3147:
25:
3008:. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the Platonic solids have only triangles, squares and pentagons for faces, and these are all constructible with a ruler and compass, there exist ruler-and-compass methods for drawing these fold-out nets. The same applies to star polyhedra, although here we must be careful to make the net for only the visible outer surface.
142:
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even imagine building a model of this fold-out net, as one draws a polyhedron's fold-out net on a piece of paper. Sadly, we could never do the necessary folding of the 3-dimensional structure to obtain the 4-dimensional polytope because of the constraints of the physical universe. Another way to "draw" the higher-dimensional shapes in 3 dimensions is via some kind of projection, for example, the analogue of either
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2794:. Certain restrictions are imposed on the set that are similar to properties satisfied by the classical regular polytopes (including the Platonic solids). The restrictions, however, are loose enough that regular tessellations, hemicubes, and even objects as strange as the 11-cell or stranger, are all examples of regular polytopes.
3094:) has some examples of such orthographic projections. Note that immersing even 4-dimensional polychora directly into two dimensions is quite confusing. Easier to understand are 3-d models of the projections. Such models are occasionally found in science museums or mathematics departments of universities (such as that of the
2801:
of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the corresponding faces of the geometric realisation. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations.
2663:
In the first part of the 20th century, Coxeter and Petrie discovered three infinite structures {4, 6}, {6, 4} and {6, 6}. They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron — all the vertices, edges and faces are alike, all the angles
2537:
contains 8 cubical cells. It consists of two cubes in parallel hyperplanes with corresponding vertices cross-connected in such a way that the 8 cross-edges are equal in length and orthogonal to the 12+12 edges situated on each cube. The corresponding faces of the two cubes are connected to form the
2761:
This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G,
3081:
The second approach is to embed the higher-dimensional objects in three-dimensional space, using methods analogous to the ways in which three-dimensional objects are drawn on the plane. For example, the fold out nets mentioned in the previous section have higher-dimensional equivalents. One might
3011:
If this net is drawn on cardboard, or similar foldable material (for example, sheet metal), the net may be cut out, folded along the uncut edges, joined along the appropriate cut edges, and so forming the polyhedron for which the net was designed. For a given polyhedron there may be many fold-out
2986:
gave what amount to ruler-and-compass constructions for the five
Platonic solids. However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead one to question what exactly it means to "construct" a regular polyhedron. (One could ask the same
2687:
would be a zig-zag. It seems to satisfy the definition of a regular polygon — all the edges are the same length, all the angles are the same, and the figure has no loose ends (because they can never be reached). More importantly, perhaps, there are symmetries of the zig-zag that can map any
3122:
technology. To understand how this might work, imagine what one would see if space were filled with cubes. The viewer would be inside one of the cubes, and would be able to see cubes in front of, behind, above, below, to the left and right of himself. If one could travel in these directions, one
2827:
Any classical regular polytope has an abstract equivalent which is regular, obtained by taking the set of faces. But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes do not retain information about angles and edge lengths, for example. And a regular
2379:
3077:
The first approach, suitable for four dimensions, uses four-dimensional stereography. Depth in a third dimension is represented with horizontal relative displacement, depth in a fourth dimension with vertical relative displacement between the left and right images of the stereograph.
3073:
In higher dimensions, it becomes harder to say what one means by "constructing" the objects. Clearly, in a 3-dimensional universe, it is impossible to build a physical model of an object having 4 or more dimensions. There are several approaches normally taken to overcome this matter.
3106:
into a kind of four dimensional object, where the fourth dimension is taken to be time. In this way, we can see (if not fully grasp) the full four-dimensional structure of the four-dimensional regular polytopes, via such cutaway cross sections. This is analogous to the way a
2599:
This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry.
3157:
Locally, this space seems like the one we are familiar with, and therefore, a virtual-reality system could, in principle, be programmed to allow exploration of these "tessellations", that is, of the 4-dimensional regular polytopes. The mathematics department at
2959:. Constructing some regular polygons in this way is very simple (the easiest is perhaps the equilateral triangle), some are more complex, and some are impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the
2550:
by joining the 8 vertices of each of its cubical faces to an additional vertex to form the four-dimensional analogue of a pyramid. Both figures, as well as other 4-dimensional figures, can be directly visualised and depicted using 4-dimensional stereographs.
2258:
Our understanding remained static for many centuries after Euclid. The subsequent history of the regular polytopes can be characterised by a gradual broadening of the basic concept, allowing more and more objects to be considered among their number.
3177:
is known. This is because of an important theorem in the study of abstract regular polytopes, providing a technique that allows the abstract regular polytope to be constructed from its symmetry group in a standard and straightforward manner.
3101:
The intersection of a four (or higher) dimensional regular polytope with a three-dimensional hyperplane will be a polytope (not necessarily regular). If the hyperplane is moved through the shape, the three-dimensional slices can be combined,
2035:
they do not contain. The remaining thing to check is that any two hyperplanes with adjacent numbers cannot be orthogonal, whereas hyperplanes with non-adjacent numbers are orthogonal. This can be done using induction (since all facets of
651:
of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original symbol written backwards: {3, 3} is self-dual, {3, 4} is dual to {4, 3}, {4, 3, 3} to {3, 3, 4} and so on.
2973:
in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical.
3019:
provides sets of plastic triangles, squares, pentagons and hexagons that can be joined edge-to-edge in a large number of different ways. A child playing with such a toy could re-discover the
Platonic solids (or the
2783:
By 1994 GrĂźnbaum was considering polytopes abstractly as combinatorial sets of points or vertices, and was unconcerned whether faces were planar. As he and others refined these ideas, such sets came to be called
2664:
are the same, and the figure has no free edges. Nowadays, they are called infinite polyhedra or apeirohedra. The regular tilings of the plane {4, 4}, {3, 6} and {6, 3} can also be regarded as infinite polyhedra.
3456:
2616:
So for example, the cube is regular because if we choose a vertex of the cube, and one of the three edges it is on, and one of the two faces containing the edge, then this triplet, known as a
2382:
A 3D projection of a rotating tesseract. This tesseract is initially oriented so that all edges are parallel to one of the four coordinate space axes. The rotation takes place in the xw plane.
474:
The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below.
3062:
3027:
In theory, almost any material may be used to construct regular polyhedra. They may be carved out of wood, modeled out of wire, formed from stained glass. The imagination is the limit.
2750:
object whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the
3004:
The
English word "construct" has the connotation of systematically building the thing constructed. The most common way presented to construct a regular polyhedron is via a
2762:
and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners.
3639:
Schläfli, L. (1858). "On the multiple integral âŤ^ n dxdy... dz, whose limits are p_1= a_1x+ b_1y+âŚ+ h_1z> 0, p_2> 0,..., p_n> 0, and x^ 2+ y^ 2+âŚ+ z^ 2< 1".
1609:
3118:
Another way a three-dimensional viewer can comprehend the structure of a four-dimensional polytope is through being "immersed" in the object, perhaps via some form of
1561:
2033:
1953:
1827:
1807:
1440:, but not every finite Coxeter groups may be realised as the isometry group of a regular polytope. Regular polytopes are in bijection with Coxeter groups with linear
2406:, pp. 143â144) for more details. Schläfli called such a figure a "polyschem" (in English, "polyscheme" or "polyschema"). The term "polytope" was introduced by
2710:
has its x, y, z, etc. coordinates as complex numbers. This effectively doubles the number of dimensions. A polytope constructed in such a unitary space is called a
1715:
1677:
1639:
1514:
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2120:
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of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension
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2074:
2054:
2013:
1993:
1973:
1933:
1913:
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of a regular polytope is the dual of the dual polytope's facet. For example, the vertex figure of {3, 3, 4} is {3, 4}, the dual of which is {4, 3} — a
2566:. From the mathematical point of view, however, these objects have the same aesthetic qualities as their more familiar two and three-dimensional relatives.
1725:
The bijection between regular polytopes and
Coxeter groups with linear Coxeter-Dynkin diagram can be understood as follows. Consider a regular polytope
3015:
Numerous children's toys, generally aimed at the teen or pre-teen age bracket, allow experimentation with regular polygons and polyhedra. For example,
2790:. An abstract polytope is defined as a partially ordered set (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by
2622:, (vertex, edge, face) can be mapped to any other such flag by a suitable symmetry of the cube. Thus we can define a regular polytope very succinctly:
2834:
are regular in the abstract world, for example, whereas only those having equal angles and edges of equal length are regular in the classical world.
2706:
has a real part, which is the bit we are all familiar with, and an imaginary part, which is a multiple of the square root of minus one. A complex
429:
and above, the simplex, hypercube and orthoplex are the only regular polytopes. There are no exceptional regular polytopes in these dimensions.
3626:
Schläfli, L. (1855). "RÊduction d'une intÊgrale multiple, qui comprend l'arc de cercle et l'aire du triangle sphÊrique comme cas particuliers".
2487:
In five and more dimensions, there are exactly three regular polytopes, which correspond to the tetrahedron, cube and octahedron: these are the
2263:(Bradwardinus) was the first to record a serious study of star polygons. Various star polyhedra appear in Renaissance art, but it was not until
3115:
of some sort, however, even a simple animation such as the one shown can already give some limited insight into the structure of the polytope.
2688:
pair of a vertex and attached edge to any other. Since then, other regular apeirogons and higher apeirotopes have continued to be discovered.
2608:-polytope is regular if any set consisting of a vertex, an edge containing it, a 2-dimensional face containing the edge, and so on up to
492:
in the 19th century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.
2765:
The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively curved (elliptic) hyperspace.
3323:
3159:
3111:
reassembles two-dimensional images to form a 3-dimensional representation of the organs being scanned. The ideal would be an animated
3016:
2420:
is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six
4329:
2939:
be regular itself – for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons.
2824:
The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes.
2402:. Between 1880 and 1900, Schläfli's results were rediscovered independently by at least nine other mathematicians — see
3721:
3554:
3533:
3261:
3173:
Normally, for abstract regular polytopes, a mathematician considers that the object is "constructed" if the structure of its
3289:
460:
3764:
3577:. Problèmes Combinatoires et ThĂŠorie des Graphes, Colloquium Internationale CNRS, Orsay. Vol. 260. pp. 191â197.
3377:
3338:
749:
3701:
3586:. Mathematical and physical sciences, NATO Advanced Study Institute. Vol. 440. Kluwer Academic. pp. 43â70.
3489:
2390:, examined and characterised the regular polytopes in higher dimensions. His efforts were first published in full in
681:, i.e. reads the same forwards and backwards, then the polyhedron is self-dual. The self-dual regular polytopes are:
68:
46:
39:
2758:). The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12.
2595:− 1)-dimensional faces are all regular and congruent, and whose vertex figures are all regular and congruent.
2530:
3095:
1444:(without branch point) and an increasing numbering of the nodes. Reversing the numbering gives the dual polytope.
3220:
737:
2738:
is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.
3678:
3591:
3512:
3371:
3150:
741:
2671:
issued a call to the geometric community to consider more abstract types of regular polytopes that he called
2353:
2346:
2272:
2268:
761:
4346:
3306:
2148:
757:
753:
2476:, can be seen as a transitional form between the hypercube and 16-cell, analogous to the way that the
2414:
some twenty years later. The term "polyhedroids" was also used in earlier literature (Hilbert, 1952).
3210:
2935:
Unlike the case for
Euclidean polytopes, an abstract polytope with regular facets and vertex figures
2500:
2421:
2308:
2292:
463:
to one another. Indeed, symmetry groups are sometimes named after regular polytopes, for example the
433:
2122:
such that adjacent numbers are linked by at least one edge and non-adjacent numbers are not linked.
3787:
3479:
3305:
Some interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron are available
2956:
2955:
The traditional way to construct a regular polygon, or indeed any other figure on the plane, is by
1524:(both can be distinguished by the increasing numbering of the nodes of the Coxeter-Dynkin diagram),
715:
33:
3757:
3170:
with dodecahedra. Such a tessellation forms an example of an infinite abstract regular polytope.
3087:
3083:
2658:
2135:
The earliest surviving mathematical treatment of regular polygons and polyhedra comes to us from
1766:
407:
228:
395:⼠3. The triangle is the 2-simplex. The square is both the 2-hypercube and the 2-orthoplex. The
3146:
2576:
A regular polyhedron is a polyhedron whose faces are all congruent regular polygons, and whose
2299:
1570:
1441:
426:
411:
50:
3705:
3249:
1530:
4301:
4294:
4287:
3123:
could explore the array of cubes, and gain an understanding of its geometrical structure. An
2970:
2056:
are again regular polytopes). Therefore, the
Coxeter-Dynkin diagram of the isometry group of
2018:
1938:
1812:
1792:
745:
369:
3826:
3804:
3792:
3582:
GrĂźnbaum, B. (1993). "Polyhedra with hollow faces". In
Bisztriczky, T.; et al. (eds.).
3127:
is not a polytope in the traditional sense. In fact, it is a tessellation of 3-dimensional (
3958:
3905:
2776:
independently discovered the same shape. He had earlier discovered a similar polytope, the
2735:
2481:
2157:
1693:
1655:
1617:
1492:
1458:
944:
901:
814:
726:
468:
464:
8:
4313:
4212:
3962:
2634:
2518:
had developed the theory of regular polytopes in four and higher dimensions, such as the
2099:
1832:
2573:
A regular polygon is a polygon whose edges are all equal and whose angles are all equal.
4182:
4132:
4082:
4039:
4009:
3969:
3932:
3750:
3606:
3501:
3199:
2612:−1 dimensions, can be mapped to any other such set by a symmetry of the polytope.
2569:
At the start of the 20th century, the definition of a regular polytope was as follows.
2504:
2411:
2360:
2280:
2260:
2079:
2059:
2039:
1998:
1978:
1958:
1918:
1898:
1878:
1858:
1772:
1748:
1728:
551:
2668:
2515:
2387:
511:}. So an equilateral triangle is {3}, a square {4}, and so on indefinitely. A regular
489:
483:
318:
106:
4321:
3737:
3717:
3690:
3674:
3587:
3550:
3529:
3508:
3485:
3257:
3021:
2847:
2818:
2786:
2725:
2675:. He developed the theory of polystromata, showing examples of new objects he called
2555:
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2284:
1338:
1250:
1169:
1106:
1022:
571:
415:
311:
307:
280:
276:
3012:
nets. For example, there are 11 for the cube, and over 900000 for the dodecahedron.
2533:, since they retain the familiar symmetry of their lower-dimensional analogues. The
235:, thus giving it the highest degree of symmetry. In particular, all its elements or
3890:
3879:
3868:
3857:
3848:
3839:
3778:
3467:
3167:
3140:
3131:) space. However, a 4-polytope can be considered a tessellation of a 3-dimensional
3054:
3005:
2994:
2773:
2712:
2697:
2529:
The latter are difficult (though not impossible) to visualise through a process of
2291:
proved the list complete in 1812. These polyhedra are known as collectively as the
1517:
932:
352:
299:
163:
3254:
Hypergraphics: Visualizing
Complex Relationships In Arts, Science, And Technololgy
3915:
3900:
3544:
3523:
3342:
3293:
3286:
3256:. AAAS Selected Symposium. Vol. 24. Taylor & Francis. pp. 109â145.
3128:
3124:
3119:
2942:
The classical vertex figure will, however, be a realisation of the abstract one.
2747:
2618:
2288:
2264:
1564:
686:
660:
574:
are {3, 3} {3, 4} {4, 3} {3, 5} {5, 3} {3, 5/2} {5/2, 3} {5, 5/2} and {5/2, 5}. {
500:
384:
306:. These two conditions are sufficient to ensure that all faces are alike and all
272:
232:
176:
155:
129:
102:
3336:
4265:
3601:
3471:
3174:
3039:
2703:
2684:
2407:
2178:
2166:
2140:
2136:
1521:
1433:
711:
671:
497:
380:
358:
224:
202:
736:
Hyperbolic tilings and honeycombs (tilings {p,p} with p>4 in 2 dimensions,
141:
4340:
4282:
4170:
4163:
4156:
4120:
4113:
4106:
4070:
4063:
3355:
3215:
3132:
2707:
2638:
2637:
of the classical regular polytopes were generalised into what are now called
2577:
2511:
2477:
2324:
2317:
2276:
2162:
1437:
1282:
656:
648:
580:
376:
simultaneously serves as the 1-simplex, the 1-hypercube and the 1-orthoplex.
303:
1356:(i.e. upwards and downwards). Join the ends to the square to form a regular
1054:
849:
271:
Regular polytopes are the generalised analog in any number of dimensions of
4222:
3163:
2642:
2461:
2238:
1642:
1007:
938:
877:
809:
516:
452:
373:
180:
121:
2626:
A regular polytope is one whose symmetry group is transitive on its flags.
192:
4231:
4192:
4142:
4092:
4049:
4019:
3951:
3937:
2998:
2645:
of space or of the plane. For example, the symmetry group of an infinite
2469:
2445:
2429:
2410:, one of Schläfli's rediscoverers, in 1882, and first used in English by
2243:
2223:
1646:
957:
951:
913:
824:
819:
488:
A concise symbolic representation for regular polytopes was developed by
456:
340:
Regular polytopes are classified primarily according to their dimension.
212:
4217:
4201:
4151:
4101:
4058:
4028:
3942:
2676:
2646:
2453:
2233:
2155:
wrote a systematic study of mathematics, publishing it under the title
2144:
1402:
1385:
1357:
1255:
885:
678:
448:
419:
288:
151:
125:
3061:
4273:
4187:
4137:
4087:
4044:
4014:
3983:
3195:
Each of the
Platonic solids occurs naturally in one form or another:
3103:
3043:
3024:), especially if given a little guidance from a knowledgeable adult.
2547:
2539:
2534:
2519:
2437:
2433:
2425:
2424:. Five of them can be seen as analogous to the Platonic solids: the
1292:
1194:
1183:
1064:
1032:
967:
667:
544:
255:
2331:
4247:
4002:
3998:
3925:
3135:
space, namely, a tessellation of the surface of a four-dimensional
3112:
3108:
2990:
2680:
2631:
2465:
2457:
2338:
1684:
1680:
835:
540:
440:
291:
quality that interests both mathematicians and non-mathematicians.
220:
147:
94:
88:
1081:, and join to form a line segment. Extend a second line of length
170:
4256:
4226:
3993:
3988:
3979:
3920:
3319:
3190:
3066:
2828:
abstract polytope may not be realisable as a classical polytope.
2777:
2769:
2743:
2563:
2559:
2543:
2523:
2473:
2449:
2394:, six years posthumously, although parts of it were published in
1718:
1483:
1447:
The classification of finite
Coxeter groups, which goes back to (
1391:
1260:
859:
704:
697:
347:
310:
are alike. Note, however, that this definition does not work for
98:
3162:
has a number of pictures of what one would see if embedded in a
2630:
In the 20th century, some important developments were made. The
2378:
2165:. His work concluded with mathematical descriptions of the five
1014:
1000:
4196:
4146:
4096:
4053:
4023:
3974:
3910:
3658:
Denkschriften der Schweizerischen Naturforschenden Gesellschaft
3374:(1935), "The complete enumeration of finite groups of the form
3136:
2208:
2152:
801:
794:
787:
780:
198:
115:
2386:
It was not until the 19th century that a Swiss mathematician,
2215:
2187:
1451:), therefore implies the classification of regular polytopes:
1242:
1235:
1228:
842:
287:). The strong symmetry of the regular polytopes gives them an
3035:
2641:. Coxeter groups also include the symmetry groups of regular
2201:
2194:
1895:(the face they are the barycenter of). The isometry group of
1275:
1047:
343:
Three classes of regular polytopes exist in every dimension:
3049:
2484:
are transitional forms between the cube and the octahedron.
523:
times around its centre is denoted by the fractional value {
459:. Two distinct regular polytopes with the same symmetry are
3946:
3656:
Schläfli, L. (1901). "Theorie der vielfachen Kontinuität".
2730:
2441:
2228:
1268:
1176:
1141:
1040:
1027:
444:
284:
1769:. The fundamental domain of the isometry group action on
439:
Regular polytopes can be further classified according to
3568:. Translated by Heath, T. L. Cambridge University Press.
3354:
Other examples may be found on the web (see for example
2805:
The theory has since been further developed, largely by
2817:
Regularity has a related, though different meaning for
2510:
By the end of the 19th century, mathematicians such as
183:, an infinite polytope, represented by Schläfli symbol
3153:, {5,3,4}, of hyperbolic space projected into 3-space.
2837:
2809:, but other researchers have also made contributions.
1348:
of the same length and centered on 'O', orthogonal to
365:
Any other regular polytope is said to be exceptional.
3380:
2920:-face which contains all other faces. Note that each
2821:, since angles and lengths of edges have no meaning.
2102:
2082:
2062:
2042:
2021:
2001:
1981:
1961:
1941:
1921:
1901:
1881:
1861:
1835:
1815:
1795:
1775:
1751:
1731:
1696:
1658:
1620:
1573:
1533:
1495:
1461:
730:
154:, a four-dimensional polytope, with 120 dodecahedral
2928: ⼠0 of the original polytope becomes an (
2866:itself. More formally, it is the abstract section
2812:
2652:
2492:
1427:
642:
128:, a three-dimensional polytope, with 12 pentagonal
3700:
3689:
3605:
3500:
3450:
2496:
2161:, which built up a logical theory of geometry and
2114:
2088:
2068:
2048:
2027:
2007:
1987:
1967:
1947:
1927:
1907:
1887:
1867:
1847:
1821:
1801:
1781:
1757:
1737:
1709:
1671:
1633:
1603:
1555:
1508:
1474:
1116:respectively from each corner, orthogonal to both
3641:Quarterly Journal of Pure and Applied Mathematics
3451:{\displaystyle r_{i}^{2}=(r_{i}r_{j})^{k_{ij}}=1}
2932: − 1)-face of the vertex figure.
2275:in 1619 that he realised these two were regular.
1649:(again the numbering allows to distinguish them),
1301:. Extend a line in opposite directions to points
988:
335:
4338:
3090:projection. Coxeter's famous book on polytopes (
2151:(c. 417 BC â 369 BC) described all five. Later,
2130:
1216:
3687:
3600:
2858:is the set of all abstract faces which contain
2806:
2373:
1371:. Their names are, in order of dimensionality:
3584:POLYTOPES: abstract, convex, and computational
3181:
2768:A few years after GrĂźnbaum's discovery of the
2253:
1875:by the dimension of the corresponding face of
3758:
3696:. Translated by Dresden, Arnold. P Noordhoff.
2015:hyperplanes can be numbered by the vertex of
1432:Regular polytopes can be classified by their
908:in a third, orthogonal, dimension a distance
611:}. The vertex figure of the 4-polytope is a {
562:faces joining around a vertex is denoted by {
3528:(2nd ed.). Cambridge University Press.
2691:
2554:Harder still to imagine are the more modern
2499:. Descriptions of these may be found in the
1975:(since the barycenter of the whole polytope
1809:in the barycentric subdivision. The simplex
599:cells joining around an edge is denoted by {
298:dimensions may be defined as having regular
201:can be shown in this orthogonal projection (
3575:Regularity of Graphs, Complexes and Designs
2797:A geometric polytope is understood to be a
912:from all three, and join to form a regular
451:share the same symmetry, as do the regular
317:A regular polytope can be represented by a
3765:
3751:
3671:Geometrical and Structural Crystallography
3065:An animated cut-away cross-section of the
2967:equal to 7, 9, 11, 13, 14, 18, 19, 21,...
1222:Graphs of the 2-orthoplex to 4-orthoplex.
1158:. Their names are, in order of dimension:
927:. Their names are, in order of dimension:
80:
3243:
3241:
3186:For examples of polygons in nature, see:
2987:question about the polygons, of course.)
1855:vertices which can be numbered from 0 to
1482:, the symmetric group, gives the regular
729:, {4,3,...,3,4}. These may be treated as
674:in any dimension are dual to each other.
69:Learn how and when to remove this message
3655:
3638:
3625:
3581:
3572:
3542:
3145:
3060:
3048:
3034:
2989:
2850:. The vertex figure of a given abstract
2755:
2729:
2422:regular convex polytopes in 4 dimensions
2399:
2395:
2391:
2377:
2306:
2176:
32:This article includes a list of general
16:Polytope with highest degree of symmetry
4330:List of regular polytopes and compounds
3616:
3521:
3498:
3477:
3370:
3247:
3091:
2916:is the maximal face, i.e. the notional
2417:
2403:
2125:
1448:
4339:
3250:"Visual Comprehension in n-Dimensions"
3238:
2653:Apeirotopes — infinite polytopes
2488:
1935:reflections around the hyperplanes of
774:Graphs of the 1-simplex to 4-simplex.
197:The 256 vertices and 1024 edges of an
3740:- List of abstract regular polytopes.
3668:
2719:
414:, there are several more exceptional
3738:The Atlas of Small Regular Polytopes
3030:
2507:, partially discovered by Schläfli.
768:
101:, a two-dimensional polytope with 5
18:
3716:. Courier Dover. pp. 159â192.
3652:(1860) pp54–68, 97–108.
2846:is also defined differently for an
2838:Vertex figure of abstract polytopes
2147:knew of at least three of them and
1360:. And so on for higher dimensions.
1147:. And so on for higher dimensions.
916:. And so on for higher dimensions.
718:({5/2,5,5/2}) in 4 dimensions.
587:A regular 4-polytope having cells {
477:
294:Classically, a regular polytope in
13:
2683:many faces. A simple example of a
2679:, that is, regular polytopes with
2022:
1942:
1816:
1796:
38:it lacks sufficient corresponding
14:
4358:
3731:
2538:remaining 6 cubical faces of the
2300:Regular polyhedron § History
1995:is fixed by any isometry). These
158:, represented by Schläfli symbol
132:, represented by Schläfli symbol
3710:Introduction to the Geometry of
3151:A regular dodecahedral honeycomb
2945:
2813:Regularity of abstract polytopes
2503:. Also of interest are the star
2337:
2330:
2323:
2316:
2214:
2207:
2200:
2193:
2186:
1428:Classification by Coxeter groups
1401:5. Regular triacontakaiditeron (
1281:
1274:
1267:
1241:
1234:
1227:
1124:(i.e. upwards). Mark new points
1053:
1046:
1039:
1013:
1006:
999:
994:Graphs of the 2-cube to 4-cube.
848:
841:
834:
800:
793:
786:
779:
643:Duality of the regular polytopes
191:
169:
140:
114:
87:
23:
3688:Van der Waerden, B. L. (1954).
3364:
3335:Some of these may be viewed at
3221:Bartel Leendert van der Waerden
3619:A Short History Of Mathematics
3549:. Cambridge University Press.
3507:. Cambridge University Press.
3423:
3399:
3348:
3329:
3312:
3299:
3279:
3270:
3252:. In Brisson, David W. (ed.).
2580:are all congruent and regular.
1550:
1544:
1220:
992:
989:Measure polytopes (hypercubes)
772:
707:in 4 dimensions, {3,4,3}.
336:Classification and description
329:and regular vertex figures as
1:
3612:. Cambridge University Press.
3226:
3096:UniversitĂŠ Libre de Bruxelles
2807:McMullen & Schulte (2002)
2742:GrĂźnbaum also discovered the
2649:would be the Coxeter group .
2591:-dimensional polytope whose (
2131:Convex polygons and polyhedra
1955:containing the vertex number
1217:Cross polytopes (orthoplexes)
279:or the regular pentagon) and
2977:
2854:-polytope at a given vertex
2374:Higher-dimensional polytopes
2336:
2329:
2322:
2315:
2273:great stellated dodecahedron
2269:small stellated dodecahedron
2213:
2206:
2199:
2192:
2185:
2096:vertices numbered from 0 to
1381:2. Square (regular tetragon)
1240:
1233:
1226:
1012:
1005:
998:
876:from it, and join to form a
799:
792:
785:
778:
622:A regular 5-polytope is an {
383:, there are infinitely many
7:
3563:
3543:Cromwell, Peter R. (1999).
3248:Brisson, David W. (2019) .
3204:
3182:Regular polytopes in nature
2950:
2254:Star polygons and polyhedra
1390:4. Regular hexadecachoron (
10:
4363:
4319:
3746:
3706:"X. The Regular Polytopes"
3608:Abstract Regular Polytopes
3318:Instructions for building
3197:
3188:
3053:A perspective projection (
2723:
2695:
2656:
2556:abstract regular polytopes
2297:
1337:. Join the ends to form a
1290:
1062:
888:, dimension at a distance
857:
677:If the Schläfli symbol is
481:
138:
85:
82:Regular polytope examples
3525:Regular Complex Polytopes
3503:Regular Complex Polytopes
3211:List of regular polytopes
2692:Regular complex polytopes
2501:list of regular polytopes
2139:mathematicians. The five
1604:{\displaystyle n=3,4,...}
1112:. Extend lines of length
1073:. Extend a line to point
434:list of regular polytopes
190:
3628:Journal de MathĂŠmatiques
3472:10.1112/jlms/s1-10.37.21
3231:
2957:compass and straightedge
2546:can be derived from the
2428:(or pentachoron) to the
2309:Kepler-Poinsot polyhedra
2293:Kepler-Poinsot polyhedra
1789:is given by any simplex
1556:{\displaystyle I_{2}(n)}
716:grand stellated 120-cell
3673:(2nd ed.). Wiley.
3484:(3rd ed.). Dover.
3125:infinite array of cubes
2659:Regular skew polyhedron
2028:{\displaystyle \Delta }
1948:{\displaystyle \Delta }
1822:{\displaystyle \Delta }
1802:{\displaystyle \Delta }
1767:barycentric subdivision
892:from both, and join to
325:with regular facets as
53:more precise citations.
3621:. The Riverside Press.
3604:; Schulte, S. (2002).
3452:
3154:
3070:
3058:
3046:
3001:
2780:(Coxeter 1982, 1984).
2739:
2448:(or hexadecachoron or
2383:
2116:
2090:
2070:
2050:
2029:
2009:
1989:
1969:
1949:
1929:
1909:
1889:
1869:
1849:
1823:
1803:
1783:
1759:
1739:
1711:
1673:
1635:
1605:
1557:
1510:
1476:
1442:Coxeter-Dynkin diagram
469:icosahedral symmetries
403:⼠5 are exceptional.
3669:Smith, J. V. (1982).
3573:GrĂźnbaum, B. (1976).
3453:
3149:
3064:
3052:
3038:
2993:
2963:-sided polygons with
2746:, a four-dimensional
2733:
2583:And so on, a regular
2381:
2117:
2091:
2071:
2051:
2030:
2010:
1990:
1970:
1950:
1930:
1910:
1890:
1870:
1850:
1824:
1804:
1784:
1760:
1740:
1721:, which is self-dual.
1712:
1710:{\displaystyle F_{4}}
1674:
1672:{\displaystyle H_{4}}
1636:
1634:{\displaystyle H_{3}}
1606:
1558:
1511:
1509:{\displaystyle B_{n}}
1477:
1475:{\displaystyle A_{n}}
1186:(regular octachoron)
760:in 4 dimensions, and
507:sides is denoted by {
387:, namely the regular
323:{a, b, c, ..., y, z},
302:(-faces) and regular
3617:Sanford, V. (1930).
3460:J. London Math. Soc.
3378:
3322:models may be found
2482:rhombic dodecahedron
2143:were known to them.
2126:History of discovery
2100:
2080:
2060:
2040:
2019:
1999:
1979:
1959:
1939:
1919:
1915:is generated by the
1899:
1879:
1859:
1833:
1813:
1793:
1773:
1749:
1729:
1694:
1656:
1618:
1571:
1531:
1493:
1459:
1197:(regular decateron)
1179:(regular hexahedron)
1097:, and likewise from
945:Equilateral triangle
902:equilateral triangle
399:-sided polygons for
4314:pentagonal polytope
4213:Uniform 10-polytope
3773:Fundamental convex
3395:
2677:regular apeirotopes
2531:dimensional analogy
2505:regular 4-polytopes
2115:{\displaystyle n-1}
1848:{\displaystyle n+1}
1436:. These are finite
1321:apart. Draw a line
1297:Begin with a point
1223:
1069:Begin with a point
995:
864:Begin with a point
775:
725:-dimensional cubic
443:. For example, the
391:-sided polygon for
83:
4183:Uniform 9-polytope
4133:Uniform 8-polytope
4083:Uniform 7-polytope
4040:Uniform 6-polytope
4010:Uniform 5-polytope
3970:Uniform polychoron
3933:Uniform polyhedron
3781:in dimensions 2â10
3702:D.M.Y. Sommerville
3448:
3381:
3341:2011-07-17 at the
3292:2007-10-28 at the
3285:See, for example,
3200:Regular polyhedron
3155:
3071:
3059:
3047:
3022:Archimedean solids
3002:
2819:abstract polytopes
2787:abstract polytopes
2740:
2720:Abstract polytopes
2472:. The sixth, the
2412:Alicia Boole Stott
2384:
2361:Great dodecahedron
2281:great dodecahedron
2261:Thomas Bradwardine
2112:
2086:
2066:
2046:
2025:
2005:
1985:
1965:
1945:
1925:
1905:
1885:
1865:
1845:
1819:
1799:
1779:
1755:
1735:
1707:
1669:
1641:gives the regular
1631:
1601:
1553:
1527:Exceptional types
1506:
1472:
1333:and orthogonal to
1221:
1172:(regular tetragon)
993:
773:
731:infinite polytopes
584:of the polyhedron.
552:regular polyhedron
331:{b, c, ..., y, z}.
327:{a, b, c, ..., y},
312:abstract polytopes
283:(for example, the
275:(for example, the
81:
4347:Regular polytopes
4335:
4334:
4322:Polytope families
3779:uniform polytopes
3723:978-0-486-84248-6
3692:Science Awakening
3556:978-0-521-66405-9
3535:978-0-521-39490-1
3481:Regular Polytopes
3478:— (1973) .
3372:Coxeter, H. S. M.
3287:Euclid's Elements
3263:978-0-429-70681-3
3139:(a 4-dimensional
3031:Higher dimensions
2848:abstract polytope
2726:Abstract polytope
2493:measure polytopes
2489:regular simplices
2371:
2370:
2366:Great icosahedron
2285:great icosahedron
2249:
2248:
2089:{\displaystyle n}
2069:{\displaystyle P}
2049:{\displaystyle P}
2008:{\displaystyle n}
1988:{\displaystyle P}
1968:{\displaystyle n}
1928:{\displaystyle n}
1908:{\displaystyle P}
1888:{\displaystyle P}
1868:{\displaystyle n}
1782:{\displaystyle P}
1758:{\displaystyle n}
1738:{\displaystyle P}
1690:Exceptional type
1652:Exceptional type
1614:Exceptional type
1289:
1288:
1152:measure polytopes
1061:
1060:
921:regular simplices
856:
855:
769:Regular simplices
764:in 5 dimensions).
756:in 3 dimensions,
572:regular polyhedra
416:regular polyhedra
281:regular polyhedra
209:
208:
162:(shown here as a
105:, represented by
79:
78:
71:
4354:
4326:Regular polytope
3887:
3876:
3865:
3824:
3767:
3760:
3753:
3744:
3743:
3727:
3697:
3695:
3684:
3665:
3648:
3635:
3622:
3613:
3611:
3597:
3578:
3569:
3560:
3539:
3522:— (1991).
3518:
3506:
3499:— (1974).
3495:
3474:
3457:
3455:
3454:
3449:
3441:
3440:
3439:
3438:
3421:
3420:
3411:
3410:
3394:
3389:
3358:
3352:
3346:
3333:
3327:
3316:
3310:
3303:
3297:
3283:
3277:
3274:
3268:
3267:
3245:
3168:hyperbolic space
3141:spherical tiling
3055:Schlegel diagram
2971:Constructibility
2774:H. S. M. Coxeter
2713:complex polytope
2698:Complex polytope
2587:-polytope is an
2341:
2334:
2327:
2320:
2304:
2303:
2218:
2211:
2204:
2197:
2190:
2174:
2173:
2121:
2119:
2118:
2113:
2095:
2093:
2092:
2087:
2075:
2073:
2072:
2067:
2055:
2053:
2052:
2047:
2034:
2032:
2031:
2026:
2014:
2012:
2011:
2006:
1994:
1992:
1991:
1986:
1974:
1972:
1971:
1966:
1954:
1952:
1951:
1946:
1934:
1932:
1931:
1926:
1914:
1912:
1911:
1906:
1894:
1892:
1891:
1886:
1874:
1872:
1871:
1866:
1854:
1852:
1851:
1846:
1828:
1826:
1825:
1820:
1808:
1806:
1805:
1800:
1788:
1786:
1785:
1780:
1764:
1762:
1761:
1756:
1744:
1742:
1741:
1736:
1716:
1714:
1713:
1708:
1706:
1705:
1678:
1676:
1675:
1670:
1668:
1667:
1640:
1638:
1637:
1632:
1630:
1629:
1610:
1608:
1607:
1602:
1565:regular polygons
1562:
1560:
1559:
1554:
1543:
1542:
1518:measure polytope
1515:
1513:
1512:
1507:
1505:
1504:
1481:
1479:
1478:
1473:
1471:
1470:
1285:
1278:
1271:
1245:
1238:
1231:
1224:
1085:, orthogonal to
1057:
1050:
1043:
1017:
1010:
1003:
996:
947:(regular trigon)
852:
845:
838:
804:
797:
790:
783:
776:
714:({5,5/2,5}) and
687:regular polygons
478:Schläfli symbols
447:and the regular
385:regular polygons
353:Measure polytope
332:
328:
324:
297:
273:regular polygons
267:
253:
249:
239:-faces (for all
238:
217:regular polytope
195:
186:
173:
164:Schlegel diagram
161:
144:
135:
118:
111:
91:
84:
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
4362:
4361:
4357:
4356:
4355:
4353:
4352:
4351:
4337:
4336:
4305:
4298:
4291:
4174:
4167:
4160:
4124:
4117:
4110:
4074:
4067:
3901:Regular polygon
3894:
3885:
3878:
3874:
3867:
3863:
3854:
3845:
3838:
3834:
3822:
3816:
3812:
3800:
3782:
3771:
3734:
3724:
3681:
3594:
3564:Euclid (1956).
3557:
3536:
3515:
3492:
3431:
3427:
3426:
3422:
3416:
3412:
3406:
3402:
3390:
3385:
3379:
3376:
3375:
3367:
3362:
3361:
3353:
3349:
3343:Wayback Machine
3334:
3330:
3317:
3313:
3304:
3300:
3294:Wayback Machine
3284:
3280:
3275:
3271:
3264:
3246:
3239:
3234:
3229:
3207:
3202:
3193:
3184:
3120:virtual reality
3057:) for tesseract
3033:
2980:
2953:
2948:
2915:
2902:
2877:
2842:The concept of
2840:
2815:
2728:
2722:
2700:
2694:
2669:Branko GrĂźnbaum
2661:
2655:
2516:Ludwig Schläfli
2497:cross polytopes
2400:Schläfli (1858)
2396:Schläfli (1855)
2392:Schläfli (1901)
2388:Ludwig Schläfli
2376:
2355:
2354:Great stellated
2348:
2347:Small stellated
2302:
2289:Augustin Cauchy
2279:discovered the
2265:Johannes Kepler
2256:
2179:Platonic solids
2167:Platonic solids
2141:Platonic solids
2133:
2128:
2101:
2098:
2097:
2081:
2078:
2077:
2061:
2058:
2057:
2041:
2038:
2037:
2020:
2017:
2016:
2000:
1997:
1996:
1980:
1977:
1976:
1960:
1957:
1956:
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1937:
1936:
1920:
1917:
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1900:
1897:
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1814:
1811:
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1794:
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1790:
1774:
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1770:
1750:
1747:
1746:
1730:
1727:
1726:
1701:
1697:
1695:
1692:
1691:
1663:
1659:
1657:
1654:
1653:
1625:
1621:
1619:
1616:
1615:
1572:
1569:
1568:
1538:
1534:
1532:
1529:
1528:
1500:
1496:
1494:
1491:
1490:
1466:
1462:
1460:
1457:
1456:
1430:
1424:
1416:-orthoplex has
1378:1. Line segment
1365:cross polytopes
1295:
1219:
1165:1. Line segment
1067:
991:
862:
771:
672:cross polytopes
645:
543:, so a regular
501:regular polygon
490:Ludwig Schläfli
486:
484:Schläfli symbol
480:
427:five dimensions
412:four dimensions
348:Regular simplex
338:
330:
326:
322:
319:Schläfli symbol
295:
259:
251:
240:
236:
196:
184:
177:cubic honeycomb
174:
159:
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107:Schläfli symbol
92:
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45:Please help to
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3732:External links
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3326:, for example.
3311:
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3276:Coxeter (1974)
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3198:Main article:
3189:Main article:
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3175:symmetry group
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2937:may or may not
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2724:Main article:
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2704:complex number
2696:Main article:
2693:
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2685:skew apeirogon
2657:Main article:
2654:
2651:
2639:Coxeter groups
2628:
2627:
2614:
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2597:
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2581:
2578:vertex figures
2574:
2418:Coxeter (1973)
2408:Reinhold Hoppe
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1434:isometry group
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1363:These are the
1344:. Draw a line
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482:Main article:
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381:two dimensions
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359:Cross polytope
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225:symmetry group
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203:Petrie polygon
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3216:Johnson solid
3214:
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3209:
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3133:non-Euclidean
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2756:GrĂźnbaum 1976
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2708:Hilbert space
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2667:In the 1960s
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2643:tessellations
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2512:Arthur Cayley
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2478:cuboctahedron
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2404:Coxeter (1973
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2277:Louis Poinsot
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2163:number theory
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2137:ancient Greek
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2002:
1982:
1962:
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1768:
1765:and take its
1752:
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1519:
1516:, gives the
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904:. Mark point
903:
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884:in a second,
883:
880:. Mark point
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868:. Mark point
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657:vertex figure
653:
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638:}. And so on.
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432:See also the
430:
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370:one dimension
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35:
30:
21:
20:
4325:
4309:
4278:
4269:
4261:
4252:
4243:
4223:10-orthoplex
3959:Dodecahedron
3880:
3869:
3858:
3849:
3840:
3831:
3827:
3817:
3809:
3805:
3797:
3793:
3774:
3711:
3709:
3691:
3670:
3661:
3657:
3649:
3644:
3640:
3631:
3627:
3618:
3607:
3602:McMullen, P.
3583:
3574:
3565:
3545:
3524:
3502:
3480:
3466:(1): 21â25,
3463:
3459:
3365:Bibliography
3350:
3331:
3314:
3301:
3281:
3272:
3253:
3194:
3185:
3172:
3164:tessellation
3156:
3117:
3100:
3092:Coxeter 1973
3084:orthographic
3080:
3076:
3072:
3026:
3014:
3010:
3006:fold-out net
3003:
2983:
2981:
2969:
2964:
2960:
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2934:
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2917:
2912:
2908:
2906:
2899:
2895:
2891:
2887:
2883:
2879:
2874:
2870:
2863:
2862:, including
2859:
2855:
2851:
2843:
2841:
2832:All polygons
2831:
2830:
2826:
2823:
2816:
2804:
2798:
2796:
2791:
2785:
2782:
2767:
2764:
2760:
2751:
2741:
2711:
2701:
2673:polystromata
2672:
2666:
2662:
2629:
2617:
2615:
2609:
2605:
2598:
2592:
2588:
2584:
2568:
2558:such as the
2553:
2528:
2509:
2486:
2462:dodecahedron
2416:
2385:
2356:dodecahedron
2349:dodecahedron
2307:
2267:studied the
2257:
2239:Dodecahedron
2177:
2156:
2134:
1724:
1643:dodecahedron
1449:Coxeter 1935
1446:
1431:
1423:
1417:
1413:
1406:
1395:
1368:
1364:
1362:
1353:
1349:
1345:
1341:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1296:
1209:
1205:
1198:
1187:
1155:
1151:
1149:
1144:
1140:to form the
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:, to form a
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1077:at distance
1074:
1070:
1068:
984:+1 vertices.
981:
977:
970:
960:
939:Line segment
924:
920:
918:
909:
905:
897:
893:
889:
881:
878:line segment
873:
869:
865:
863:
810:Line segment
722:
721:All regular
703:The regular
693:
692:All regular
676:
665:
654:
646:
635:
631:
627:
623:
616:
612:
608:
604:
600:
596:
592:
588:
579:
575:
570:}. The nine
567:
563:
559:
555:
536:
532:
528:
524:
520:
519:which winds
517:star polygon
512:
508:
504:
487:
473:
453:dodecahedron
438:
431:
424:
405:
400:
396:
392:
388:
378:
374:line segment
367:
364:
342:
339:
321:of the form
316:
293:
270:
264:
260:
246:
242:
229:transitively
216:
210:
181:tessellation
122:dodecahedron
65:
56:
37:
4232:10-demicube
4193:9-orthoplex
4143:8-orthoplex
4093:7-orthoplex
4050:6-orthoplex
4020:5-orthoplex
3975:Pentachoron
3963:Icosahedron
3938:Tetrahedron
3088:perspective
2999:icosahedron
2799:realization
2792:containment
2470:icosahedron
2446:4-orthoplex
2430:tetrahedron
2244:Icosahedron
2224:Tetrahedron
1647:icosahedron
1409:5-orthoplex
1398:4-orthoplex
1384:3. Regular
1369:orthoplexes
1325:of length 2
1309:a distance
966:5. Regular
958:pentachoron
956:4. Regular
952:tetrahedron
950:3. Regular
914:tetrahedron
900:to form an
825:Pentachoron
820:Tetrahedron
762:{3,3,4,3,3}
679:palindromic
465:tetrahedral
457:icosahedron
420:4-polytopes
361:(Orthoplex)
355:(Hypercube)
213:mathematics
51:introducing
4218:10-simplex
4202:9-demicube
4152:8-demicube
4102:7-demicube
4059:6-demicube
4029:5-demicube
3943:Octahedron
3714:Dimensions
3680:0471861685
3647:: 269â301.
3634:: 359â394.
3593:0792330161
3514:052120125X
3227:References
2681:infinitely
2647:chessboard
2464:, and the
2454:octahedron
2234:Octahedron
2149:Theaetetus
2145:Pythagoras
1717:gives the
1679:gives the
1403:Pentacross
1386:octahedron
1358:octahedron
1256:Octahedron
1208:-cube has
1156:hypercubes
886:orthogonal
727:honeycombs
449:octahedron
175:A regular
152:polychoron
146:A regular
126:polyhedron
120:A regular
93:A regular
34:references
4266:orthoplex
4188:9-simplex
4138:8-simplex
4088:7-simplex
4045:6-simplex
4015:5-simplex
3984:Tesseract
3704:(2020) .
3546:Polyhedra
3129:Euclidean
3044:tesseract
2982:Euclid's
2978:Polyhedra
2748:self-dual
2548:tesseract
2540:tesseract
2535:tesseract
2520:tesseract
2452:) to the
2440:) to the
2438:tesseract
2434:hypercube
2426:4-simplex
2107:−
2023:Δ
1943:Δ
1817:Δ
1797:Δ
1563:give the
1420:vertices.
1293:Orthoplex
1212:vertices.
1195:Penteract
1184:Tesseract
1065:Hypercube
1033:Tesseract
973:5-simplex
968:hexateron
963:4-simplex
925:simplexes
758:{5,3,3,5}
698:simplexes
578:} is the
547:is {5/2}.
545:pentagram
531:}, where
289:aesthetic
256:dimension
59:July 2014
4341:Category
4320:Topics:
4283:demicube
4248:polytope
4242:Uniform
4003:600-cell
3999:120-cell
3952:Demicube
3926:Pentagon
3906:Triangle
3664:: 1â237.
3566:Elements
3339:Archived
3290:Archived
3205:See also
3113:hologram
3109:CAT scan
3104:animated
2984:Elements
2951:Polygons
2736:Hemicube
2632:symmetry
2522:and the
2480:and the
2466:600-cell
2458:120-cell
2271:and the
2158:Elements
1685:600-cell
1683:and the
1681:120-cell
1520:and the
1375:0. Point
1162:0. Point
1145:ABCDEFGH
815:Triangle
541:co-prime
441:symmetry
308:vertices
250:, where
221:polytope
185:{4,3,4}.
160:{5,3,3}.
148:120-cell
95:pentagon
4257:simplex
4227:10-cube
3994:24-cell
3980:16-cell
3921:Hexagon
3775:regular
3320:origami
3191:Polygon
3067:24-cell
2924:-face,
2778:57-cell
2770:11-cell
2744:11-cell
2564:11-cell
2562:or the
2560:57-cell
2544:24-cell
2524:24-cell
2474:24-cell
2468:to the
2460:to the
2450:16-cell
1719:24-cell
1484:simplex
1412:... An
1392:16-cell
1261:16-cell
1204:... An
1089:, from
976:... An
860:Simplex
831:
754:{3,6,3}
750:{6,3,6}
746:{3,5,3}
742:{5,3,5}
738:{4,4,4}
705:24-cell
668:measure
595:} with
558:} with
515:-sided
503:having
254:is the
231:on its
99:polygon
47:improve
4197:9-cube
4147:8-cube
4097:7-cube
4054:6-cube
4024:5-cube
3911:Square
3788:Family
3720:
3677:
3590:
3553:
3532:
3511:
3488:
3260:
3137:sphere
3017:klikko
2907:where
2754:face (
2635:groups
2542:. The
2456:, the
2444:, the
2432:, the
2153:Euclid
1567:(with
1339:square
1251:Square
1201:5-cube
1190:4-cube
1170:Square
1107:square
1023:Square
752:, and
689:, {a}.
498:convex
372:, the
300:facets
277:square
223:whose
199:8-cube
134:{5,3}.
36:, but
3916:p-gon
3232:Notes
1489:Type
1455:Type
1317:and 2
1313:from
933:Point
408:three
233:flags
227:acts
219:is a
179:is a
156:cells
150:is a
130:faces
124:is a
103:edges
97:is a
4274:cube
3947:Cube
3777:and
3718:ISBN
3675:ISBN
3588:ISBN
3551:ISBN
3530:ISBN
3509:ISBN
3486:ISBN
3324:here
3307:here
3258:ISBN
3160:UIUC
3042:for
2997:for
2752:same
2734:The
2619:flag
2514:and
2495:and
2442:cube
2436:(or
2398:and
2283:and
2229:Cube
2076:has
1829:has
1645:and
1352:and
1342:ACBD
1305:and
1177:Cube
1142:cube
1120:and
1110:ABCD
1028:Cube
896:and
710:The
685:All
670:and
666:The
661:cell
655:The
649:dual
647:The
539:are
535:and
467:and
461:dual
455:and
445:cube
418:and
410:and
285:cube
241:0 â¤
215:, a
110:{5}.
3823:(p)
3468:doi
3458:",
3166:of
3143:).
3098:).
3086:or
3040:Net
2995:Net
2882:= {
2604:An
1367:or
1346:EOF
1323:COD
1193:5.
1182:4.
1175:3.
1168:2.
1154:or
1101:to
1093:to
943:2.
937:1.
931:0.
923:or
425:In
422:.
406:In
379:In
368:In
211:In
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744:.
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982:n
978:n
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906:D
898:B
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