The 24-cell contains 18 helical
8690:. It also contains two kinds of
8141:
7656:
7651:
7646:
7641:
7636:
7631:
7626:
7350:
7169:
7162:
7120:
7113:
7065:
6922:) are the long diameters of the
6881:. The opposite vertex is 180Β° =
6853:. Another 8 vertices lie 120Β° =
4944:
4937:
4930:
4923:
4916:
4909:
4339:
4334:
4329:
4324:
4319:
4314:
4309:
4300:
4295:
4290:
4285:
4280:
4275:
4270:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
4222:
4217:
4212:
4207:
4202:
4197:
4192:
4183:
4178:
4173:
4168:
4163:
4158:
4153:
4144:
4139:
4134:
4129:
4124:
4119:
4114:
1926:
1919:
1912:
1905:
1898:
1891:
1321:
1316:
1311:
1306:
1301:
1296:
1291:
1282:
1277:
1272:
1267:
1262:
1257:
1252:
1243:
1238:
1233:
1228:
1223:
1218:
1213:
1204:
1199:
1194:
1189:
1184:
1179:
1174:
1165:
1160:
1155:
1150:
1145:
1140:
1135:
1126:
1121:
1116:
1111:
1106:
1101:
1096:
878:
873:
868:
863:
858:
853:
848:
833:
828:
823:
818:
813:
808:
803:
566:(four-dimensional analogue of a
405:
386:
381:
376:
371:
363:
358:
353:
348:
343:
335:
330:
325:
320:
315:
307:
302:
297:
292:
287:
282:
277:
269:
264:
259:
254:
249:
244:
239:
96:
30:
33377:Journal of Mathematical Physics
32362:, The reason for antiparticles.
32067:, p. 120, Β§7.2.: "... any
29573:to the corresponding invariant
27478:
27341:
26517:
26507:
26498:
26489:
26400:
26355:
26250:
26208:
26146:
26015:edges joins a pair of parallel
26002:
25881:
25872:
25717:
25657:
25468:
25383:
25261:
23654:is displaced to another vertex
23501:would be the first to consider
23219:
23201:
18717:elongated hexagonal dipyramidal
18587:Second layer of meridian cells
9281:Characteristics of the 24-cell
8742:
8697:
8691:
8223:tesseractic honeycomb {4,3,3,4}
7468:
7250:edges apart on a great circle.
5652:{\displaystyle {\tfrac {1}{2}}}
5620:{\displaystyle {\tfrac {1}{2}}}
5588:{\displaystyle {\tfrac {1}{4}}}
5071:
2053:
21:Prothrombin complex concentrate
33312:Delft University of Technology
33208:10.1088/1751-8113/43/10/105304
33116:Delft University of Technology
33066:. Springer New York. pp.
32969:Geometries and Transformations
32930:(1970), "Twisted Honeycombs",
32312:Perez-Gracia & Thomas 2017
32280:Perez-Gracia & Thomas 2017
32268:Perez-Gracia & Thomas 2017
32256:Perez-Gracia & Thomas 2017
31952:
31936:
31696:, p. 6, Β§3.2 Theorem 3.4.
31366:
31360:
31305:
31299:
31240:
31234:
31159:
31153:
31094:
31088:
31029:
31023:
30961:
30955:
30806:
30800:
30724:
30678:
30671:The representative coordinate
30649:
30603:
30585:
30564:
30518:
30498:
30452:
30377:
30371:
30309:
30303:
30148:
30142:
30115:
30053:
29963:
29957:
29898:
29892:
29830:
29824:
29799:fibration as 4 great hexagons.
29742:
29675:
29669:
29653:fibration as 4 great hexagons.
29642:
29513:
29453:
29435:
29414:
29354:
29334:
29310:
28557:, three black and three white
28373:
28296:in the plane by folding it at
28009:denoted {6/2}=2{3} or hexagram
27920:that is Clifford parallel to P
27861:that is Clifford parallel to P
27577:rotation between two vertices
27566:rotation between two vertices
27508:Left and right are not colors:
27467:
26086:above or below the hyperplane.
25821:
25606:radially equilateral tesseract
25043:hexagon chord) apart, not one
23076:
18659:
18644:
18556:First layer of meridian cells
18495:
18487:
18371:
18365:
18257:
18251:
18190:
18184:
18139:
18133:
18085:
18079:
18037:
18031:
17983:
17977:
17924:takes the 4 hexagon planes of
17889:
17883:
17807:
17801:
17695:
17668:
17602:
17578:
17490:
17484:
17326:
17302:
17239:
17215:
17130:
17124:
16955:
16931:
16868:
16844:
16759:
16753:
16598:
16571:
16505:
16481:
16393:
16387:
16215:
16155:
16092:
16068:
15983:
15977:
15822:
15776:
15713:
15667:
15582:
15576:
15421:
15369:
15303:
15257:
15169:
15163:
15005:
14953:
14887:
14841:
14753:
14747:
14589:
14543:
14480:
14434:
14349:
14343:
14174:
14147:
14081:
14021:
13933:
13927:
13755:
13731:
13668:
13608:
13523:
13517:
13362:
13290:
13224:
13164:
13076:
13070:
12912:
12852:
12789:
12729:
12644:
12638:
12469:
12406:
12340:
12280:
12192:
12186:
12014:
11945:
11882:
11822:
11737:
11731:
11579:
11510:
11490:
11430:
11323:
11317:
11302:pictured to the corresponding
11284:
11283:Pictured are sets of disjoint
11228:(the 8 vertices of one of the
11187:
8885:
8735:all have a common center). An
8671:
8595:vertex of its six vertices by
8523:
8347:
8209:Radially equilateral honeycomb
7486:
7232:
768:
738:
711:
1:
33338:"Conformal Villarceau Rotors"
33331:. Cambridge University Press.
33026:(also under Icositetrachoron)
32783:
30438:
30124:characteristic of the 24-cell
30123:
30120:characteristic of the 16-cell
29746:
29646:
29439:
29300:
28793:
28554:
28490:
27752:edge apart (at every pair of
25359:4-dimensional Euclidean space
24978:
23529:unit radius coordinate system
19564:. With itself, it can form a
19398:
19075:The regular complex polytope
18491:
18434:The 24-cell is bounded by 24
18429:
11421:
8859:
8788:discrete hexagonal fibrations
8787:
8683:
8414:Double rotations come in two
8246:
8213:The dual tessellation of the
8128:
7023:
5202:
5131:
2113:
33306:Mebius, Johan (July 2015) .
33253:"Symmetries and the 24-cell"
33147:Adv. Appl. Clifford Algebras
33108:van Ittersum, Clara (2020).
33076:10.1007/978-0-387-92714-5_20
33041:The Geometry of Art and Life
32919:10.1016/0898-1221(89)90148-X
32294:, pp. 30β32, (Paper 3)
32157:Koca, Al-Ajmi & Koc 2007
31459:
30589:
28565:. Each of these chiral skew
28559:isoclinic hexagram geodesics
28534:joining the 24 cell centers.
28292:A strip of paper can form a
27823:is 120 degrees away along a
27819:, the first vertex reached V
25410:between any two vertices is
24982:
19383:
19369:
19165:
18987:
18797:
18752:
18648:
18570:Non-meridian / interstitial
10662:characteristic 4-orthoscheme
10654:
10652:
10650:
10648:
10608:
10557:
10555:
10553:
10551:
10511:
10460:
10458:
10456:
10454:
10414:
10363:
10361:
10359:
10357:
10334:
10241:
10236:
10204:
10199:
10159:
10078:
10073:
10041:
10036:
9996:
9915:
9910:
9878:
9873:
9833:
9740:
9735:
9703:
9698:
9658:
9621:
9616:
9584:
9579:
9539:
9502:
9497:
9465:
9460:
9420:
9366:
9361:
9329:
9324:
9301:
9287:
9179:
9172:
9165:
9158:
9151:
9135:
9099:
9069:
8675:
8240:
8098:Hurwitz integral quaternions
7992:, with the 6 vertices of an
7615:
7454:
7196:
6978:
6743:
3942:
2184:
7:
32945:"The Story of the 120-Cell"
32902:"Trisecting an Orthoscheme"
32875:(Paper 24) H.S.M. Coxeter,
32868:(Paper 23) H.S.M. Coxeter,
32861:(Paper 22) H.S.M. Coxeter,
32382:and performed by physicist
32360:Feynman & Weinberg 1987
32009:{\displaystyle \gamma _{4}}
31958:{\displaystyle \gamma _{4}}
31764:integer vertices (16-cell).
31712:angles are defined between
29759:circulate on that isocline.
29747:great hexagon central plane
29647:great hexagon central plane
29131:{\displaystyle \pm {q_{n}}}
28316:{\displaystyle 60^{\circ }}
28021:
27811:Departing from any vertex V
27428:characteristic orthoschemes
26675:occur around an axis line.
25954:4-dimensional great circles
23053:
18715:parallel projection has an
18119:{\displaystyle R_{-q8,-q7}}
18017:{\displaystyle R_{-q7,-q8}}
17947:to the 4 hexagon planes of
17869:{\displaystyle \pm {q_{n}}}
17837:{\displaystyle {R_{ql,qr}}}
10599:{\displaystyle _{3}R^{4}/l}
10561:
10502:{\displaystyle _{2}R^{4}/l}
10464:
10405:{\displaystyle _{1}R^{4}/l}
10367:
10325:{\displaystyle _{0}R^{4}/l}
10287:
10150:{\displaystyle _{2}R^{3}/l}
10112:
9987:{\displaystyle _{1}R^{3}/l}
9949:
9824:{\displaystyle _{0}R^{3}/l}
9786:
9655:
9536:
9417:
9298:
9285:
8618:Two planes are also called
8614:Clifford parallel polytopes
8219:16-cell honeycomb {3,3,4,3}
8215:24-cell honeycomb {3,4,3,3}
8065:Quaternionic interpretation
7973:The 24 root vectors of the
6728:, which in the language of
3727:
2341:{\displaystyle {\sqrt {2}}}
2314:{\displaystyle {\sqrt {2}}}
2287:{\displaystyle {\sqrt {2}}}
2260:{\displaystyle {\sqrt {2}}}
2233:{\displaystyle {\sqrt {2}}}
2206:{\displaystyle {\sqrt {2}}}
675:
574:{3,4,3}. It is also called
50:. The specific problem is:
10:
34666:
34622:
34049:
33912:grand stellated dodecaplex
33868:great stellated dodecaplex
33481:
33231:Cambridge University Press
32855:(Paper 3) H.S.M. Coxeter,
32036:{\displaystyle \beta _{4}}
31716:-dimensional subspaces.)".
31615:Waegell & Aravind 2009
31397:{\displaystyle R_{q1,-q1}}
31190:{\displaystyle R_{q2,-q2}}
30992:{\displaystyle R_{q6,-q4}}
30837:{\displaystyle R_{q6,-q6}}
30340:{\displaystyle R_{q7,-q1}}
29994:{\displaystyle R_{q7,-q7}}
29861:{\displaystyle R_{q7,-q8}}
28020:edges belong to different
26736:+2 dimensions is exactly 2
26732:of the sphere embedded in
26677:Four dimensional rotations
26476:characteristic orthoscheme
26290:square great circles, the
26281:16 hexagonal great circles
25198:of the 24-cell is a cube.)
23699:16-cell is an orthonormal
23495:discovered the quaternions
23135:, and the two-dimensional
22745:
21899:
21173:
20161:
19582:
19397:
18949:
18878:
18759:
17701:{\displaystyle (-1,0,0,0)}
17521:{\displaystyle R_{q1,-q1}}
16604:{\displaystyle (0,0,0,-1)}
16424:{\displaystyle R_{q2,-q2}}
15200:{\displaystyle R_{q6,-q4}}
14784:{\displaystyle R_{q6,-q6}}
14180:{\displaystyle (-1,0,0,0)}
13964:{\displaystyle R_{q7,-q1}}
13107:{\displaystyle R_{q7,-q7}}
12223:{\displaystyle R_{q7,-q8}}
11597:
11267:Chiral symmetry operations
9274:Characteristic orthoscheme
9073:Five ways of looking at a
9063:to each other the way the
8805:helical hexagram isoclines
8247:regular convex 4-polytopes
8136:convex regular 4-polytopes
7680:, by reflection of that 4-
7520:
3952:Regular convex 4-polytopes
928:Regular convex 4-polytopes
716:
698:convex regular 4-polytopes
18:
33648:
33644:
33633:
33491:
33487:
33476:
33355:10.1007/s00006-019-0960-5
33320:10.13140/RG.2.1.3310.3205
33159:10.1007/s00006-016-0683-9
32824:Regular Complex Polytopes
32708:) with its corresponding
32504:Tyrrell & Semple 1971
32456:Tyrrell & Semple 1971
32444:Tyrrell & Semple 1971
31655:Tyrrell & Semple 1971
31333:{\displaystyle R_{q1,q1}}
31268:{\displaystyle R_{q2,q1}}
31122:{\displaystyle R_{q2,q7}}
31057:{\displaystyle R_{q4,q4}}
30405:{\displaystyle R_{q6,q6}}
30176:{\displaystyle R_{q7,q1}}
30114:carries the two kinds of
30112:hybrid isoclinic rotation
29926:{\displaystyle R_{q7,q7}}
29703:{\displaystyle R_{q7,q8}}
29340:{\displaystyle (0,0,1,0)}
28758:of the 24-cell's 12 axes.
28636:orthogonal cuboctahedron.
27352:chiralities, with no net
27248:tesseract double rotation
26123:) or 60 degrees apart in
26052:One way to visualize the
25050:edge (60Β°) apart. By the
24312:covering all 24 vertices.
23613:plane. Notice that the 8
23465:covering all 24 vertices.
22781:
22772:
22762:
22757:
20162:24-cell family polytopes
19576:Related uniform polytopes
18968:
18957:
18950:
18826:
18790:
18755:
18617:
18592:
18544:
18486:There is another related
18399:{\displaystyle R_{q7,q8}}
18285:{\displaystyle R_{q7,q8}}
18218:{\displaystyle R_{q7,q8}}
18167:{\displaystyle R_{q7,q8}}
18065:{\displaystyle R_{q8,q7}}
17917:{\displaystyle R_{q7,q8}}
17661:
17616:
17608:{\displaystyle (1,0,0,0)}
17571:
17529:
17477:
17467:{\displaystyle ^{q1,-q1}}
17423:
17332:{\displaystyle (1,0,0,0)}
17295:
17253:
17245:{\displaystyle (1,0,0,0)}
17208:
17166:
17158:{\displaystyle R_{q1,q1}}
17117:
17066:
16961:{\displaystyle (1,0,0,0)}
16924:
16882:
16874:{\displaystyle (0,0,0,1)}
16837:
16795:
16787:{\displaystyle R_{q2,q1}}
16746:
16695:
16564:
16519:
16511:{\displaystyle (0,0,0,1)}
16474:
16432:
16380:
16370:{\displaystyle ^{q2,-q2}}
16326:
16148:
16106:
16098:{\displaystyle (0,0,0,1)}
16061:
16019:
16011:{\displaystyle R_{q2,q7}}
15970:
15919:
15769:
15727:
15660:
15618:
15610:{\displaystyle R_{q4,q4}}
15569:
15518:
15362:
15317:
15250:
15208:
15156:
15146:{\displaystyle ^{q6,-q4}}
15102:
14946:
14901:
14834:
14792:
14740:
14730:{\displaystyle ^{q6,-q6}}
14686:
14536:
14494:
14427:
14385:
14377:{\displaystyle R_{q6,q6}}
14336:
14285:
14140:
14095:
14014:
13972:
13920:
13910:{\displaystyle ^{q7,-q1}}
13866:
13761:{\displaystyle (1,0,0,0)}
13724:
13682:
13601:
13559:
13551:{\displaystyle R_{q7,q1}}
13510:
13459:
13283:
13238:
13157:
13115:
13063:
13053:{\displaystyle ^{q7,-q7}}
13009:
12845:
12803:
12722:
12680:
12672:{\displaystyle R_{q7,q7}}
12631:
12580:
12399:
12354:
12273:
12231:
12179:
12169:{\displaystyle ^{q7,-q8}}
12125:
11938:
11896:
11815:
11773:
11765:{\displaystyle R_{q7,q8}}
11724:
11673:
11646:
11621:
11618:
11503:to the vertex coordinate
11351:{\displaystyle R_{q7,q8}}
10931:of the octahedron), plus
9280:
9072:
8688:isoclinic helix hexagrams
8680:isoclinic helix octagrams
8165:by regular 24-cells, the
8030:; its size has the ratio
7965:group, as shown in this F
7559:Tetrahedral constructions
7492:
7078:
7064:
7053:
6775:lengths from each other:
3987:
3971:
3950:
3817:coordinates of the form:
969:
953:
926:
556:four-dimensional geometry
532:
510:
498:
470:
458:
446:
436:
426:
413:
395:
232:
121:
116:Convex regular 4-polytope
111:
95:
90:
33638:
31836:, p. 29, (Paper 3)
31522:, p. 25, (Paper 3)
30054:rotations of the 24-cell
30020:rotations of the 16-cell
29272:{\displaystyle -{q_{n}}}
29213:{\displaystyle -{q_{n}}}
28380:together constitute the
28007:regular compound polygon
26826:4-dimensionally diagonal
26688:of its orthogonal edges
26299:square (tesseract) faces
26203:its configuration matrix
23233:radius coordinate system
23131:, the three-dimensional
23092:grand stellated 120-cell
23070:
21157:
21150:
21143:
21136:
21129:
21122:
21115:
21108:
21101:
21094:
21088:
21079:
21072:
21065:
21058:
21051:
21044:
21037:
21021:
21007:
20993:
20986:
20972:
20965:
20958:
20949:
20942:
20935:
20928:
20921:
20914:
20907:
20900:
20893:
20886:
20880:
20871:
20864:
20857:
20850:
20843:
20836:
20829:
20822:
20815:
20808:
20802:
20793:
20786:
20779:
20772:
20765:
20758:
20751:
20744:
20737:
20730:
20723:
19570:compound of two 24-cells
19562:grand stellated 120-cell
19014:Related complex polygons
18483:). See the table below.
18423:Octacube steel sculpture
17107:{\displaystyle ^{q1,q1}}
16736:{\displaystyle ^{q2,q1}}
15960:{\displaystyle ^{q2,q7}}
15559:{\displaystyle ^{q4,q4}}
14326:{\displaystyle ^{q6,q6}}
13500:{\displaystyle ^{q7,q1}}
12621:{\displaystyle ^{q7,q7}}
11714:{\displaystyle ^{q7,q8}}
11230:three inscribed 16-cells
9236:connect all 24-cells in
9007:diameter chords form an
8771:octahedron in the ring.
8658:
8106:binary tetrahedral group
8081:When interpreted as the
8075:binary tetrahedral group
8049:, which is generated by
7969:Coxeter plane projection
7481:of interior 4-polytopes
7233:triangular great circles
6988:Stereographic projection
3941:radius coordinates used
31907:, pp. 73β79, Β§4.2.
30930:fibration as 6 squares.
30915:skew octagram isoclines
28460:Just as each face of a
28139:characteristic rotation
28072:, and isoclines on the
28040:Isoclinic geodesics or
27932:are adjacent vertices,
27905:is 120 degrees beyond V
27846:is 120 degrees beyond V
27479:cell rings of octahedra
27415:enantiomorphous forms:
25822:chord of a great circle
25575:. In Euclidean 4-space
24674: (β
23927: (β
19020:regular complex polygon
18962:cells and one set of 8
18905:Rectified demitesseract
18887:with cells colored by D
18731:Perspective projections
11206:720Β° isoclinic rotation
9219:compounds of 5 24-cells
8129:unit radius coordinates
7024:hexagonal great circles
3932:unit radius coordinates
607:, being constructed of
33940:great grand dodecaplex
33039:Ghyka, Matila (1977).
32906:Computers Math. Applic
32037:
32010:
31959:
31781:, p. 150, Gosset.
31398:
31334:
31269:
31191:
31123:
31058:
30993:
30888:{12/3}=3{4} dodecagram
30871:
30838:
30778:
30755:
30731:
30656:
30571:
30505:
30439:unit radius 4-polytope
30406:
30341:
30281:
30280:{\displaystyle \pm q1}
30255:
30227:
30177:
30078:
30044:
29995:
29927:
29862:
29775:{12/2}=2{6} dodecagram
29732:{12/4}=4{3} dodecagram
29704:
29604:{12/4}=4{3} dodecagram
29520:
29421:
29341:
29301:unit radius 4-polytope
29279:planes are 180Β° apart.
29273:
29241:
29214:
29182:
29155:
29132:
29075:
29074:{\displaystyle \pm q8}
29049:
29048:{\displaystyle \pm q7}
28889:of the characteristic
28887:Coxeter-Dynkin diagram
28353:
28317:
28294:flattened MΓΆbius strip
28251:
28250:{\displaystyle 2\pi r}
28225:
28224:{\displaystyle 2\pi r}
28131:
28130:{\displaystyle 2\pi r}
28105:
28104:{\displaystyle 2\pi r}
27395:face-bonded cell pairs
26480:Coxeter-Dynkin diagram
26240:
26008:Each pair of parallel
25598:
25569:
25545:in that curved 3-space
25529:
25497:
25366:
25036:hexagon edges (or one
24506: (
24302:
24276:
24253:
24227:
24208:
23759: (
23647:4-dimensional diagonal
23561: (
23455:
23454:{\displaystyle \pm q3}
23429:
23428:{\displaystyle \pm q2}
23403:
23402:{\displaystyle \pm q1}
23381:
23271: (
23237: (
23195:configuration matrices
21174:B4 symmetry polytopes
20214:runcitruncated 24-cell
20202:cantitruncated 24-cell
19142:
18740:perspective projection
18679:
18643:One can also follow a
18628:
18426:
18400:
18352:
18329:
18286:
18219:
18168:
18120:
18066:
18018:
17964:
17941:
17918:
17870:
17844:each quaternion group
17838:
17702:
17652:
17651:{\displaystyle ^{-q1}}
17609:
17562:
17522:
17468:
17333:
17286:
17246:
17199:
17159:
17108:
16962:
16915:
16875:
16828:
16788:
16737:
16605:
16555:
16554:{\displaystyle ^{-q2}}
16512:
16465:
16425:
16371:
16222:
16139:
16099:
16052:
16012:
15961:
15829:
15760:
15720:
15651:
15611:
15560:
15428:
15353:
15352:{\displaystyle ^{-q4}}
15310:
15241:
15201:
15147:
15012:
14937:
14936:{\displaystyle ^{-q6}}
14894:
14825:
14785:
14731:
14596:
14527:
14487:
14418:
14378:
14327:
14181:
14131:
14130:{\displaystyle ^{-q1}}
14088:
14005:
13965:
13911:
13762:
13715:
13675:
13592:
13552:
13501:
13369:
13274:
13273:{\displaystyle ^{-q7}}
13231:
13148:
13108:
13054:
12919:
12836:
12796:
12713:
12673:
12622:
12476:
12390:
12389:{\displaystyle ^{-q8}}
12347:
12264:
12224:
12170:
12021:
11929:
11889:
11806:
11766:
11715:
11664:
11639:
11586:
11497:
11414:
11391:
11352:
11177:
11144:
11111:
11078:
11045:
11012:
10979:
10946:
10920:
10887:
10854:
10817:
10784:
10751:
10674:Coxeter-Dynkin diagram
10672:is represented by the
10640:
10600:
10543:
10503:
10446:
10406:
10349:
10326:
10265:
10228:
10191:
10151:
10102:
10065:
10028:
9988:
9939:
9902:
9865:
9825:
9764:
9727:
9690:
9645:
9608:
9571:
9526:
9489:
9452:
9395:
9353:
9316:
8752:
8391:
8326:
8281:through 60 degrees (a
8123:Viewed as the 24 unit
8096:. This is the ring of
8078:
8041:of the 24-cell is the
7970:
7938:
7732:
7623:characteristic simplex
7042:Orthogonal projections
6991:
6768:
6704:
6654:
6604:
6581:
6558:
6520:
6446:
6363:
6300:
6249:
6226:
6177:
6114:
6038:
5975:
5924:
5901:
5850:
5787:
5740:
5693:
5653:
5621:
5589:
5552:
5498:
5460:
5437:
5414:
5381:
5336:
5313:
5290:
5267:
5244:
5221:
3909:
3805:
3707:
3657:
3607:
3584:
3561:
3523:
3456:
3380:
3319:
3270:
3234:
3176:
3113:
3037:
2967:
2916:
2893:
2849:
2786:
2739:
2692:
2669:
2629:
2589:
2544:
2509:
2469:
2436:
2403:
2380:
2342:
2315:
2288:
2261:
2234:
2207:
898:and is inscribed in a
793:
633:. The 24-cell and the
551:
223:
172:
32993:(Manuscript ed.)
32376:Philippine wine dance
32111:, p. 78, Β§4.2.5.
32038:
32011:
31960:
31854:completely orthogonal
31399:
31335:
31270:
31192:
31124:
31059:
30994:
30869:
30839:
30779:
30756:
30732:
30657:
30572:
30506:
30433:by one instance of a
30407:
30342:
30282:
30256:
30254:{\displaystyle -{q1}}
30228:
30178:
30079:
30077:{\displaystyle F_{4}}
30045:
30043:{\displaystyle B_{4}}
29996:
29928:
29863:
29705:
29521:
29422:
29342:
29295:by one instance of a
29274:
29242:
29240:{\displaystyle q_{n}}
29215:
29183:
29181:{\displaystyle q_{n}}
29156:
29133:
29076:
29050:
28899:generating point ring
28866:characteristic angles
28354:
28318:
28252:
28226:
28132:
28106:
27468:great circle polygons
27338:Clifford displacement
27279:completely orthogonal
26918:and reserve the term
26822:Clifford displacement
26703:dimensional analogies
26339:completely orthogonal
26234:
26180:completely orthogonal
26121:completely orthogonal
25766:completely orthogonal
25599:
25570:
25530:
25498:
25363:completely orthogonal
25353:spanned by a twisted
25341:
24983:square central planes
24911:completely orthogonal
24303:
24277:
24254:
24228:
24206:
23609:is the square in the
23456:
23430:
23404:
23379:
23355:completely orthogonal
23314:completely orthogonal
23297:is the square in the
20221:omnitruncated 24-cell
19143:
18960:rectified tetrahedral
18953:rectified tetrahedral
18677:
18651:geodesics along four
18626:
18421:
18401:
18353:
18330:
18287:
18220:
18169:
18121:
18067:
18019:
17965:
17942:
17919:
17871:
17839:
17703:
17653:
17610:
17563:
17561:{\displaystyle ^{q1}}
17523:
17469:
17334:
17287:
17285:{\displaystyle ^{q1}}
17247:
17200:
17198:{\displaystyle ^{q1}}
17160:
17109:
16963:
16916:
16914:{\displaystyle ^{q1}}
16876:
16829:
16827:{\displaystyle ^{q2}}
16789:
16738:
16606:
16556:
16513:
16466:
16464:{\displaystyle ^{q2}}
16426:
16372:
16223:
16140:
16138:{\displaystyle ^{q7}}
16100:
16053:
16051:{\displaystyle ^{q2}}
16013:
15962:
15830:
15761:
15759:{\displaystyle ^{q4}}
15721:
15652:
15650:{\displaystyle ^{q4}}
15612:
15561:
15429:
15354:
15311:
15242:
15240:{\displaystyle ^{q6}}
15202:
15148:
15013:
14938:
14895:
14826:
14824:{\displaystyle ^{q6}}
14786:
14732:
14597:
14528:
14526:{\displaystyle ^{q6}}
14488:
14419:
14417:{\displaystyle ^{q6}}
14379:
14328:
14182:
14132:
14089:
14006:
14004:{\displaystyle ^{q7}}
13966:
13912:
13763:
13716:
13714:{\displaystyle ^{q1}}
13676:
13593:
13591:{\displaystyle ^{q7}}
13553:
13502:
13370:
13275:
13232:
13149:
13147:{\displaystyle ^{q7}}
13109:
13055:
12920:
12837:
12835:{\displaystyle ^{q7}}
12797:
12714:
12712:{\displaystyle ^{q7}}
12674:
12623:
12477:
12391:
12348:
12265:
12263:{\displaystyle ^{q7}}
12225:
12171:
12022:
11930:
11928:{\displaystyle ^{q8}}
11890:
11807:
11805:{\displaystyle ^{q7}}
11767:
11716:
11665:
11640:
11587:
11498:
11415:
11392:
11353:
11285:great circle polygons
11243:Tracing the orbit of
11238:generates the 24-cell
11178:
11145:
11112:
11079:
11046:
11013:
10980:
10947:
10921:
10888:
10855:
10826:characteristic angles
10818:
10785:
10752:
10641:
10601:
10544:
10504:
10447:
10407:
10350:
10327:
10266:
10229:
10192:
10152:
10103:
10066:
10029:
9989:
9940:
9903:
9866:
9826:
9765:
9728:
9691:
9646:
9609:
9572:
9527:
9490:
9453:
9396:
9354:
9317:
9250:edges to each other.
8750:
8684:great circle hexagons
8468:Clifford displacement
8385:
8363:completely orthogonal
8320:
8072:
7957:
7939:
7730:
7345:completely orthogonal
6986:
6766:
6705:
6655:
6605:
6582:
6559:
6521:
6447:
6364:
6301:
6250:
6227:
6178:
6115:
6039:
5976:
5925:
5902:
5851:
5788:
5741:
5694:
5654:
5622:
5590:
5553:
5499:
5461:
5438:
5415:
5382:
5337:
5314:
5291:
5268:
5245:
5222:
3923:, these are the unit
3921:Viewed as quaternions
3910:
3813:and 16 vertices with
3806:
3722:completely orthogonal
3708:
3658:
3608:
3585:
3562:
3524:
3457:
3381:
3320:
3271:
3235:
3177:
3114:
3038:
2968:
2917:
2894:
2850:
2787:
2740:
2693:
2670:
2630:
2590:
2545:
2510:
2470:
2437:
2404:
2381:
2343:
2316:
2289:
2262:
2235:
2208:
917:. Such polytopes are
794:
547:
224:
173:
33784:stellated dodecaplex
33251:(23 December 2021).
32020:
31993:
31923:
31486:, p. 249, 11.5.
31357:
31296:
31231:
31150:
31085:
31020:
30952:
30797:
30765:
30742:
30675:
30600:
30588:, in this example a
30586:great circle polygon
30515:
30449:
30368:
30300:
30265:
30237:
30214:
30139:
30061:
30027:
29954:
29889:
29821:
29666:
29450:
29438:, in this example a
29436:great circle polygon
29351:
29307:
29251:
29224:
29192:
29165:
29142:
29110:
29059:
29033:
28333:
28300:
28235:
28209:
28115:
28089:
27451:. Things which have
26542:tetrahedral pyramids
26244:rhombic dodecahedron
26237:rhombic dodecahedron
25579:
25550:
25510:
25478:
25408:Pythagorean distance
24928:Visualize the three
24286:
24263:
24237:
24214:
24097:is a hexagon on the
23661:(120Β°) away, moving
23651:Pythagorean distance
23439:
23413:
23387:
23141:Radially equilateral
23060:Octacube (sculpture)
19123:
18688:rhombic dodecahedral
18670:Parallel projections
18593:Southern Hemisphere
18545:Northern Hemisphere
18362:
18339:
18316:
18248:
18181:
18130:
18076:
18028:
17974:
17951:
17928:
17880:
17848:
17798:
17794:In a rotation class
17665:
17630:
17575:
17543:
17481:
17437:
17299:
17267:
17212:
17180:
17121:
17080:
16928:
16896:
16841:
16809:
16750:
16709:
16568:
16533:
16478:
16446:
16384:
16340:
16152:
16120:
16065:
16033:
15974:
15933:
15773:
15741:
15664:
15632:
15573:
15532:
15366:
15331:
15254:
15222:
15160:
15116:
14950:
14915:
14838:
14806:
14744:
14700:
14540:
14508:
14431:
14399:
14340:
14299:
14144:
14109:
14018:
13986:
13924:
13880:
13728:
13696:
13605:
13573:
13514:
13473:
13287:
13252:
13161:
13129:
13067:
13023:
12849:
12817:
12726:
12694:
12635:
12594:
12403:
12368:
12277:
12245:
12183:
12139:
11942:
11910:
11819:
11787:
11728:
11687:
11651:
11626:
11507:
11427:
11401:
11378:
11314:
11156:
11123:
11090:
11057:
11024:
10991:
10958:
10936:
10929:characteristic radii
10899:
10866:
10833:
10796:
10763:
10730:
10613:
10566:
10516:
10469:
10419:
10372:
10339:
10292:
10246:
10209:
10164:
10117:
10083:
10046:
10001:
9954:
9920:
9883:
9838:
9791:
9745:
9708:
9663:
9626:
9589:
9544:
9507:
9470:
9425:
9371:
9334:
9306:
9214:(4 different ways).
8676:great circle squares
8448:remarkably symmetric
8253:of their underlying
8112:root lattice is the
7823:
7814:configuration matrix
7755:, alluded to above.
7534:tetrahedral pyramids
7532:chords to remove 16
7450:rhombic dodecahedron
7197:square great circles
6667:
6617:
6594:
6571:
6533:
6464:
6376:
6313:
6262:
6239:
6190:
6132:
6051:
5988:
5937:
5914:
5863:
5805:
5753:
5706:
5666:
5634:
5602:
5570:
5511:
5473:
5450:
5427:
5394:
5354:
5326:
5303:
5280:
5257:
5234:
5211:
3823:
3764:
3670:
3620:
3597:
3574:
3536:
3474:
3393:
3332:
3283:
3247:
3189:
3131:
3050:
2980:
2929:
2906:
2862:
2804:
2752:
2705:
2682:
2642:
2602:
2562:
2522:
2482:
2449:
2416:
2393:
2360:
2328:
2301:
2274:
2247:
2220:
2193:
919:radially equilateral
735:
651:rhombic dodecahedron
183:
133:
106:(vertices and edges)
58:improve this article
46:to meet Knowledge's
34650:Regular 4-polytopes
34617:pentagonal polytope
34516:Uniform 10-polytope
34076:Fundamental convex
33470:Regular 4-polytopes
33389:2007JMP....48k3514K
33336:Dorst, Leo (2019).
33289:2010Symm....2.1423M
33030:Klitzing, Richard.
32765:, pp. 265β266.
32753:with illustrations.
32634:Kim & Rote 2016
32492:Kim & Rote 2016
32480:Kim & Rote 2016
32432:Kim & Rote 2016
32420:Kim & Rote 2016
32324:Kim & Rote 2016
32244:Kim & Rote 2016
32087:-space enclosed by
31706:Kim & Rote 2016
31667:Kim & Rote 2016
29106:A quaternion group
28893:orthoscheme is the
28573:, a helical circle
28137:circumference. The
27481:. Some things have
27418:isoclinic rotations
27371:the squares of the
27048:, characterized by
27041:rotation in 4-space
26962:double rotation as
26950:double rotation as
26812:, also known as an
25361:, and could lie in
25118:cubic cells (as in
25115:dimensional analogy
25081:16-cell, and a 60Β°
25022:rotation by 60Β° in
24884:completely disjoint
24364:on the chessboard,
24301:{\displaystyle -q8}
24252:{\displaystyle -q7}
20196:bitruncated 24-cell
20191:cantellated 24-cell
19515:bitruncated 24-cell
19489:uniform 4-polytopes
19483:Related 4-polytopes
19168:
18744:tetrakis hexahedral
11192:The 24-cell can be
10715:tetrahedral pyramid
8629:completely disjoint
8497:edge lengths away (
8477:The 24-cell in the
8442:Isoclinic rotations
8125:Hurwitz quaternions
8012:simple Lie groups.
7688:Cubic constructions
7266:isoclinic rotations
7231:chords occur in 32
7195:chords occur in 18
7045:
3925:Hurwitz quaternions
721:The 24-cell is the
34486:Uniform 9-polytope
34436:Uniform 8-polytope
34386:Uniform 7-polytope
34343:Uniform 6-polytope
34313:Uniform 5-polytope
34273:Uniform polychoron
34236:Uniform polyhedron
34084:in dimensions 2β10
33728:stellated 120-cell
33595:hecatonicosachoron
33427:2007-07-15 at the
33412:24-cell animations
33298:10.3390/sym2031423
33012:Weisstein, Eric W.
32952:Notices of the AMS
32384:Richard P. Feynman
32348:The Plattner Story
32033:
32006:
31955:
31394:
31342:identity operation
31330:
31265:
31187:
31119:
31054:
30989:
30872:
30834:
30777:{\displaystyle q6}
30774:
30754:{\displaystyle q6}
30751:
30727:
30710:
30693:
30652:
30635:
30618:
30567:
30550:
30533:
30501:
30496:
30479:
30402:
30337:
30277:
30251:
30226:{\displaystyle q1}
30223:
30173:
30074:
30040:
29991:
29923:
29858:
29788:isoclinic rotation
29700:
29634:isoclinic rotation
29630:hexagram isoclines
29563:isoclinic rotation
29542:isoclinic rotation
29516:
29511:
29496:
29481:
29466:
29417:
29412:
29397:
29382:
29367:
29337:
29269:
29237:
29210:
29178:
29154:{\displaystyle q7}
29151:
29128:
29071:
29045:
28909:-1)-elements into
28782:Reye configuration
28349:
28313:
28247:
28221:
28189:isoclinic rotation
28127:
28101:
28068:All isoclines are
27058:isoclinic rotation
26849:as the plane moves
26814:isoclinic rotation
26671:Three dimensional
26662:, and unit radius.
26241:
26235:Construction of a
26143:a 60 degree angle.
25978:isoclinic rotation
25970:Villarceau circles
25923:isoclinic geodesic
25867:isoclinic rotation
25652:isoclinic rotation
25594:
25565:
25525:
25493:
25369:Clifford parallels
25367:
25258:-distant vertices.
24748:
24586:
24310:discrete fibration
24298:
24275:{\displaystyle q8}
24272:
24249:
24226:{\displaystyle q7}
24223:
24209:
24001:
23839:
23668:β 0.866 (half the
23643:isoclinic rotation
23575:
23463:discrete fibration
23451:
23425:
23399:
23382:
23281:
23251:
20208:runcinated 24-cell
19587:uniform polychora
19166:
19138:
18719:envelope, and the
18680:
18629:
18427:
18396:
18351:{\displaystyle q8}
18348:
18328:{\displaystyle q7}
18325:
18282:
18242:isoclinic rotation
18215:
18164:
18116:
18062:
18014:
17963:{\displaystyle q8}
17960:
17940:{\displaystyle q7}
17937:
17914:
17866:
17834:
17698:
17648:
17605:
17558:
17518:
17464:
17329:
17282:
17242:
17195:
17155:
17104:
16958:
16911:
16871:
16824:
16784:
16733:
16601:
16551:
16508:
16461:
16421:
16367:
16218:
16213:
16198:
16183:
16168:
16135:
16095:
16048:
16008:
15957:
15825:
15820:
15803:
15756:
15716:
15711:
15694:
15647:
15607:
15556:
15424:
15419:
15399:
15349:
15306:
15289:
15272:
15237:
15197:
15143:
15008:
14991:
14971:
14933:
14890:
14873:
14856:
14821:
14781:
14727:
14592:
14575:
14558:
14523:
14483:
14466:
14449:
14414:
14374:
14323:
14177:
14127:
14084:
14079:
14064:
14049:
14034:
14001:
13961:
13907:
13758:
13711:
13671:
13666:
13651:
13636:
13621:
13588:
13548:
13497:
13365:
13360:
13342:
13324:
13306:
13270:
13227:
13222:
13207:
13192:
13177:
13144:
13104:
13050:
12915:
12910:
12895:
12880:
12865:
12832:
12792:
12787:
12772:
12757:
12742:
12709:
12669:
12618:
12472:
12467:
12452:
12437:
12422:
12386:
12343:
12338:
12323:
12308:
12293:
12260:
12220:
12166:
12017:
12012:
11994:
11976:
11958:
11925:
11885:
11880:
11865:
11850:
11835:
11802:
11762:
11711:
11663:{\displaystyle qr}
11660:
11638:{\displaystyle ql}
11635:
11582:
11577:
11559:
11541:
11523:
11493:
11488:
11473:
11458:
11443:
11413:{\displaystyle q8}
11410:
11390:{\displaystyle q7}
11387:
11348:
11273:symmetry operation
11173:
11170:
11140:
11137:
11107:
11104:
11074:
11071:
11041:
11038:
11008:
11005:
10975:
10972:
10942:
10916:
10913:
10883:
10880:
10850:
10847:
10828:π, π, π), plus
10813:
10810:
10780:
10777:
10747:
10744:
10636:
10627:
10596:
10539:
10530:
10499:
10442:
10433:
10402:
10345:
10322:
10261:
10259:
10224:
10222:
10187:
10178:
10147:
10098:
10096:
10061:
10059:
10024:
10015:
9984:
9935:
9933:
9898:
9896:
9861:
9852:
9821:
9760:
9758:
9723:
9721:
9686:
9677:
9641:
9639:
9604:
9602:
9567:
9558:
9522:
9520:
9485:
9483:
9448:
9439:
9391:
9389:
9349:
9347:
9312:
9253:Two vertices four
9100:Discrete fibration
8992:describes how the
8765:isoclinic rotation
8753:
8737:isoclinic rotation
8392:
8370:square or a great
8333:. The plane is an
8327:
8307:isoclinic rotation
8290:Planes of rotation
8279:isoclinic rotation
8257:which is known as
8134:Vertices of other
8079:
7971:
7934:
7928:
7924:
7808:As a configuration
7733:
7668:fundamental region
7504:edges to remove 8
7178:Dihedral symmetry
7129:Dihedral symmetry
7040:
7022:edges occur in 16
6992:
6769:
6726:Reye configuration
6700:
6692:
6650:
6642:
6600:
6577:
6554:
6546:
6516:
6498:
6479:
6442:
6430:
6359:
6347:
6296:
6284:
6245:
6222:
6210:
6173:
6161:
6110:
6098:
6034:
6022:
5971:
5958:
5923:{\displaystyle 24}
5920:
5897:
5884:
5846:
5834:
5783:
5774:
5736:
5727:
5689:
5680:
5649:
5647:
5617:
5615:
5585:
5583:
5548:
5540:
5494:
5486:
5456:
5433:
5410:
5377:
5368:
5332:
5309:
5286:
5263:
5240:
5217:
5146:irregular hexagons
3905:
3898:
3880:
3862:
3844:
3801:
3703:
3695:
3653:
3645:
3603:
3580:
3557:
3549:
3519:
3489:
3452:
3440:
3376:
3364:
3315:
3303:
3266:
3230:
3218:
3172:
3160:
3109:
3097:
3033:
3021:
2963:
2950:
2915:{\displaystyle 48}
2912:
2889:
2845:
2833:
2782:
2773:
2735:
2726:
2688:
2665:
2657:
2625:
2617:
2585:
2577:
2540:
2505:
2497:
2465:
2432:
2399:
2376:
2338:
2311:
2284:
2257:
2230:
2203:
2128:irregular hexagons
789:
564:regular 4-polytope
552:
219:
213:
168:
162:
34638:
34637:
34625:Polytope families
34082:uniform polytopes
34044:
34043:
34040:
34039:
34036:
34035:
34031:
34030:
33629:
33628:
33625:
33624:
33620:
33619:
33397:10.1063/1.2809467
33085:978-0-387-92713-8
33050:978-0-486-23542-4
32991:Uniform Polytopes
32979:978-1-107-10340-5
32849:978-0-471-01003-6
32811:Regular Polytopes
32727:24-cell honeycomb
32404:Villarceau circle
32109:van Ittersum 2020
31975:van Ittersum 2020
31905:van Ittersum 2020
31824:
31820:
31406:central inversion
30709:
30705:
30692:
30688:
30634:
30630:
30617:
30613:
30549:
30545:
30532:
30528:
30495:
30491:
30478:
30474:
29510:
29495:
29480:
29465:
29411:
29396:
29381:
29366:
28371:Clifford parallel
28341:
28167:Hopf fiber bundle
27958:Clifford polygon.
27939:apart. The three
27715:are 60Β° apart in
26350:edges are visible
26133:Clifford parallel
25632:Clifford parallel
25344:Clifford parallel
25140:Clifford parallel
24827:
24806:
24788:
24767:
24745:
24727:
24709:
24691:
24671:
24650:
24629:
24608:
24583:
24565:
24544:
24526:
24101:axis. Unlike the
24080:
24059:
24041:
24020:
23998:
23980:
23962:
23944:
23924:
23903:
23882:
23861:
23836:
23818:
23797:
23779:
23510:quaternion as an
23051:
23050:
22741:
22740:
22592:
22583:
22572:
22561:
22552:
22541:
22530:
22519:
21750:
21741:
21730:
21719:
21710:
21699:
21688:
21677:
21166:
21165:
20318:
20309:
20300:
20289:
20278:
20267:
20256:
20242:
20223:
20216:
20204:
20186:rectified 24-cell
20176:truncated 24-cell
20157:
20156:
19566:polytope compound
19508:rectified 24-cell
19501:truncated 24-cell
19480:
19479:
19011:
19010:
18910:Rectified 16-cell
18845:Related polytopes
18842:
18841:
18834:
18796:
18795:
18621:
18620:
17792:
17791:
16212:
16197:
16182:
16167:
15819:
15815:
15802:
15798:
15710:
15706:
15693:
15689:
15418:
15414:
15398:
15394:
15288:
15284:
15271:
15267:
14990:
14986:
14970:
14966:
14872:
14868:
14855:
14851:
14574:
14570:
14557:
14553:
14465:
14461:
14448:
14444:
14078:
14063:
14048:
14033:
13665:
13650:
13635:
13620:
13359:
13341:
13323:
13305:
13221:
13206:
13191:
13176:
12909:
12894:
12879:
12864:
12786:
12771:
12756:
12741:
12466:
12451:
12436:
12421:
12337:
12322:
12307:
12292:
12011:
11993:
11975:
11957:
11879:
11864:
11849:
11834:
11576:
11558:
11540:
11522:
11487:
11472:
11457:
11442:
11422:vertex coordinate
11218:geodesic isocline
11183:
11171:
11169:
11150:
11138:
11136:
11117:
11105:
11103:
11084:
11072:
11070:
11051:
11039:
11037:
11018:
11006:
11004:
10985:
10973:
10971:
10952:
10945:{\displaystyle 1}
10926:
10914:
10912:
10893:
10881:
10879:
10860:
10848:
10846:
10823:
10811:
10809:
10790:
10778:
10776:
10757:
10745:
10743:
10658:
10657:
10646:
10628:
10626:
10606:
10549:
10531:
10529:
10509:
10452:
10434:
10432:
10412:
10355:
10348:{\displaystyle 1}
10332:
10271:
10258:
10239:
10234:
10221:
10202:
10197:
10179:
10177:
10157:
10108:
10095:
10076:
10071:
10058:
10039:
10034:
10016:
10014:
9994:
9945:
9932:
9913:
9908:
9895:
9876:
9871:
9853:
9851:
9831:
9770:
9757:
9738:
9733:
9720:
9701:
9696:
9678:
9676:
9651:
9638:
9619:
9614:
9601:
9582:
9577:
9559:
9557:
9532:
9519:
9500:
9495:
9482:
9463:
9458:
9440:
9438:
9401:
9388:
9364:
9359:
9346:
9327:
9322:
9315:{\displaystyle 1}
9271:
9270:
9238:24-chord circuits
9221:, isoclines with
8811:vertex of a skew
8331:plane of rotation
8167:24-cell honeycomb
8161:of 4-dimensional
8057:roots. This is a
7606:
7186:
7185:
7075:Dihedral symmetry
7035:Clifford parallel
7003:geodesic distance
6886:
6873:
6845:
6825:
6759:Hypercubic chords
6719:Clifford parallel
6714:
6713:
6710:
6691:
6685:
6677:
6660:
6641:
6635:
6627:
6610:
6603:{\displaystyle 2}
6587:
6580:{\displaystyle 1}
6564:
6545:
6526:
6497:
6493:
6478:
6474:
6452:
6429:
6426:
6404:
6369:
6346:
6330:
6306:
6283:
6279:
6255:
6248:{\displaystyle 8}
6232:
6209:
6183:
6160:
6154:
6120:
6097:
6081:
6079:
6044:
6021:
6005:
5981:
5959:
5957:
5930:
5907:
5885:
5883:
5856:
5833:
5827:
5793:
5775:
5773:
5746:
5728:
5726:
5699:
5681:
5679:
5659:
5646:
5627:
5614:
5595:
5582:
5558:
5539:
5536:
5504:
5485:
5466:
5459:{\displaystyle 1}
5443:
5436:{\displaystyle 1}
5420:
5402:
5387:
5369:
5367:
5342:
5335:{\displaystyle 1}
5319:
5312:{\displaystyle 1}
5296:
5289:{\displaystyle 1}
5273:
5266:{\displaystyle 1}
5250:
5243:{\displaystyle 1}
5227:
5220:{\displaystyle 1}
5127:4 rectangles x 4
4349:Mirror dihedrals
3897:
3879:
3861:
3843:
3717:
3716:
3713:
3694:
3688:
3680:
3663:
3644:
3638:
3630:
3613:
3606:{\displaystyle 8}
3590:
3583:{\displaystyle 4}
3567:
3548:
3529:
3501:
3488:
3484:
3462:
3439:
3421:
3386:
3363:
3325:
3302:
3276:
3258:
3240:
3217:
3211:
3182:
3159:
3153:
3119:
3096:
3080:
3078:
3043:
3020:
3002:
2973:
2951:
2949:
2922:
2899:
2877:
2855:
2832:
2826:
2792:
2774:
2772:
2745:
2727:
2725:
2698:
2691:{\displaystyle 1}
2675:
2656:
2652:
2635:
2616:
2612:
2595:
2576:
2572:
2550:
2515:
2496:
2492:
2475:
2457:
2442:
2424:
2409:
2402:{\displaystyle 2}
2386:
2368:
2348:
2336:
2321:
2309:
2294:
2282:
2267:
2255:
2240:
2228:
2213:
2201:
2109:4 rectangles x 4
1331:Mirror dihedrals
691:24-cell honeycomb
662:24-cell honeycomb
649:and its dual the
629:. The 24-cell is
591:icosatetrahedroid
542:
541:
86:
85:
78:
48:quality standards
39:This article may
34657:
34629:Regular polytope
34190:
34179:
34168:
34127:
34070:
34063:
34056:
34047:
34046:
34020:
34018:
34017:
34014:
34011:
33992:
33990:
33989:
33986:
33983:
33964:
33962:
33961:
33958:
33955:
33936:
33934:
33933:
33930:
33927:
33908:
33906:
33905:
33902:
33899:
33892:
33890:
33889:
33886:
33883:
33864:
33862:
33861:
33858:
33855:
33840:grand dodecaplex
33836:
33834:
33833:
33830:
33827:
33812:great dodecaplex
33808:
33806:
33805:
33802:
33799:
33780:
33778:
33777:
33774:
33771:
33752:
33750:
33749:
33746:
33743:
33651:
33650:
33646:
33645:
33635:
33634:
33580:icositetrachoron
33494:
33493:
33489:
33488:
33478:
33477:
33463:
33456:
33449:
33440:
33439:
33400:
33367:
33357:
33332:
33323:
33302:
33300:
33283:(3): 1423β1449.
33267:
33265:
33263:
33244:
33219:
33201:
33180:
33170:
33144:
33134:
33132:
33119:
33104:
33102:
33089:
33065:
33054:
33035:
33025:
33024:
33006:
33005:(Ph.D. ed.)
32994:
32982:
32959:
32949:
32939:
32923:
32921:
32893:
32852:
32841:
32827:
32815:
32801:
32800:. Johann Planck.
32792:Kepler, Johannes
32778:
32772:
32766:
32760:
32754:
32736:
32730:
32691:
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32495:
32489:
32483:
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32465:
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32435:
32429:
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32417:
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32333:
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32315:
32309:
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32253:
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32241:
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32192:
32186:
32177:
32166:
32160:
32154:
32148:
32142:
32136:
32130:
32124:
32123:, p. 18-21.
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32007:
32005:
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31984:
31978:
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31950:
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31527:
31517:
31511:
31505:
31499:
31493:
31487:
31481:
31475:
31469:
31454:
31445:enantiomorphous
31427:
31421:
31414:
31413:
31404:rotation is the
31403:
31401:
31400:
31395:
31393:
31392:
31351:
31345:
31340:rotation is the
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28873:
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28836:
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28819:
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28797:
28778:
28772:
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28704:
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28697:
28690:
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28682:
28681:
28669:
28658:
28655:
28646:
28643:
28637:
28620:
28614:
28610:
28604:
28601:
28595:
28588:
28582:
28561:run through the
28550:
28535:
28531:
28525:
28523:
28522:
28516:
28515:
28508:
28502:
28482:
28473:
28458:
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28404:
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28342:
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28322:
28320:
28319:
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28311:
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28281:
28275:
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28248:
28230:
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28181:
28170:
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28136:
28134:
28133:
28128:
28110:
28108:
28107:
28102:
28083:
28077:
28066:
28053:
28038:
28029:
28019:
28018:
28005:instance of the
27986:
27973:
27965:
27959:
27945:
27944:
27938:
27937:
27915:
27914:
27865:. (Notice that V
27856:
27855:
27838:is inclined to P
27829:
27828:
27809:
27798:
27796:
27795:
27785:
27784:
27778:
27777:
27770:
27757:
27751:
27750:
27736:
27735:
27705:
27699:
27692:
27691:
27685:
27684:
27670:
27669:
27663:
27662:
27645:
27644:
27638:
27637:
27628:
27627:
27612:
27606:
27595:
27589:
27583:
27582:
27572:
27571:
27561:
27560:
27553:
27544:
27532:
27511:
27503:
27497:
27390:black and white:
27363:
27357:
27334:Clifford polygon
27330:
27299:
27275:
27264:
27257:
27251:
27244:
27238:
27236:
27235:
27229:
27228:
27214:
27213:
27203:
27202:
27196:
27195:
27189:
27188:
27182:
27173:
27163:
27162:
27152:
27151:
27120:
27109:
27078:
27065:
27037:
27031:
27020:
27003:
26936:
26923:
26912:
26911:
26879:
26860:
26840:
26829:
26803:
26788:
26781:simple rotations
26767:
26761:
26699:
26693:
26690:at the same time
26669:
26663:
26661:
26660:
26654:
26652:
26651:
26648:
26645:
26644:
26643:
26631:
26625:
26621:
26615:
26602:
26596:
26585:
26574:
26571:
26570:
26564:
26563:
26556:
26545:
26538:
26532:
26530:
26529:
26521:
26515:
26511:
26505:
26502:
26496:
26493:
26487:
26484:generating point
26448:
26439:
26437:
26436:
26429:
26416:
26413:
26412:
26404:
26398:
26394:
26381:
26377:
26366:
26363:Kepler's drawing
26359:
26353:
26349:
26348:
26330:
26329:
26321:
26320:
26314:
26313:
26307:
26306:
26298:
26297:
26289:
26288:
26277:
26268:
26264:
26258:
26254:
26248:
26228:
26222:
26212:
26206:
26198:
26187:
26176:
26163:
26159:
26153:
26150:
26144:
26108:
26087:
26050:
26037:
26035:
26034:
26028:
26027:
26021:
26020:
26014:
26013:
26006:
26000:
25997:Clifford polygon
25927:simple geodesics
25919:
25888:
25885:
25879:
25876:
25870:
25842:
25829:
25781:
25770:
25762:
25753:
25745:
25732:
25721:
25715:
25713:
25712:
25706:
25705:
25699:
25698:
25692:
25691:
25684:
25683:
25677:
25676:
25669:
25668:
25661:
25655:
25628:
25609:
25603:
25601:
25600:
25595:
25593:
25592:
25587:
25574:
25572:
25571:
25566:
25564:
25563:
25558:
25534:
25532:
25531:
25526:
25524:
25523:
25518:
25502:
25500:
25499:
25494:
25492:
25491:
25486:
25472:
25466:
25454:
25439:
25437:
25436:
25430:
25429:
25423:
25422:
25416:
25415:
25404:
25395:
25387:
25381:
25365:rotation planes.
25335:
25296:
25294:
25293:
25287:
25286:
25280:
25279:
25273:
25272:
25265:
25259:
25257:
25256:
25250:
25249:
25243:
25242:
25236:
25235:
25229:
25228:
25221:
25208:
25205:
25199:
25192:
25183:
25181:
25180:
25170:
25169:
25155:
25154:
25148:
25147:
25132:
25123:
25107:
25094:
25064:
25063:
25049:
25048:
25042:
25041:
25035:
25034:
25014:symmetric: four
24991:
24990:
24972:
24971:
24965:
24964:
24958:
24957:
24951:
24950:
24926:
24915:
24906:
24891:
24880:
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24860:
24854:
24853:
24841:
24837:
24833:
24826:
24824:
24823:
24820:
24817:
24811:
24810:
24805:
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24802:
24799:
24796:
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24781:
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24729:
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24633:
24628:
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23878:
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23776:
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23772:
23769:
23763:
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23725:
23710:
23696:
23695:
23687:
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23592:
23588:
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23554:
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23479:
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23457:
23452:
23434:
23432:
23431:
23426:
23408:
23406:
23405:
23400:
23373:
23358:
23346:
23331:
23327:
23318:
23309:
23303:
23294:
23290:
23286:
23282:
23278:
23274:
23268:
23264:
23260:
23256:
23252:
23248:
23244:
23240:
23232:
23231:
23223:
23217:
23216:the same radius.
23205:
23199:
23186:
23180:
23179:is the pentagon.
23171:{3,3,5} and the
23153:
23144:
23124:
23099:
23080:
23043:
23032:
23021:
23010:
22999:
22988:
22977:
22959:
22948:
22937:
22926:
22915:
22904:
22893:
22881:
22874:
22867:
22860:
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22846:
22839:
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22737:
22730:
22723:
22716:
22709:
22702:
22695:
22688:
22681:
22666:
22659:
22652:
22645:
22638:
22631:
22624:
22617:
22610:
22590:
22581:
22570:
22559:
22550:
22539:
22528:
22517:
22500:
22499:
22498:
22494:
22493:
22489:
22488:
22484:
22483:
22479:
22478:
22474:
22473:
22469:
22468:
22461:
22460:
22459:
22455:
22454:
22450:
22449:
22445:
22444:
22440:
22439:
22435:
22434:
22430:
22429:
22422:
22421:
22420:
22416:
22415:
22411:
22410:
22406:
22405:
22401:
22400:
22393:
22392:
22391:
22387:
22386:
22382:
22381:
22377:
22376:
22372:
22371:
22367:
22366:
22362:
22361:
22354:
22353:
22352:
22348:
22347:
22343:
22342:
22338:
22337:
22333:
22332:
22325:
22324:
22323:
22319:
22318:
22314:
22313:
22309:
22308:
22304:
22303:
22299:
22298:
22294:
22293:
22286:
22285:
22284:
22280:
22279:
22275:
22274:
22270:
22269:
22265:
22264:
22260:
22259:
22255:
22254:
22247:
22246:
22245:
22241:
22240:
22236:
22235:
22231:
22230:
22226:
22225:
22218:
22217:
22216:
22212:
22211:
22207:
22206:
22202:
22201:
22197:
22196:
22192:
22191:
22187:
22186:
22179:
22178:
22177:
22173:
22172:
22168:
22167:
22163:
22162:
22158:
22157:
22150:
22149:
22148:
22144:
22143:
22139:
22138:
22134:
22133:
22129:
22128:
22124:
22123:
22119:
22118:
22111:
22110:
22109:
22105:
22104:
22100:
22099:
22095:
22094:
22090:
22089:
22082:
22081:
22080:
22076:
22075:
22071:
22070:
22066:
22065:
22061:
22060:
22056:
22055:
22051:
22050:
22043:
22042:
22041:
22037:
22036:
22032:
22031:
22027:
22026:
22022:
22021:
22014:
22013:
22012:
22008:
22007:
22003:
22002:
21998:
21997:
21993:
21992:
21988:
21987:
21983:
21982:
21895:
21888:
21881:
21874:
21867:
21860:
21853:
21846:
21839:
21824:
21817:
21810:
21803:
21796:
21789:
21782:
21775:
21768:
21748:
21739:
21728:
21717:
21708:
21697:
21686:
21675:
21658:
21657:
21656:
21652:
21651:
21647:
21646:
21642:
21641:
21637:
21636:
21632:
21631:
21627:
21626:
21619:
21618:
21617:
21613:
21612:
21608:
21607:
21603:
21602:
21598:
21597:
21593:
21592:
21588:
21587:
21580:
21579:
21578:
21574:
21573:
21569:
21568:
21564:
21563:
21559:
21558:
21554:
21553:
21549:
21548:
21541:
21540:
21539:
21535:
21534:
21530:
21529:
21525:
21524:
21520:
21519:
21512:
21511:
21510:
21506:
21505:
21501:
21500:
21496:
21495:
21491:
21490:
21486:
21485:
21481:
21480:
21473:
21472:
21471:
21467:
21466:
21462:
21461:
21457:
21456:
21452:
21451:
21447:
21446:
21442:
21441:
21434:
21433:
21432:
21428:
21427:
21423:
21422:
21418:
21417:
21413:
21412:
21408:
21407:
21403:
21402:
21395:
21394:
21393:
21389:
21388:
21384:
21383:
21379:
21378:
21374:
21373:
21369:
21368:
21364:
21363:
21356:
21355:
21354:
21350:
21349:
21345:
21344:
21340:
21339:
21335:
21334:
21327:
21326:
21325:
21321:
21320:
21316:
21315:
21311:
21310:
21306:
21305:
21301:
21300:
21296:
21295:
21288:
21287:
21286:
21282:
21281:
21277:
21276:
21272:
21271:
21267:
21266:
21262:
21261:
21257:
21256:
21171:
21170:
21162:
21155:
21148:
21141:
21134:
21127:
21120:
21113:
21106:
21099:
21084:
21077:
21070:
21063:
21056:
21049:
21033:
21026:
21019:
21012:
21005:
20998:
20991:
20984:
20977:
20970:
20954:
20947:
20940:
20933:
20926:
20919:
20912:
20905:
20898:
20891:
20876:
20869:
20862:
20855:
20848:
20841:
20834:
20827:
20820:
20813:
20798:
20791:
20784:
20777:
20770:
20763:
20756:
20749:
20742:
20735:
20719:
20718:
20717:
20713:
20712:
20708:
20707:
20703:
20702:
20698:
20697:
20693:
20692:
20688:
20687:
20680:
20679:
20678:
20674:
20673:
20669:
20668:
20664:
20663:
20659:
20658:
20654:
20653:
20649:
20648:
20641:
20640:
20639:
20635:
20634:
20630:
20629:
20625:
20624:
20620:
20619:
20615:
20614:
20610:
20609:
20602:
20601:
20600:
20596:
20595:
20591:
20590:
20586:
20585:
20581:
20580:
20576:
20575:
20571:
20570:
20563:
20562:
20561:
20557:
20556:
20552:
20551:
20547:
20546:
20542:
20541:
20537:
20536:
20532:
20531:
20524:
20523:
20522:
20518:
20517:
20513:
20512:
20508:
20507:
20503:
20502:
20498:
20497:
20493:
20492:
20485:
20484:
20483:
20479:
20478:
20474:
20473:
20469:
20468:
20464:
20463:
20459:
20458:
20454:
20453:
20446:
20445:
20444:
20440:
20439:
20435:
20434:
20430:
20429:
20425:
20424:
20420:
20419:
20415:
20414:
20407:
20406:
20405:
20401:
20400:
20396:
20395:
20391:
20390:
20386:
20385:
20381:
20380:
20376:
20375:
20368:
20367:
20366:
20362:
20361:
20357:
20356:
20352:
20351:
20347:
20346:
20342:
20341:
20337:
20336:
20316:
20307:
20298:
20287:
20276:
20265:
20254:
20240:
20219:
20212:
20200:
20159:
20158:
20083:
20076:
20069:
20062:
20055:
20048:
20041:
20034:
20025:
20024:
20023:
20019:
20018:
20014:
20013:
20009:
20008:
20002:
20001:
20000:
19996:
19995:
19991:
19990:
19986:
19985:
19981:
19980:
19973:
19972:
19971:
19967:
19966:
19962:
19961:
19957:
19956:
19950:
19949:
19948:
19944:
19943:
19939:
19938:
19934:
19933:
19929:
19928:
19921:
19920:
19919:
19915:
19914:
19910:
19909:
19905:
19904:
19898:
19897:
19896:
19892:
19891:
19887:
19886:
19882:
19881:
19877:
19876:
19869:
19868:
19867:
19863:
19862:
19858:
19857:
19853:
19852:
19846:
19845:
19844:
19840:
19839:
19835:
19834:
19830:
19829:
19825:
19824:
19817:
19816:
19815:
19811:
19810:
19806:
19805:
19801:
19800:
19796:
19795:
19789:
19788:
19787:
19783:
19782:
19778:
19777:
19773:
19772:
19768:
19767:
19760:
19759:
19758:
19754:
19753:
19749:
19748:
19744:
19743:
19739:
19738:
19732:
19731:
19730:
19726:
19725:
19721:
19720:
19716:
19715:
19711:
19710:
19703:
19702:
19701:
19697:
19696:
19692:
19691:
19687:
19686:
19682:
19681:
19675:
19674:
19673:
19669:
19668:
19664:
19663:
19659:
19658:
19654:
19653:
19646:
19645:
19644:
19640:
19639:
19635:
19634:
19630:
19629:
19625:
19624:
19618:
19617:
19616:
19612:
19611:
19607:
19606:
19602:
19601:
19597:
19596:
19580:
19579:
19475:
19474:
19473:
19469:
19468:
19464:
19463:
19449:
19441:
19440:
19439:
19435:
19434:
19430:
19429:
19415:
19406:
19365:
19364:
19363:
19359:
19358:
19354:
19353:
19338:
19337:
19336:
19332:
19331:
19327:
19326:
19311:
19310:
19309:
19305:
19304:
19300:
19299:
19295:
19294:
19290:
19289:
19285:
19284:
19280:
19279:
19266:
19265:
19264:
19260:
19259:
19255:
19254:
19239:
19238:
19237:
19233:
19232:
19228:
19227:
19212:
19211:
19210:
19206:
19205:
19201:
19200:
19196:
19195:
19191:
19190:
19186:
19185:
19181:
19180:
19169:
19147:
19145:
19144:
19139:
19137:
19136:
19131:
19118:
19117:
19116:
19112:
19111:
19107:
19106:
19100:
19099:
19098:
19094:
19093:
19089:
19088:
19064:
19063:
19062:
19058:
19057:
19053:
19052:
19046:
19045:
19044:
19040:
19039:
19035:
19034:
19007:
19000:
18993:
18951:Three sets of 8
18946:
18914:Regular 24-cell
18876:
18875:
18836:
18835:
18816:
18805:
18798:
18784:
18774:
18765:
18753:
18657:
18656:
18519:Number of Cells
18513:
18512:
18482:
18481:
18478:
18405:
18403:
18402:
18397:
18395:
18394:
18357:
18355:
18354:
18349:
18334:
18332:
18331:
18326:
18307:
18305:
18304:
18301:
18298:
18291:
18289:
18288:
18283:
18281:
18280:
18224:
18222:
18221:
18216:
18214:
18213:
18173:
18171:
18170:
18165:
18163:
18162:
18125:
18123:
18122:
18117:
18115:
18114:
18071:
18069:
18068:
18063:
18061:
18060:
18023:
18021:
18020:
18015:
18013:
18012:
17969:
17967:
17966:
17961:
17946:
17944:
17943:
17938:
17923:
17921:
17920:
17915:
17913:
17912:
17875:
17873:
17872:
17867:
17865:
17864:
17863:
17843:
17841:
17840:
17835:
17833:
17832:
17831:
17785:
17784:
17774:
17772:
17771:
17768:
17765:
17754:
17753:
17743:
17741:
17740:
17737:
17734:
17723:
17722:
17707:
17705:
17704:
17699:
17657:
17655:
17654:
17649:
17647:
17646:
17625:
17614:
17612:
17611:
17606:
17567:
17565:
17564:
17559:
17557:
17556:
17538:
17527:
17525:
17524:
17519:
17517:
17516:
17473:
17471:
17470:
17465:
17463:
17462:
17432:
17416:
17415:
17405:
17403:
17402:
17399:
17396:
17385:
17384:
17374:
17372:
17371:
17368:
17365:
17354:
17353:
17338:
17336:
17335:
17330:
17291:
17289:
17288:
17283:
17281:
17280:
17262:
17251:
17249:
17248:
17243:
17204:
17202:
17201:
17196:
17194:
17193:
17175:
17164:
17162:
17161:
17156:
17154:
17153:
17113:
17111:
17110:
17105:
17103:
17102:
17075:
17059:
17058:
17048:
17046:
17045:
17042:
17039:
17028:
17027:
17017:
17015:
17014:
17011:
17008:
16997:
16996:
16986:
16984:
16983:
16980:
16977:
16967:
16965:
16964:
16959:
16920:
16918:
16917:
16912:
16910:
16909:
16891:
16880:
16878:
16877:
16872:
16833:
16831:
16830:
16825:
16823:
16822:
16804:
16793:
16791:
16790:
16785:
16783:
16782:
16742:
16740:
16739:
16734:
16732:
16731:
16704:
16688:
16687:
16677:
16675:
16674:
16671:
16668:
16657:
16656:
16646:
16644:
16643:
16640:
16637:
16626:
16625:
16610:
16608:
16607:
16602:
16560:
16558:
16557:
16552:
16550:
16549:
16528:
16517:
16515:
16514:
16509:
16470:
16468:
16467:
16462:
16460:
16459:
16441:
16430:
16428:
16427:
16422:
16420:
16419:
16376:
16374:
16373:
16368:
16366:
16365:
16335:
16319:
16318:
16308:
16306:
16305:
16302:
16299:
16288:
16287:
16277:
16275:
16274:
16271:
16268:
16257:
16256:
16246:
16244:
16243:
16240:
16237:
16227:
16225:
16224:
16219:
16214:
16205:
16199:
16190:
16184:
16175:
16169:
16160:
16144:
16142:
16141:
16136:
16134:
16133:
16115:
16104:
16102:
16101:
16096:
16057:
16055:
16054:
16049:
16047:
16046:
16028:
16017:
16015:
16014:
16009:
16007:
16006:
15966:
15964:
15963:
15958:
15956:
15955:
15928:
15912:
15911:
15901:
15899:
15898:
15895:
15892:
15881:
15880:
15870:
15868:
15867:
15864:
15861:
15850:
15849:
15834:
15832:
15831:
15826:
15821:
15811:
15810:
15804:
15794:
15793:
15765:
15763:
15762:
15757:
15755:
15754:
15736:
15725:
15723:
15722:
15717:
15712:
15702:
15701:
15695:
15685:
15684:
15656:
15654:
15653:
15648:
15646:
15645:
15627:
15616:
15614:
15613:
15608:
15606:
15605:
15565:
15563:
15562:
15557:
15555:
15554:
15527:
15511:
15510:
15494:
15493:
15483:
15481:
15480:
15477:
15474:
15463:
15462:
15452:
15450:
15449:
15446:
15443:
15433:
15431:
15430:
15425:
15420:
15410:
15409:
15400:
15390:
15389:
15358:
15356:
15355:
15350:
15348:
15347:
15326:
15315:
15313:
15312:
15307:
15290:
15280:
15279:
15273:
15263:
15262:
15246:
15244:
15243:
15238:
15236:
15235:
15217:
15206:
15204:
15203:
15198:
15196:
15195:
15152:
15150:
15149:
15144:
15142:
15141:
15111:
15095:
15094:
15084:
15082:
15081:
15078:
15075:
15064:
15063:
15053:
15051:
15050:
15047:
15044:
15033:
15032:
15017:
15015:
15014:
15009:
14992:
14982:
14981:
14972:
14962:
14961:
14942:
14940:
14939:
14934:
14932:
14931:
14910:
14899:
14897:
14896:
14891:
14874:
14864:
14863:
14857:
14847:
14846:
14830:
14828:
14827:
14822:
14820:
14819:
14801:
14790:
14788:
14787:
14782:
14780:
14779:
14736:
14734:
14733:
14728:
14726:
14725:
14695:
14679:
14678:
14668:
14666:
14665:
14662:
14659:
14648:
14647:
14637:
14635:
14634:
14631:
14628:
14617:
14616:
14601:
14599:
14598:
14593:
14576:
14566:
14565:
14559:
14549:
14548:
14532:
14530:
14529:
14524:
14522:
14521:
14503:
14492:
14490:
14489:
14484:
14467:
14457:
14456:
14450:
14440:
14439:
14423:
14421:
14420:
14415:
14413:
14412:
14394:
14383:
14381:
14380:
14375:
14373:
14372:
14332:
14330:
14329:
14324:
14322:
14321:
14294:
14278:
14277:
14267:
14265:
14264:
14261:
14258:
14247:
14246:
14236:
14234:
14233:
14230:
14227:
14216:
14215:
14205:
14203:
14202:
14199:
14196:
14186:
14184:
14183:
14178:
14136:
14134:
14133:
14128:
14126:
14125:
14104:
14093:
14091:
14090:
14085:
14080:
14071:
14065:
14056:
14050:
14041:
14035:
14026:
14010:
14008:
14007:
14002:
14000:
13999:
13981:
13970:
13968:
13967:
13962:
13960:
13959:
13916:
13914:
13913:
13908:
13906:
13905:
13875:
13859:
13858:
13848:
13846:
13845:
13842:
13839:
13828:
13827:
13817:
13815:
13814:
13811:
13808:
13797:
13796:
13786:
13784:
13783:
13780:
13777:
13767:
13765:
13764:
13759:
13720:
13718:
13717:
13712:
13710:
13709:
13691:
13680:
13678:
13677:
13672:
13667:
13658:
13652:
13643:
13637:
13628:
13622:
13613:
13597:
13595:
13594:
13589:
13587:
13586:
13568:
13557:
13555:
13554:
13549:
13547:
13546:
13506:
13504:
13503:
13498:
13496:
13495:
13468:
13452:
13451:
13441:
13439:
13438:
13435:
13432:
13421:
13420:
13410:
13408:
13407:
13404:
13401:
13390:
13389:
13374:
13372:
13371:
13366:
13361:
13352:
13343:
13334:
13325:
13316:
13307:
13298:
13279:
13277:
13276:
13271:
13269:
13268:
13247:
13236:
13234:
13233:
13228:
13223:
13214:
13208:
13199:
13193:
13184:
13178:
13169:
13153:
13151:
13150:
13145:
13143:
13142:
13124:
13113:
13111:
13110:
13105:
13103:
13102:
13059:
13057:
13056:
13051:
13049:
13048:
13018:
13002:
13001:
12991:
12989:
12988:
12985:
12982:
12971:
12970:
12960:
12958:
12957:
12954:
12951:
12940:
12939:
12924:
12922:
12921:
12916:
12911:
12902:
12896:
12887:
12881:
12872:
12866:
12857:
12841:
12839:
12838:
12833:
12831:
12830:
12812:
12801:
12799:
12798:
12793:
12788:
12779:
12773:
12764:
12758:
12749:
12743:
12734:
12718:
12716:
12715:
12710:
12708:
12707:
12689:
12678:
12676:
12675:
12670:
12668:
12667:
12627:
12625:
12624:
12619:
12617:
12616:
12589:
12573:
12572:
12562:
12560:
12559:
12556:
12553:
12542:
12541:
12531:
12529:
12528:
12525:
12522:
12511:
12510:
12500:
12498:
12497:
12494:
12491:
12481:
12479:
12478:
12473:
12468:
12459:
12453:
12444:
12438:
12429:
12423:
12414:
12395:
12393:
12392:
12387:
12385:
12384:
12363:
12352:
12350:
12349:
12344:
12339:
12330:
12324:
12315:
12309:
12300:
12294:
12285:
12269:
12267:
12266:
12261:
12259:
12258:
12240:
12229:
12227:
12226:
12221:
12219:
12218:
12175:
12173:
12172:
12167:
12165:
12164:
12134:
12118:
12117:
12107:
12105:
12104:
12101:
12098:
12087:
12086:
12076:
12074:
12073:
12070:
12067:
12056:
12055:
12045:
12043:
12042:
12039:
12036:
12026:
12024:
12023:
12018:
12013:
12004:
11995:
11986:
11977:
11968:
11959:
11950:
11934:
11932:
11931:
11926:
11924:
11923:
11905:
11894:
11892:
11891:
11886:
11881:
11872:
11866:
11857:
11851:
11842:
11836:
11827:
11811:
11809:
11808:
11803:
11801:
11800:
11782:
11771:
11769:
11768:
11763:
11761:
11760:
11720:
11718:
11717:
11712:
11710:
11709:
11682:
11669:
11667:
11666:
11661:
11644:
11642:
11641:
11636:
11595:
11594:
11591:
11589:
11588:
11583:
11578:
11569:
11560:
11551:
11542:
11533:
11524:
11515:
11502:
11500:
11499:
11494:
11489:
11480:
11474:
11465:
11459:
11450:
11444:
11435:
11419:
11417:
11416:
11411:
11396:
11394:
11393:
11388:
11373:
11371:
11370:
11367:
11364:
11357:
11355:
11354:
11349:
11347:
11346:
11257:
11256:
11182:
11180:
11179:
11174:
11172:
11162:
11160:
11152:
11149:
11147:
11146:
11141:
11139:
11129:
11127:
11119:
11116:
11114:
11113:
11108:
11106:
11096:
11094:
11086:
11083:
11081:
11080:
11075:
11073:
11063:
11061:
11053:
11050:
11048:
11047:
11042:
11040:
11030:
11028:
11020:
11017:
11015:
11014:
11009:
11007:
10997:
10995:
10987:
10984:
10982:
10981:
10976:
10974:
10964:
10962:
10954:
10951:
10949:
10948:
10943:
10932:
10925:
10923:
10922:
10917:
10915:
10905:
10903:
10895:
10892:
10890:
10889:
10884:
10882:
10872:
10870:
10862:
10859:
10857:
10856:
10851:
10849:
10839:
10837:
10829:
10822:
10820:
10819:
10814:
10812:
10802:
10800:
10792:
10789:
10787:
10786:
10781:
10779:
10769:
10767:
10759:
10756:
10754:
10753:
10748:
10746:
10736:
10734:
10726:
10712:
10711:
10710:
10706:
10705:
10701:
10700:
10696:
10695:
10691:
10690:
10686:
10685:
10681:
10680:
10666:irregular 5-cell
10645:
10643:
10642:
10637:
10629:
10619:
10617:
10609:
10605:
10603:
10602:
10597:
10592:
10587:
10586:
10577:
10576:
10562:
10548:
10546:
10545:
10540:
10532:
10522:
10520:
10512:
10508:
10506:
10505:
10500:
10495:
10490:
10489:
10480:
10479:
10465:
10451:
10449:
10448:
10443:
10435:
10425:
10423:
10415:
10411:
10409:
10408:
10403:
10398:
10393:
10392:
10383:
10382:
10368:
10354:
10352:
10351:
10346:
10335:
10331:
10329:
10328:
10323:
10318:
10313:
10312:
10303:
10302:
10288:
10270:
10268:
10267:
10262:
10260:
10251:
10242:
10237:
10233:
10231:
10230:
10225:
10223:
10214:
10205:
10200:
10196:
10194:
10193:
10188:
10180:
10170:
10168:
10160:
10156:
10154:
10153:
10148:
10143:
10138:
10137:
10128:
10127:
10113:
10107:
10105:
10104:
10099:
10097:
10088:
10079:
10074:
10070:
10068:
10067:
10062:
10060:
10051:
10042:
10037:
10033:
10031:
10030:
10025:
10017:
10007:
10005:
9997:
9993:
9991:
9990:
9985:
9980:
9975:
9974:
9965:
9964:
9950:
9944:
9942:
9941:
9936:
9934:
9925:
9916:
9911:
9907:
9905:
9904:
9899:
9897:
9888:
9879:
9874:
9870:
9868:
9867:
9862:
9854:
9844:
9842:
9834:
9830:
9828:
9827:
9822:
9817:
9812:
9811:
9802:
9801:
9787:
9769:
9767:
9766:
9761:
9759:
9750:
9741:
9736:
9732:
9730:
9729:
9724:
9722:
9713:
9704:
9699:
9695:
9693:
9692:
9687:
9679:
9669:
9667:
9659:
9650:
9648:
9647:
9642:
9640:
9631:
9622:
9617:
9613:
9611:
9610:
9605:
9603:
9594:
9585:
9580:
9576:
9574:
9573:
9568:
9560:
9550:
9548:
9540:
9531:
9529:
9528:
9523:
9521:
9512:
9503:
9498:
9494:
9492:
9491:
9486:
9484:
9475:
9466:
9461:
9457:
9455:
9454:
9449:
9441:
9431:
9429:
9421:
9400:
9398:
9397:
9392:
9390:
9384:
9376:
9367:
9362:
9358:
9356:
9355:
9350:
9348:
9339:
9330:
9325:
9321:
9319:
9318:
9313:
9302:
9278:
9277:
9266:
9265:
9259:
9258:
9249:
9248:
9235:
9234:
9209:
9208:
9195:
9194:
9184:
9177:
9170:
9163:
9156:
9070:
9046:
9045:
9039:
9038:
9028:
9027:
9020:
9019:
9006:
9005:
8998:
8997:
8987:
8986:
8965:
8964:
8954:
8953:
8947:
8946:
8923:Clifford polygon
8901:
8900:
8894:
8893:
8883:
8882:
8875:
8874:
8868:
8867:
8857:
8856:
8850:
8849:
8831:
8830:
8824:
8823:
8729:
8728:
8722:
8721:
8715:
8714:
8708:
8707:
8633:double rotations
8560:
8559:
8546:
8545:
8539:
8538:
8532:
8531:
8517:
8516:
8510:
8509:
8503:
8502:
8378:Double rotations
8344:axis of rotation
8313:Simple rotations
8036:
8035:
7983:simple Lie group
7943:
7941:
7940:
7935:
7933:
7932:
7925:
7802:
7801:
7795:
7794:
7788:
7787:
7781:
7780:
7741:
7740:
7723:
7722:
7712:
7711:
7664:irregular 5-cell
7661:
7660:
7659:
7655:
7654:
7650:
7649:
7645:
7644:
7640:
7639:
7635:
7634:
7630:
7629:
7613:
7612:
7605:
7603:
7602:
7599:
7596:
7595:
7588:
7586:
7585:
7576:
7575:
7569:
7568:
7549:
7548:
7543:edges, exposing
7542:
7541:
7531:
7530:
7503:
7502:
7439:
7437:
7436:
7433:
7430:
7423:
7421:
7420:
7417:
7414:
7407:
7405:
7404:
7401:
7398:
7391:
7389:
7388:
7385:
7382:
7342:
7341:
7316:
7315:
7309:
7308:
7302:
7301:
7295:
7294:
7288:
7287:
7281:
7280:
7270:simple rotations
7259:
7258:
7249:
7248:
7241:
7240:
7230:
7229:
7219:
7218:
7212:
7211:
7205:
7204:
7194:
7193:
7173:
7166:
7124:
7117:
7069:
7046:
7039:
7032:
7031:
7021:
7020:
7011:
7010:
6974:
6973:
6967:
6966:
6960:
6959:
6953:
6952:
6946:
6945:
6939:
6938:
6932:
6931:
6921:
6920:
6914:
6913:
6907:
6906:
6900:
6899:
6885:
6882:
6880:
6879:
6872:
6870:
6869:
6866:
6863:
6862:
6854:
6852:
6851:
6844:
6842:
6841:
6838:
6835:
6834:
6827:
6824:
6822:
6821:
6818:
6815:
6814:
6807:
6802:
6801:
6795:
6794:
6788:
6787:
6781:
6780:
6754:
6753:
6732:is written as 12
6709:
6707:
6706:
6701:
6693:
6687:
6686:
6683:
6678:
6675:
6672:
6663:
6659:
6657:
6656:
6651:
6643:
6637:
6636:
6633:
6628:
6625:
6622:
6613:
6609:
6607:
6606:
6601:
6590:
6586:
6584:
6583:
6578:
6567:
6563:
6561:
6560:
6555:
6547:
6538:
6529:
6525:
6523:
6522:
6517:
6509:
6508:
6503:
6499:
6489:
6488:
6480:
6470:
6469:
6460:
6451:
6449:
6448:
6443:
6435:
6431:
6428:
6427:
6422:
6420:
6419:
6406:
6405:
6400:
6388:
6372:
6368:
6366:
6365:
6360:
6352:
6348:
6345:
6344:
6343:
6326:
6325:
6309:
6305:
6303:
6302:
6297:
6289:
6285:
6275:
6274:
6258:
6254:
6252:
6251:
6246:
6235:
6231:
6229:
6228:
6223:
6215:
6211:
6202:
6186:
6182:
6180:
6179:
6174:
6166:
6162:
6156:
6155:
6150:
6144:
6128:
6119:
6117:
6116:
6111:
6103:
6099:
6096:
6095:
6094:
6080:
6075:
6064:
6063:
6047:
6043:
6041:
6040:
6035:
6027:
6023:
6020:
6019:
6018:
6001:
6000:
5984:
5980:
5978:
5977:
5972:
5964:
5960:
5950:
5948:
5933:
5929:
5927:
5926:
5921:
5910:
5906:
5904:
5903:
5898:
5890:
5886:
5876:
5874:
5859:
5855:
5853:
5852:
5847:
5839:
5835:
5829:
5828:
5823:
5817:
5801:
5792:
5790:
5789:
5784:
5776:
5769:
5768:
5759:
5757:
5749:
5745:
5743:
5742:
5737:
5729:
5722:
5721:
5712:
5710:
5702:
5698:
5696:
5695:
5690:
5682:
5672:
5670:
5662:
5658:
5656:
5655:
5650:
5648:
5639:
5630:
5626:
5624:
5623:
5618:
5616:
5607:
5598:
5594:
5592:
5591:
5586:
5584:
5575:
5566:
5557:
5555:
5554:
5549:
5541:
5538:
5537:
5532:
5530:
5529:
5516:
5507:
5503:
5501:
5500:
5495:
5487:
5478:
5469:
5465:
5463:
5462:
5457:
5446:
5442:
5440:
5439:
5434:
5423:
5419:
5417:
5416:
5411:
5403:
5398:
5390:
5386:
5384:
5383:
5378:
5370:
5360:
5358:
5350:
5341:
5339:
5338:
5333:
5322:
5318:
5316:
5315:
5310:
5299:
5295:
5293:
5292:
5287:
5276:
5272:
5270:
5269:
5264:
5253:
5249:
5247:
5246:
5241:
5230:
5226:
5224:
5223:
5218:
5207:
5096:675 in 120-cell
5093:120 in 120-cell
5042:120 dodecahedra
4996:1200 triangular
4971:600 tetrahedral
4968:120 icosahedral
4948:
4941:
4934:
4927:
4920:
4913:
4901:
4899:
4898:
4895:
4892:
4886:
4884:
4883:
4880:
4877:
4871:
4869:
4868:
4865:
4862:
4856:
4854:
4853:
4850:
4847:
4841:
4839:
4838:
4835:
4832:
4826:
4824:
4823:
4820:
4817:
4809:
4807:
4806:
4803:
4800:
4794:
4792:
4791:
4788:
4785:
4779:
4777:
4776:
4773:
4770:
4764:
4762:
4761:
4758:
4755:
4749:
4747:
4746:
4743:
4740:
4734:
4732:
4731:
4728:
4725:
4717:
4715:
4714:
4711:
4708:
4702:
4700:
4699:
4696:
4693:
4687:
4685:
4684:
4681:
4678:
4672:
4670:
4669:
4666:
4663:
4657:
4655:
4654:
4651:
4648:
4642:
4640:
4639:
4636:
4633:
4625:
4623:
4622:
4619:
4616:
4610:
4608:
4607:
4604:
4601:
4595:
4593:
4592:
4589:
4586:
4580:
4578:
4577:
4574:
4571:
4565:
4563:
4562:
4559:
4556:
4550:
4548:
4547:
4544:
4541:
4533:
4531:
4530:
4527:
4524:
4518:
4516:
4515:
4512:
4509:
4503:
4501:
4500:
4497:
4494:
4488:
4486:
4485:
4482:
4479:
4473:
4471:
4470:
4467:
4464:
4458:
4456:
4455:
4452:
4449:
4441:
4439:
4438:
4435:
4432:
4426:
4424:
4423:
4420:
4417:
4411:
4409:
4408:
4405:
4402:
4396:
4394:
4393:
4390:
4387:
4381:
4379:
4378:
4375:
4372:
4366:
4364:
4363:
4360:
4357:
4344:
4343:
4342:
4338:
4337:
4333:
4332:
4328:
4327:
4323:
4322:
4318:
4317:
4313:
4312:
4305:
4304:
4303:
4299:
4298:
4294:
4293:
4289:
4288:
4284:
4283:
4279:
4278:
4274:
4273:
4266:
4265:
4264:
4260:
4259:
4255:
4254:
4250:
4249:
4245:
4244:
4240:
4239:
4235:
4234:
4227:
4226:
4225:
4221:
4220:
4216:
4215:
4211:
4210:
4206:
4205:
4201:
4200:
4196:
4195:
4188:
4187:
4186:
4182:
4181:
4177:
4176:
4172:
4171:
4167:
4166:
4162:
4161:
4157:
4156:
4149:
4148:
4147:
4143:
4142:
4138:
4137:
4133:
4132:
4128:
4127:
4123:
4122:
4118:
4117:
3948:
3947:
3940:
3939:
3914:
3912:
3911:
3906:
3904:
3900:
3899:
3890:
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2212:
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2209:
2204:
2202:
2197:
2189:
2078:675 in 120-cell
2075:120 in 120-cell
2024:120 dodecahedra
1978:1200 triangular
1953:600 tetrahedral
1950:120 icosahedral
1930:
1923:
1916:
1909:
1902:
1895:
1883:
1881:
1880:
1877:
1874:
1868:
1866:
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1378:
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1300:
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1280:
1276:
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1266:
1265:
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1256:
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1247:
1246:
1242:
1241:
1237:
1236:
1232:
1231:
1227:
1226:
1222:
1221:
1217:
1216:
1209:
1208:
1207:
1203:
1202:
1198:
1197:
1193:
1192:
1188:
1187:
1183:
1182:
1178:
1177:
1170:
1169:
1168:
1164:
1163:
1159:
1158:
1154:
1153:
1149:
1148:
1144:
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1139:
1138:
1131:
1130:
1129:
1125:
1124:
1120:
1119:
1115:
1114:
1110:
1109:
1105:
1104:
1100:
1099:
936:
935:
924:
923:
908:
907:
897:
896:
883:
882:
881:
877:
876:
872:
871:
867:
866:
862:
861:
857:
856:
852:
851:
838:
837:
836:
832:
831:
827:
826:
822:
821:
817:
816:
812:
811:
807:
806:
798:
796:
795:
790:
785:
784:
779:
642:three dimensions
583:icositetrachoron
409:
391:
390:
389:
385:
384:
380:
379:
375:
374:
368:
367:
366:
362:
361:
357:
356:
352:
351:
347:
346:
340:
339:
338:
334:
333:
329:
328:
324:
323:
319:
318:
312:
311:
310:
306:
305:
301:
300:
296:
295:
291:
290:
286:
285:
281:
280:
274:
273:
272:
268:
267:
263:
262:
258:
257:
253:
252:
248:
247:
243:
242:
228:
226:
225:
220:
218:
214:
177:
175:
174:
169:
167:
163:
103:Schlegel diagram
100:
88:
87:
81:
74:
70:
67:
61:
34:
33:
26:
34665:
34664:
34660:
34659:
34658:
34656:
34655:
34654:
34640:
34639:
34608:
34601:
34594:
34477:
34470:
34463:
34427:
34420:
34413:
34377:
34370:
34204:Regular polygon
34197:
34188:
34181:
34177:
34170:
34166:
34157:
34148:
34141:
34137:
34125:
34119:
34115:
34103:
34085:
34074:
34045:
34032:
34027:
34015:
34012:
34009:
34008:
34006:
33999:
33996:grand tetraplex
33987:
33984:
33981:
33980:
33978:
33971:
33968:great icosaplex
33959:
33956:
33953:
33952:
33950:
33943:
33931:
33928:
33925:
33924:
33922:
33915:
33903:
33900:
33897:
33896:
33894:
33887:
33884:
33881:
33880:
33878:
33871:
33859:
33856:
33853:
33852:
33850:
33843:
33831:
33828:
33825:
33824:
33822:
33815:
33803:
33800:
33797:
33796:
33794:
33787:
33775:
33772:
33769:
33768:
33766:
33759:
33747:
33744:
33741:
33740:
33738:
33727:
33720:
33713:
33711:
33704:
33697:
33695:
33688:
33686:
33679:
33672:
33665:
33663:
33656:
33640:
33621:
33616:
33601:
33586:
33571:
33556:
33541:
33483:
33472:
33467:
33429:Wayback Machine
33408:
33403:
33261:
33259:
33241:
33142:
33086:
33051:
32999:Johnson, Norman
32987:Johnson, Norman
32980:
32964:Johnson, Norman
32947:
32928:Coxeter, H.S.M.
32898:Coxeter, H.S.M.
32886:Coxeter, H.S.M.
32850:
32832:Coxeter, H.S.M.
32820:Coxeter, H.S.M.
32806:Coxeter, H.S.M.
32786:
32781:
32773:
32769:
32761:
32757:
32747:Hopf fibrations
32737:
32733:
32714:Petrie polygons
32692:
32688:
32680:
32676:
32668:
32664:
32656:
32652:
32644:
32640:
32632:
32628:
32620:
32616:
32608:
32604:
32596:
32592:
32584:
32580:
32566:
32557:
32553:
32547:
32538:
32534:
32527:
32523:
32514:
32510:
32502:
32498:
32490:
32486:
32478:
32474:
32466:
32462:
32454:
32450:
32442:
32438:
32430:
32426:
32418:
32414:
32398:
32394:
32370:
32366:
32358:
32354:
32334:
32330:
32322:
32318:
32310:
32306:
32290:
32286:
32278:
32274:
32266:
32262:
32254:
32250:
32242:
32238:
32215:
32211:
32206:
32199:
32195:
32187:
32180:
32167:
32163:
32155:
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32143:
32139:
32131:
32127:
32119:
32115:
32107:
32103:
32063:
32059:
32051:
32047:
32027:
32023:
32021:
32018:
32017:
32000:
31996:
31994:
31991:
31990:
31985:
31981:
31973:
31969:
31946:
31942:
31930:
31926:
31924:
31921:
31920:
31915:
31911:
31903:
31899:
31891:
31884:
31879:
31875:
31871:
31867:
31863:
31859:
31848:
31844:
31832:
31828:
31813:
31809:
31801:
31797:
31789:
31785:
31777:
31768:
31761:
31757:
31753:
31748:
31744:
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31732:
31724:
31720:
31704:
31700:
31692:
31688:
31680:
31673:
31665:
31661:
31653:
31649:
31641:
31637:
31625:
31621:
31613:
31609:
31601:
31597:
31589:
31585:
31577:
31573:
31561:
31554:
31546:
31542:
31534:
31530:
31518:
31514:
31506:
31502:
31494:
31490:
31482:
31478:
31470:
31466:
31462:
31457:
31439:right hand rule
31428:
31424:
31411:
31409:
31373:
31369:
31358:
31355:
31354:
31352:
31348:
31312:
31308:
31297:
31294:
31293:
31291:
31287:
31278:
31276:
31247:
31243:
31232:
31229:
31228:
31226:
31222:
31217:
31213:
31200:
31198:
31166:
31162:
31151:
31148:
31147:
31145:
31141:
31132:
31130:
31101:
31097:
31086:
31083:
31082:
31080:
31076:
31067:
31065:
31036:
31032:
31021:
31018:
31017:
31015:
31011:
31002:
31000:
30968:
30964:
30953:
30950:
30949:
30947:
30943:
30938:
30934:
30904:
30902:
30893:
30891:
30885:
30881:
30864:
30860:
30847:
30845:
30813:
30809:
30798:
30795:
30794:
30792:
30788:
30766:
30763:
30762:
30743:
30740:
30739:
30698:
30681:
30676:
30673:
30672:
30670:
30666:
30623:
30606:
30601:
30598:
30597:
30580:
30578:
30538:
30521:
30516:
30513:
30512:
30484:
30467:
30450:
30447:
30446:
30445:orientation is
30428:
30424:
30415:
30413:
30384:
30380:
30369:
30366:
30365:
30363:
30359:
30350:
30348:
30316:
30312:
30301:
30298:
30297:
30295:
30291:
30266:
30263:
30262:
30243:
30238:
30235:
30234:
30215:
30212:
30211:
30209:
30205:
30199:
30195:
30186:
30184:
30155:
30151:
30140:
30137:
30136:
30134:
30130:
30109:
30100:
30094:
30090:
30068:
30064:
30062:
30059:
30058:
30034:
30030:
30028:
30025:
30024:
30017:
30013:
30004:
30002:
29970:
29966:
29955:
29952:
29951:
29949:
29945:
29936:
29934:
29905:
29901:
29890:
29887:
29886:
29884:
29880:
29871:
29869:
29837:
29833:
29822:
29819:
29818:
29816:
29812:
29807:
29803:
29772:
29763:
29757:
29753:
29743:great triangles
29737:
29735:
29729:
29722:
29713:
29711:
29682:
29678:
29667:
29664:
29663:
29661:
29657:
29643:great triangles
29620:
29618:
29609:
29607:
29601:
29594:
29589:
29585:
29560:
29556:
29539:
29530:
29501:
29486:
29471:
29456:
29451:
29448:
29447:
29430:
29428:
29402:
29387:
29372:
29357:
29352:
29349:
29348:
29308:
29305:
29304:
29290:
29283:
29262:
29258:
29257:
29252:
29249:
29248:
29247:planes and the
29231:
29227:
29225:
29222:
29221:
29203:
29199:
29198:
29193:
29190:
29189:
29172:
29168:
29166:
29163:
29162:
29143:
29140:
29139:
29121:
29117:
29116:
29111:
29108:
29107:
29105:
29086:
29060:
29057:
29056:
29034:
29031:
29030:
29028:
29021:
29015:
29011:
29002:
28988:
28986:
28972:
28970:
28967:
28960:
28951:
28947:
28941:
28937:
28885:-polytope, the
28880:
28876:
28859:
28852:
28843:
28839:
28831:
28829:
28824:
28822:
28816:
28814:
28807:
28800:
28791:
28787:
28779:
28775:
28766:
28762:
28752:
28745:
28736:
28734:
28732:
28728:
28716:
28714:
28709:
28707:
28702:
28700:
28695:
28693:
28691:
28687:
28679:
28677:
28670:
28661:
28656:
28649:
28644:
28640:
28621:
28617:
28613:rotation alone.
28611:
28607:
28602:
28598:
28589:
28585:
28551:
28538:
28532:
28528:
28520:
28518:
28513:
28511:
28509:
28505:
28483:
28476:
28459:
28455:
28450:
28446:
28437:
28433:
28419:
28415:
28405:
28396:
28368:
28364:
28336:
28334:
28331:
28330:
28329:
28307:
28303:
28301:
28298:
28297:
28291:
28287:
28282:
28278:
28273:
28265:
28261:
28236:
28233:
28232:
28210:
28207:
28206:
28194:simple rotation
28182:
28173:
28155:
28146:
28116:
28113:
28112:
28090:
28087:
28086:
28084:
28080:
28067:
28056:
28050:great 2-spheres
28039:
28032:
28016:
28014:
28012:
27991:
27987:
27976:
27966:
27962:
27957:
27953:
27949:
27942:
27940:
27935:
27933:
27931:
27927:
27923:
27919:
27912:
27910:
27908:
27904:
27896:
27892:
27888:
27884:
27880:
27876:
27872:
27868:
27864:
27860:
27853:
27851:
27850:along a second
27849:
27845:
27841:
27837:
27833:
27826:
27824:
27822:
27818:
27814:
27810:
27801:
27793:
27791:
27782:
27780:
27775:
27773:
27771:
27760:
27748:
27746:
27744:
27740:
27737:chords apart, P
27733:
27731:
27726:
27722:
27714:
27710:
27706:
27702:
27689:
27687:
27682:
27680:
27678:
27674:
27667:
27665:
27660:
27658:
27653:
27649:
27642:
27640:
27635:
27633:
27625:
27623:
27621:
27617:
27613:
27609:
27604:
27600:
27596:
27592:
27580:
27578:
27569:
27567:
27558:
27556:
27554:
27547:
27533:
27514:
27504:
27500:
27368:black or white:
27364:
27360:
27331:
27302:
27276:
27267:
27258:
27254:
27245:
27241:
27233:
27231:
27226:
27224:
27211:
27209:
27200:
27198:
27193:
27191:
27186:
27184:
27183:
27176:
27160:
27158:
27156:
27149:
27147:
27141:
27121:
27112:
27080:The adjectives
27079:
27068:
27054:simple rotation
27046:double rotation
27038:
27034:
27021:
27006:
26937:
26926:
26909:
26907:
26880:
26863:
26841:
26832:
26804:
26791:
26776:double rotation
26768:
26764:
26700:
26696:
26670:
26666:
26658:
26656:
26649:
26646:
26641:
26639:
26638:
26637:
26635:
26632:
26628:
26622:
26618:
26603:
26599:
26586:
26577:
26568:
26566:
26561:
26559:
26557:
26548:
26539:
26535:
26527:
26525:
26522:
26518:
26512:
26508:
26503:
26499:
26494:
26490:
26449:
26442:
26434:
26432:
26431:Each of the 72
26430:
26419:
26410:
26408:
26405:
26401:
26395:
26384:
26378:
26369:
26360:
26356:
26346:
26344:
26327:
26325:
26318:
26316:
26311:
26309:
26304:
26302:
26295:
26293:
26286:
26284:
26278:
26271:
26265:
26261:
26255:
26251:
26229:
26225:
26213:
26209:
26199:
26190:
26177:
26166:
26160:
26156:
26151:
26147:
26109:
26090:
26051:
26040:
26032:
26030:
26025:
26023:
26018:
26016:
26011:
26009:
26007:
26003:
25920:
25891:
25886:
25882:
25877:
25873:
25859:
25843:
25832:
25782:
25773:
25763:
25756:
25746:
25735:
25722:
25718:
25710:
25708:
25703:
25701:
25696:
25694:
25689:
25687:
25681:
25679:
25674:
25672:
25666:
25664:
25662:
25658:
25629:
25612:
25588:
25583:
25582:
25580:
25577:
25576:
25559:
25554:
25553:
25551:
25548:
25547:
25519:
25514:
25513:
25511:
25508:
25507:
25487:
25482:
25481:
25479:
25476:
25475:
25473:
25469:
25455:
25442:
25434:
25432:
25427:
25425:
25420:
25418:
25413:
25411:
25405:
25398:
25388:
25384:
25372:circles on the
25336:
25299:
25291:
25289:
25284:
25282:
25277:
25275:
25270:
25268:
25266:
25262:
25254:
25252:
25247:
25245:
25240:
25238:
25233:
25231:
25226:
25224:
25222:
25211:
25206:
25202:
25193:
25186:
25178:
25176:
25167:
25165:
25152:
25150:
25145:
25143:
25133:
25126:
25108:
25097:
25093:of each other.)
25061:
25059:
25046:
25044:
25039:
25037:
25032:
25030:
24988:
24986:
24969:
24967:
24962:
24960:
24955:
24953:
24948:
24946:
24927:
24918:
24907:
24894:
24881:
24866:
24858:
24856:
24851:
24849:
24843:
24839:
24835:
24831:
24829:
24821:
24818:
24815:
24814:
24812:
24808:
24800:
24797:
24794:
24793:
24791:
24782:
24779:
24776:
24775:
24773:
24769:
24761:
24758:
24755:
24754:
24752:
24747:
24739:
24736:
24733:
24732:
24730:
24721:
24718:
24715:
24714:
24712:
24703:
24700:
24697:
24696:
24694:
24685:
24682:
24679:
24678:
24676:
24673:
24665:
24662:
24659:
24658:
24656:
24652:
24644:
24641:
24638:
24637:
24635:
24631:
24623:
24620:
24617:
24616:
24614:
24610:
24602:
24599:
24596:
24595:
24593:
24589:
24585:
24577:
24574:
24571:
24570:
24568:
24559:
24556:
24553:
24552:
24550:
24546:
24538:
24535:
24532:
24531:
24529:
24520:
24517:
24514:
24513:
24511:
24507:
24505:
24501:
24497:
24493:
24489:
24487:
24482:
24480:
24478:
24471:
24463:
24461:
24459:
24452:
24444:
24442:
24437:
24435:
24430:
24428:
24423:
24421:
24419:
24394:
24367:
24365:
24360:
24358:
24339:
24316:
24287:
24284:
24283:
24264:
24261:
24260:
24238:
24235:
24234:
24215:
24212:
24211:
24201:
24174:
24156:
24145:
24133:
24131:
24130:geodesics: two
24125:
24123:
24121:
24112:
24104:
24102:
24096:
24092:
24088:
24084:
24082:
24074:
24071:
24068:
24067:
24065:
24061:
24053:
24050:
24047:
24046:
24044:
24035:
24032:
24029:
24028:
24026:
24022:
24014:
24011:
24008:
24007:
24005:
24000:
23992:
23989:
23986:
23985:
23983:
23974:
23971:
23968:
23967:
23965:
23956:
23953:
23950:
23949:
23947:
23938:
23935:
23932:
23931:
23929:
23926:
23918:
23915:
23912:
23911:
23909:
23905:
23897:
23894:
23891:
23890:
23888:
23884:
23876:
23873:
23870:
23869:
23867:
23863:
23855:
23852:
23849:
23848:
23846:
23842:
23838:
23830:
23827:
23824:
23823:
23821:
23812:
23809:
23806:
23805:
23803:
23799:
23791:
23788:
23785:
23784:
23782:
23773:
23770:
23767:
23766:
23764:
23760:
23758:
23754:
23750:
23746:
23742:
23740:
23726:
23713:
23693:
23691:
23688:
23679:
23671:
23669:
23664:
23662:
23657:
23655:
23640:
23621:
23608:
23604:
23600:
23596:
23594:
23590:
23586:
23582:
23578:
23574:
23570:
23566:
23562:
23560:
23556:
23552:
23548:
23544:
23542:
23534:
23532:
23522:
23518:
23480:
23469:
23440:
23437:
23436:
23414:
23411:
23410:
23388:
23385:
23384:
23374:
23361:
23347:
23334:
23328:
23321:
23310:
23306:
23296:
23292:
23288:
23284:
23280:
23276:
23272:
23270:
23266:
23262:
23258:
23254:
23250:
23246:
23242:
23238:
23236:
23235:. For example:
23229:
23227:
23224:
23220:
23206:
23202:
23187:
23183:
23154:
23147:
23125:
23102:
23081:
23077:
23073:
23056:
23044:
23033:
23022:
23011:
23000:
22989:
22978:
22969:
22960:
22949:
22938:
22927:
22916:
22905:
22894:
22674:
22601:
22593:
22584:
22575:
22573:
22564:
22562:
22553:
22544:
22542:
22533:
22531:
22522:
22520:
22507:
22496:
22491:
22486:
22481:
22476:
22471:
22466:
22464:
22457:
22452:
22447:
22442:
22437:
22432:
22427:
22425:
22418:
22413:
22408:
22403:
22398:
22396:
22394:
22389:
22384:
22379:
22374:
22369:
22364:
22359:
22357:
22350:
22345:
22340:
22335:
22330:
22328:
22326:
22321:
22316:
22311:
22306:
22301:
22296:
22291:
22289:
22282:
22277:
22272:
22267:
22262:
22257:
22252:
22250:
22243:
22238:
22233:
22228:
22223:
22221:
22219:
22214:
22209:
22204:
22199:
22194:
22189:
22184:
22182:
22175:
22170:
22165:
22160:
22155:
22153:
22151:
22146:
22141:
22136:
22131:
22126:
22121:
22116:
22114:
22107:
22102:
22097:
22092:
22087:
22085:
22083:
22078:
22073:
22068:
22063:
22058:
22053:
22048:
22046:
22039:
22034:
22029:
22024:
22019:
22017:
22015:
22010:
22005:
22000:
21995:
21990:
21985:
21980:
21978:
21973:
21964:
21957:
21950:
21943:
21936:
21929:
21922:
21915:
21832:
21759:
21751:
21742:
21733:
21731:
21722:
21720:
21711:
21702:
21700:
21691:
21689:
21680:
21678:
21665:
21654:
21649:
21644:
21639:
21634:
21629:
21624:
21622:
21615:
21610:
21605:
21600:
21595:
21590:
21585:
21583:
21576:
21571:
21566:
21561:
21556:
21551:
21546:
21544:
21537:
21532:
21527:
21522:
21517:
21515:
21513:
21508:
21503:
21498:
21493:
21488:
21483:
21478:
21476:
21469:
21464:
21459:
21454:
21449:
21444:
21439:
21437:
21430:
21425:
21420:
21415:
21410:
21405:
21400:
21398:
21391:
21386:
21381:
21376:
21371:
21366:
21361:
21359:
21352:
21347:
21342:
21337:
21332:
21330:
21328:
21323:
21318:
21313:
21308:
21303:
21298:
21293:
21291:
21284:
21279:
21274:
21269:
21264:
21259:
21254:
21252:
21247:
21238:
21231:
21224:
21217:
21210:
21203:
21196:
21189:
21092:
21041:
20962:
20884:
20806:
20726:
20715:
20710:
20705:
20700:
20695:
20690:
20685:
20683:
20676:
20671:
20666:
20661:
20656:
20651:
20646:
20644:
20637:
20632:
20627:
20622:
20617:
20612:
20607:
20605:
20598:
20593:
20588:
20583:
20578:
20573:
20568:
20566:
20559:
20554:
20549:
20544:
20539:
20534:
20529:
20527:
20520:
20515:
20510:
20505:
20500:
20495:
20490:
20488:
20481:
20476:
20471:
20466:
20461:
20456:
20451:
20449:
20442:
20437:
20432:
20427:
20422:
20417:
20412:
20410:
20403:
20398:
20393:
20388:
20383:
20378:
20373:
20371:
20364:
20359:
20354:
20349:
20344:
20339:
20334:
20332:
20327:
20319:
20310:
20301:
20292:
20290:
20281:
20279:
20270:
20268:
20259:
20257:
20245:
20243:
20230:
20152:
20145:
20138:
20131:
20124:
20120:
20113:
20109:
20102:
20098:
20091:
20021:
20016:
20011:
20006:
20004:
20003:
19998:
19993:
19988:
19983:
19978:
19976:
19969:
19964:
19959:
19954:
19952:
19951:
19946:
19941:
19936:
19931:
19926:
19924:
19917:
19912:
19907:
19902:
19900:
19899:
19894:
19889:
19884:
19879:
19874:
19872:
19865:
19860:
19855:
19850:
19848:
19847:
19842:
19837:
19832:
19827:
19822:
19820:
19813:
19808:
19803:
19798:
19793:
19791:
19790:
19785:
19780:
19775:
19770:
19765:
19763:
19756:
19751:
19746:
19741:
19736:
19734:
19733:
19728:
19723:
19718:
19713:
19708:
19706:
19699:
19694:
19689:
19684:
19679:
19677:
19676:
19671:
19666:
19661:
19656:
19651:
19649:
19642:
19637:
19632:
19627:
19622:
19620:
19619:
19614:
19609:
19604:
19599:
19594:
19592:
19586:
19578:
19543:snub octahedron
19519:cell-transitive
19485:
19471:
19466:
19461:
19459:
19457:
19453:
19450:
19437:
19432:
19427:
19425:
19423:
19419:
19416:
19407:
19361:
19356:
19351:
19349:
19347:
19344:
19334:
19329:
19324:
19322:
19320:
19317:
19307:
19302:
19297:
19292:
19287:
19282:
19277:
19275:
19262:
19257:
19252:
19250:
19248:
19244:
19235:
19230:
19225:
19223:
19221:
19217:
19208:
19203:
19198:
19193:
19188:
19183:
19178:
19176:
19162:
19159:
19155:
19151:
19132:
19127:
19126:
19124:
19121:
19120:
19114:
19109:
19104:
19102:
19096:
19091:
19086:
19084:
19082:
19078:
19071:
19068:
19060:
19055:
19050:
19048:
19042:
19037:
19032:
19030:
19028:
19024:
19016:
18983:
18937:, , order 1152
18936:
18929:
18922:
18898:
18894:
18890:
18872:
18864:
18852:
18847:
18837:
18828:
18817:
18806:
18785:
18775:
18766:
18733:
18672:
18660:described above
18654:
18652:
18502:cross-section.
18496:described above
18479:
18476:
18474:
18432:
18416:
18378:
18374:
18363:
18360:
18359:
18340:
18337:
18336:
18317:
18314:
18313:
18302:
18299:
18296:
18295:
18293:
18264:
18260:
18249:
18246:
18245:
18235:right rotations
18197:
18193:
18182:
18179:
18178:
18146:
18142:
18131:
18128:
18127:
18092:
18088:
18077:
18074:
18073:
18044:
18040:
18029:
18026:
18025:
17990:
17986:
17975:
17972:
17971:
17952:
17949:
17948:
17929:
17926:
17925:
17896:
17892:
17881:
17878:
17877:
17859:
17855:
17854:
17849:
17846:
17845:
17815:
17811:
17810:
17799:
17796:
17795:
17782:
17780:
17769:
17766:
17763:
17762:
17760:
17751:
17749:
17738:
17735:
17732:
17731:
17729:
17720:
17718:
17666:
17663:
17662:
17658:
17636:
17633:
17631:
17628:
17627:
17626:
17620:
17576:
17573:
17572:
17568:
17549:
17546:
17544:
17541:
17540:
17539:
17533:
17497:
17493:
17482:
17479:
17478:
17474:
17443:
17440:
17438:
17435:
17434:
17433:
17427:
17413:
17411:
17400:
17397:
17394:
17393:
17391:
17382:
17380:
17369:
17366:
17363:
17362:
17360:
17351:
17349:
17300:
17297:
17296:
17292:
17273:
17270:
17268:
17265:
17264:
17263:
17257:
17213:
17210:
17209:
17205:
17186:
17183:
17181:
17178:
17177:
17176:
17170:
17137:
17133:
17122:
17119:
17118:
17114:
17086:
17083:
17081:
17078:
17077:
17076:
17070:
17056:
17054:
17043:
17040:
17037:
17036:
17034:
17025:
17023:
17012:
17009:
17006:
17005:
17003:
16994:
16992:
16981:
16978:
16975:
16974:
16972:
16929:
16926:
16925:
16921:
16902:
16899:
16897:
16894:
16893:
16892:
16886:
16842:
16839:
16838:
16834:
16815:
16812:
16810:
16807:
16806:
16805:
16799:
16766:
16762:
16751:
16748:
16747:
16743:
16715:
16712:
16710:
16707:
16706:
16705:
16699:
16685:
16683:
16672:
16669:
16666:
16665:
16663:
16654:
16652:
16641:
16638:
16635:
16634:
16632:
16623:
16621:
16569:
16566:
16565:
16561:
16539:
16536:
16534:
16531:
16530:
16529:
16523:
16479:
16476:
16475:
16471:
16452:
16449:
16447:
16444:
16443:
16442:
16436:
16400:
16396:
16385:
16382:
16381:
16377:
16346:
16343:
16341:
16338:
16337:
16336:
16330:
16316:
16314:
16303:
16300:
16297:
16296:
16294:
16285:
16283:
16272:
16269:
16266:
16265:
16263:
16254:
16252:
16241:
16238:
16235:
16234:
16232:
16203:
16188:
16173:
16158:
16153:
16150:
16149:
16145:
16126:
16123:
16121:
16118:
16117:
16116:
16110:
16066:
16063:
16062:
16058:
16039:
16036:
16034:
16031:
16030:
16029:
16023:
15990:
15986:
15975:
15972:
15971:
15967:
15939:
15936:
15934:
15931:
15930:
15929:
15923:
15909:
15907:
15896:
15893:
15890:
15889:
15887:
15878:
15876:
15865:
15862:
15859:
15858:
15856:
15847:
15845:
15808:
15791:
15774:
15771:
15770:
15766:
15747:
15744:
15742:
15739:
15738:
15737:
15731:
15699:
15682:
15665:
15662:
15661:
15657:
15638:
15635:
15633:
15630:
15629:
15628:
15622:
15589:
15585:
15574:
15571:
15570:
15566:
15538:
15535:
15533:
15530:
15529:
15528:
15522:
15508:
15506:
15491:
15489:
15478:
15475:
15472:
15471:
15469:
15460:
15458:
15447:
15444:
15441:
15440:
15438:
15407:
15387:
15367:
15364:
15363:
15359:
15337:
15334:
15332:
15329:
15328:
15327:
15321:
15277:
15260:
15255:
15252:
15251:
15247:
15228:
15225:
15223:
15220:
15219:
15218:
15212:
15176:
15172:
15161:
15158:
15157:
15153:
15122:
15119:
15117:
15114:
15113:
15112:
15106:
15092:
15090:
15079:
15076:
15073:
15072:
15070:
15061:
15059:
15048:
15045:
15042:
15041:
15039:
15030:
15028:
14979:
14959:
14951:
14948:
14947:
14943:
14921:
14918:
14916:
14913:
14912:
14911:
14905:
14861:
14844:
14839:
14836:
14835:
14831:
14812:
14809:
14807:
14804:
14803:
14802:
14796:
14760:
14756:
14745:
14742:
14741:
14737:
14706:
14703:
14701:
14698:
14697:
14696:
14690:
14676:
14674:
14663:
14660:
14657:
14656:
14654:
14645:
14643:
14632:
14629:
14626:
14625:
14623:
14614:
14612:
14563:
14546:
14541:
14538:
14537:
14533:
14514:
14511:
14509:
14506:
14505:
14504:
14498:
14454:
14437:
14432:
14429:
14428:
14424:
14405:
14402:
14400:
14397:
14396:
14395:
14389:
14356:
14352:
14341:
14338:
14337:
14333:
14305:
14302:
14300:
14297:
14296:
14295:
14289:
14275:
14273:
14262:
14259:
14256:
14255:
14253:
14244:
14242:
14231:
14228:
14225:
14224:
14222:
14213:
14211:
14200:
14197:
14194:
14193:
14191:
14145:
14142:
14141:
14137:
14115:
14112:
14110:
14107:
14106:
14105:
14099:
14069:
14054:
14039:
14024:
14019:
14016:
14015:
14011:
13992:
13989:
13987:
13984:
13983:
13982:
13976:
13940:
13936:
13925:
13922:
13921:
13917:
13886:
13883:
13881:
13878:
13877:
13876:
13870:
13856:
13854:
13843:
13840:
13837:
13836:
13834:
13825:
13823:
13812:
13809:
13806:
13805:
13803:
13794:
13792:
13781:
13778:
13775:
13774:
13772:
13729:
13726:
13725:
13721:
13702:
13699:
13697:
13694:
13693:
13692:
13686:
13656:
13641:
13626:
13611:
13606:
13603:
13602:
13598:
13579:
13576:
13574:
13571:
13570:
13569:
13563:
13530:
13526:
13515:
13512:
13511:
13507:
13479:
13476:
13474:
13471:
13470:
13469:
13463:
13449:
13447:
13436:
13433:
13430:
13429:
13427:
13418:
13416:
13405:
13402:
13399:
13398:
13396:
13387:
13385:
13350:
13332:
13314:
13296:
13288:
13285:
13284:
13280:
13258:
13255:
13253:
13250:
13249:
13248:
13242:
13212:
13197:
13182:
13167:
13162:
13159:
13158:
13154:
13135:
13132:
13130:
13127:
13126:
13125:
13119:
13083:
13079:
13068:
13065:
13064:
13060:
13029:
13026:
13024:
13021:
13020:
13019:
13013:
12999:
12997:
12986:
12983:
12980:
12979:
12977:
12968:
12966:
12955:
12952:
12949:
12948:
12946:
12937:
12935:
12900:
12885:
12870:
12855:
12850:
12847:
12846:
12842:
12823:
12820:
12818:
12815:
12814:
12813:
12807:
12777:
12762:
12747:
12732:
12727:
12724:
12723:
12719:
12700:
12697:
12695:
12692:
12691:
12690:
12684:
12651:
12647:
12636:
12633:
12632:
12628:
12600:
12597:
12595:
12592:
12591:
12590:
12584:
12570:
12568:
12557:
12554:
12551:
12550:
12548:
12539:
12537:
12526:
12523:
12520:
12519:
12517:
12508:
12506:
12495:
12492:
12489:
12488:
12486:
12457:
12442:
12427:
12412:
12404:
12401:
12400:
12396:
12374:
12371:
12369:
12366:
12365:
12364:
12358:
12328:
12313:
12298:
12283:
12278:
12275:
12274:
12270:
12251:
12248:
12246:
12243:
12242:
12241:
12235:
12199:
12195:
12184:
12181:
12180:
12176:
12145:
12142:
12140:
12137:
12136:
12135:
12129:
12115:
12113:
12102:
12099:
12096:
12095:
12093:
12084:
12082:
12071:
12068:
12065:
12064:
12062:
12053:
12051:
12040:
12037:
12034:
12033:
12031:
12002:
11984:
11966:
11948:
11943:
11940:
11939:
11935:
11916:
11913:
11911:
11908:
11907:
11906:
11900:
11870:
11855:
11840:
11825:
11820:
11817:
11816:
11812:
11793:
11790:
11788:
11785:
11784:
11783:
11777:
11744:
11740:
11729:
11726:
11725:
11721:
11693:
11690:
11688:
11685:
11684:
11683:
11677:
11652:
11649:
11648:
11627:
11624:
11623:
11619:Rotation class
11609:
11604:symmetry group
11602:of the 24-cell
11567:
11549:
11531:
11513:
11508:
11505:
11504:
11478:
11463:
11448:
11433:
11428:
11425:
11424:
11402:
11399:
11398:
11379:
11376:
11375:
11368:
11365:
11362:
11361:
11359:
11330:
11326:
11315:
11312:
11311:
11269:
11254:
11252:
11217:
11190:
11159:
11157:
11154:
11153:
11126:
11124:
11121:
11120:
11093:
11091:
11088:
11087:
11060:
11058:
11055:
11054:
11027:
11025:
11022:
11021:
10994:
10992:
10989:
10988:
10961:
10959:
10956:
10955:
10937:
10934:
10933:
10902:
10900:
10897:
10896:
10869:
10867:
10864:
10863:
10836:
10834:
10831:
10830:
10799:
10797:
10794:
10793:
10766:
10764:
10761:
10760:
10733:
10731:
10728:
10727:
10708:
10703:
10698:
10693:
10688:
10683:
10678:
10676:
10616:
10614:
10611:
10610:
10588:
10582:
10578:
10572:
10569:
10567:
10564:
10563:
10519:
10517:
10514:
10513:
10491:
10485:
10481:
10475:
10472:
10470:
10467:
10466:
10422:
10420:
10417:
10416:
10394:
10388:
10384:
10378:
10375:
10373:
10370:
10369:
10340:
10337:
10336:
10314:
10308:
10304:
10298:
10295:
10293:
10290:
10289:
10249:
10247:
10244:
10243:
10212:
10210:
10207:
10206:
10167:
10165:
10162:
10161:
10139:
10133:
10129:
10123:
10120:
10118:
10115:
10114:
10086:
10084:
10081:
10080:
10049:
10047:
10044:
10043:
10004:
10002:
9999:
9998:
9976:
9970:
9966:
9960:
9957:
9955:
9952:
9951:
9923:
9921:
9918:
9917:
9886:
9884:
9881:
9880:
9841:
9839:
9836:
9835:
9813:
9807:
9803:
9797:
9794:
9792:
9789:
9788:
9748:
9746:
9743:
9742:
9711:
9709:
9706:
9705:
9666:
9664:
9661:
9660:
9629:
9627:
9624:
9623:
9592:
9590:
9587:
9586:
9547:
9545:
9542:
9541:
9510:
9508:
9505:
9504:
9473:
9471:
9468:
9467:
9428:
9426:
9423:
9422:
9377:
9374:
9372:
9369:
9368:
9337:
9335:
9332:
9331:
9307:
9304:
9303:
9276:
9263:
9261:
9256:
9254:
9246:
9244:
9232:
9230:
9228:
9206:
9204:
9192:
9190:
9146:
9140:
9132:
9124:
9116:
9105:Diameter chords
9090:Petrie polygons
9043:
9041:
9036:
9034:
9025:
9023:
9017:
9015:
9012:
9003:
9001:
8995:
8993:
8984:
8982:
8962:
8960:
8951:
8949:
8944:
8942:
8931:
8906:
8898:
8896:
8891:
8889:
8880:
8878:
8872:
8870:
8865:
8863:
8854:
8852:
8847:
8845:
8828:
8826:
8821:
8819:
8817:
8801:
8761:simple rotation
8745:
8726:
8724:
8719:
8717:
8712:
8710:
8705:
8703:
8700:
8672:geodesic fibers
8661:
8616:
8588:
8557:
8555:
8543:
8541:
8536:
8534:
8529:
8527:
8514:
8512:
8507:
8505:
8500:
8498:
8444:
8432:right hand rule
8400:double rotation
8388:double rotation
8380:
8358:simple rotation
8323:simple rotation
8315:
8303:simple rotation
8299:double rotation
8292:
8283:double rotation
8271:
8243:
8211:
8195:lattice packing
8180:SchlΓ€fli symbol
8176:
8172:
8163:Euclidean space
8155:
8144:
8119:
8111:
8088:
8067:
8056:
8048:
8033:
8031:
8029:
8024:
8011:
8007:
7978:
7968:
7963:
7952:
7927:
7926:
7923:
7922:
7917:
7912:
7907:
7901:
7900:
7895:
7890:
7885:
7879:
7878:
7873:
7868:
7863:
7857:
7856:
7851:
7846:
7841:
7834:
7827:
7826:
7824:
7821:
7820:
7810:
7799:
7797:
7792:
7790:
7785:
7783:
7778:
7776:
7761:
7738:
7736:
7720:
7718:
7709:
7707:
7703:
7690:
7678:
7657:
7652:
7647:
7642:
7637:
7632:
7627:
7625:
7610:
7608:
7600:
7597:
7593:
7592:
7591:
7589:
7583:
7581:
7573:
7571:
7566:
7564:
7561:
7546:
7544:
7539:
7537:
7528:
7526:
7523:
7500:
7498:
7495:
7471:
7434:
7431:
7428:
7427:
7425:
7418:
7415:
7412:
7411:
7409:
7402:
7399:
7396:
7395:
7393:
7386:
7383:
7380:
7379:
7377:
7370:
7353:
7339:
7337:
7313:
7311:
7306:
7304:
7299:
7297:
7292:
7290:
7285:
7283:
7278:
7276:
7256:
7254:
7246:
7244:
7238:
7236:
7227:
7225:
7216:
7214:
7209:
7207:
7202:
7200:
7191:
7189:
7154:
7150:
7144:
7104:
7100:
7093:
7089:
7057:
7044:of the 24-cell
7029:
7027:
7018:
7016:
7008:
7006:
6981:
6971:
6969:
6964:
6962:
6957:
6955:
6950:
6948:
6943:
6941:
6936:
6934:
6929:
6927:
6918:
6916:
6911:
6909:
6904:
6902:
6897:
6895:
6883:
6877:
6875:
6867:
6864:
6860:
6858:
6857:
6855:
6849:
6847:
6839:
6836:
6832:
6831:
6830:
6828:
6819:
6816:
6812:
6811:
6810:
6808:
6799:
6797:
6792:
6790:
6785:
6783:
6778:
6776:
6761:
6751:
6749:
6746:
6739:
6735:
6721:to each other.
6682:
6674:
6673:
6670:
6668:
6665:
6664:
6632:
6624:
6623:
6620:
6618:
6615:
6614:
6595:
6592:
6591:
6572:
6569:
6568:
6536:
6534:
6531:
6530:
6504:
6486:
6482:
6481:
6467:
6465:
6462:
6461:
6421:
6415:
6411:
6407:
6399:
6389:
6386:
6382:
6377:
6374:
6373:
6339:
6335:
6331:
6323:
6319:
6314:
6311:
6310:
6272:
6268:
6263:
6260:
6259:
6240:
6237:
6236:
6200:
6196:
6191:
6188:
6187:
6149:
6145:
6142:
6138:
6133:
6130:
6129:
6090:
6086:
6082:
6074:
6061:
6057:
6052:
6049:
6048:
6014:
6010:
6006:
5998:
5994:
5989:
5986:
5985:
5947:
5943:
5938:
5935:
5934:
5915:
5912:
5911:
5873:
5869:
5864:
5861:
5860:
5822:
5818:
5815:
5811:
5806:
5803:
5802:
5764:
5760:
5756:
5754:
5751:
5750:
5717:
5713:
5709:
5707:
5704:
5703:
5669:
5667:
5664:
5663:
5637:
5635:
5632:
5631:
5605:
5603:
5600:
5599:
5573:
5571:
5568:
5567:
5531:
5525:
5521:
5520:
5514:
5512:
5509:
5508:
5476:
5474:
5471:
5470:
5451:
5448:
5447:
5428:
5425:
5424:
5397:
5395:
5392:
5391:
5357:
5355:
5352:
5351:
5327:
5324:
5323:
5304:
5301:
5300:
5281:
5278:
5277:
5258:
5255:
5254:
5235:
5232:
5231:
5212:
5209:
5208:
5154:Petrie polygons
5084:10-dodecahedron
5039:600 tetrahedra
5016:1200 triangles
4993:720 pentagonal
4962:16 tetrahedral
4896:
4893:
4890:
4889:
4887:
4881:
4878:
4875:
4874:
4872:
4866:
4863:
4860:
4859:
4857:
4851:
4848:
4845:
4844:
4842:
4836:
4833:
4830:
4829:
4827:
4821:
4818:
4815:
4814:
4812:
4804:
4801:
4798:
4797:
4795:
4789:
4786:
4783:
4782:
4780:
4774:
4771:
4768:
4767:
4765:
4759:
4756:
4753:
4752:
4750:
4744:
4741:
4738:
4737:
4735:
4729:
4726:
4723:
4722:
4720:
4712:
4709:
4706:
4705:
4703:
4697:
4694:
4691:
4690:
4688:
4682:
4679:
4676:
4675:
4673:
4667:
4664:
4661:
4660:
4658:
4652:
4649:
4646:
4645:
4643:
4637:
4634:
4631:
4630:
4628:
4620:
4617:
4614:
4613:
4611:
4605:
4602:
4599:
4598:
4596:
4590:
4587:
4584:
4583:
4581:
4575:
4572:
4569:
4568:
4566:
4560:
4557:
4554:
4553:
4551:
4545:
4542:
4539:
4538:
4536:
4528:
4525:
4522:
4521:
4519:
4513:
4510:
4507:
4506:
4504:
4498:
4495:
4492:
4491:
4489:
4483:
4480:
4477:
4476:
4474:
4468:
4465:
4462:
4461:
4459:
4453:
4450:
4447:
4446:
4444:
4436:
4433:
4430:
4429:
4427:
4421:
4418:
4415:
4414:
4412:
4406:
4403:
4400:
4399:
4397:
4391:
4388:
4385:
4384:
4382:
4376:
4373:
4370:
4369:
4367:
4361:
4358:
4355:
4354:
4352:
4340:
4335:
4330:
4325:
4320:
4315:
4310:
4308:
4301:
4296:
4291:
4286:
4281:
4276:
4271:
4269:
4262:
4257:
4252:
4247:
4242:
4237:
4232:
4230:
4223:
4218:
4213:
4208:
4203:
4198:
4193:
4191:
4184:
4179:
4174:
4169:
4164:
4159:
4154:
4152:
4145:
4140:
4135:
4130:
4125:
4120:
4115:
4113:
4109:Coxeter mirrors
4084:SchlΓ€fli symbol
4076:
4070:
4062:
4056:
4048:
4046:
4038:
4032:
4024:
4018:
4010:
4004:
3992:
3984:
3976:
3968:
3937:
3935:
3888:
3870:
3852:
3834:
3830:
3826:
3824:
3821:
3820:
3771:
3767:
3765:
3762:
3761:
3742:
3740:
3732:The 24-cell is
3730:
3685:
3677:
3676:
3673:
3671:
3668:
3667:
3635:
3627:
3626:
3623:
3621:
3618:
3617:
3598:
3595:
3594:
3575:
3572:
3571:
3539:
3537:
3534:
3533:
3507:
3496:
3492:
3491:
3477:
3475:
3472:
3471:
3432:
3428:
3424:
3416:
3406:
3403:
3399:
3394:
3391:
3390:
3356:
3352:
3348:
3342:
3338:
3333:
3330:
3329:
3293:
3289:
3284:
3281:
3280:
3253:
3248:
3245:
3244:
3206:
3202:
3199:
3195:
3190:
3187:
3186:
3148:
3144:
3141:
3137:
3132:
3129:
3128:
3089:
3085:
3081:
3073:
3060:
3056:
3051:
3048:
3047:
3013:
3009:
3005:
2997:
2993:
2990:
2986:
2981:
2978:
2977:
2939:
2935:
2930:
2927:
2926:
2907:
2904:
2903:
2872:
2868:
2863:
2860:
2859:
2821:
2817:
2814:
2810:
2805:
2802:
2801:
2763:
2759:
2755:
2753:
2750:
2749:
2716:
2712:
2708:
2706:
2703:
2702:
2683:
2680:
2679:
2645:
2643:
2640:
2639:
2605:
2603:
2600:
2599:
2565:
2563:
2560:
2559:
2523:
2520:
2519:
2485:
2483:
2480:
2479:
2452:
2450:
2447:
2446:
2419:
2417:
2414:
2413:
2394:
2391:
2390:
2363:
2361:
2358:
2357:
2331:
2329:
2326:
2325:
2304:
2302:
2299:
2298:
2277:
2275:
2272:
2271:
2250:
2248:
2245:
2244:
2223:
2221:
2218:
2217:
2196:
2194:
2191:
2190:
2136:Petrie polygons
2066:10-dodecahedron
2021:600 tetrahedra
1998:1200 triangles
1975:720 pentagonal
1944:16 tetrahedral
1878:
1875:
1872:
1871:
1869:
1863:
1860:
1857:
1856:
1854:
1848:
1845:
1842:
1841:
1839:
1833:
1830:
1827:
1826:
1824:
1818:
1815:
1812:
1811:
1809:
1803:
1800:
1797:
1796:
1794:
1786:
1783:
1780:
1779:
1777:
1771:
1768:
1765:
1764:
1762:
1756:
1753:
1750:
1749:
1747:
1741:
1738:
1735:
1734:
1732:
1726:
1723:
1720:
1719:
1717:
1711:
1708:
1705:
1704:
1702:
1694:
1691:
1688:
1687:
1685:
1679:
1676:
1673:
1672:
1670:
1664:
1661:
1658:
1657:
1655:
1649:
1646:
1643:
1642:
1640:
1634:
1631:
1628:
1627:
1625:
1619:
1616:
1613:
1612:
1610:
1602:
1599:
1596:
1595:
1593:
1587:
1584:
1581:
1580:
1578:
1572:
1569:
1566:
1565:
1563:
1557:
1554:
1551:
1550:
1548:
1542:
1539:
1536:
1535:
1533:
1527:
1524:
1521:
1520:
1518:
1510:
1507:
1504:
1503:
1501:
1495:
1492:
1489:
1488:
1486:
1480:
1477:
1474:
1473:
1471:
1465:
1462:
1459:
1458:
1456:
1450:
1447:
1444:
1443:
1441:
1435:
1432:
1429:
1428:
1426:
1418:
1415:
1412:
1411:
1409:
1403:
1400:
1397:
1396:
1394:
1388:
1385:
1382:
1381:
1379:
1373:
1370:
1367:
1366:
1364:
1358:
1355:
1352:
1351:
1349:
1343:
1340:
1337:
1336:
1334:
1322:
1317:
1312:
1307:
1302:
1297:
1292:
1290:
1283:
1278:
1273:
1268:
1263:
1258:
1253:
1251:
1244:
1239:
1234:
1229:
1224:
1219:
1214:
1212:
1205:
1200:
1195:
1190:
1185:
1180:
1175:
1173:
1166:
1161:
1156:
1151:
1146:
1141:
1136:
1134:
1127:
1122:
1117:
1112:
1107:
1102:
1097:
1095:
1091:Coxeter mirrors
1066:SchlΓ€fli symbol
1058:
1052:
1044:
1038:
1030:
1028:
1020:
1014:
1006:
1000:
992:
986:
974:
966:
958:
950:
933:
931:
905:
903:
894:
892:
879:
874:
869:
864:
859:
854:
849:
847:
834:
829:
824:
819:
814:
809:
804:
802:
780:
775:
774:
736:
733:
732:
719:
714:
678:
579:
572:SchlΓ€fli symbol
493:
489:
487:
483:
480:
387:
382:
377:
372:
370:
364:
359:
354:
349:
344:
342:
341:
336:
331:
326:
321:
316:
314:
308:
303:
298:
293:
288:
283:
278:
276:
275:
270:
265:
260:
255:
250:
245:
240:
238:
234:Coxeter diagram
212:
211:
205:
204:
198:
197:
190:
186:
184:
181:
180:
178:
161:
160:
148:
147:
140:
136:
134:
131:
130:
128:
123:SchlΓ€fli symbol
107:
105:
82:
71:
65:
62:
55:
35:
31:
24:
17:
12:
11:
5:
34663:
34653:
34652:
34636:
34635:
34620:
34619:
34610:
34606:
34599:
34592:
34588:
34579:
34562:
34553:
34542:
34541:
34539:
34537:
34532:
34523:
34518:
34512:
34511:
34509:
34507:
34502:
34493:
34488:
34482:
34481:
34479:
34475:
34468:
34461:
34457:
34452:
34443:
34438:
34432:
34431:
34429:
34425:
34418:
34411:
34407:
34402:
34393:
34388:
34382:
34381:
34379:
34375:
34368:
34364:
34359:
34350:
34345:
34339:
34338:
34336:
34334:
34329:
34320:
34315:
34309:
34308:
34299:
34294:
34289:
34280:
34275:
34269:
34268:
34259:
34257:
34252:
34243:
34238:
34232:
34231:
34226:
34221:
34216:
34211:
34206:
34200:
34199:
34195:
34191:
34186:
34175:
34164:
34155:
34146:
34139:
34133:
34123:
34117:
34111:
34105:
34099:
34093:
34087:
34086:
34075:
34073:
34072:
34065:
34058:
34050:
34042:
34041:
34038:
34037:
34034:
34033:
34029:
34028:
34026:
34025:
34022:
34002:
34000:
33998:
33997:
33994:
33974:
33972:
33970:
33969:
33966:
33946:
33944:
33942:
33941:
33938:
33918:
33916:
33914:
33913:
33910:
33874:
33872:
33870:
33869:
33866:
33846:
33844:
33842:
33841:
33838:
33818:
33816:
33814:
33813:
33810:
33790:
33788:
33786:
33785:
33782:
33762:
33760:
33758:
33757:
33754:
33734:
33731:
33730:
33723:
33716:
33707:
33700:
33691:
33682:
33675:
33668:
33659:
33649:
33642:
33641:
33631:
33630:
33627:
33626:
33623:
33622:
33618:
33617:
33615:
33614:
33611:
33610:hexacosichoron
33608:
33604:
33602:
33600:
33599:
33596:
33593:
33589:
33587:
33585:
33584:
33581:
33578:
33574:
33572:
33570:
33569:
33566:
33565:hexadecachoron
33563:
33559:
33557:
33555:
33554:
33551:
33548:
33544:
33542:
33540:
33539:
33536:
33533:
33529:
33526:
33525:
33520:
33515:
33510:
33505:
33500:
33492:
33485:
33484:
33474:
33473:
33466:
33465:
33458:
33451:
33443:
33437:
33436:
33431:
33419:
33414:
33407:
33406:External links
33404:
33402:
33401:
33383:(11): 113514.
33368:
33333:
33324:
33303:
33268:
33245:
33239:
33220:
33192:(10): 105304.
33181:
33135:
33120:
33105:
33090:
33084:
33055:
33049:
33036:
33027:
33007:
32995:
32983:
32978:
32960:
32940:
32924:
32912:(1β3): 59β71.
32894:
32882:
32881:
32880:
32873:
32866:
32859:
32848:
32828:
32816:
32802:
32787:
32785:
32782:
32780:
32779:
32767:
32755:
32743:Clifford torus
32731:
32686:
32674:
32662:
32650:
32638:
32626:
32614:
32602:
32590:
32578:
32564:
32551:
32545:
32532:
32525:
32521:
32508:
32496:
32484:
32472:
32460:
32448:
32436:
32424:
32412:
32408:Hopf fibration
32392:
32364:
32352:
32328:
32316:
32304:
32284:
32272:
32260:
32248:
32236:
32209:
32204:
32193:
32178:
32173:Thorold Gosset
32161:
32149:
32145:Stillwell 2001
32137:
32125:
32121:Stillwell 2001
32113:
32101:
32057:
32045:
32030:
32026:
32003:
31999:
31979:
31967:
31954:
31949:
31945:
31941:
31938:
31933:
31929:
31909:
31897:
31895:, p. 181.
31882:
31877:
31873:
31869:
31865:
31861:
31857:
31842:
31826:
31807:
31795:
31783:
31766:
31742:
31730:
31718:
31698:
31686:
31682:Stillwell 2001
31671:
31659:
31647:
31643:Stillwell 2001
31635:
31619:
31607:
31595:
31583:
31571:
31552:
31540:
31528:
31512:
31500:
31488:
31476:
31463:
31461:
31458:
31456:
31455:
31431:right rotation
31422:
31391:
31388:
31385:
31382:
31379:
31376:
31372:
31368:
31365:
31362:
31346:
31327:
31324:
31321:
31318:
31315:
31311:
31307:
31304:
31301:
31285:
31262:
31259:
31256:
31253:
31250:
31246:
31242:
31239:
31236:
31220:
31211:
31184:
31181:
31178:
31175:
31172:
31169:
31165:
31161:
31158:
31155:
31139:
31116:
31113:
31110:
31107:
31104:
31100:
31096:
31093:
31090:
31074:
31051:
31048:
31045:
31042:
31039:
31035:
31031:
31028:
31025:
31009:
30986:
30983:
30980:
30977:
30974:
30971:
30967:
30963:
30960:
30957:
30941:
30932:
30879:
30858:
30831:
30828:
30825:
30822:
30819:
30816:
30812:
30808:
30805:
30802:
30786:
30773:
30770:
30750:
30747:
30726:
30723:
30720:
30717:
30714:
30708:
30704:
30697:
30691:
30687:
30680:
30664:
30651:
30648:
30645:
30642:
30639:
30633:
30629:
30622:
30616:
30612:
30605:
30594:Hopf fibration
30566:
30563:
30560:
30557:
30554:
30548:
30544:
30537:
30531:
30527:
30520:
30500:
30494:
30490:
30483:
30477:
30473:
30466:
30463:
30460:
30457:
30454:
30435:distinct chord
30422:
30399:
30396:
30393:
30390:
30387:
30383:
30379:
30376:
30373:
30357:
30334:
30331:
30328:
30325:
30322:
30319:
30315:
30311:
30308:
30305:
30289:
30276:
30273:
30270:
30249:
30246:
30242:
30222:
30219:
30203:
30193:
30170:
30167:
30164:
30161:
30158:
30154:
30150:
30147:
30144:
30128:
30116:central planes
30098:
30088:
30084:symmetry group
30071:
30067:
30050:symmetry group
30037:
30033:
30011:
29988:
29985:
29982:
29979:
29976:
29973:
29969:
29965:
29962:
29959:
29943:
29920:
29917:
29914:
29911:
29908:
29904:
29900:
29897:
29894:
29878:
29855:
29852:
29849:
29846:
29843:
29840:
29836:
29832:
29829:
29826:
29810:
29801:
29761:
29751:
29720:
29697:
29694:
29691:
29688:
29685:
29681:
29677:
29674:
29671:
29655:
29592:
29583:
29579:Rotation class
29554:
29528:
29515:
29509:
29506:
29500:
29494:
29491:
29485:
29479:
29476:
29470:
29464:
29461:
29455:
29444:Hopf fibration
29416:
29410:
29407:
29401:
29395:
29392:
29386:
29380:
29377:
29371:
29365:
29362:
29356:
29336:
29333:
29330:
29327:
29324:
29321:
29318:
29315:
29312:
29297:distinct chord
29281:
29265:
29261:
29256:
29234:
29230:
29206:
29202:
29197:
29175:
29171:
29150:
29147:
29124:
29120:
29115:
29084:
29070:
29067:
29064:
29044:
29041:
29038:
29019:
29009:
28958:
28945:
28935:
28931:symmetry group
28901:. The regular
28881:For a regular
28874:
28850:
28837:
28798:
28789:
28785:
28773:
28760:
28743:
28726:
28685:
28659:
28647:
28638:
28615:
28605:
28596:
28583:
28555:great hexagons
28536:
28526:
28503:
28474:
28453:
28444:
28431:
28420:All isoclinic
28413:
28394:
28386:Hopf fibration
28362:
28348:
28345:
28340:
28310:
28306:
28285:
28276:
28271:
28259:
28246:
28243:
28240:
28220:
28217:
28214:
28171:
28144:
28142:circumference.
28126:
28123:
28120:
28100:
28097:
28094:
28078:
28054:
28030:
28010:
27989:
27974:
27960:
27955:
27951:
27947:
27929:
27925:
27921:
27917:
27909:along a third
27906:
27902:
27894:
27890:
27886:
27882:
27881:lies in both P
27878:
27874:
27870:
27866:
27862:
27858:
27847:
27843:
27839:
27835:
27831:
27820:
27816:
27812:
27799:
27758:
27742:
27738:
27724:
27720:
27712:
27708:
27700:
27676:
27672:
27651:
27647:
27619:
27615:
27607:
27602:
27598:
27590:
27545:
27512:
27498:
27425:which include
27423:chiral objects
27393:e.g. adjacent
27358:
27300:
27265:
27252:
27239:
27174:
27154:
27139:
27124:Petrie polygon
27122:The 24-cell's
27110:
27066:
27032:
27004:
26924:
26861:
26859:plane rotates.
26830:
26789:
26762:
26717:and dimension
26709:and dimension
26694:
26664:
26626:
26616:
26597:
26575:
26546:
26533:
26516:
26506:
26497:
26488:
26464:right triangle
26440:
26417:
26399:
26382:
26367:
26354:
26269:
26259:
26249:
26223:
26207:
26188:
26164:
26154:
26145:
26088:
26038:
26001:
25974:Clifford torus
25889:
25880:
25871:
25857:
25847:Hopf fibration
25830:
25771:
25754:
25733:
25716:
25656:
25637:Hopf fibration
25610:
25591:
25586:
25562:
25557:
25522:
25517:
25490:
25485:
25467:
25440:
25396:
25382:
25297:
25260:
25209:
25200:
25184:
25124:
25095:
24916:
24892:
24882:Polytopes are
24864:
24469:
24450:
24392:
24387:different path
24314:
24297:
24294:
24291:
24271:
24268:
24248:
24245:
24242:
24222:
24219:
24172:
24143:
24110:
23711:
23677:
23619:
23516:
23467:
23450:
23447:
23444:
23424:
23421:
23418:
23398:
23395:
23392:
23359:
23332:
23319:
23304:
23218:
23200:
23181:
23177:Petrie polygon
23145:
23100:
23096:great 120-cell
23074:
23072:
23069:
23068:
23067:
23062:
23055:
23052:
23049:
23048:
23037:
23026:
23015:
23004:
22993:
22982:
22971:
22965:
22964:
22953:
22942:
22931:
22920:
22909:
22898:
22887:
22883:
22882:
22875:
22868:
22861:
22854:
22847:
22840:
22833:
22829:
22828:
22822:
22817:
22812:
22807:
22802:
22797:
22792:
22784:
22783:
22780:
22777:
22774:
22771:
22767:
22766:
22761:
22756:
22752:
22751:
22750:,3} polytopes
22739:
22738:
22731:
22724:
22717:
22710:
22703:
22696:
22689:
22682:
22675:
22672:
22668:
22667:
22660:
22653:
22646:
22639:
22632:
22625:
22618:
22611:
22604:
22596:
22595:
22589:
22586:
22580:
22577:
22569:
22566:
22558:
22555:
22549:
22546:
22538:
22535:
22527:
22524:
22516:
22513:
22510:
22502:
22501:
22462:
22423:
22355:
22287:
22248:
22180:
22112:
22044:
21976:
21968:
21967:
21960:
21956:runcitruncated
21953:
21949:cantitruncated
21946:
21939:
21932:
21925:
21918:
21911:
21906:
21902:
21901:
21897:
21896:
21889:
21882:
21875:
21868:
21861:
21854:
21847:
21840:
21833:
21830:
21826:
21825:
21818:
21811:
21804:
21797:
21790:
21783:
21776:
21769:
21762:
21754:
21753:
21747:
21744:
21738:
21735:
21727:
21724:
21716:
21713:
21707:
21704:
21696:
21693:
21685:
21682:
21674:
21671:
21668:
21660:
21659:
21620:
21581:
21542:
21474:
21435:
21396:
21357:
21289:
21250:
21242:
21241:
21234:
21230:runcitruncated
21227:
21223:cantitruncated
21220:
21213:
21206:
21199:
21192:
21185:
21180:
21176:
21175:
21164:
21163:
21156:
21149:
21142:
21135:
21128:
21121:
21114:
21107:
21100:
21093:
21090:
21086:
21085:
21078:
21071:
21064:
21057:
21050:
21043:
21039:
21035:
21034:
21027:
21020:
21013:
21006:
20999:
20992:
20985:
20978:
20971:
20964:
20960:
20956:
20955:
20948:
20941:
20934:
20927:
20920:
20913:
20906:
20899:
20892:
20885:
20882:
20878:
20877:
20870:
20863:
20856:
20849:
20842:
20835:
20828:
20821:
20814:
20807:
20804:
20800:
20799:
20792:
20785:
20778:
20771:
20764:
20757:
20750:
20743:
20736:
20729:
20721:
20720:
20681:
20642:
20603:
20564:
20525:
20486:
20447:
20408:
20369:
20330:
20322:
20321:
20315:
20312:
20306:
20303:
20297:
20294:
20286:
20283:
20275:
20272:
20264:
20261:
20253:
20250:
20247:
20239:
20236:
20233:
20225:
20224:
20217:
20210:
20205:
20198:
20193:
20188:
20183:
20178:
20173:
20168:
20164:
20163:
20155:
20154:
20153:s{3}=s{3,4,3}
20147:
20146:t{3}=t{3,4,3}
20140:
20139:r{3}=r{3,4,3}
20133:
20126:
20122:
20115:
20111:
20104:
20100:
20093:
20085:
20084:
20077:
20070:
20063:
20056:
20049:
20042:
20035:
20027:
20026:
19974:
19922:
19870:
19818:
19761:
19704:
19647:
19589:
19588:
19584:
19577:
19574:
19558:great 120-cell
19523:
19522:
19511:
19504:
19484:
19481:
19478:
19477:
19455:
19451:
19443:
19421:
19417:
19409:
19400:
19396:
19395:
19392:
19389:
19386:
19382:
19381:
19378:
19375:
19372:
19368:
19367:
19345:
19342:
19340:
19318:
19315:
19313:
19272:
19268:
19267:
19246:
19242:
19240:
19219:
19215:
19213:
19173:
19160:
19157:
19153:
19149:
19135:
19130:
19080:
19076:
19069:
19066:
19026:
19022:
19015:
19012:
19009:
19008:
19001:
18994:
18986:
18985:
18975:
18974:
18969:One set of 24
18967:
18958:One set of 16
18956:
18948:
18947:
18939:
18938:
18934:
18931:
18930:, , order 384
18927:
18924:
18923:, , order 192
18920:
18916:
18915:
18912:
18907:
18901:
18900:
18896:
18892:
18888:
18870:
18862:
18851:
18848:
18846:
18843:
18840:
18839:
18824:
18823:
18809:
18808:
18794:
18793:
18788:
18787:
18778:
18768:
18758:
18757:
18732:
18729:
18671:
18668:
18633:Hopf fibration
18619:
18618:
18616:
18613:
18609:
18608:
18605:
18602:
18599:
18595:
18594:
18591:
18588:
18585:
18582:
18578:
18577:
18574:
18571:
18568:
18565:
18561:
18560:
18557:
18554:
18551:
18547:
18546:
18543:
18540:
18537:
18534:
18530:
18529:
18526:
18523:
18520:
18517:
18463:hyperspherical
18431:
18428:
18415:
18412:
18393:
18390:
18387:
18384:
18381:
18377:
18373:
18370:
18367:
18347:
18344:
18324:
18321:
18279:
18276:
18273:
18270:
18267:
18263:
18259:
18256:
18253:
18231:left rotations
18212:
18209:
18206:
18203:
18200:
18196:
18192:
18189:
18186:
18161:
18158:
18155:
18152:
18149:
18145:
18141:
18138:
18135:
18113:
18110:
18107:
18104:
18101:
18098:
18095:
18091:
18087:
18084:
18081:
18059:
18056:
18053:
18050:
18047:
18043:
18039:
18036:
18033:
18011:
18008:
18005:
18002:
17999:
17996:
17993:
17989:
17985:
17982:
17979:
17959:
17956:
17936:
17933:
17911:
17908:
17905:
17902:
17899:
17895:
17891:
17888:
17885:
17862:
17858:
17853:
17830:
17827:
17824:
17821:
17818:
17814:
17809:
17806:
17803:
17790:
17789:
17786:
17778:
17775:
17758:
17755:
17747:
17744:
17727:
17724:
17716:
17713:
17709:
17708:
17697:
17694:
17691:
17688:
17685:
17682:
17679:
17676:
17673:
17670:
17660:
17645:
17642:
17639:
17635:
17615:
17604:
17601:
17598:
17595:
17592:
17589:
17586:
17583:
17580:
17570:
17555:
17552:
17548:
17528:
17515:
17512:
17509:
17506:
17503:
17500:
17496:
17492:
17489:
17486:
17476:
17461:
17458:
17455:
17452:
17449:
17446:
17442:
17421:
17420:
17417:
17409:
17406:
17389:
17386:
17378:
17375:
17358:
17355:
17347:
17344:
17340:
17339:
17328:
17325:
17322:
17319:
17316:
17313:
17310:
17307:
17304:
17294:
17279:
17276:
17272:
17252:
17241:
17238:
17235:
17232:
17229:
17226:
17223:
17220:
17217:
17207:
17192:
17189:
17185:
17165:
17152:
17149:
17146:
17143:
17140:
17136:
17132:
17129:
17126:
17116:
17101:
17098:
17095:
17092:
17089:
17085:
17064:
17063:
17060:
17052:
17049:
17032:
17029:
17021:
17018:
17001:
16998:
16990:
16987:
16969:
16968:
16957:
16954:
16951:
16948:
16945:
16942:
16939:
16936:
16933:
16923:
16908:
16905:
16901:
16881:
16870:
16867:
16864:
16861:
16858:
16855:
16852:
16849:
16846:
16836:
16821:
16818:
16814:
16794:
16781:
16778:
16775:
16772:
16769:
16765:
16761:
16758:
16755:
16745:
16730:
16727:
16724:
16721:
16718:
16714:
16693:
16692:
16689:
16681:
16678:
16661:
16658:
16650:
16647:
16630:
16627:
16619:
16616:
16612:
16611:
16600:
16597:
16594:
16591:
16588:
16585:
16582:
16579:
16576:
16573:
16563:
16548:
16545:
16542:
16538:
16518:
16507:
16504:
16501:
16498:
16495:
16492:
16489:
16486:
16483:
16473:
16458:
16455:
16451:
16431:
16418:
16415:
16412:
16409:
16406:
16403:
16399:
16395:
16392:
16389:
16379:
16364:
16361:
16358:
16355:
16352:
16349:
16345:
16324:
16323:
16320:
16312:
16309:
16292:
16289:
16281:
16278:
16261:
16258:
16250:
16247:
16229:
16228:
16217:
16211:
16208:
16202:
16196:
16193:
16187:
16181:
16178:
16172:
16166:
16163:
16157:
16147:
16132:
16129:
16125:
16105:
16094:
16091:
16088:
16085:
16082:
16079:
16076:
16073:
16070:
16060:
16045:
16042:
16038:
16018:
16005:
16002:
15999:
15996:
15993:
15989:
15985:
15982:
15979:
15969:
15954:
15951:
15948:
15945:
15942:
15938:
15917:
15916:
15913:
15905:
15902:
15885:
15882:
15874:
15871:
15854:
15851:
15843:
15840:
15836:
15835:
15824:
15818:
15814:
15807:
15801:
15797:
15790:
15787:
15784:
15781:
15778:
15768:
15753:
15750:
15746:
15726:
15715:
15709:
15705:
15698:
15692:
15688:
15681:
15678:
15675:
15672:
15669:
15659:
15644:
15641:
15637:
15617:
15604:
15601:
15598:
15595:
15592:
15588:
15584:
15581:
15578:
15568:
15553:
15550:
15547:
15544:
15541:
15537:
15516:
15515:
15512:
15504:
15501:
15498:
15495:
15487:
15484:
15467:
15464:
15456:
15453:
15435:
15434:
15423:
15417:
15413:
15406:
15403:
15397:
15393:
15386:
15383:
15380:
15377:
15374:
15371:
15361:
15346:
15343:
15340:
15336:
15316:
15305:
15302:
15299:
15296:
15293:
15287:
15283:
15276:
15270:
15266:
15259:
15249:
15234:
15231:
15227:
15207:
15194:
15191:
15188:
15185:
15182:
15179:
15175:
15171:
15168:
15165:
15155:
15140:
15137:
15134:
15131:
15128:
15125:
15121:
15100:
15099:
15096:
15088:
15085:
15068:
15065:
15057:
15054:
15037:
15034:
15026:
15023:
15019:
15018:
15007:
15004:
15001:
14998:
14995:
14989:
14985:
14978:
14975:
14969:
14965:
14958:
14955:
14945:
14930:
14927:
14924:
14920:
14900:
14889:
14886:
14883:
14880:
14877:
14871:
14867:
14860:
14854:
14850:
14843:
14833:
14818:
14815:
14811:
14791:
14778:
14775:
14772:
14769:
14766:
14763:
14759:
14755:
14752:
14749:
14739:
14724:
14721:
14718:
14715:
14712:
14709:
14705:
14684:
14683:
14680:
14672:
14669:
14652:
14649:
14641:
14638:
14621:
14618:
14610:
14607:
14603:
14602:
14591:
14588:
14585:
14582:
14579:
14573:
14569:
14562:
14556:
14552:
14545:
14535:
14520:
14517:
14513:
14493:
14482:
14479:
14476:
14473:
14470:
14464:
14460:
14453:
14447:
14443:
14436:
14426:
14411:
14408:
14404:
14384:
14371:
14368:
14365:
14362:
14359:
14355:
14351:
14348:
14345:
14335:
14320:
14317:
14314:
14311:
14308:
14304:
14283:
14282:
14279:
14271:
14268:
14251:
14248:
14240:
14237:
14220:
14217:
14209:
14206:
14188:
14187:
14176:
14173:
14170:
14167:
14164:
14161:
14158:
14155:
14152:
14149:
14139:
14124:
14121:
14118:
14114:
14094:
14083:
14077:
14074:
14068:
14062:
14059:
14053:
14047:
14044:
14038:
14032:
14029:
14023:
14013:
13998:
13995:
13991:
13971:
13958:
13955:
13952:
13949:
13946:
13943:
13939:
13935:
13932:
13929:
13919:
13904:
13901:
13898:
13895:
13892:
13889:
13885:
13864:
13863:
13860:
13852:
13849:
13832:
13829:
13821:
13818:
13801:
13798:
13790:
13787:
13769:
13768:
13757:
13754:
13751:
13748:
13745:
13742:
13739:
13736:
13733:
13723:
13708:
13705:
13701:
13681:
13670:
13664:
13661:
13655:
13649:
13646:
13640:
13634:
13631:
13625:
13619:
13616:
13610:
13600:
13585:
13582:
13578:
13558:
13545:
13542:
13539:
13536:
13533:
13529:
13525:
13522:
13519:
13509:
13494:
13491:
13488:
13485:
13482:
13478:
13457:
13456:
13453:
13445:
13442:
13425:
13422:
13414:
13411:
13394:
13391:
13383:
13380:
13376:
13375:
13364:
13358:
13355:
13349:
13346:
13340:
13337:
13331:
13328:
13322:
13319:
13313:
13310:
13304:
13301:
13295:
13292:
13282:
13267:
13264:
13261:
13257:
13237:
13226:
13220:
13217:
13211:
13205:
13202:
13196:
13190:
13187:
13181:
13175:
13172:
13166:
13156:
13141:
13138:
13134:
13114:
13101:
13098:
13095:
13092:
13089:
13086:
13082:
13078:
13075:
13072:
13062:
13047:
13044:
13041:
13038:
13035:
13032:
13028:
13007:
13006:
13003:
12995:
12992:
12975:
12972:
12964:
12961:
12944:
12941:
12933:
12930:
12926:
12925:
12914:
12908:
12905:
12899:
12893:
12890:
12884:
12878:
12875:
12869:
12863:
12860:
12854:
12844:
12829:
12826:
12822:
12802:
12791:
12785:
12782:
12776:
12770:
12767:
12761:
12755:
12752:
12746:
12740:
12737:
12731:
12721:
12706:
12703:
12699:
12679:
12666:
12663:
12660:
12657:
12654:
12650:
12646:
12643:
12640:
12630:
12615:
12612:
12609:
12606:
12603:
12599:
12578:
12577:
12574:
12566:
12563:
12546:
12543:
12535:
12532:
12515:
12512:
12504:
12501:
12483:
12482:
12471:
12465:
12462:
12456:
12450:
12447:
12441:
12435:
12432:
12426:
12420:
12417:
12411:
12408:
12398:
12383:
12380:
12377:
12373:
12353:
12342:
12336:
12333:
12327:
12321:
12318:
12312:
12306:
12303:
12297:
12291:
12288:
12282:
12272:
12257:
12254:
12250:
12230:
12217:
12214:
12211:
12208:
12205:
12202:
12198:
12194:
12191:
12188:
12178:
12163:
12160:
12157:
12154:
12151:
12148:
12144:
12123:
12122:
12119:
12111:
12108:
12091:
12088:
12080:
12077:
12060:
12057:
12049:
12046:
12028:
12027:
12016:
12010:
12007:
12001:
11998:
11992:
11989:
11983:
11980:
11974:
11971:
11965:
11962:
11956:
11953:
11947:
11937:
11922:
11919:
11915:
11895:
11884:
11878:
11875:
11869:
11863:
11860:
11854:
11848:
11845:
11839:
11833:
11830:
11824:
11814:
11799:
11796:
11792:
11772:
11759:
11756:
11753:
11750:
11747:
11743:
11739:
11736:
11733:
11723:
11708:
11705:
11702:
11699:
11696:
11692:
11671:
11670:
11659:
11656:
11645:
11634:
11631:
11620:
11617:
11613:
11612:
11607:
11581:
11575:
11572:
11566:
11563:
11557:
11554:
11548:
11545:
11539:
11536:
11530:
11527:
11521:
11518:
11512:
11492:
11486:
11483:
11477:
11471:
11468:
11462:
11456:
11453:
11447:
11441:
11438:
11432:
11409:
11406:
11386:
11383:
11345:
11342:
11339:
11336:
11333:
11329:
11325:
11322:
11319:
11296:rotation class
11289:Hopf fibration
11268:
11265:
11215:
11189:
11186:
11168:
11165:
11135:
11132:
11102:
11099:
11069:
11066:
11036:
11033:
11003:
11000:
10970:
10967:
10941:
10911:
10908:
10878:
10875:
10845:
10842:
10808:
10805:
10775:
10772:
10742:
10739:
10656:
10655:
10653:
10651:
10649:
10647:
10635:
10632:
10625:
10622:
10607:
10595:
10591:
10585:
10581:
10575:
10571:
10559:
10558:
10556:
10554:
10552:
10550:
10538:
10535:
10528:
10525:
10510:
10498:
10494:
10488:
10484:
10478:
10474:
10462:
10461:
10459:
10457:
10455:
10453:
10441:
10438:
10431:
10428:
10413:
10401:
10397:
10391:
10387:
10381:
10377:
10365:
10364:
10362:
10360:
10358:
10356:
10344:
10333:
10321:
10317:
10311:
10307:
10301:
10297:
10285:
10284:
10282:
10280:
10278:
10276:
10273:
10272:
10257:
10254:
10240:
10235:
10220:
10217:
10203:
10198:
10186:
10183:
10176:
10173:
10158:
10146:
10142:
10136:
10132:
10126:
10122:
10110:
10109:
10094:
10091:
10077:
10072:
10057:
10054:
10040:
10035:
10023:
10020:
10013:
10010:
9995:
9983:
9979:
9973:
9969:
9963:
9959:
9947:
9946:
9931:
9928:
9914:
9909:
9894:
9891:
9877:
9872:
9860:
9857:
9850:
9847:
9832:
9820:
9816:
9810:
9806:
9800:
9796:
9784:
9783:
9781:
9779:
9777:
9775:
9772:
9771:
9756:
9753:
9739:
9734:
9719:
9716:
9702:
9697:
9685:
9682:
9675:
9672:
9657:
9653:
9652:
9637:
9634:
9620:
9615:
9600:
9597:
9583:
9578:
9566:
9563:
9556:
9553:
9538:
9534:
9533:
9518:
9515:
9501:
9496:
9481:
9478:
9464:
9459:
9447:
9444:
9437:
9434:
9419:
9415:
9414:
9412:
9410:
9408:
9406:
9403:
9402:
9387:
9383:
9380:
9365:
9360:
9345:
9342:
9328:
9323:
9311:
9300:
9296:
9295:
9292:
9289:
9286:
9283:
9282:
9275:
9272:
9269:
9268:
9251:
9241:
9226:
9215:
9197:
9186:
9185:
9178:
9171:
9164:
9157:
9149:
9148:
9145:{24/12}={12/2}
9144:
9141:
9138:
9133:
9130:
9125:
9122:
9117:
9114:
9108:
9107:
9102:
9097:
9092:
9087:
9081:
9080:
9061:of the 24-cell
9010:
8990:Boundary cells
8930:
8927:
8910:Petrie polygon
8904:
8886:great triangle
8815:
8800:
8797:
8784:Hopf fibration
8744:
8741:
8699:
8696:
8660:
8657:
8615:
8612:
8586:
8451:transformation
8443:
8440:
8379:
8376:
8314:
8311:
8291:
8288:
8270:
8267:
8242:
8239:
8210:
8207:
8174:
8170:
8153:
8143:
8140:
8117:
8109:
8102:group of units
8086:
8066:
8063:
8059:solvable group
8054:
8046:
8039:symmetry group
8027:
8022:
8009:
8005:
7976:
7966:
7961:
7951:
7948:
7931:
7921:
7918:
7916:
7913:
7911:
7908:
7906:
7903:
7902:
7899:
7896:
7894:
7891:
7889:
7886:
7884:
7881:
7880:
7877:
7874:
7872:
7869:
7867:
7864:
7862:
7859:
7858:
7855:
7852:
7850:
7847:
7845:
7842:
7840:
7837:
7836:
7833:
7832:
7830:
7809:
7806:
7760:
7759:Boundary cells
7757:
7745:demihypercubes
7713:(120Β°) apart.
7702:
7699:
7689:
7686:
7676:
7672:symmetry group
7616:three 16-cells
7560:
7557:
7522:
7519:
7506:cubic pyramids
7494:
7491:
7470:
7467:
7446:cubic pyramids
7369:
7366:
7352:
7349:
7184:
7183:
7181:
7179:
7175:
7174:
7167:
7160:
7156:
7155:
7152:
7148:
7145:
7142:
7139:
7138:Coxeter plane
7135:
7134:
7132:
7130:
7126:
7125:
7118:
7111:
7107:
7106:
7102:
7098:
7095:
7091:
7087:
7084:
7083:Coxeter plane
7080:
7079:
7077:
7071:
7070:
7063:
7059:
7058:
7055:
7052:
7001:polygons. The
6980:
6977:
6760:
6757:
6745:
6742:
6737:
6733:
6730:configurations
6712:
6711:
6699:
6696:
6690:
6681:
6661:
6649:
6646:
6640:
6631:
6611:
6599:
6588:
6576:
6565:
6553:
6550:
6544:
6541:
6527:
6515:
6512:
6507:
6502:
6496:
6492:
6485:
6477:
6473:
6458:
6454:
6453:
6441:
6438:
6434:
6425:
6418:
6414:
6410:
6403:
6398:
6395:
6392:
6385:
6381:
6370:
6358:
6355:
6351:
6342:
6338:
6334:
6329:
6322:
6318:
6307:
6295:
6292:
6288:
6282:
6278:
6271:
6267:
6256:
6244:
6233:
6221:
6218:
6214:
6208:
6205:
6199:
6195:
6184:
6172:
6169:
6165:
6159:
6153:
6148:
6141:
6137:
6126:
6122:
6121:
6109:
6106:
6102:
6093:
6089:
6085:
6078:
6073:
6070:
6067:
6060:
6056:
6045:
6033:
6030:
6026:
6017:
6013:
6009:
6004:
5997:
5993:
5982:
5970:
5967:
5963:
5956:
5953:
5946:
5942:
5931:
5919:
5908:
5896:
5893:
5889:
5882:
5879:
5872:
5868:
5857:
5845:
5842:
5838:
5832:
5826:
5821:
5814:
5810:
5799:
5795:
5794:
5782:
5779:
5772:
5767:
5763:
5747:
5735:
5732:
5725:
5720:
5716:
5700:
5688:
5685:
5678:
5675:
5660:
5645:
5642:
5628:
5613:
5610:
5596:
5581:
5578:
5564:
5560:
5559:
5547:
5544:
5535:
5528:
5524:
5519:
5505:
5493:
5490:
5484:
5481:
5467:
5455:
5444:
5432:
5421:
5409:
5406:
5401:
5388:
5376:
5373:
5366:
5363:
5348:
5344:
5343:
5331:
5320:
5308:
5297:
5285:
5274:
5262:
5251:
5239:
5228:
5216:
5205:
5199:
5198:
5191:
5184:
5177:
5170:
5163:
5156:
5150:
5149:
5142:
5135:
5128:
5125:
5118:
5116:
5114:Great polygons
5110:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5087:
5086:
5080:
5078:30-tetrahedron
5074:
5068:
5062:
5056:
5050:
5044:
5043:
5040:
5037:
5034:
5031:
5030:16 tetrahedra
5028:
5025:
5021:
5020:
5019:720 pentagons
5017:
5014:
5011:
5008:
5005:
5002:
4998:
4997:
4994:
4991:
4990:96 triangular
4988:
4987:32 triangular
4985:
4982:
4981:10 triangular
4979:
4973:
4972:
4969:
4966:
4963:
4960:
4957:
4956:5 tetrahedral
4954:
4950:
4949:
4942:
4935:
4928:
4921:
4914:
4907:
4903:
4902:
4810:
4718:
4626:
4534:
4442:
4350:
4346:
4345:
4306:
4267:
4228:
4189:
4150:
4111:
4105:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4080:
4079:
4065:
4051:
4041:
4027:
4013:
3999:
3995:
3994:
3990:
3986:
3982:
3978:
3974:
3970:
3966:
3962:
3960:Symmetry group
3956:
3955:
3903:
3896:
3893:
3887:
3884:
3878:
3875:
3869:
3866:
3860:
3857:
3851:
3848:
3842:
3839:
3833:
3829:
3799:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3770:
3729:
3726:
3715:
3714:
3702:
3699:
3693:
3684:
3664:
3652:
3649:
3643:
3634:
3614:
3602:
3591:
3579:
3568:
3556:
3553:
3547:
3544:
3530:
3518:
3515:
3510:
3505:
3500:
3495:
3487:
3483:
3468:
3464:
3463:
3451:
3448:
3444:
3435:
3431:
3427:
3420:
3415:
3412:
3409:
3402:
3398:
3387:
3375:
3372:
3368:
3359:
3355:
3351:
3347:
3341:
3337:
3326:
3314:
3311:
3307:
3301:
3298:
3292:
3288:
3277:
3265:
3262:
3257:
3252:
3241:
3229:
3226:
3222:
3216:
3210:
3205:
3198:
3194:
3183:
3171:
3168:
3164:
3158:
3152:
3147:
3140:
3136:
3125:
3121:
3120:
3108:
3105:
3101:
3092:
3088:
3084:
3077:
3072:
3069:
3066:
3059:
3055:
3044:
3032:
3029:
3025:
3016:
3012:
3008:
3001:
2996:
2989:
2985:
2974:
2962:
2959:
2955:
2948:
2945:
2938:
2934:
2923:
2911:
2900:
2888:
2885:
2881:
2876:
2871:
2867:
2856:
2844:
2841:
2837:
2831:
2825:
2820:
2813:
2809:
2798:
2794:
2793:
2781:
2778:
2771:
2766:
2762:
2746:
2734:
2731:
2724:
2719:
2715:
2699:
2687:
2676:
2664:
2661:
2655:
2651:
2636:
2624:
2621:
2615:
2611:
2596:
2584:
2581:
2575:
2571:
2556:
2552:
2551:
2539:
2536:
2533:
2530:
2527:
2516:
2504:
2501:
2495:
2491:
2476:
2464:
2461:
2456:
2443:
2431:
2428:
2423:
2410:
2398:
2387:
2375:
2372:
2367:
2354:
2350:
2349:
2335:
2322:
2308:
2295:
2281:
2268:
2254:
2241:
2227:
2214:
2200:
2187:
2181:
2180:
2173:
2166:
2159:
2152:
2145:
2138:
2132:
2131:
2124:
2117:
2110:
2107:
2100:
2098:
2096:Great polygons
2092:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2069:
2068:
2062:
2060:30-tetrahedron
2056:
2050:
2044:
2038:
2032:
2026:
2025:
2022:
2019:
2016:
2013:
2012:16 tetrahedra
2010:
2007:
2003:
2002:
2001:720 pentagons
1999:
1996:
1993:
1990:
1987:
1984:
1980:
1979:
1976:
1973:
1972:96 triangular
1970:
1969:32 triangular
1967:
1964:
1963:10 triangular
1961:
1955:
1954:
1951:
1948:
1945:
1942:
1939:
1938:5 tetrahedral
1936:
1932:
1931:
1924:
1917:
1910:
1903:
1896:
1889:
1885:
1884:
1792:
1700:
1608:
1516:
1424:
1332:
1328:
1327:
1288:
1249:
1210:
1171:
1132:
1093:
1087:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1062:
1061:
1047:
1033:
1023:
1009:
995:
981:
977:
976:
972:
968:
964:
960:
956:
952:
948:
944:
942:Symmetry group
938:
937:
788:
783:
778:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
718:
715:
713:
710:
677:
674:
605:polyoctahedron
577:
568:Platonic solid
562:is the convex
540:
539:
536:
530:
529:
512:
508:
507:
502:
496:
495:
491:
485:
482:, , order 1152
478:
474:
468:
467:
462:
460:Petrie polygon
456:
455:
450:
444:
443:
440:
434:
433:
430:
424:
423:
417:
411:
410:
399:
393:
392:
236:
230:
229:
217:
210:
207:
206:
203:
200:
199:
196:
193:
192:
189:
166:
159:
156:
153:
150:
149:
146:
143:
142:
139:
125:
119:
118:
113:
109:
108:
101:
93:
92:
84:
83:
53:subarticle(s).
38:
36:
29:
15:
9:
6:
4:
3:
2:
34662:
34651:
34648:
34647:
34645:
34634:
34630:
34626:
34621:
34618:
34614:
34611:
34609:
34602:
34595:
34589:
34587:
34583:
34580:
34578:
34574:
34570:
34566:
34563:
34561:
34557:
34554:
34552:
34548:
34544:
34543:
34540:
34538:
34536:
34533:
34531:
34527:
34524:
34522:
34519:
34517:
34514:
34513:
34510:
34508:
34506:
34503:
34501:
34497:
34494:
34492:
34489:
34487:
34484:
34483:
34480:
34478:
34471:
34464:
34458:
34456:
34453:
34451:
34447:
34444:
34442:
34439:
34437:
34434:
34433:
34430:
34428:
34421:
34414:
34408:
34406:
34403:
34401:
34397:
34394:
34392:
34389:
34387:
34384:
34383:
34380:
34378:
34371:
34365:
34363:
34360:
34358:
34354:
34351:
34349:
34346:
34344:
34341:
34340:
34337:
34335:
34333:
34330:
34328:
34324:
34321:
34319:
34316:
34314:
34311:
34310:
34307:
34303:
34300:
34298:
34295:
34293:
34292:Demitesseract
34290:
34288:
34284:
34281:
34279:
34276:
34274:
34271:
34270:
34267:
34263:
34260:
34258:
34256:
34253:
34251:
34247:
34244:
34242:
34239:
34237:
34234:
34233:
34230:
34227:
34225:
34222:
34220:
34217:
34215:
34212:
34210:
34207:
34205:
34202:
34201:
34198:
34192:
34189:
34185:
34178:
34174:
34167:
34163:
34158:
34154:
34149:
34145:
34140:
34138:
34136:
34132:
34122:
34118:
34116:
34114:
34110:
34106:
34104:
34102:
34098:
34094:
34092:
34089:
34088:
34083:
34079:
34071:
34066:
34064:
34059:
34057:
34052:
34051:
34048:
34023:
34004:
34003:
34001:
33995:
33976:
33975:
33973:
33967:
33948:
33947:
33945:
33939:
33920:
33919:
33917:
33911:
33876:
33875:
33873:
33867:
33848:
33847:
33845:
33839:
33820:
33819:
33817:
33811:
33792:
33791:
33789:
33783:
33764:
33763:
33761:
33755:
33736:
33735:
33733:
33732:
33729:
33724:
33722:
33717:
33715:
33708:
33706:
33701:
33699:
33692:
33690:
33683:
33681:
33676:
33674:
33669:
33667:
33660:
33658:
33653:
33652:
33647:
33643:
33636:
33632:
33612:
33609:
33606:
33605:
33603:
33597:
33594:
33591:
33590:
33588:
33582:
33579:
33576:
33575:
33573:
33567:
33564:
33561:
33560:
33558:
33552:
33549:
33546:
33545:
33543:
33537:
33534:
33531:
33530:
33528:
33527:
33524:
33521:
33519:
33516:
33514:
33511:
33509:
33506:
33504:
33501:
33499:
33496:
33495:
33490:
33486:
33479:
33475:
33471:
33464:
33459:
33457:
33452:
33450:
33445:
33444:
33441:
33435:
33432:
33430:
33426:
33423:
33420:
33418:
33415:
33413:
33410:
33409:
33398:
33394:
33390:
33386:
33382:
33378:
33374:
33369:
33365:
33361:
33356:
33351:
33347:
33343:
33339:
33334:
33330:
33325:
33321:
33317:
33313:
33309:
33304:
33299:
33294:
33290:
33286:
33282:
33278:
33274:
33269:
33258:
33254:
33250:
33246:
33242:
33240:0-521-08042-8
33236:
33232:
33228:
33227:
33221:
33217:
33213:
33209:
33205:
33200:
33195:
33191:
33187:
33182:
33178:
33174:
33169:
33164:
33160:
33156:
33152:
33148:
33141:
33136:
33131:
33126:
33121:
33117:
33113:
33112:
33106:
33101:
33096:
33091:
33087:
33081:
33077:
33073:
33069:
33064:
33063:
33062:Shaping Space
33056:
33052:
33046:
33042:
33037:
33033:
33028:
33022:
33021:
33016:
33013:
33008:
33004:
33000:
32996:
32992:
32988:
32984:
32981:
32975:
32971:
32970:
32965:
32961:
32957:
32953:
32946:
32941:
32937:
32933:
32929:
32925:
32920:
32915:
32911:
32907:
32903:
32899:
32895:
32891:
32887:
32883:
32878:
32874:
32871:
32867:
32864:
32860:
32858:
32854:
32853:
32851:
32845:
32840:
32839:
32833:
32829:
32825:
32821:
32817:
32813:
32812:
32807:
32803:
32799:
32798:
32793:
32789:
32788:
32776:
32771:
32764:
32763:Banchoff 2013
32759:
32752:
32748:
32744:
32740:
32739:Banchoff 2013
32735:
32728:
32723:
32719:
32715:
32711:
32707:
32703:
32699:
32695:
32690:
32683:
32678:
32671:
32666:
32659:
32654:
32647:
32642:
32635:
32630:
32623:
32618:
32611:
32606:
32599:
32594:
32587:
32582:
32575:
32571:
32567:
32560:
32555:
32548:
32541:
32536:
32529:
32517:
32512:
32505:
32500:
32493:
32488:
32481:
32476:
32469:
32464:
32457:
32452:
32445:
32440:
32433:
32428:
32421:
32416:
32409:
32405:
32401:
32396:
32389:
32385:
32381:
32377:
32373:
32368:
32361:
32356:
32349:
32345:
32341:
32337:
32332:
32325:
32320:
32313:
32308:
32301:
32297:
32293:
32288:
32281:
32276:
32269:
32264:
32257:
32252:
32245:
32240:
32233:
32230:
32226:
32222:
32218:
32213:
32202:
32197:
32190:
32185:
32183:
32174:
32170:
32165:
32158:
32153:
32147:, p. 22.
32146:
32141:
32134:
32129:
32122:
32117:
32110:
32105:
32098:
32094:
32090:
32086:
32082:
32079:-dimensional
32078:
32074:
32070:
32066:
32061:
32054:
32049:
32028:
32024:
32001:
31997:
31988:
31983:
31977:, p. 79.
31976:
31971:
31947:
31943:
31939:
31931:
31927:
31918:
31913:
31906:
31901:
31894:
31889:
31887:
31855:
31851:
31846:
31839:
31835:
31830:
31816:
31811:
31804:
31799:
31792:
31787:
31780:
31775:
31773:
31771:
31751:
31746:
31739:
31734:
31727:
31722:
31715:
31711:
31707:
31702:
31695:
31690:
31684:, p. 24.
31683:
31678:
31676:
31668:
31663:
31656:
31651:
31645:, p. 17.
31644:
31639:
31632:
31628:
31623:
31616:
31611:
31604:
31599:
31592:
31587:
31580:
31575:
31568:
31564:
31559:
31557:
31549:
31544:
31537:
31532:
31525:
31521:
31516:
31509:
31504:
31498:, p. 68.
31497:
31492:
31485:
31480:
31473:
31468:
31464:
31452:
31448:
31444:
31440:
31436:
31435:left rotation
31432:
31426:
31419:
31407:
31389:
31386:
31383:
31380:
31377:
31374:
31370:
31363:
31350:
31343:
31325:
31322:
31319:
31316:
31313:
31309:
31302:
31289:
31260:
31257:
31254:
31251:
31248:
31244:
31237:
31224:
31215:
31208:
31182:
31179:
31176:
31173:
31170:
31167:
31163:
31156:
31143:
31114:
31111:
31108:
31105:
31102:
31098:
31091:
31078:
31049:
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31043:
31040:
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31033:
31026:
31013:
30984:
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30965:
30958:
30945:
30936:
30929:
30924:
30920:
30916:
30911:
30900:
30889:
30883:
30876:
30868:
30862:
30855:
30829:
30826:
30823:
30820:
30817:
30814:
30810:
30803:
30790:
30771:
30768:
30748:
30745:
30721:
30718:
30715:
30712:
30706:
30702:
30695:
30689:
30685:
30668:
30646:
30643:
30640:
30637:
30631:
30627:
30620:
30614:
30610:
30595:
30591:
30587:
30577:designates a
30561:
30558:
30555:
30552:
30546:
30542:
30535:
30529:
30525:
30492:
30488:
30481:
30475:
30471:
30464:
30461:
30458:
30455:
30444:
30440:
30436:
30432:
30426:
30397:
30394:
30391:
30388:
30385:
30381:
30374:
30361:
30332:
30329:
30326:
30323:
30320:
30317:
30313:
30306:
30293:
30274:
30271:
30268:
30247:
30244:
30240:
30220:
30217:
30207:
30197:
30168:
30165:
30162:
30159:
30156:
30152:
30145:
30132:
30125:
30121:
30117:
30113:
30107:
30105:
30103:
30092:
30085:
30069:
30065:
30055:
30051:
30035:
30031:
30021:
30015:
29986:
29983:
29980:
29977:
29974:
29971:
29967:
29960:
29947:
29918:
29915:
29912:
29909:
29906:
29902:
29895:
29882:
29853:
29850:
29847:
29844:
29841:
29838:
29834:
29827:
29814:
29805:
29798:
29793:
29789:
29784:
29780:
29776:
29770:
29768:
29766:
29755:
29748:
29744:
29733:
29727:
29725:
29695:
29692:
29689:
29686:
29683:
29679:
29672:
29659:
29652:
29648:
29644:
29639:
29635:
29631:
29627:
29616:
29605:
29599:
29597:
29587:
29580:
29576:
29572:
29568:
29564:
29558:
29551:
29547:
29543:
29537:
29535:
29533:
29507:
29504:
29498:
29492:
29489:
29483:
29477:
29474:
29468:
29462:
29459:
29445:
29441:
29440:great hexagon
29437:
29427:designates a
29408:
29405:
29399:
29393:
29390:
29384:
29378:
29375:
29369:
29363:
29360:
29331:
29328:
29325:
29322:
29319:
29316:
29313:
29302:
29298:
29294:
29288:
29286:
29263:
29259:
29254:
29232:
29228:
29204:
29200:
29195:
29173:
29169:
29148:
29145:
29122:
29118:
29113:
29103:
29101:
29099:
29097:
29095:
29093:
29091:
29089:
29068:
29065:
29062:
29042:
29039:
29036:
29026:
29024:
29013:
29006:
29000:
28996:
28992:
28984:
28980:
28976:
28965:
28963:
28955:
28949:
28939:
28932:
28928:
28924:
28920:
28916:
28912:
28908:
28904:
28900:
28896:
28892:
28888:
28884:
28878:
28871:
28867:
28863:
28857:
28855:
28847:
28841:
28811:
28805:
28803:
28795:
28783:
28777:
28770:
28764:
28757:
28750:
28748:
28730:
28723:
28722:chord diagram
28689:
28674:
28673:deterministic
28668:
28666:
28664:
28654:
28652:
28642:
28634:
28629:
28625:
28619:
28609:
28600:
28593:
28587:
28580:
28579:fiber bundles
28576:
28572:
28568:
28564:
28560:
28556:
28549:
28547:
28545:
28543:
28541:
28530:
28507:
28500:
28496:
28492:
28488:
28481:
28479:
28471:
28467:
28463:
28457:
28448:
28441:
28435:
28428:
28423:
28417:
28409:
28403:
28401:
28399:
28391:
28390:warp and woof
28387:
28383:
28379:
28378:fiber bundles
28375:
28372:
28366:
28346:
28343:
28338:
28327:
28308:
28304:
28295:
28289:
28280:
28269:
28263:
28244:
28241:
28238:
28218:
28215:
28212:
28204:
28200:
28195:
28190:
28186:
28180:
28178:
28176:
28168:
28164:
28159:
28153:
28151:
28149:
28140:
28124:
28121:
28118:
28098:
28095:
28092:
28082:
28075:
28071:
28065:
28063:
28061:
28059:
28051:
28047:
28043:
28037:
28035:
28027:
28023:
28013:. Successive
28008:
28004:
27999:
27995:
27985:
27983:
27981:
27979:
27970:
27964:
27900:
27808:
27806:
27804:
27789:
27769:
27767:
27765:
27763:
27755:
27745:are just one
27730:
27718:
27704:
27696:
27657:
27632:
27611:
27594:
27587:
27576:
27565:
27552:
27550:
27542:
27538:
27531:
27529:
27527:
27525:
27523:
27521:
27519:
27517:
27509:
27502:
27494:
27490:
27486:
27485:
27480:
27476:
27475:
27470:
27469:
27464:
27460:
27456:
27455:
27450:
27446:
27442:
27441:
27440:fiber bundles
27436:
27435:
27430:
27429:
27424:
27420:
27419:
27414:
27413:
27412:right or left
27408:
27404:
27400:
27396:
27392:
27391:
27386:
27382:
27378:
27374:
27370:
27369:
27362:
27355:
27351:
27347:
27343:
27339:
27335:
27329:
27327:
27325:
27323:
27321:
27319:
27317:
27315:
27313:
27311:
27309:
27307:
27305:
27297:
27293:
27289:
27285:
27280:
27274:
27272:
27270:
27262:
27256:
27249:
27243:
27222:
27218:
27207:
27206:4-orthoscheme
27181:
27179:
27171:
27167:
27145:
27137:
27133:
27129:
27125:
27119:
27117:
27115:
27106:
27101:
27097:
27092:
27087:
27083:
27077:
27075:
27073:
27071:
27063:
27059:
27055:
27051:
27047:
27042:
27036:
27029:
27025:
27019:
27017:
27015:
27013:
27011:
27009:
27001:
26997:
26993:
26989:
26985:
26981:
26977:
26973:
26969:
26965:
26961:
26957:
26953:
26949:
26945:
26941:
26935:
26933:
26931:
26929:
26921:
26917:
26905:
26901:
26897:
26893:
26889:
26885:
26878:
26876:
26874:
26872:
26870:
26868:
26866:
26858:
26854:
26850:
26846:
26839:
26837:
26835:
26827:
26823:
26819:
26815:
26811:
26809:
26802:
26800:
26798:
26796:
26794:
26786:
26782:
26778:
26777:
26771:
26766:
26759:
26755:
26751:
26747:
26743:
26739:
26735:
26731:
26727:
26725:
26720:
26716:
26712:
26708:
26704:
26698:
26691:
26687:
26683:
26678:
26674:
26668:
26630:
26620:
26613:
26608:
26601:
26594:
26590:
26584:
26582:
26580:
26555:
26553:
26551:
26543:
26537:
26520:
26510:
26501:
26492:
26485:
26481:
26477:
26473:
26469:
26465:
26461:
26457:
26453:
26447:
26445:
26428:
26426:
26424:
26422:
26403:
26393:
26391:
26389:
26387:
26376:
26374:
26372:
26364:
26358:
26351:
26340:
26336:
26332:
26300:
26282:
26276:
26274:
26263:
26253:
26245:
26238:
26233:
26227:
26221:
26217:
26211:
26204:
26197:
26195:
26193:
26185:
26181:
26175:
26173:
26171:
26169:
26158:
26149:
26142:
26138:
26134:
26130:
26126:
26122:
26118:
26114:
26107:
26105:
26103:
26101:
26099:
26097:
26095:
26093:
26084:
26080:
26076:
26072:
26067:
26063:
26059:
26056:-dimensional
26055:
26049:
26047:
26045:
26043:
26005:
25998:
25995:
25991:
25990:winding route
25987:
25983:
25979:
25975:
25971:
25967:
25963:
25959:
25955:
25951:
25950:screw threads
25947:
25943:
25938:
25937:shortest path
25934:
25933:
25928:
25924:
25918:
25916:
25914:
25912:
25910:
25908:
25906:
25904:
25902:
25900:
25898:
25896:
25894:
25884:
25875:
25868:
25864:
25863:
25853:
25848:
25841:
25839:
25837:
25835:
25827:
25823:
25819:
25815:
25811:
25807:
25802:
25798:
25794:
25790:
25786:
25780:
25778:
25776:
25767:
25761:
25759:
25750:
25744:
25742:
25740:
25738:
25730:
25727:(the central
25726:
25725:cuboctahedron
25720:
25660:
25653:
25649:
25648:
25642:
25638:
25633:
25627:
25625:
25623:
25621:
25619:
25617:
25615:
25607:
25589:
25560:
25546:
25542:
25538:
25537:cubic pyramid
25520:
25506:
25488:
25471:
25464:
25459:
25458:vertex figure
25453:
25451:
25449:
25447:
25445:
25409:
25403:
25401:
25392:
25386:
25379:
25375:
25370:
25364:
25360:
25356:
25352:
25348:
25347:great circles
25345:
25340:
25334:
25332:
25330:
25328:
25326:
25324:
25322:
25320:
25318:
25316:
25314:
25312:
25310:
25308:
25306:
25304:
25302:
25264:
25220:
25218:
25216:
25214:
25204:
25197:
25196:vertex figure
25191:
25189:
25174:
25163:
25159:
25141:
25137:
25131:
25129:
25121:
25116:
25112:
25106:
25104:
25102:
25100:
25092:
25088:
25084:
25080:
25076:
25072:
25068:
25057:
25053:
25029:
25025:
25021:
25017:
25013:
25009:
25007:
25002:
24998:
24996:
24984:
24980:
24976:
24944:
24939:
24938:isoclinically
24935:
24931:
24925:
24923:
24921:
24912:
24905:
24903:
24901:
24899:
24897:
24889:
24886:if all their
24885:
24879:
24877:
24875:
24873:
24871:
24869:
24847:
24476:
24474:
24457:
24455:
24417:
24415:
24413:
24411:
24409:
24407:
24405:
24403:
24401:
24399:
24397:
24388:
24383:
24378:
24374:
24356:
24351:
24347:
24343:
24337:
24335:
24333:
24331:
24329:
24327:
24325:
24323:
24321:
24319:
24311:
24295:
24292:
24289:
24269:
24266:
24246:
24243:
24240:
24220:
24217:
24205:
24199:
24197:
24195:
24193:
24191:
24189:
24187:
24185:
24183:
24181:
24179:
24177:
24169:
24168:cubic pyramid
24164:
24160:
24159:cuboctahedron
24154:
24152:
24150:
24148:
24140:
24139:cubic pyramid
24119:
24117:
24115:
24100:
23738:
23734:
23730:
23724:
23722:
23720:
23718:
23716:
23708:
23703:
23702:
23686:
23684:
23682:
23652:
23648:
23644:
23638:
23636:
23634:
23632:
23630:
23628:
23626:
23624:
23616:
23612:
23540:
23530:
23526:
23520:
23513:
23508:
23504:
23500:
23496:
23492:
23488:
23484:
23478:
23476:
23474:
23472:
23464:
23448:
23445:
23442:
23422:
23419:
23416:
23396:
23393:
23390:
23378:
23372:
23370:
23368:
23366:
23364:
23356:
23352:
23345:
23343:
23341:
23339:
23337:
23326:
23324:
23315:
23308:
23300:
23234:
23222:
23214:
23210:
23204:
23196:
23192:
23185:
23178:
23174:
23170:
23166:
23162:
23158:
23152:
23150:
23142:
23138:
23134:
23133:cuboctahedron
23130:
23123:
23121:
23119:
23117:
23115:
23113:
23111:
23109:
23107:
23105:
23097:
23093:
23089:
23085:
23079:
23075:
23066:
23063:
23061:
23058:
23057:
23047:
23042:
23038:
23036:
23031:
23027:
23025:
23020:
23016:
23014:
23009:
23005:
23003:
22998:
22994:
22992:
22987:
22983:
22981:
22976:
22972:
22967:
22966:
22963:
22958:
22954:
22952:
22947:
22943:
22941:
22936:
22932:
22930:
22925:
22921:
22919:
22914:
22910:
22908:
22903:
22899:
22897:
22892:
22888:
22885:
22884:
22880:
22876:
22873:
22869:
22866:
22862:
22859:
22855:
22852:
22848:
22845:
22841:
22838:
22834:
22831:
22830:
22827:
22826:{3,∞,3}
22823:
22821:
22818:
22816:
22813:
22811:
22808:
22806:
22803:
22801:
22798:
22796:
22793:
22790:
22786:
22785:
22778:
22775:
22769:
22768:
22765:
22760:
22754:
22753:
22749:
22744:
22736:
22732:
22729:
22725:
22722:
22718:
22715:
22711:
22708:
22704:
22701:
22697:
22694:
22690:
22687:
22683:
22680:
22676:
22670:
22669:
22665:
22661:
22658:
22654:
22651:
22647:
22644:
22640:
22637:
22633:
22630:
22626:
22623:
22619:
22616:
22612:
22609:
22605:
22603:
22598:
22597:
22587:
22578:
22567:
22556:
22547:
22536:
22525:
22514:
22511:
22509:
22504:
22503:
22463:
22424:
22356:
22288:
22249:
22181:
22113:
22045:
21977:
21975:
21970:
21969:
21966:
21963:omnitruncated
21961:
21959:
21954:
21952:
21947:
21945:
21940:
21938:
21933:
21931:
21926:
21924:
21919:
21917:
21912:
21910:
21907:
21904:
21903:
21898:
21894:
21890:
21887:
21883:
21880:
21876:
21873:
21869:
21866:
21862:
21859:
21855:
21852:
21848:
21845:
21841:
21838:
21834:
21828:
21827:
21823:
21819:
21816:
21812:
21809:
21805:
21802:
21798:
21795:
21791:
21788:
21784:
21781:
21777:
21774:
21770:
21767:
21763:
21761:
21756:
21755:
21745:
21736:
21725:
21714:
21705:
21694:
21683:
21672:
21669:
21667:
21662:
21661:
21621:
21582:
21543:
21475:
21436:
21397:
21358:
21290:
21251:
21249:
21244:
21243:
21240:
21237:omnitruncated
21235:
21233:
21228:
21226:
21221:
21219:
21214:
21212:
21207:
21205:
21200:
21198:
21193:
21191:
21186:
21184:
21181:
21178:
21177:
21172:
21169:
21161:
21154:
21147:
21140:
21133:
21126:
21119:
21112:
21105:
21098:
21087:
21083:
21076:
21069:
21062:
21055:
21048:
21036:
21032:
21025:
21018:
21011:
21004:
20997:
20990:
20983:
20976:
20969:
20957:
20953:
20946:
20939:
20932:
20925:
20918:
20911:
20904:
20897:
20890:
20879:
20875:
20868:
20861:
20854:
20847:
20840:
20833:
20826:
20819:
20812:
20801:
20797:
20790:
20783:
20776:
20769:
20762:
20755:
20748:
20741:
20734:
20728:
20722:
20682:
20643:
20604:
20565:
20526:
20487:
20448:
20409:
20370:
20331:
20329:
20324:
20323:
20313:
20304:
20295:
20284:
20273:
20262:
20251:
20248:
20237:
20234:
20232:
20227:
20226:
20222:
20218:
20215:
20211:
20209:
20206:
20203:
20199:
20197:
20194:
20192:
20189:
20187:
20184:
20182:
20179:
20177:
20174:
20172:
20169:
20166:
20165:
20160:
20151:
20148:
20144:
20141:
20137:
20134:
20130:
20127:
20119:
20116:
20108:
20105:
20097:
20094:
20090:
20087:
20086:
20082:
20078:
20075:
20071:
20068:
20064:
20061:
20057:
20054:
20050:
20047:
20043:
20040:
20036:
20033:
20029:
20028:
19975:
19923:
19871:
19819:
19762:
19705:
19648:
19591:
19590:
19581:
19573:
19571:
19567:
19563:
19559:
19555:
19551:
19546:
19544:
19540:
19536:
19532:
19528:
19520:
19516:
19512:
19509:
19505:
19502:
19498:
19497:
19496:
19494:
19490:
19448:
19444:
19414:
19410:
19405:
19401:
19393:
19390:
19387:
19384:
19379:
19376:
19373:
19370:
19341:
19314:
19312:, order 1152
19273:
19270:
19269:
19241:
19214:
19174:
19171:
19170:
19164:
19133:
19073:
19021:
19006:
19002:
18999:
18995:
18992:
18988:
18982:
18981:
18980:Vertex figure
18976:
18972:
18965:
18961:
18954:
18945:
18940:
18932:
18925:
18918:
18917:
18913:
18911:
18908:
18906:
18903:
18902:
18886:
18882:
18877:
18874:
18868:
18860:
18858:
18825:
18821:
18815:
18810:
18804:
18800:
18799:
18789:
18783:
18779:
18773:
18769:
18764:
18760:
18754:
18751:
18747:
18745:
18741:
18738:
18728:
18726:
18722:
18718:
18714:
18709:
18707:
18703:
18702:cuboctahedral
18699:
18694:
18692:
18689:
18685:
18676:
18667:
18663:
18661:
18650:
18646:
18641:
18637:
18634:
18625:
18614:
18611:
18610:
18606:
18603:
18600:
18597:
18596:
18589:
18586:
18583:
18580:
18579:
18575:
18572:
18569:
18566:
18563:
18562:
18558:
18555:
18552:
18549:
18548:
18541:
18538:
18535:
18532:
18531:
18527:
18524:
18521:
18518:
18515:
18514:
18511:
18509:
18503:
18501:
18500:cuboctahedron
18497:
18493:
18489:
18484:
18472:
18468:
18464:
18460:
18459:circumference
18456:
18452:
18448:
18444:
18440:
18437:
18424:
18420:
18414:Visualization
18411:
18409:
18391:
18388:
18385:
18382:
18379:
18375:
18368:
18345:
18342:
18322:
18319:
18311:
18277:
18274:
18271:
18268:
18265:
18261:
18254:
18243:
18238:
18236:
18232:
18228:
18210:
18207:
18204:
18201:
18198:
18194:
18187:
18175:
18159:
18156:
18153:
18150:
18147:
18143:
18136:
18111:
18108:
18105:
18102:
18099:
18096:
18093:
18089:
18082:
18057:
18054:
18051:
18048:
18045:
18041:
18034:
18009:
18006:
18003:
18000:
17997:
17994:
17991:
17987:
17980:
17957:
17954:
17934:
17931:
17909:
17906:
17903:
17900:
17897:
17893:
17886:
17860:
17856:
17851:
17828:
17825:
17822:
17819:
17816:
17812:
17804:
17787:
17779:
17776:
17759:
17756:
17748:
17745:
17728:
17725:
17717:
17714:
17711:
17710:
17692:
17689:
17686:
17683:
17680:
17677:
17674:
17671:
17643:
17640:
17637:
17634:
17624:
17619:
17618:{24/12}=12{2}
17599:
17596:
17593:
17590:
17587:
17584:
17581:
17553:
17550:
17547:
17537:
17532:
17531:{24/12}=12{2}
17513:
17510:
17507:
17504:
17501:
17498:
17494:
17487:
17459:
17456:
17453:
17450:
17447:
17444:
17441:
17431:
17426:
17425:{24/12}=12{2}
17422:
17418:
17410:
17407:
17390:
17387:
17379:
17376:
17359:
17356:
17348:
17345:
17342:
17341:
17323:
17320:
17317:
17314:
17311:
17308:
17305:
17277:
17274:
17271:
17261:
17256:
17255:{24/12}=12{2}
17236:
17233:
17230:
17227:
17224:
17221:
17218:
17190:
17187:
17184:
17174:
17169:
17168:{24/12}=12{2}
17150:
17147:
17144:
17141:
17138:
17134:
17127:
17099:
17096:
17093:
17090:
17087:
17084:
17074:
17069:
17065:
17061:
17053:
17050:
17033:
17030:
17022:
17019:
17002:
16999:
16991:
16988:
16971:
16970:
16952:
16949:
16946:
16943:
16940:
16937:
16934:
16906:
16903:
16900:
16890:
16885:
16884:{24/12}=12{2}
16865:
16862:
16859:
16856:
16853:
16850:
16847:
16819:
16816:
16813:
16803:
16798:
16797:{24/12}=12{2}
16779:
16776:
16773:
16770:
16767:
16763:
16756:
16728:
16725:
16722:
16719:
16716:
16713:
16703:
16698:
16697:{24/12}=12{2}
16694:
16690:
16682:
16679:
16662:
16659:
16651:
16648:
16631:
16628:
16620:
16617:
16614:
16613:
16595:
16592:
16589:
16586:
16583:
16580:
16577:
16574:
16546:
16543:
16540:
16537:
16527:
16522:
16502:
16499:
16496:
16493:
16490:
16487:
16484:
16456:
16453:
16450:
16440:
16435:
16416:
16413:
16410:
16407:
16404:
16401:
16397:
16390:
16362:
16359:
16356:
16353:
16350:
16347:
16344:
16334:
16329:
16328:{24/12}=12{2}
16325:
16321:
16313:
16310:
16293:
16290:
16282:
16279:
16262:
16259:
16251:
16248:
16231:
16230:
16209:
16206:
16200:
16194:
16191:
16185:
16179:
16176:
16170:
16164:
16161:
16130:
16127:
16124:
16114:
16109:
16089:
16086:
16083:
16080:
16077:
16074:
16071:
16043:
16040:
16037:
16027:
16022:
16003:
16000:
15997:
15994:
15991:
15987:
15980:
15952:
15949:
15946:
15943:
15940:
15937:
15927:
15922:
15918:
15914:
15906:
15903:
15886:
15883:
15875:
15872:
15855:
15852:
15844:
15841:
15838:
15837:
15816:
15812:
15805:
15799:
15795:
15788:
15785:
15782:
15779:
15751:
15748:
15745:
15735:
15730:
15707:
15703:
15696:
15690:
15686:
15679:
15676:
15673:
15670:
15642:
15639:
15636:
15626:
15621:
15602:
15599:
15596:
15593:
15590:
15586:
15579:
15551:
15548:
15545:
15542:
15539:
15536:
15526:
15521:
15517:
15513:
15505:
15502:
15499:
15496:
15488:
15485:
15468:
15465:
15457:
15454:
15437:
15436:
15415:
15411:
15404:
15401:
15395:
15391:
15384:
15381:
15378:
15375:
15372:
15344:
15341:
15338:
15335:
15325:
15320:
15300:
15297:
15294:
15291:
15285:
15281:
15274:
15268:
15264:
15232:
15229:
15226:
15216:
15211:
15192:
15189:
15186:
15183:
15180:
15177:
15173:
15166:
15138:
15135:
15132:
15129:
15126:
15123:
15120:
15110:
15105:
15104:{24/9}=3{8/3}
15101:
15097:
15089:
15086:
15069:
15066:
15058:
15055:
15038:
15035:
15027:
15024:
15021:
15020:
15002:
14999:
14996:
14993:
14987:
14983:
14976:
14973:
14967:
14963:
14956:
14928:
14925:
14922:
14919:
14909:
14904:
14884:
14881:
14878:
14875:
14869:
14865:
14858:
14852:
14848:
14816:
14813:
14810:
14800:
14795:
14776:
14773:
14770:
14767:
14764:
14761:
14757:
14750:
14722:
14719:
14716:
14713:
14710:
14707:
14704:
14694:
14689:
14688:{24/12}=12{2}
14685:
14681:
14673:
14670:
14653:
14650:
14642:
14639:
14622:
14619:
14611:
14608:
14605:
14604:
14586:
14583:
14580:
14577:
14571:
14567:
14560:
14554:
14550:
14518:
14515:
14512:
14502:
14497:
14477:
14474:
14471:
14468:
14462:
14458:
14451:
14445:
14441:
14409:
14406:
14403:
14393:
14388:
14369:
14366:
14363:
14360:
14357:
14353:
14346:
14318:
14315:
14312:
14309:
14306:
14303:
14293:
14288:
14284:
14280:
14272:
14269:
14252:
14249:
14241:
14238:
14221:
14218:
14210:
14207:
14190:
14189:
14171:
14168:
14165:
14162:
14159:
14156:
14153:
14150:
14122:
14119:
14116:
14113:
14103:
14098:
14075:
14072:
14066:
14060:
14057:
14051:
14045:
14042:
14036:
14030:
14027:
13996:
13993:
13990:
13980:
13975:
13956:
13953:
13950:
13947:
13944:
13941:
13937:
13930:
13902:
13899:
13896:
13893:
13890:
13887:
13884:
13874:
13869:
13868:{24/8}=4{6/2}
13865:
13861:
13853:
13850:
13833:
13830:
13822:
13819:
13802:
13799:
13791:
13788:
13771:
13770:
13752:
13749:
13746:
13743:
13740:
13737:
13734:
13706:
13703:
13700:
13690:
13685:
13662:
13659:
13653:
13647:
13644:
13638:
13632:
13629:
13623:
13617:
13614:
13583:
13580:
13577:
13567:
13562:
13543:
13540:
13537:
13534:
13531:
13527:
13520:
13492:
13489:
13486:
13483:
13480:
13477:
13467:
13462:
13458:
13454:
13446:
13443:
13426:
13423:
13415:
13412:
13395:
13392:
13384:
13381:
13378:
13377:
13356:
13353:
13347:
13344:
13338:
13335:
13329:
13326:
13320:
13317:
13311:
13308:
13302:
13299:
13293:
13265:
13262:
13259:
13256:
13246:
13241:
13240:{24/8}=4{6/2}
13218:
13215:
13209:
13203:
13200:
13194:
13188:
13185:
13179:
13173:
13170:
13139:
13136:
13133:
13123:
13118:
13099:
13096:
13093:
13090:
13087:
13084:
13080:
13073:
13045:
13042:
13039:
13036:
13033:
13030:
13027:
13017:
13012:
13011:{24/12}=12{2}
13008:
13004:
12996:
12993:
12976:
12973:
12965:
12962:
12945:
12942:
12934:
12931:
12928:
12927:
12906:
12903:
12897:
12891:
12888:
12882:
12876:
12873:
12867:
12861:
12858:
12827:
12824:
12821:
12811:
12806:
12783:
12780:
12774:
12768:
12765:
12759:
12753:
12750:
12744:
12738:
12735:
12704:
12701:
12698:
12688:
12683:
12664:
12661:
12658:
12655:
12652:
12648:
12641:
12613:
12610:
12607:
12604:
12601:
12598:
12588:
12583:
12579:
12575:
12567:
12564:
12547:
12544:
12536:
12533:
12516:
12513:
12505:
12502:
12485:
12484:
12463:
12460:
12454:
12448:
12445:
12439:
12433:
12430:
12424:
12418:
12415:
12409:
12381:
12378:
12375:
12372:
12362:
12357:
12334:
12331:
12325:
12319:
12316:
12310:
12304:
12301:
12295:
12289:
12286:
12255:
12252:
12249:
12239:
12234:
12215:
12212:
12209:
12206:
12203:
12200:
12196:
12189:
12161:
12158:
12155:
12152:
12149:
12146:
12143:
12133:
12128:
12124:
12120:
12112:
12109:
12092:
12089:
12081:
12078:
12061:
12058:
12050:
12047:
12030:
12029:
12008:
12005:
11999:
11996:
11990:
11987:
11981:
11978:
11972:
11969:
11963:
11960:
11954:
11951:
11920:
11917:
11914:
11904:
11899:
11898:{24/8}=4{6/2}
11876:
11873:
11867:
11861:
11858:
11852:
11846:
11843:
11837:
11831:
11828:
11797:
11794:
11791:
11781:
11776:
11757:
11754:
11751:
11748:
11745:
11741:
11734:
11706:
11703:
11700:
11697:
11694:
11691:
11681:
11676:
11675:{24/8}=4{6/2}
11672:
11657:
11654:
11647:Right planes
11632:
11629:
11615:
11614:
11611:
11610:
11601:
11596:
11593:
11573:
11570:
11564:
11561:
11555:
11552:
11546:
11543:
11537:
11534:
11528:
11525:
11519:
11516:
11484:
11481:
11475:
11469:
11466:
11460:
11454:
11451:
11445:
11439:
11436:
11423:
11407:
11404:
11384:
11381:
11343:
11340:
11337:
11334:
11331:
11327:
11320:
11309:
11305:
11301:
11297:
11292:
11290:
11286:
11281:
11279:
11274:
11264:
11262:
11250:
11246:
11241:
11239:
11235:
11232:) performing
11231:
11227:
11223:
11219:
11211:
11207:
11203:
11199:
11195:
11185:
11166:
11163:
11133:
11130:
11100:
11097:
11067:
11064:
11034:
11031:
11001:
10998:
10968:
10965:
10939:
10930:
10909:
10906:
10876:
10873:
10843:
10840:
10827:
10806:
10803:
10773:
10770:
10740:
10737:
10722:
10720:
10717:based on the
10716:
10675:
10671:
10667:
10663:
10633:
10630:
10623:
10620:
10593:
10589:
10583:
10579:
10573:
10570:
10560:
10536:
10533:
10526:
10523:
10496:
10492:
10486:
10482:
10476:
10473:
10463:
10439:
10436:
10429:
10426:
10399:
10395:
10389:
10385:
10379:
10376:
10366:
10342:
10319:
10315:
10309:
10305:
10299:
10296:
10286:
10283:
10281:
10279:
10277:
10275:
10274:
10255:
10252:
10218:
10215:
10184:
10181:
10174:
10171:
10144:
10140:
10134:
10130:
10124:
10121:
10111:
10092:
10089:
10055:
10052:
10021:
10018:
10011:
10008:
9981:
9977:
9971:
9967:
9961:
9958:
9948:
9929:
9926:
9892:
9889:
9858:
9855:
9848:
9845:
9818:
9814:
9808:
9804:
9798:
9795:
9785:
9782:
9780:
9778:
9776:
9774:
9773:
9754:
9751:
9717:
9714:
9683:
9680:
9673:
9670:
9654:
9635:
9632:
9598:
9595:
9564:
9561:
9554:
9551:
9535:
9516:
9513:
9479:
9476:
9445:
9442:
9435:
9432:
9416:
9413:
9411:
9409:
9407:
9405:
9404:
9385:
9381:
9378:
9343:
9340:
9309:
9297:
9284:
9279:
9252:
9242:
9239:
9224:
9223:golden chords
9220:
9216:
9213:
9202:
9201:skew polygons
9198:
9188:
9187:
9183:
9176:
9169:
9162:
9155:
9150:
9147:
9142:
9137:
9134:
9129:
9126:
9121:
9118:
9113:
9110:
9109:
9106:
9103:
9101:
9098:
9096:
9095:In a 600-cell
9093:
9091:
9088:
9086:
9083:
9082:
9079:
9076:
9071:
9068:
9066:
9062:
9058:
9053:
9050:
9049:720Β° octagram
9030:
9013:
8991:
8980:
8975:
8973:
8967:
8958:
8940:
8936:
8926:
8924:
8918:
8915:
8911:
8887:
8861:
8860:great hexagon
8842:
8838:
8836:
8814:
8810:
8806:
8796:
8793:
8789:
8785:
8780:
8778:
8772:
8770:
8766:
8762:
8757:
8749:
8740:
8738:
8734:
8695:
8693:
8689:
8685:
8681:
8677:
8673:
8668:
8666:
8656:
8652:
8648:
8646:
8642:
8638:
8634:
8630:
8624:
8621:
8611:
8609:
8605:
8600:
8598:
8594:
8590:
8582:
8577:
8573:
8569:
8565:
8553:
8550:
8525:
8521:
8496:
8492:
8487:
8486:vice versa).
8484:
8480:
8475:
8473:
8469:
8465:
8461:
8456:
8455:isoclinically
8452:
8449:
8439:
8435:
8433:
8429:
8425:
8421:
8417:
8412:
8410:
8406:
8401:
8397:
8389:
8384:
8375:
8373:
8368:
8364:
8359:
8354:
8352:
8350:
8345:
8340:
8336:
8335:invariant set
8332:
8324:
8319:
8310:
8308:
8304:
8300:
8296:
8287:
8284:
8280:
8276:
8266:
8264:
8260:
8256:
8252:
8248:
8238:
8234:
8230:
8226:
8224:
8220:
8216:
8206:
8204:
8200:
8196:
8192:
8187:
8185:
8181:
8168:
8164:
8160:
8156:
8149:
8148:Voronoi cells
8142:Voronoi cells
8139:
8137:
8132:
8130:
8126:
8121:
8115:
8107:
8103:
8099:
8095:
8091:
8084:
8076:
8071:
8062:
8060:
8052:
8044:
8040:
8025:
8018:
8013:
8003:
7999:
7998:cuboctahedron
7995:
7991:
7987:
7984:
7980:
7964:
7956:
7947:
7944:
7929:
7919:
7914:
7909:
7904:
7897:
7892:
7887:
7882:
7875:
7870:
7865:
7860:
7853:
7848:
7843:
7838:
7828:
7818:
7815:
7805:
7774:
7769:
7765:
7756:
7754:
7748:
7746:
7729:
7725:
7714:
7698:
7694:
7685:
7683:
7679:
7673:
7669:
7666:which is the
7665:
7624:
7619:
7617:
7578:
7556:
7554:
7535:
7518:
7516:
7511:
7507:
7490:
7488:
7484:
7480:
7476:
7466:
7462:
7460:
7459:cuboctahedron
7456:
7451:
7447:
7443:
7375:
7365:
7363:
7359:
7351:Constructions
7348:
7346:
7334:
7332:
7327:
7323:
7322:cuboctahedron
7318:
7273:
7271:
7267:
7261:
7251:
7234:
7222:
7198:
7182:
7180:
7177:
7176:
7172:
7168:
7165:
7161:
7158:
7157:
7146:
7140:
7137:
7136:
7133:
7131:
7128:
7127:
7123:
7119:
7116:
7112:
7109:
7108:
7096:
7085:
7082:
7081:
7076:
7073:
7072:
7068:
7061:
7060:
7051:
7050:Coxeter plane
7048:
7047:
7043:
7038:
7036:
7025:
7013:
7004:
7000:
6997:
6989:
6985:
6976:
6925:
6893:
6888:
6804:
6774:
6765:
6756:
6741:
6731:
6727:
6722:
6720:
6697:
6694:
6688:
6679:
6662:
6647:
6644:
6638:
6629:
6612:
6597:
6589:
6574:
6566:
6551:
6548:
6542:
6539:
6528:
6513:
6510:
6505:
6500:
6494:
6490:
6483:
6475:
6471:
6459:
6456:
6455:
6439:
6436:
6432:
6423:
6416:
6412:
6408:
6401:
6396:
6393:
6390:
6383:
6379:
6371:
6356:
6353:
6349:
6340:
6336:
6332:
6327:
6320:
6316:
6308:
6293:
6290:
6286:
6280:
6276:
6269:
6265:
6257:
6242:
6234:
6219:
6216:
6212:
6206:
6203:
6197:
6193:
6185:
6170:
6167:
6163:
6157:
6151:
6146:
6139:
6135:
6127:
6124:
6123:
6107:
6104:
6100:
6091:
6087:
6083:
6076:
6071:
6068:
6065:
6058:
6054:
6046:
6031:
6028:
6024:
6015:
6011:
6007:
6002:
5995:
5991:
5983:
5968:
5965:
5961:
5954:
5951:
5944:
5940:
5932:
5917:
5909:
5894:
5891:
5887:
5880:
5877:
5870:
5866:
5858:
5843:
5840:
5836:
5830:
5824:
5819:
5812:
5808:
5800:
5797:
5796:
5780:
5777:
5770:
5765:
5761:
5748:
5733:
5730:
5723:
5718:
5714:
5701:
5686:
5683:
5676:
5673:
5661:
5643:
5640:
5629:
5611:
5608:
5597:
5579:
5576:
5565:
5563:Short radius
5562:
5561:
5545:
5542:
5533:
5526:
5522:
5517:
5506:
5491:
5488:
5482:
5479:
5468:
5453:
5445:
5430:
5422:
5407:
5404:
5399:
5389:
5374:
5371:
5364:
5361:
5349:
5346:
5345:
5329:
5321:
5306:
5298:
5283:
5275:
5260:
5252:
5237:
5229:
5214:
5206:
5204:
5201:
5200:
5196:
5192:
5189:
5185:
5182:
5178:
5175:
5171:
5168:
5164:
5161:
5157:
5155:
5152:
5151:
5147:
5143:
5140:
5136:
5133:
5129:
5126:
5123:
5119:
5117:
5115:
5112:
5111:
5108:10 600-cells
5107:
5104:
5101:
5098:
5095:
5092:
5089:
5088:
5085:
5081:
5079:
5075:
5073:
5069:
5067:
5063:
5061:
5060:8-tetrahedron
5057:
5055:
5054:5-tetrahedron
5051:
5049:
5046:
5045:
5041:
5038:
5036:24 octahedra
5035:
5032:
5029:
5027:5 tetrahedra
5026:
5023:
5022:
5018:
5015:
5013:96 triangles
5012:
5009:
5007:32 triangles
5006:
5004:10 triangles
5003:
5000:
4999:
4995:
4992:
4989:
4986:
4983:
4980:
4978:
4975:
4974:
4970:
4967:
4964:
4961:
4959:8 octahedral
4958:
4955:
4952:
4951:
4947:
4943:
4940:
4936:
4933:
4929:
4926:
4922:
4919:
4915:
4912:
4908:
4905:
4904:
4811:
4719:
4627:
4535:
4443:
4351:
4348:
4347:
4307:
4268:
4229:
4190:
4151:
4112:
4110:
4107:
4106:
4102:
4099:
4096:
4093:
4090:
4087:
4085:
4082:
4081:
4078:
4075:
4069:
4066:
4064:
4061:
4055:
4052:
4050:
4045:
4042:
4040:
4037:
4031:
4028:
4026:
4023:
4017:
4014:
4012:
4009:
4003:
4000:
3997:
3996:
3993:
3985:
3979:
3977:
3969:
3963:
3961:
3958:
3957:
3953:
3949:
3946:
3944:
3933:
3928:
3926:
3922:
3918:
3915:
3901:
3894:
3891:
3885:
3882:
3876:
3873:
3867:
3864:
3858:
3855:
3849:
3846:
3840:
3837:
3831:
3827:
3818:
3816:
3811:
3797:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3768:
3759:
3758:coordinates:
3757:
3752:
3749:
3737:
3735:
3725:
3723:
3700:
3697:
3691:
3682:
3665:
3650:
3647:
3641:
3632:
3615:
3600:
3592:
3577:
3569:
3554:
3551:
3545:
3542:
3531:
3516:
3513:
3508:
3503:
3498:
3493:
3485:
3481:
3469:
3466:
3465:
3449:
3446:
3442:
3433:
3429:
3425:
3418:
3413:
3410:
3407:
3400:
3396:
3388:
3373:
3370:
3366:
3357:
3353:
3349:
3345:
3339:
3335:
3327:
3312:
3309:
3305:
3299:
3296:
3290:
3286:
3278:
3263:
3260:
3255:
3250:
3242:
3227:
3224:
3220:
3214:
3208:
3203:
3196:
3192:
3184:
3169:
3166:
3162:
3156:
3150:
3145:
3138:
3134:
3126:
3123:
3122:
3106:
3103:
3099:
3090:
3086:
3082:
3075:
3070:
3067:
3064:
3057:
3053:
3045:
3030:
3027:
3023:
3014:
3010:
3006:
2999:
2994:
2987:
2983:
2975:
2960:
2957:
2953:
2946:
2943:
2936:
2932:
2924:
2909:
2901:
2886:
2883:
2879:
2874:
2869:
2865:
2857:
2842:
2839:
2835:
2829:
2823:
2818:
2811:
2807:
2799:
2796:
2795:
2779:
2776:
2769:
2764:
2760:
2747:
2732:
2729:
2722:
2717:
2713:
2700:
2685:
2677:
2662:
2659:
2653:
2649:
2637:
2622:
2619:
2613:
2609:
2597:
2582:
2579:
2573:
2569:
2557:
2555:Short radius
2554:
2553:
2537:
2534:
2531:
2528:
2525:
2517:
2502:
2499:
2493:
2489:
2477:
2462:
2459:
2454:
2444:
2429:
2426:
2421:
2411:
2396:
2388:
2373:
2370:
2365:
2355:
2352:
2351:
2333:
2323:
2306:
2296:
2279:
2269:
2252:
2242:
2225:
2215:
2198:
2188:
2186:
2183:
2182:
2178:
2174:
2171:
2167:
2164:
2160:
2157:
2153:
2150:
2146:
2143:
2139:
2137:
2134:
2133:
2129:
2125:
2122:
2118:
2115:
2111:
2108:
2105:
2101:
2099:
2097:
2094:
2093:
2090:10 600-cells
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2070:
2067:
2063:
2061:
2057:
2055:
2051:
2049:
2045:
2043:
2042:8-tetrahedron
2039:
2037:
2036:5-tetrahedron
2033:
2031:
2028:
2027:
2023:
2020:
2018:24 octahedra
2017:
2014:
2011:
2009:5 tetrahedra
2008:
2005:
2004:
2000:
1997:
1995:96 triangles
1994:
1991:
1989:32 triangles
1988:
1986:10 triangles
1985:
1982:
1981:
1977:
1974:
1971:
1968:
1965:
1962:
1960:
1957:
1956:
1952:
1949:
1946:
1943:
1941:8 octahedral
1940:
1937:
1934:
1933:
1929:
1925:
1922:
1918:
1915:
1911:
1908:
1904:
1901:
1897:
1894:
1890:
1887:
1886:
1793:
1701:
1609:
1517:
1425:
1333:
1330:
1329:
1289:
1250:
1211:
1172:
1133:
1094:
1092:
1089:
1088:
1084:
1081:
1078:
1075:
1072:
1069:
1067:
1064:
1063:
1060:
1057:
1051:
1048:
1046:
1043:
1037:
1034:
1032:
1027:
1024:
1022:
1019:
1013:
1010:
1008:
1005:
999:
996:
994:
991:
985:
982:
979:
978:
975:
967:
961:
959:
951:
945:
943:
940:
939:
929:
925:
922:
920:
916:
915:cuboctahedron
912:
901:
889:
887:
846:
842:
799:
786:
781:
771:
765:
762:
759:
756:
753:
750:
747:
744:
741:
730:
728:
724:
709:
707:
703:
699:
694:
692:
688:
684:
673:
671:
667:
666:parallelotope
663:
659:
654:
652:
648:
647:cuboctahedron
643:
638:
636:
632:
628:
624:
623:vertex figure
620:
615:
613:
610:
606:
602:
601:hyper-diamond
598:
597:
592:
588:
584:
580:
573:
569:
565:
561:
557:
550:
546:
537:
535:
534:Uniform index
531:
528:
524:
520:
516:
513:
509:
506:
503:
501:
497:
494:, , order 192
488:, , order 384
481:
475:
473:
472:Coxeter group
469:
466:
463:
461:
457:
454:
451:
449:
448:Vertex figure
445:
441:
439:
435:
431:
429:
425:
422:
418:
416:
412:
408:
404:
400:
398:
394:
237:
235:
231:
215:
208:
201:
194:
187:
164:
157:
154:
151:
144:
137:
126:
124:
120:
117:
114:
110:
104:
99:
94:
89:
80:
77:
69:
59:
54:
49:
45:
44:
37:
28:
27:
22:
34612:
34581:
34572:
34564:
34555:
34546:
34526:10-orthoplex
34296:
34262:Dodecahedron
34183:
34172:
34161:
34152:
34143:
34134:
34130:
34120:
34112:
34108:
34100:
34096:
33512:
33380:
33376:
33345:
33341:
33328:
33307:
33280:
33276:
33260:. Retrieved
33257:gregegan.net
33256:
33225:
33189:
33185:
33150:
33146:
33110:
33061:
33040:
33018:
33002:
32990:
32968:
32955:
32951:
32935:
32931:
32909:
32905:
32889:
32876:
32869:
32862:
32856:
32837:
32823:
32810:
32796:
32775:Coxeter 1991
32770:
32758:
32734:
32717:
32698:group theory
32694:Coxeter 1970
32689:
32677:
32670:Coxeter 1973
32665:
32658:Coxeter 1973
32653:
32646:Coxeter 1973
32641:
32629:
32622:Coxeter 1973
32617:
32610:Coxeter 1973
32605:
32598:Coxeter 1973
32593:
32586:Coxeter 1973
32581:
32562:
32559:Coxeter 1973
32554:
32543:
32540:Coxeter 1973
32535:
32519:
32516:Coxeter 1973
32511:
32499:
32487:
32475:
32468:Coxeter 1973
32463:
32451:
32439:
32427:
32415:
32395:
32379:
32375:
32367:
32355:
32347:
32336:Coxeter 1973
32331:
32319:
32307:
32295:
32292:Coxeter 1995
32287:
32275:
32263:
32251:
32239:
32231:
32228:
32224:
32220:
32217:Coxeter 1973
32212:
32196:
32189:Coxeter 1973
32169:Coxeter 1973
32164:
32152:
32140:
32128:
32116:
32104:
32099:-1)-spaces."
32096:
32092:
32088:
32084:
32080:
32076:
32072:
32068:
32065:Coxeter 1973
32060:
32053:Coxeter 1973
32048:
31987:Coxeter 1973
31982:
31970:
31917:Coxeter 1973
31912:
31900:
31850:Coxeter 1973
31845:
31837:
31834:Coxeter 1995
31829:
31815:Coxeter 1973
31810:
31803:Coxeter 1973
31798:
31791:Coxeter 1973
31786:
31779:Coxeter 1973
31745:
31738:Coxeter 1973
31733:
31726:Coxeter 1973
31721:
31713:
31709:
31701:
31689:
31662:
31650:
31638:
31630:
31627:Coxeter 1973
31622:
31610:
31603:Coxeter 1973
31598:
31591:Coxeter 1973
31586:
31579:Coxeter 1973
31574:
31566:
31563:Coxeter 1973
31548:Coxeter 1973
31543:
31536:Coxeter 1968
31531:
31523:
31520:Coxeter 1995
31515:
31508:Coxeter 1973
31503:
31491:
31484:Johnson 2018
31479:
31472:Coxeter 1973
31467:
31450:
31446:
31434:
31430:
31425:
31417:
31405:
31349:
31341:
31288:
31223:
31214:
31206:
31142:
31077:
31012:
30944:
30935:
30927:
30922:
30882:
30861:
30853:
30789:
30667:
30590:great square
30511:. Thus e.g.
30442:
30430:
30425:
30360:
30292:
30206:
30196:
30131:
30111:
30091:
30014:
29946:
29881:
29813:
29804:
29796:
29791:
29754:
29658:
29650:
29637:
29586:
29578:
29574:
29570:
29566:
29557:
29550:Right planes
29549:
29545:
29292:
29012:
29004:
28998:
28994:
28990:
28982:
28978:
28974:
28948:
28938:
28929:-polytope's
28926:
28922:
28918:
28914:
28910:
28906:
28902:
28894:
28890:
28882:
28877:
28870:golden ratio
28865:
28862:Coxeter 1973
28840:
28776:
28768:
28763:
28755:
28729:
28688:
28672:
28641:
28632:
28627:
28623:
28618:
28608:
28599:
28586:
28570:
28529:
28506:
28486:
28456:
28447:
28439:
28434:
28426:
28421:
28416:
28407:
28381:
28374:great circle
28369:Each set of
28365:
28326:aspect ratio
28288:
28279:
28267:
28262:
28203:great circle
28202:
28198:
28184:
28138:
28081:
28049:
28045:
28041:
27998:screw thread
27994:MΓΆbius strip
27968:
27963:
27898:
27788:MΓΆbius strip
27753:
27728:
27716:
27703:
27694:
27655:
27630:
27610:
27593:
27585:
27574:
27563:
27507:
27501:
27492:
27488:
27483:
27482:
27472:
27466:
27462:
27458:
27453:
27452:
27438:
27432:
27426:
27422:
27416:
27411:
27410:
27402:
27398:
27394:
27389:
27388:
27384:
27380:
27376:
27367:
27366:
27361:
27349:
27346:MΓΆbius strip
27333:
27296:16-cell axis
27291:
27287:
27260:
27255:
27242:
27216:
27169:
27165:
27104:
27099:
27095:
27085:
27081:
27061:
27057:
27053:
27049:
27045:
27035:
27027:
27023:
26999:
26992:equal-angled
26991:
26987:
26983:
26979:
26975:
26971:
26967:
26963:
26959:
26955:
26951:
26947:
26943:
26939:
26920:great circle
26919:
26915:
26903:
26899:
26895:
26891:
26887:
26883:
26856:
26852:
26848:
26844:
26817:
26810:displacement
26806:
26784:
26780:
26774:
26765:
26757:
26753:
26749:
26745:
26741:
26737:
26733:
26730:surface area
26729:
26726:-sphere rule
26723:
26718:
26714:
26710:
26706:
26697:
26689:
26685:
26667:
26629:
26619:
26611:
26606:
26600:
26592:
26588:
26536:
26519:
26509:
26500:
26491:
26483:
26472:mirror walls
26471:
26402:
26357:
26342:
26323:
26291:
26280:
26262:
26252:
26247:(tesseract).
26239:from a cube.
26226:
26210:
26157:
26148:
26140:
26136:
26128:
26124:
26116:
26112:
26082:
26078:
26074:
26065:
26061:
26053:
26004:
25996:
25984:bent into a
25965:
25957:
25953:
25941:
25936:
25930:
25926:
25922:
25883:
25874:
25861:
25851:
25825:
25817:
25813:
25809:
25800:
25797:great circle
25784:
25769:orthogonal).
25719:
25659:
25646:
25640:
25544:
25540:
25470:
25462:
25390:
25385:
25263:
25203:
25172:
25161:
25157:
25086:
25082:
25078:
25074:
25070:
25066:
25055:
25027:
25023:
25019:
25005:
25000:
24994:
24974:
24942:
24910:
24888:element sets
24887:
24883:
24845:
24840:
24836:
24832:
24809:
24770:
24653:
24632:
24611:
24590:
24547:
24508:
24502:
24498:
24494:
24490:
24386:
24381:
24376:
24372:
24349:
24162:
24098:
24093:
24089:
24085:
24062:
24023:
23906:
23885:
23864:
23843:
23800:
23761:
23755:
23751:
23747:
23743:
23737:half-integer
23736:
23732:
23728:
23706:
23700:
23614:
23610:
23605:
23601:
23597:
23591:
23587:
23583:
23579:
23571:
23567:
23563:
23557:
23553:
23549:
23545:
23538:
23528:
23524:
23519:
23511:
23353:if they are
23350:
23307:
23298:
23293:
23289:
23285:
23277:
23273:
23267:
23263:
23259:
23255:
23247:
23243:
23239:
23226:
23221:
23212:
23208:
23203:
23190:
23184:
23167:{5, 3}, the
23165:dodecahedron
23163:{3, 5}, the
23140:
23078:
22799:
22788:
22779:Paracompact
22747:
21167:
20181:snub 24-cell
20170:
20132:{3}={3,4,3}
20128:
19547:
19537:produces an
19531:snub 24-cell
19527:golden ratio
19524:
19486:
19163:, order 72.
19074:
19072:, order 96.
19017:
18978:
18884:
18855:
18853:
18820:stereoscopic
18748:
18737:vertex-first
18736:
18734:
18720:
18712:
18710:
18705:
18697:
18695:
18684:vertex-first
18683:
18681:
18664:
18645:great circle
18642:
18638:
18630:
18522:Description
18504:
18488:great circle
18485:
18455:great circle
18433:
18239:
18234:
18230:
18176:
17793:
15921:{24/2}=2{12}
13461:{24/2}=2{12}
12127:{24/2}=2{12}
11622:Left planes
11605:
11307:
11304:right planes
11303:
11299:
11295:
11293:
11282:
11270:
11244:
11242:
11233:
11221:
11209:
11202:intersecting
11201:
11197:
11191:
10928:
10825:
10723:
10669:
10659:
9060:
9056:
9054:
9031:
8979:4-cell rings
8976:
8968:
8956:
8932:
8922:
8919:
8913:
8843:
8839:
8808:
8802:
8791:
8781:
8773:
8769:non-adjacent
8768:
8758:
8754:
8743:6-cell rings
8701:
8698:4-cell rings
8669:
8662:
8653:
8649:
8644:
8640:
8628:
8625:
8619:
8617:
8601:
8592:
8580:
8575:
8567:
8563:
8494:
8490:
8488:
8478:
8476:
8470:, after its
8467:
8463:
8459:
8445:
8436:
8423:
8419:
8413:
8408:
8404:
8395:
8393:
8366:
8355:
8348:
8343:
8338:
8330:
8328:
8306:
8302:
8298:
8293:
8272:
8244:
8235:
8231:
8227:
8212:
8199:hyperspheres
8188:
8183:
8159:tessellation
8145:
8133:
8122:
8090:root lattice
8080:
8014:
7972:
7945:
7819:
7811:
7770:
7766:
7762:
7749:
7734:
7715:
7704:
7695:
7691:
7620:
7579:
7562:
7524:
7496:
7472:
7469:Diminishings
7463:
7371:
7354:
7335:
7331:fiber bundle
7325:
7319:
7274:
7268:rather than
7262:
7252:
7223:
7187:
7014:
6999:great circle
6993:
6889:
6805:
6770:
6747:
6723:
6715:
5347:Edge length
5105:25 24-cells
5072:6-octahedron
4074:dodecahedron
4071:
4057:
4047:
4043:
4033:
4019:
4005:
3954:of radius 1
3931:
3929:
3919:
3916:
3819:
3815:half-integer
3814:
3812:
3760:
3755:
3753:
3747:
3738:
3731:
3718:
2353:Edge length
2087:25 24-cells
2054:6-octahedron
1056:dodecahedron
1053:
1039:
1029:
1025:
1015:
1001:
987:
918:
890:
800:
731:
727:permutations
720:
695:
686:
682:
679:
655:
639:
616:
604:
600:
594:
590:
586:
582:
575:
559:
553:
72:
63:
56:Please help
51:
40:
34535:10-demicube
34496:9-orthoplex
34446:8-orthoplex
34396:7-orthoplex
34353:6-orthoplex
34323:5-orthoplex
34278:Pentachoron
34266:Icosahedron
34241:Tetrahedron
33726:great grand
33712:icosahedral
33703:great grand
33655:icosahedral
33568:4-orthoplex
33535:pentachoron
33168:2117/113067
33153:: 523β538.
32958:(1): 17β25.
32386:during his
32372:Mebius 2015
32344:H. G. Wells
32176:honeycombs.
32093:hyperplanes
31893:Kepler 1619
31821:and Fig 8.2
31694:Copher 2019
30910:MΓΆbius loop
29783:MΓΆbius loop
29626:MΓΆbius loop
29575:Right plane
29546:Left planes
28794:16 hexagons
28563:6-cell ring
28158:MΓΆbius loop
28026:6-cell ring
27698:geodesics).
26851:, rotating
26624:honeycomb).
26452:orthoscheme
26058:hyperplanes
25986:MΓΆbius loop
25980:. They are
23507:polyschemes
23198:4-polytope.
23161:icosahedron
23046:{∞,3}
22962:{3,∞}
22782:Noncompact
21942:bitruncated
21928:cantellated
21216:bitruncated
21202:cantellated
19539:icosahedron
19517:, which is
19394:24 3-edges
19366:, order 72
19339:, order 96
18725:antiprismic
18666:fibration.
18604:South Pole
18539:North Pole
18525:Colatitude
18310:orientation
18126:. The name
17068:{24/1}={24}
16521:{24/6}=6{4}
16434:{24/6}=6{4}
16108:{24/4}=4{6}
16021:{24/6}=6{4}
15729:{24/6}=6{4}
15620:{24/6}=6{4}
15520:{24/1}={24}
15319:{24/6}=6{4}
15210:{24/6}=6{4}
14903:{24/6}=6{4}
14794:{24/6}=6{4}
14496:{24/6}=6{4}
14387:{24/6}=6{4}
14287:{24/1}={24}
14097:{24/6}=6{4}
13974:{24/4}=4{6}
13684:{24/6}=6{4}
13561:{24/4}=4{6}
13117:{24/4}=4{6}
12805:{24/4}=4{6}
12682:{24/4}=4{6}
12582:{24/1}={24}
12356:{24/4}=4{6}
12233:{24/4}=4{6}
11775:{24/4}=4{6}
11300:left planes
11278:reflections
11261:6-cell ring
11188:Reflections
8777:orientation
8665:intertwined
8604:MΓΆbius ring
8572:orientation
8396:constrained
8083:quaternions
8051:reflections
8017:root system
7990:hyperplanes
7979:root system
7682:orthoscheme
7303:apart. The
7289:apart. The
5203:Long radius
5099:2 16-cells
5010:24 squares
4965:24 cubical
4060:icosahedron
4008:tetrahedron
2185:Long radius
2081:2 16-cells
1992:24 squares
1947:24 cubical
1042:icosahedron
990:tetrahedron
723:convex hull
712:Coordinates
129:r{3,3,4} =
60:if you can.
34521:10-simplex
34505:9-demicube
34455:8-demicube
34405:7-demicube
34362:6-demicube
34332:5-demicube
34246:Octahedron
33598:dodecaplex
33310:(Thesis).
33262:10 October
33249:Egan, Greg
33130:1603.07269
33114:(Thesis).
33100:1903.06971
32784:References
32572:, half of
32524:is {12}, h
32400:Dorst 2019
31496:Ghyka 1977
31451:directions
30443:cell-first
30431:top vertex
29571:Left plane
29544:, all the
29293:top vertex
28954:polychoron
28470:hyperplane
28466:polychoron
28462:polyhedron
27756:vertices).
27541:chessboard
27496:isoclines.
27474:fibrations
27471:and their
27373:chessboard
27292:right pair
27136:chessboard
27132:dodecagram
27126:is a skew
27052:angles. A
26958:, and the
26740:times the
26458:irregular
26301:, and 144
26060:is as the
25962:fibrations
25729:hyperplane
25162:fibrations
24355:chessboard
23697:apart). A
23487:quaternion
23275:0, β1, β1,
22576:tr{3,3,4}
22565:2t{3,3,4}
22545:rr{3,3,4}
21935:runcinated
21734:tr{4,3,3}
21723:2t{4,3,3}
21703:rr{4,3,3}
21209:runcinated
20293:tr{3,4,3}
20282:2t{3,4,3}
20271:rr{3,4,3}
19535:octahedron
19493:truncation
19388:96 2-edges
18971:octahedral
18964:octahedral
18867:octahedral
18727:envelope.
18721:face-first
18713:edge-first
18698:cell-first
18494:geodesics
18471:South Pole
18467:North Pole
18436:octahedral
18430:Cell rings
15154:4π
{8/3}
13918:4π
{6/2}
11722:4π
{6/2}
11298:takes the
11200:number of
9225:of length
9212:dodecagons
9120:Dodecagons
8858:edges are
8692:cell rings
8686:and their
8678:and their
8472:discoverer
8251:expression
8043:Weyl group
7994:octahedron
7775:, 16-cell
7771:As we saw
7753:4-pyramids
6924:hypercubes
6457:4-Content
5181:dodecagons
5102:3 8-cells
5090:Inscribed
4984:24 square
4103:{5, 3, 3}
4100:{3, 3, 5}
4097:{3, 4, 3}
4094:{4, 3, 3}
4091:{3, 3, 4}
4088:{3, 3, 3}
4077:600-point
4063:120-point
4022:octahedron
3467:4-Content
2163:dodecagons
2084:3 8-cells
2072:Inscribed
1966:24 square
1085:{5, 3, 3}
1082:{3, 3, 5}
1079:{3, 4, 3}
1076:{4, 3, 3}
1073:{3, 3, 4}
1070:{3, 3, 3}
1059:600-point
1045:120-point
1004:octahedron
930:of radius
902:of radius
886:octahedron
841:rectifying
619:octahedral
609:octahedral
511:Properties
66:March 2024
34569:orthoplex
34491:9-simplex
34441:8-simplex
34391:7-simplex
34348:6-simplex
34318:5-simplex
34287:Tesseract
33756:icosaplex
33696:stellated
33687:stellated
33664:stellated
33613:tetraplex
33550:tesseract
33538:4-simplex
33364:253592159
33216:118501180
33199:0911.2289
33020:MathWorld
33015:"24-Cell"
32808:(1973) .
32710:honeycomb
32528:is {12/5}
32133:Egan 2021
32025:β
31998:γ
31944:β
31928:γ
31750:Egan 2021
31460:Citations
31384:−
31177:−
30979:−
30824:−
30327:−
30269:±
30241:−
29981:−
29848:−
29779:dodecagon
29255:−
29196:−
29114:±
29063:±
29037:±
28592:Hopf link
28567:hexagrams
28491:hexagonal
28384:discrete
28344:≈
28309:∘
28242:π
28216:π
28163:Hopf link
28122:π
28096:π
28070:geodesics
28042:isoclines
27972:rotation.
27575:isoclinic
27489:isoclines
27407:chirality
27385:isoclines
27342:cell ring
27288:left pair
27128:dodecagon
27024:isoclinic
26892:same time
26845:invariant
26728:that the
26673:rotations
26162:vertices.
26129:isoclinic
25966:isoclines
25463:full size
25378:Hopf link
25006:isoclinic
24290:−
24241:−
23539:diagonals
23443:±
23417:±
23391:±
23302:features.
23159:{5}, the
23129:tesseract
22534:t{3,3,4}
22523:r{3,3,4}
21921:truncated
21914:rectified
21692:t{4,3,3}
21681:r{4,3,3}
21239:tesseract
21232:tesseract
21225:tesseract
21218:tesseract
21211:tesseract
21204:tesseract
21197:tesseract
21195:truncated
21190:tesseract
21188:rectified
21183:tesseract
20260:r{3,4,3}
20249:s{3,4,3}
20246:t{3,4,3}
20092:h{4,3,3}
19391:24 4-edge
19371:Vertices
19271:Symmetry
19175:{3,4,3},
18899:symmetry
18857:rectified
18615:24 cells
18508:tesseract
18492:hexagonal
18447:tesseract
18408:hexagrams
18106:−
18094:−
18004:−
17992:−
17852:±
17672:−
17638:−
17508:−
17454:−
16593:−
16541:−
16411:−
16357:−
15968:4π
{12}
15405:−
15385:−
15339:−
15187:−
15133:−
14977:−
14957:−
14923:−
14771:−
14717:−
14151:−
14117:−
13951:−
13897:−
13508:4π
{12}
13348:−
13330:−
13312:−
13294:−
13260:−
13094:−
13040:−
12410:−
12376:−
12210:−
12177:4π
{12}
12156:−
12000:−
11982:−
11964:−
11616:Isocline
11600:rotations
11565:−
11547:−
11529:−
10631:≈
10534:≈
10437:≈
10253:π
10216:π
10182:≈
10090:π
10053:π
9927:π
9890:π
9856:≈
9752:π
9715:π
9681:≈
9633:π
9596:π
9514:π
9477:π
9443:≈
9382:π
9341:π
9294:dihedral
9085:Edge path
9011:8{4}=4{2}
8957:isoclinic
8620:isoclinic
8483:chirality
8275:honeycomb
8241:Rotations
8189:The unit
7515:tesseract
7483:inscribed
7442:tesseract
6979:Geodesics
6744:Triangles
6695:≈
6680:×
6645:≈
6630:×
6549:≈
6511:≈
6437:≈
6413:ϕ
6354:≈
6337:ϕ
6291:≈
6217:≈
6168:≈
6105:≈
6088:ϕ
6029:≈
6012:ϕ
5966:≈
5892:≈
5841:≈
5778:≈
5762:ϕ
5731:≈
5715:ϕ
5684:≈
5543:≈
5523:ϕ
5489:≈
5483:ϕ
5405:≈
5372:≈
4953:Vertices
4049:24-point
4039:16-point
3886:±
3868:±
3850:±
3832:±
3773:±
3748:inscribed
3734:self-dual
3698:≈
3683:×
3648:≈
3633:×
3552:≈
3514:≈
3447:≈
3430:ϕ
3371:≈
3354:ϕ
3261:≈
3225:≈
3167:≈
3104:≈
3087:ϕ
3028:≈
3011:ϕ
2958:≈
2884:≈
2840:≈
2777:≈
2761:ϕ
2730:≈
2714:ϕ
2660:≈
2620:≈
2580:≈
2535:≈
2532:ϕ
2529:−
2500:≈
2494:ϕ
2460:≈
2427:≈
2371:≈
1935:Vertices
1031:24-point
1021:16-point
913:, or the
772:∈
751:±
742:±
702:tesseract
658:tesselate
635:tesseract
631:self-dual
581:, or the
527:isohedral
505:Self-dual
465:dodecagon
34644:Category
34623:Topics:
34586:demicube
34551:polytope
34545:Uniform
34306:600-cell
34302:120-cell
34255:Demicube
34229:Pentagon
34209:Triangle
33721:600-cell
33714:120-cell
33705:120-cell
33698:120-cell
33689:120-cell
33680:120-cell
33673:120-cell
33666:120-cell
33657:120-cell
33583:octaplex
33523:600-cell
33518:120-cell
33425:Archived
33277:Symmetry
33177:12350382
33001:(1966),
32989:(1991),
32966:(2018),
32900:(1989).
32888:(1968).
32822:(1991),
32794:(1619).
32706:3-sphere
32702:polytope
32549:is {12}.
32380:Binasuan
32300:120-cell
30899:octagram
29615:hexagram
29581:column).
29567:Isocline
28571:isocline
28440:generate
28199:isocline
28185:isocline
28074:3-sphere
27790:that is
27695:adjacent
27646:apart. V
27622:are two
27449:600-cell
27409:come in
27383:and the
27381:vertices
27237:β 0.866.
26972:directly
26916:isocline
26884:isocline
26818:sideways
26808:Clifford
26514:24-cell.
26216:120-cell
26119:angles (
25932:geodesic
25862:isocline
25856:octagram
25793:3-sphere
25789:2-sphere
25647:isocline
25505:3-sphere
25374:3-sphere
25351:3-sphere
25020:diagonal
25012:3-sphere
25008:rotation
24997:rotation
24979:hexagons
24930:16-cells
24862:squares.
23499:SchlΓ€fli
23491:Hamilton
23173:120-cell
23169:600-cell
23157:pentagon
23094:and the
23054:See also
22776:Compact
22600:Schlegel
22594:{3,3,4}
22585:{3,3,4}
22554:{3,3,4}
22512:{3,3,4}
22506:SchlΓ€fli
21758:Schlegel
21752:{4,3,3}
21743:{4,3,3}
21712:{4,3,3}
21670:{4,3,3}
21664:SchlΓ€fli
20725:Schlegel
20320:{3,4,3}
20311:{3,4,3}
20302:{3,4,3}
20235:{3,4,3}
20229:SchlΓ€fli
20125:{4,3,3}
20114:{4,3,3}
20103:{4,3,3}
19487:Several
18861:, with B
18691:envelope
18584:8 cells
18576:Equator
18567:6 cells
18553:8 cells
18516:Layer #
18451:120-cell
18449:and the
18233:and its
17659:2π
{2}
17569:2π
{2}
17475:2π
{2}
17293:2π
{2}
17206:2π
{2}
17115:0π
{1}
16922:2π
{2}
16835:2π
{2}
16744:4π
{2}
16562:2π
{4}
16472:2π
{4}
16378:4π
{2}
16146:2π
{6}
16059:2π
{4}
15767:2π
{4}
15658:2π
{4}
15567:4π
{1}
15360:2π
{4}
15248:2π
{4}
14944:2π
{4}
14832:2π
{4}
14738:4π
{2}
14534:2π
{4}
14425:2π
{4}
14334:4π
{1}
14138:2π
{4}
14012:2π
{6}
13722:2π
{4}
13599:2π
{6}
13281:2π
{6}
13155:2π
{6}
13061:4π
{2}
12843:2π
{6}
12720:2π
{6}
12629:4π
{1}
12397:2π
{6}
12271:2π
{6}
11936:2π
{6}
11813:2π
{6}
11308:isocline
11249:isocline
11214:hexagram
9112:16-cells
9009:octagram
8935:octagram
8914:two sets
8813:hexagram
8589:geodesic
8585:hexagram
8552:hexagram
8526:. Three
8524:triangle
8428:threaded
8367:directly
8255:symmetry
8116:of the F
8019:of type
7362:600-cell
7360:and the
6996:geodesic
5174:octagons
5160:pentagon
5139:decagons
5132:hexagons
5033:8 cubes
4068:120-cell
4054:600-cell
4025:8-point
4011:5-point
3728:Hexagons
2156:octagons
2142:pentagon
2121:decagons
2114:hexagons
2015:8 cubes
1050:120-cell
1036:600-cell
1007:8-point
993:5-point
900:3-sphere
676:Geometry
670:zonotope
596:octacube
587:octaplex
523:isotoxal
519:isogonal
438:Vertices
41:require
34560:simplex
34530:10-cube
34297:24-cell
34283:16-cell
34224:Hexagon
34078:regular
34019:
34007:
33991:
33979:
33963:
33951:
33935:
33923:
33907:
33895:
33891:
33879:
33863:
33851:
33835:
33823:
33807:
33795:
33779:
33767:
33751:
33739:
33607:{3,3,5}
33592:{5,3,3}
33577:{3,4,3}
33562:{3,3,4}
33547:{4,3,3}
33532:{3,3,3}
33513:24-cell
33508:16-cell
33385:Bibcode
33285:Bibcode
32718:twisted
32081:simplex
31410:√
31277:√
31199:√
31131:√
31066:√
31001:√
30903:√
30892:√
30846:√
30579:√
30414:√
30349:√
30185:√
30003:√
29935:√
29870:√
29736:√
29712:√
29619:√
29608:√
29429:√
29082:planes.
28989:where 2
28973:where 2
28925:of the
28921:is the
28830:√
28823:√
28815:√
28735:√
28715:√
28708:√
28701:√
28694:√
28678:√
28575:winding
28519:√
28512:√
28499:torsion
28022:8-cells
28015:√
27941:√
27934:√
27911:√
27889:. But P
27852:√
27825:√
27792:√
27781:√
27774:√
27754:nearest
27747:√
27732:√
27688:√
27681:√
27666:√
27659:√
27641:√
27634:√
27624:√
27579:√
27568:√
27557:√
27537:bishops
27454:neither
27445:16-cell
27354:torsion
27232:√
27225:√
27210:√
27199:√
27192:√
27185:√
27159:√
27148:√
27144:bishops
26908:√
26756:times 2
26657:√
26653:
26640:√
26636:
26567:√
26560:√
26526:√
26460:simplex
26433:√
26409:√
26345:√
26326:√
26317:√
26310:√
26303:√
26294:√
26285:√
26077:points
26071:simplex
26031:√
26024:√
26017:√
26010:√
25982:helices
25972:on the
25946:spirals
25935:is the
25801:equator
25791:on the
25749:16-cell
25747:In the
25709:√
25702:√
25695:√
25688:√
25680:√
25673:√
25665:√
25433:√
25426:√
25419:√
25412:√
25406:If the
25355:annulus
25349:on the
25290:√
25283:√
25276:√
25269:√
25253:√
25246:√
25239:√
25232:√
25225:√
25177:√
25166:√
25151:√
25144:√
25136:16-cell
25060:√
25045:√
25038:√
25031:√
24987:√
24968:√
24961:√
24954:√
24947:√
24943:surface
24914:origin.
24857:√
24850:√
24825:
24813:
24804:
24792:
24786:
24774:
24765:
24753:
24743:
24731:
24725:
24713:
24707:
24695:
24689:
24677:
24669:
24657:
24648:
24636:
24627:
24615:
24606:
24594:
24581:
24569:
24563:
24551:
24542:
24530:
24524:
24512:
24481:√
24462:√
24443:√
24436:√
24429:√
24422:√
24373:missing
24366:√
24359:√
24342:bishops
24132:√
24124:√
24103:√
24078:
24066:
24057:
24045:
24039:
24027:
24018:
24006:
23996:
23984:
23978:
23966:
23960:
23948:
23942:
23930:
23922:
23910:
23901:
23889:
23880:
23868:
23859:
23847:
23834:
23822:
23816:
23804:
23795:
23783:
23777:
23765:
23733:integer
23692:√
23670:√
23663:√
23656:√
23615:integer
23533:√
23349:or (4)
23228:√
23191:rounder
23137:hexagon
23088:simplex
23084:polygon
22970:figure
22820:{3,8,3}
22815:{3,7,3}
22810:{3,6,3}
22805:{3,5,3}
22800:{3,4,3}
22795:{3,3,3}
22773:Finite
22602:diagram
22591:0,1,2,3
22574:{3,3,4}
22563:{3,3,4}
22543:{3,3,4}
22532:{3,3,4}
22521:{3,3,4}
21974:diagram
21972:Coxeter
21965:16-cell
21958:16-cell
21951:16-cell
21944:16-cell
21937:16-cell
21930:16-cell
21923:16-cell
21916:16-cell
21909:16-cell
21900:
21760:diagram
21749:0,1,2,3
21732:{4,3,3}
21721:{4,3,3}
21701:{4,3,3}
21690:{4,3,3}
21679:{4,3,3}
21248:diagram
21246:Coxeter
20727:diagram
20328:diagram
20326:Coxeter
20317:0,1,2,3
20291:{3,4,3}
20280:{3,4,3}
20269:{3,4,3}
20258:{3,4,3}
20244:{3,4,3}
20171:24-cell
20150:sr{3,3}
20143:tr{3,3}
20136:rr{3,3}
20118:2t{3,3}
20096:2r{3,3}
19554:simplex
19550:polygon
18966:cells.
18895:, and F
18885:24-cell
18883:of the
18859:16-cell
18658:chords
18653:√
18601:1 cell
18536:1 cell
18528:Region
18457:with a
18306:
18294:
17788:1.414~
17781:√
17773:
17761:
17757:1.414~
17750:√
17742:
17730:
17719:√
17419:1.414~
17412:√
17404:
17392:
17388:1.414~
17381:√
17373:
17361:
17350:√
17062:1.414~
17055:√
17047:
17035:
17031:1.414~
17024:√
17016:
17004:
17000:1.414~
16993:√
16985:
16973:
16691:1.414~
16684:√
16676:
16664:
16660:1.414~
16653:√
16645:
16633:
16622:√
16315:√
16307:
16295:
16291:1.414~
16284:√
16276:
16264:
16253:√
16245:
16233:
15915:1.414~
15908:√
15900:
15888:
15884:1.414~
15877:√
15869:
15857:
15846:√
15507:√
15497:1.414~
15490:√
15482:
15470:
15466:1.414~
15459:√
15451:
15439:
15098:1.414~
15091:√
15083:
15071:
15067:1.414~
15060:√
15052:
15040:
15029:√
14682:1.414~
14675:√
14667:
14655:
14651:1.414~
14644:√
14636:
14624:
14613:√
14281:1.414~
14274:√
14266:
14254:
14243:√
14235:
14223:
14219:1.732~
14212:√
14204:
14192:
13862:1.414~
13855:√
13847:
13835:
13824:√
13816:
13804:
13793:√
13785:
13773:
13455:1.732~
13448:√
13440:
13428:
13417:√
13409:
13397:
13386:√
12998:√
12990:
12978:
12967:√
12959:
12947:
12936:√
12569:√
12561:
12549:
12538:√
12530:
12518:
12507:√
12499:
12487:
12121:1.732~
12114:√
12106:
12094:
12083:√
12075:
12063:
12059:1.732~
12052:√
12044:
12032:
11598:Proper
11372:
11360:
11253:√
9262:√
9255:√
9245:√
9231:√
9205:√
9196:edges.
9191:√
9136:Squares
9128:24-gram
9078:24-gram
9042:√
9035:√
9024:√
9016:√
9002:√
8994:√
8983:√
8961:√
8950:√
8943:√
8897:√
8890:√
8879:√
8871:√
8864:√
8853:√
8846:√
8827:√
8820:√
8725:√
8718:√
8711:√
8704:√
8597:winding
8556:√
8542:√
8535:√
8528:√
8513:√
8506:√
8499:√
8418:forms:
8349:central
8305:and an
8249:are an
8217:is the
8150:of the
8085:, the F
8032:√
8002:16-cell
7981:of the
7798:√
7791:√
7784:√
7777:√
7737:√
7719:√
7708:√
7670:of its
7609:√
7604:
7590:
7582:√
7572:√
7565:√
7553:16-cell
7545:√
7538:√
7527:√
7521:16-cell
7499:√
7473:We can
7438:
7426:
7422:
7410:
7406:
7394:
7390:
7378:
7374:16-cell
7338:√
7317:apart.
7312:√
7305:√
7298:√
7291:√
7284:√
7277:√
7255:√
7245:√
7237:√
7226:√
7215:√
7208:√
7201:√
7190:√
7028:√
7017:√
7007:√
6970:√
6963:√
6956:√
6949:√
6942:√
6935:√
6928:√
6917:√
6910:√
6903:√
6896:√
6876:√
6871:
6856:
6848:√
6843:
6829:
6823:
6809:
6798:√
6791:√
6784:√
6777:√
6750:√
6125:Volume
5195:30-gons
5188:30-gons
5167:octagon
5122:squares
4900:
4888:
4885:
4873:
4870:
4858:
4855:
4843:
4840:
4828:
4825:
4813:
4808:
4796:
4793:
4781:
4778:
4766:
4763:
4751:
4748:
4736:
4733:
4721:
4716:
4704:
4701:
4689:
4686:
4674:
4671:
4659:
4656:
4644:
4641:
4629:
4624:
4612:
4609:
4597:
4594:
4582:
4579:
4567:
4564:
4552:
4549:
4537:
4532:
4520:
4517:
4505:
4502:
4490:
4487:
4475:
4472:
4460:
4457:
4445:
4440:
4428:
4425:
4413:
4410:
4398:
4395:
4383:
4380:
4368:
4365:
4353:
4044:24-cell
4016:16-cell
3936:√
3756:integer
3741:√
3124:Volume
2177:30-gons
2170:30-gons
2149:octagon
2104:squares
1882:
1870:
1867:
1855:
1852:
1840:
1837:
1825:
1822:
1810:
1807:
1795:
1790:
1778:
1775:
1763:
1760:
1748:
1745:
1733:
1730:
1718:
1715:
1703:
1698:
1686:
1683:
1671:
1668:
1656:
1653:
1641:
1638:
1626:
1623:
1611:
1606:
1594:
1591:
1579:
1576:
1564:
1561:
1549:
1546:
1534:
1531:
1519:
1514:
1502:
1499:
1487:
1484:
1472:
1469:
1457:
1454:
1442:
1439:
1427:
1422:
1410:
1407:
1395:
1392:
1380:
1377:
1365:
1362:
1350:
1347:
1335:
1026:24-cell
998:16-cell
932:√
911:hexagon
904:√
893:√
845:16-cell
717:Squares
706:16-cell
570:) with
560:24-cell
127:{3,4,3}
91:24-cell
43:cleanup
34500:9-cube
34450:8-cube
34400:7-cube
34357:6-cube
34327:5-cube
34214:Square
34091:Family
33553:4-cube
33503:8-cell
33498:5-cell
33482:Convex
33362:
33348:(44).
33237:
33214:
33175:
33082:
33070:β266.
33047:
32976:
32846:
32722:chiral
32574:{24/5}
32570:{12/5}
32340:MΓΆbius
31876:, ππ
31872:, ππ
31758:yellow
31447:shapes
31443:chiral
29540:In an
28997:+ 1 β€
28771:right.
28624:before
28422:planes
28408:unless
28003:skewed
27996:, its
27877:, as V
27564:simple
27403:chords
27284:chiral
27263:lines.
27261:dashed
27166:second
27091:chiral
27064:angle.
27028:simple
27022:In an
26996:Cayley
26946:: the
26900:single
26742:volume
26468:facets
26456:chiral
26335:digons
26186:plane.
26083:define
25267:Thus (
25223:Eight
25067:beyond
25056:cannot
25052:chiral
24995:simple
24838:0, β1,
24350:cannot
24122:Eight
24091:0, β1,
23641:In an
23603:0, β1,
23565:0, β1,
23435:, and
23291:1, β1,
23241:0, β1,
23086:nor a
22968:Vertex
22886:Cells
22832:Image
22755:Space
22508:symbol
21666:symbol
20231:symbol
20129:r{3,3}
20107:t{3,3}
19568:: the
19552:nor a
19541:, or "
19399:Image
19385:Edges
18973:cells
18955:cells
18879:Three
18649:square
18612:Total
18227:chiral
11222:second
10668:. The
9267:axis.
9203:of 12
9131:{24/5}
9115:3{3/8}
9067:does.
9029:axes.
8809:second
8682:, and
8637:chiral
8593:second
8522:not a
8479:double
8460:all 16
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6294:11.314
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6032:198.48
5969:41.569
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3651:15.451
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3374:47.214
3264:22.627
3228:15.085
3107:180.73
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2961:83.138
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12565:60Β°
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