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vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal
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43:
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9841:
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1300:
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349:
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9855:
1128:
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10103:
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145:
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1111:
1134:
1133:
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1114:
1113:
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1135:
1115:
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3343:
4-orthoscheme which meet at the center of a 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. If the regular 5-cell has unit radius and edge length
1171:
1174:
1173:
1169:
686:
represents the 5-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 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is
1367:
envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far
5352:
When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges, and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge. This makes the six
1537:
with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight
1379:
The face-first projection of the 5-cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D
1195:
3342:
The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell). The four edges of each
1718:
mirror-surfaced orthoscheme instance is reflected in its own facets. More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements (from 90 degrees on
1454:
in which all edges are mutually perpendicular. In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular
1175:
1353:
projection envelope. The closest vertex of the 5-cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex.
1132:
1112:
1151:
1154:
1153:
1149:
1051:
1155:
6030:
1192:
1172:
210:
1152:
6925:
7015:
545:). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem:
7323:
1066:
central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell. Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them. The characteristic
5542:
7094:
3839:, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.
1389:
The cell-first projection of the 5-cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here.
7408:
5916:
5823:
5730:
5637:
1405:
such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These
1082:
of the great digons of the 5-cell. Each of the two fibrations corresponds to a left-right pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5-cell has only two distinct period 5
7489:
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6590:
6512:
7167:
6309:
6240:
6171:
5922:
6102:
1130:
1110:
6367:
547:
Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another.
1470:
in the bounding facets of its particular characteristic orthoscheme. For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the
2770:
2599:
2440:
2257:
2130:
2015:
6835:
3140:
3043:
1880:
1699:
instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.
1345:
1190:
1170:
5451:
3330:
658:
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2644:
2302:
2175:
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1150:
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7232:
7201:
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7541:
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6795:
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of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no 5-cell vertices.
6931:
2852:
2681:
2510:
2339:
2212:
2085:
8825:
8746:
8677:
8618:
8569:
8530:
1958:
1919:
11074:
10658:
9256:
8501:
8386:
1514:(the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be
7642:
7624:
7596:
2946:
1089:(those circles through all 5 vertices), each of which acts as the single isocline of a right rotation and the single isocline of a left rotation in two different fibrations.
159:
9302:
8432:
9675:
9596:
9527:
9468:
9419:
9380:
9351:
12170:
Kaleidoscopes: Selected
Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
11069:
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10911:
10906:
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10595:
10581:
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10518:
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9695:
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9400:
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9322:
8472:
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8114:
8104:
8094:
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7997:
7987:
7977:
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7899:
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5271:
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5211:
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761:
251:
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3003:
2913:
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2400:
512:
11051:
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4827:
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4697:
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4607:
4587:
4577:
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4413:
4403:
4370:
4360:
4350:
4317:
4307:
4297:
4264:
4254:
4244:
4211:
4201:
4191:
4158:
4148:
4138:
1814:
1804:
1794:
1784:
1776:
1766:
1756:
1684:
1674:
1664:
1654:
1643:
1633:
1623:
982:
972:
905:
281:
271:
261:
7637:
7619:
7601:
9750:
9740:
9730:
9720:
9710:
9700:
9690:
9680:
9661:
9651:
9641:
9631:
9621:
9611:
9601:
9582:
9572:
9562:
9552:
9542:
9532:
9513:
9503:
9493:
9483:
9473:
9454:
9444:
9434:
9424:
9405:
9395:
9385:
9366:
9356:
8880:
8870:
8860:
8850:
8840:
8830:
8820:
8810:
8791:
8781:
8771:
8761:
8751:
8741:
8731:
8712:
8702:
8692:
8682:
8672:
8662:
8643:
8633:
8623:
8613:
8603:
8584:
8574:
8564:
8554:
8535:
8525:
8515:
8496:
8486:
7240:
11036:
10984:
10974:
10945:
10935:
10888:
10878:
10849:
10839:
10792:
10782:
10753:
10743:
10638:
10610:
10600:
10561:
10543:
10533:
10523:
10500:
10490:
10480:
10447:
10429:
10419:
10409:
10386:
10376:
10366:
10333:
10315:
10305:
10295:
10272:
10262:
10252:
9337:
9327:
8467:
8457:
8119:
8109:
8099:
8080:
8070:
8060:
8041:
8031:
8021:
8002:
7992:
7982:
7963:
7953:
7943:
7924:
7914:
7904:
7885:
7875:
7865:
7846:
7836:
7826:
7807:
7797:
7787:
5266:
5256:
5246:
5236:
5206:
5196:
5186:
5176:
5146:
5136:
5126:
5116:
5086:
5076:
5066:
5056:
5026:
5016:
5006:
4996:
4882:
4872:
4862:
4852:
4822:
4812:
4802:
4792:
4762:
4752:
4742:
4732:
4702:
4692:
4682:
4672:
4642:
4632:
4622:
4612:
4582:
4572:
4562:
4552:
4418:
4408:
4398:
4388:
4365:
4355:
4345:
4335:
4312:
4302:
4292:
4282:
4259:
4249:
4239:
4229:
4206:
4196:
4186:
4176:
4153:
4143:
4133:
4123:
1809:
1799:
1789:
1771:
1761:
1751:
1679:
1669:
1659:
1638:
1628:
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987:
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967:
920:
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853:
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833:
786:
776:
766:
276:
266:
256:
1100:
1482:), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length
5460:
11847:
7021:
5331:
7329:
5327:
1380:
viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection.
1068:
7588:
projection, with a red and blue 5-cell vertices and edges. This compound has ] symmetry, order 240. The intersection of these two 5-cells is a uniform
1545:
edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a
5829:
5736:
5643:
5550:
581:
is formed by any three points which are not all in the same line). Any such five points constitute a 5-cell, though not usually a regular 5-cell. The
1818:
just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.
7414:
12375:
6674:
6596:
6518:
1559:
diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a
7704:
6446:
11488:
1552:
diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes). The third additional edge is a
163:
7100:
11743:
constant ≈ 1.618, for which
Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.
12343:
6035:
The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a
6025:{\displaystyle \left({\tfrac {2}{\phi }},{\tfrac {2}{\phi }},{\tfrac {2}{\phi }},{\tfrac {2}{\phi }}\right)/({\tfrac {1}{\phi }}-2)}
6246:
6177:
6108:
13547:
6045:
1730:
Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme. There is a 4-orthoscheme which is the
1463:
6315:
3707:(edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is
12288:
12234:
12175:
12108:
12079:
12057:
12032:
11180:
5306:, with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges form a
663:
It is the first in the sequence of 6 convex regular 4-polytopes, in order of volume at a given radius or number of vertexes.
10588:
2783:
2738:
2612:
2567:
2408:
2270:
2225:
2143:
2098:
1983:
6803:
3108:
3011:
1848:
12982:
11477:
11901:
11482:
11472:
5383:
1714:
of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a
12089:
Miyazaki, K.; Ishii, M. (2021). "Symmetry in
Projection of 4-Dimensional Regular Polychora". In Darvas, György (ed.).
3299:
12368:
11913:
5302:
of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a
3982:
607:
185:
126:
107:
11988:
Akiyama, Jin; Hitotumatu, Sin; Sato, Ikuro (2012). "Determination of the element numbers of the regular polytopes".
12016:
11708:
11316:
11024:
3205:
79:
422:
11800:
5097:
167:
12263:
Banchoff, Thomas F. (2013). "Torus
Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.).
1533:
A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a
12640:
12244:
12159:
12134:
12127:
12067:
11784:
11467:
11174:
11169:
10024:
10014:
10004:
6036:
64:
6920:{\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}
3810:
3777:
3744:
3711:
3678:
3549:
3516:
3483:
3446:
3413:
3380:
1932:
1893:
86:
10511:
10468:
10397:
10194:
10029:
10019:
7206:
7175:
6412:
6375:
4977:
3645:
3612:
3347:
13569:
13564:
12361:
10035:
7495:
7010:{\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}
1648:
1515:
11816:
6800:
Coordinates for the vertices of another origin-centered regular 5-cell with edge length 2 and radius
6752:
11768:
11164:
4376:
3578:(the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the
2453:
2028:
683:
17:
3267:
2828:
2657:
2486:
2315:
2188:
2061:
667:
93:
12624:
11624:
11400:
10173:
10009:
9958:
9312:
8442:
7774:
7679:
5299:
5037:
4773:
1706:
instances of its characteristic orthoscheme that all meet at the regular polytope's center. The number
1092:
Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The
600:, meaning its dual polytope is 5-cell itself. Its maximal intersection with 3-dimensional space is the
9225:
8355:
671:
12178:
11889:
11697:
11686:
11305:
11294:
10962:
10923:
4432:
Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a
2921:
538:
226:
9271:
8401:
13005:
12608:
12599:
12576:
12138:
4653:
3971:
1215:
75:
585:
5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex
12975:
11965:
9136:
9131:
9111:
8266:
8261:
8241:
7318:{\displaystyle \left({\sqrt {3}},{\sqrt {5}},{\sqrt {10}},\pm {\sqrt {30}}\right)/(4{\sqrt {3}})}
5157:
4833:
3586:
3158:
3061:
2964:
2874:
2691:
2520:
2361:
1560:
1511:
1258:
490:
60:
53:
1529:. Three are left-handed and three are right handed. A left and a right meet at each square face.
1439:
154:
may contain an excessive amount of intricate detail that may interest only a particular audience
12272:
9141:
9121:
9116:
8271:
8251:
8246:
450:
11941:
13519:
13512:
13505:
11826:
11810:
11778:
11159:
9146:
9126:
9073:
8276:
8256:
8206:
7750:
5366:
4713:
4533:
3960:
3847:
There are many lower symmetry forms of the 5-cell, including these found as uniform polytope
13044:
13022:
13010:
12264:
12153:
p. 296, Table I (iii): Regular
Polytopes, three regular polytopes in n-dimensions (n≥5)
8187:
8173:
8145:
13176:
13123:
12617:
12569:
8194:
7761:
5217:
1364:
1251:
577:(as a tetrahedron is formed by any four points which are not all in the same plane, and a
8:
13531:
13430:
13180:
12325:
10712:
10705:
10698:
8166:
8159:
7651:
7570:
5537:{\displaystyle 2(\phi -1/(2-{\tfrac {1}{\phi }}))={\sqrt {\tfrac {16}{5}}}\approx 1.7888}
4270:
3977:
3949:
1451:
1075:
534:
8180:
13400:
13350:
13300:
13257:
13227:
13187:
13150:
12968:
12384:
12222:
12209:
12049:
10182:
10060:
8152:
7734:
7729:
7724:
7691:
7589:
4948:
4593:
4030:
4023:
3999:
3890:
1411:
597:
393:
11659:
11541:
11534:
11527:
11233:
11226:
11219:
11092:
10088:
10081:
10074:
7089:{\displaystyle \left({\frac {1}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ 0,\ 0\right)}
4913:
4459:
4041:
3904:
1739:
1526:
1456:
1431:
1028:
446:
233:
215:
13539:
12306:
12284:
12265:
12230:
12171:
12104:
12075:
12053:
12028:
11929:
11675:
11602:
11378:
11351:
11283:
10866:
10151:
9939:
7739:
5307:
5303:
4934:
4164:
1321:
590:
332:
100:
13543:
13108:
13097:
13086:
13075:
13066:
13057:
12996:
12992:
12276:
12096:
12020:
11997:
11613:
11591:
11427:
11389:
11367:
11118:
10216:
10162:
10140:
9980:
9879:
9763:
9210:
9202:
9194:
8893:
8340:
8332:
8324:
8132:
7719:
7714:
7562:
7403:{\displaystyle \left({\sqrt {3}},{\sqrt {5}},-{\sqrt {40}},0\right)/(4{\sqrt {3}})}
5346:
4926:
4475:
4433:
4323:
4096:
4053:
1724:
1455:
simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a
601:
415:
31:
11910:, pp. 17–20, §10 The Coxeter Classification of Four-Dimensional Point Groups.
4941:
1727:, they contain one of everything needed to construct the polytope by replication.
1688:
eight different ways, with six 4-orthoschemes surrounding each of four orthogonal
13133:
13118:
12280:
12208:
Kim, Heuna; Rote, G. (2016). "Congruence
Testing of Point Sets in 4 Dimensions".
12090:
12043:
12010:
11462:
10684:
5369:
is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (𝜙,𝜙,𝜙,𝜙), with edge length 2
5354:
4972:
4528:
4511:
4106:
4011:
1571:
1438:
face-bonded to each other, with a total of 10 edges and 10 triangular faces.) An
1435:
698:
573:
in high dimension. It is formed by any five points which are not all in the same
542:
411:
407:
403:
322:
243:
11882:
11198:
10677:
5911:{\displaystyle \left(1,1,1,{\tfrac {2}{\phi }}-3\right)/({\tfrac {1}{\phi }}-2)}
5818:{\displaystyle \left(1,1,{\tfrac {2}{\phi }}-3,1\right)/({\tfrac {1}{\phi }}-2)}
5725:{\displaystyle \left(1,{\tfrac {2}{\phi }}-3,1,1\right)/({\tfrac {1}{\phi }}-2)}
5632:{\displaystyle \left({\tfrac {2}{\phi }}-3,1,1,1\right)/({\tfrac {1}{\phi }}-2)}
13483:
12633:
12592:
12585:
12309:
12117:
11739:𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the
11499:
11191:
10670:
10053:
10046:
9927:
9919:
9059:
9051:
5311:
4955:
4497:
4217:
3966:
1344:
1243:
1211:
1093:
1079:
362:
12338:
12100:
12024:
12001:
8986:
4518:
1254:(two tetrahedra joined face-to-face) with the two opposite vertices centered.
1078:
of the 5-cell through all 5 vertices along 5 edges, so there are two discrete
1055:
13558:
13500:
13388:
13381:
13374:
13338:
13331:
13324:
13288:
13281:
9946:
9911:
9903:
9895:
9153:
9043:
9035:
9027:
9019:
8283:
7695:
7585:
4490:
4483:
4037:
3955:
3848:
1467:
1415:
1270:
388:
374:
342:
11681:
11289:
10918:
8965:
8138:
4962:
4504:
1161:
Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio
514:
polytope), the simplest possible convex 4-polytope, and is analogous to the
166:
any relevant information, and removing excessive detail that may be against
13440:
11740:
11580:
11356:
10129:
5454:
5377:
4111:
4082:
3944:
1735:
1720:
1534:
1032:
523:
7584:
The compound of two 5-cells in dual configurations can be seen in this A5
13449:
13410:
13360:
13310:
13267:
13237:
13169:
13155:
11664:
11642:
11558:
11334:
11272:
11250:
11088:
10827:
10731:
10107:
9943:
9887:
9787:
8919:
7566:
7484:{\displaystyle \left({\sqrt {3}},-{\sqrt {45}},0,0\right)/(4{\sqrt {3}})}
5322:
of the 5-cell. The pentagram's blue edges are the chords of the 5-cell's
4068:
4003:
1780:
can be dissected into 120 instances of this characteristic 4-orthoscheme
1447:
1384:
1374:
1358:
1350:
1035:
of 5 points, which is literally an illustration of the regular 5-cell in
998:
818:
570:
515:
354:
295:
11863:
4089:
13435:
13419:
13369:
13319:
13276:
13246:
13160:
11670:
11653:
11648:
11586:
11362:
11278:
11261:
10861:
10770:
10135:
9003:
7554:
6741:{\displaystyle \left(-{\sqrt {5}},-{\sqrt {5}},{\sqrt {5}},-1\right)/4}
6663:{\displaystyle \left(-{\sqrt {5}},{\sqrt {5}},-{\sqrt {5}},-1\right)/4}
6585:{\displaystyle \left({\sqrt {5}},-{\sqrt {5}},-{\sqrt {5}},-1\right)/4}
4007:
574:
566:
562:
442:
12353:
8951:
4075:
1574:
vertex (a vertex of the opposing 3-cube), which is the apex. Thus the
13491:
13405:
13355:
13305:
13262:
13232:
13201:
12417:
12314:
11520:
11212:
10067:
9950:
6507:{\displaystyle \left({\sqrt {5}},{\sqrt {5}},{\sqrt {5}},-1\right)/4}
5315:
5314:
of the 5-cell. The blue edges connect every second vertex, forming a
3938:
1235:
11597:
11373:
10146:
9819:
5290:
4436:
is represented by removing the ringed nodes of the
Coxeter diagram.
4061:
1695:
tesseract long diameters. The 4-cube can also be dissected into 384
1521:
42:
30:
For the sequence of fifth element numbers of Pascal's triangle, see
13465:
13220:
13216:
13143:
12437:
12432:
12214:
11837:
11835:
11513:
11419:
11205:
11113:
10691:
10354:
10211:
10190:
9975:
9954:
9871:
7659:
1231:
1219:
931:
885:
586:
578:
519:
434:
367:
12348:
11703:
11506:
11311:
11019:
1459:, and each triangular face is a 2-orthoscheme (a right triangle).
1349:
The vertex-first projection of the 5-cell into 3 dimensions has a
1313:
1306:
13474:
13444:
13211:
13206:
13197:
13138:
12427:
12422:
12332:
12070:(2018). "Finite Symmetry Groups, 11.5 Spherical Coxeter groups".
11154:
11084:
10283:
10186:
9847:
8979:
7162:{\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)}
3933:
1443:
1402:
558:
483:
12122:
On the
Regular and Semi-Regular Figures in Space of n Dimensions
11898:, pp. 198–202, §11.7 Regular figures and their truncations.
11856:, pp. 292–293, Table I(ii): The sixteen regular polytopes {
11832:
11692:
11300:
10957:
10822:
9840:
8972:
3475:
around its exterior right-triangle face (the edges opposite the
1363:
The edge-first projection of the 5-cell into 3 dimensions has a
453:
object bounded by five tetrahedral cells. It is also known as a
13414:
13364:
13314:
13271:
13241:
13128:
11735:) uses the greek letter 𝝓 (phi) to represent one of the three
11575:
11340:
10765:
10124:
9826:
8958:
5380:. While these coordinates are not origin-centered, subtracting
1471:
1299:
314:
307:
299:
289:
11637:
11619:
11608:
11553:
11395:
11384:
11329:
11267:
11245:
10726:
10168:
10157:
10102:
1570:. It reaches through the exact center of the tesseract to the
1050:
348:
11564:
11256:
10113:
9854:
5342:
1063:
9833:
6304:{\displaystyle \left(-1,-1,1,{\frac {-1}{\sqrt {5}}}\right)}
6235:{\displaystyle \left(-1,1,-1,{\frac {-1}{\sqrt {5}}}\right)}
6166:{\displaystyle \left(1,-1,-1,{\frac {-1}{\sqrt {5}}}\right)}
1250:
Coxeter plane projection of the regular 5-cell is that of a
702:
Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975
666:
The convex hull of two 5-cells in dual configuration is the
13164:
11974:, p. 30, §4.2. The Crystallographic regular polytopes.
11569:
11345:
10118:
5326:, the circular rotational path its vertices take during an
1702:
More generally, any regular polytope can be dissected into
1466:
of the regular polytopes, because each regular polytope is
691:-faces can be read as rows left of the diagonal, while the
6097:{\displaystyle \left(1,1,1,{\frac {-1}{\sqrt {5}}}\right)}
1516:
dissected into instances of its characteristic orthoscheme
12015:. Carbon Materials: Chemistry and Physics. Vol. 10.
11727:
11725:
6362:{\displaystyle \left(0,0,0,{\frac {4}{\sqrt {5}}}\right)}
209:
10197:{3,3,6} of hyperbolic space also has tetrahedral cells.
7658:
This compound can be seen as the 4D analogue of the 2D
11722:
7557:
in 5-space, as (distinct) permutations of (0,0,0,0,1)
7212:
7181:
6809:
6418:
6380:
6002:
5977:
5962:
5947:
5932:
5888:
5857:
5795:
5758:
5702:
5659:
5609:
5560:
5516:
5494:
5429:
3816:
3783:
3750:
3717:
3684:
3651:
3618:
3555:
3522:
3489:
3452:
3419:
3386:
3353:
3304:
3211:
3114:
3017:
2833:
2788:
2744:
2662:
2617:
2573:
2491:
2414:
2320:
2275:
2231:
2193:
2148:
2104:
2066:
1989:
1854:
610:
12166:(2nd ed.), Cambridge: Cambridge University Press
11987:
11841:
11756:
9274:
9228:
8404:
8358:
7498:
7417:
7332:
7243:
7209:
7178:
7103:
7024:
6934:
6846:
6806:
6755:
6677:
6599:
6521:
6449:
6415:
6378:
6318:
6249:
6180:
6111:
6048:
5925:
5832:
5739:
5646:
5553:
5463:
5386:
3813:
3780:
3747:
3714:
3681:
3648:
3615:
3589:
3552:
3519:
3486:
3449:
3416:
3383:
3376:, its characteristic 5-cell's ten edges have lengths
3350:
3302:
3270:
3208:
3161:
3111:
3064:
3014:
2967:
2924:
2877:
2831:
2786:
2765:{\displaystyle {\sqrt {\tfrac {1}{60}}}\approx 0.129}
2741:
2694:
2660:
2615:
2594:{\displaystyle {\sqrt {\tfrac {1}{20}}}\approx 0.224}
2570:
2523:
2489:
2456:
2435:{\displaystyle {\sqrt {\tfrac {3}{20}}}\approx 0.387}
2411:
2364:
2318:
2273:
2252:{\displaystyle {\sqrt {\tfrac {2}{15}}}\approx 0.103}
2228:
2191:
2146:
2125:{\displaystyle {\sqrt {\tfrac {1}{30}}}\approx 0.183}
2101:
2064:
2031:
2010:{\displaystyle {\sqrt {\tfrac {1}{10}}}\approx 0.316}
1986:
1935:
1896:
1851:
1740:
characteristic tetrahedron of the regular tetrahedron
493:
11953:
6830:{\displaystyle {\sqrt {\tfrac {8}{5}}}\approx 1.265}
6372:
Scaling these or the previous set of coordinates by
3135:{\displaystyle {\sqrt {\tfrac {1}{6}}}\approx 0.408}
3038:{\displaystyle {\sqrt {\tfrac {3}{8}}}\approx 0.612}
1875:{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}
1242:
Coxeter plane projection of the 5-cell is that of a
526:
with a tetrahedral base and four tetrahedral sides.
7685:
1027:All these elements of the 5-cell are enumerated in
67:. Unsourced material may be challenged and removed.
12304:
9296:
9250:
8426:
8380:
7553:and radius 1) can be more simply constructed on a
7535:
7483:
7402:
7317:
7226:
7195:
7161:
7088:
7009:
6919:
6829:
6789:
6740:
6662:
6584:
6506:
6432:
6409:origin-centered regular 5-cells with edge lengths
6395:
6361:
6303:
6234:
6165:
6096:
6024:
5910:
5817:
5724:
5631:
5536:
5445:
3830:
3797:
3764:
3731:
3698:
3665:
3632:
3599:
3569:
3536:
3503:
3466:
3433:
3400:
3367:
3324:
3278:
3231:
3191:
3134:
3094:
3037:
2997:
2940:
2907:
2846:
2809:
2764:
2724:
2675:
2638:
2593:
2553:
2504:
2467:
2434:
2394:
2333:
2296:
2251:
2206:
2169:
2124:
2079:
2042:
2009:
1952:
1913:
1874:
652:
506:
5446:{\displaystyle (1,1,1,1)/(2-{\tfrac {1}{\phi }})}
1230:Coxeter plane projects the 5-cell into a regular
13556:
12326:"4D uniform polytopes (polychora) x3o3o3o - pen"
3325:{\displaystyle {\tfrac {{\text{arc sec }}4}{2}}}
1096:is depicted in its rectangular (wrapping) form.
653:{\textstyle \arccos(-1/4)\approx 75.52^{\circ }}
11938:, p. 139, §7.9 The characteristic simplex.
12335:Marco Möller's Regular polytopes in R (German)
12255:The Theory of Uniform Polytopes and Honeycombs
11950:, p. 290, Table I(ii); "dihedral angles".
7690:The pentachoron (5-cell) is the simplest of 9
5285:
2810:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜂}}}
2639:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜂}}}
2297:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜂}}}
2170:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜂}}}
695:-figures are read as rows after the diagonal.
12976:
12369:
12088:
11806:
11793:, p. 120, §7.2. see illustration Fig 7.2
4921:( )∨( )∨( )∨( )∨( )
3232:{\displaystyle {\sqrt {\tfrac {1}{16}}}=0.25}
11922:, pp. 292–293, Table I(ii); "5-cell, 𝛼
1719:down). The characteristic simplexes are the
1418:which give rise to the various 4-polytopes.
958:
891:
824:
757:
27:Four-dimensional analogue of the tetrahedron
12147:p. 120, §7.2. see illustration Fig 7.2
12124:, Messenger of Mathematics, Macmillan, 1900
1732:characteristic 5-cell of the regular 5-cell
1085:
12983:
12969:
12376:
12362:
1045:
687:identical to its 180 degree rotation. The
12213:
3923:
1371:
1341:
1054:A 3D projection of a 5-cell performing a
214:A 3D projection of a 5-cell performing a
186:Learn how and when to remove this message
127:Learn how and when to remove this message
12262:
12225:, Heidi Burgiel, Chaim Goodman-Strauss,
11959:
7561:(0,1,1,1,1); in these positions it is a
5341:
5289:
3831:{\displaystyle {\sqrt {\tfrac {1}{16}}}}
3798:{\displaystyle {\sqrt {\tfrac {1}{60}}}}
3765:{\displaystyle {\sqrt {\tfrac {2}{15}}}}
3732:{\displaystyle {\sqrt {\tfrac {1}{30}}}}
3699:{\displaystyle {\sqrt {\tfrac {1}{16}}}}
3570:{\displaystyle {\sqrt {\tfrac {1}{60}}}}
3537:{\displaystyle {\sqrt {\tfrac {1}{20}}}}
3504:{\displaystyle {\sqrt {\tfrac {3}{20}}}}
3467:{\displaystyle {\sqrt {\tfrac {2}{15}}}}
3434:{\displaystyle {\sqrt {\tfrac {1}{30}}}}
3401:{\displaystyle {\sqrt {\tfrac {1}{10}}}}
1836:
1833:
1520:
1210:
1049:
697:
549:No solution exists in three dimensions.
13548:List of regular polytopes and compounds
12383:
12237:(Chapter 26. pp. 409: Hemicubes: 1
12207:
12158:
12133:
12066:
11971:
11947:
11935:
11919:
11907:
11895:
11869:
11853:
11790:
11762:
11732:
10185:with tetrahedral cells, along with the
7546:The vertices of a 4-simplex (with edge
7227:{\displaystyle {\sqrt {\tfrac {5}{2}}}}
7196:{\displaystyle {\sqrt {\tfrac {5}{8}}}}
6433:{\displaystyle {\sqrt {\tfrac {5}{2}}}}
6396:{\displaystyle {\tfrac {\sqrt {5}}{4}}}
5294:The 5-cell Boerdijk–Coxeter helix
3666:{\displaystyle {\sqrt {\tfrac {1}{6}}}}
3633:{\displaystyle {\sqrt {\tfrac {3}{8}}}}
3368:{\displaystyle {\sqrt {\tfrac {5}{2}}}}
541:(the four-dimensional analogues of the
14:
13557:
12198:Regular and Semi-Regular Polytopes III
12008:
11822:
9957:{5,3,3} of Euclidean 4-space, and the
5453:from each translates the 4-polytope's
1824:Characteristics of the regular 5-cell
12552:
12395:
12357:
12305:
12191:Regular and Semi-Regular Polytopes II
12041:
11774:
7536:{\displaystyle \left(-1,0,0,0\right)}
1649:dissected into 24 such 4-orthoschemes
1434:. (The 5 vertices form 5 tetrahedral
677:
12323:
12184:Regular and Semi Regular Polytopes I
6790:{\displaystyle \left(0,0,0,1\right)}
6440:. The hyperpyramid has coordinates:
1396:
138:
65:adding citations to reliable sources
36:
11872:, p. 12, §1.8. Configurations.
11842:Akiyama, Hitotumatu & Sato 2012
5347:Net of five tetrahedra (one hidden)
4918:{ }∨( )∨( )∨( )
1576:characteristic 5-cell of the 4-cube
1430:is a 5-cell where all 10 faces are
522:in two dimensions. The 5-cell is a
24:
12339:Jonathan Bowers, Regular polychora
5544:, with the following coordinates:
3582:of the regular tetrahedron), plus
2847:{\displaystyle {\tfrac {\pi }{2}}}
2676:{\displaystyle {\tfrac {\pi }{2}}}
2505:{\displaystyle {\tfrac {\pi }{2}}}
2334:{\displaystyle {\tfrac {\pi }{3}}}
2207:{\displaystyle {\tfrac {\pi }{3}}}
2080:{\displaystyle {\tfrac {\pi }{3}}}
1074:There are only two ways to make a
565:. In other words, the 5-cell is a
25:
13581:
12298:
9938:It is in the {p,3,3} sequence of
5298:A 5-cell can be constructed as a
3983:Omnitruncated 4-simplex honeycomb
1953:{\displaystyle \pi -2{\text{𝟁}}}
1914:{\displaystyle \pi -2{\text{𝜂}}}
1218:wireframe (edge projected onto a
12939:great grand stellated dodecaplex
12143:(3rd ed.). New York: Dover.
11702:
11691:
11680:
11669:
11658:
11647:
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11618:
11607:
11596:
11585:
11574:
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11552:
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11533:
11526:
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11512:
11505:
11498:
11394:
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11361:
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11339:
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11299:
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11277:
11266:
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11244:
11232:
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11204:
11197:
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11072:
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11029:
11018:
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11005:
11000:
10995:
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10948:
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10938:
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10928:
10917:
10909:
10904:
10899:
10891:
10886:
10881:
10876:
10871:
10860:
10852:
10847:
10842:
10837:
10832:
10821:
10813:
10808:
10803:
10795:
10790:
10785:
10780:
10775:
10764:
10756:
10751:
10746:
10741:
10736:
10725:
10711:
10704:
10697:
10690:
10683:
10676:
10669:
10656:
10651:
10646:
10641:
10636:
10631:
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10618:
10613:
10608:
10603:
10598:
10593:
10579:
10574:
10569:
10564:
10559:
10554:
10546:
10541:
10536:
10531:
10526:
10521:
10516:
10503:
10498:
10493:
10488:
10483:
10478:
10473:
10460:
10455:
10450:
10445:
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10417:
10412:
10407:
10402:
10389:
10384:
10379:
10374:
10369:
10364:
10359:
10346:
10341:
10336:
10331:
10326:
10318:
10313:
10308:
10303:
10298:
10293:
10288:
10275:
10270:
10265:
10260:
10255:
10250:
10245:
10167:
10156:
10145:
10134:
10123:
10112:
10101:
10087:
10080:
10073:
10066:
10059:
10052:
10045:
9853:
9846:
9839:
9832:
9825:
9818:
9753:
9748:
9743:
9738:
9733:
9728:
9723:
9718:
9713:
9708:
9703:
9698:
9693:
9688:
9683:
9678:
9673:
9664:
9659:
9654:
9649:
9644:
9639:
9634:
9629:
9624:
9619:
9614:
9609:
9604:
9599:
9594:
9585:
9580:
9575:
9570:
9565:
9560:
9555:
9550:
9545:
9540:
9535:
9530:
9525:
9516:
9511:
9506:
9501:
9496:
9491:
9486:
9481:
9476:
9471:
9466:
9457:
9452:
9447:
9442:
9437:
9432:
9427:
9422:
9417:
9408:
9403:
9398:
9393:
9388:
9383:
9378:
9369:
9364:
9359:
9354:
9349:
9340:
9335:
9330:
9325:
9320:
9251:{\displaystyle {\tilde {E}}_{8}}
8985:
8978:
8971:
8964:
8957:
8950:
8883:
8878:
8873:
8868:
8863:
8858:
8853:
8848:
8843:
8838:
8833:
8828:
8823:
8818:
8813:
8808:
8803:
8794:
8789:
8784:
8779:
8774:
8769:
8764:
8759:
8754:
8749:
8744:
8739:
8734:
8729:
8724:
8715:
8710:
8705:
8700:
8695:
8690:
8685:
8680:
8675:
8670:
8665:
8660:
8655:
8646:
8641:
8636:
8631:
8626:
8621:
8616:
8611:
8606:
8601:
8596:
8587:
8582:
8577:
8572:
8567:
8562:
8557:
8552:
8547:
8538:
8533:
8528:
8523:
8518:
8513:
8508:
8499:
8494:
8489:
8484:
8479:
8470:
8465:
8460:
8455:
8450:
8381:{\displaystyle {\tilde {E}}_{8}}
8193:
8186:
8179:
8172:
8165:
8158:
8151:
8144:
8137:
8122:
8117:
8112:
8107:
8102:
8097:
8092:
8083:
8078:
8073:
8068:
8063:
8058:
8053:
8044:
8039:
8034:
8029:
8024:
8019:
8014:
8005:
8000:
7995:
7990:
7985:
7980:
7975:
7966:
7961:
7956:
7951:
7946:
7941:
7936:
7927:
7922:
7917:
7912:
7907:
7902:
7897:
7888:
7883:
7878:
7873:
7868:
7863:
7858:
7849:
7844:
7839:
7834:
7829:
7824:
7819:
7810:
7805:
7800:
7795:
7790:
7785:
7780:
7686:Related polytopes and honeycombs
7650:
7640:
7635:
7630:
7622:
7617:
7612:
7604:
7599:
7594:
5269:
5264:
5259:
5254:
5249:
5244:
5239:
5234:
5229:
5209:
5204:
5199:
5194:
5189:
5184:
5179:
5174:
5169:
5149:
5144:
5139:
5134:
5129:
5124:
5119:
5114:
5109:
5089:
5084:
5079:
5074:
5069:
5064:
5059:
5054:
5049:
5029:
5024:
5019:
5014:
5009:
5004:
4999:
4994:
4989:
4961:
4954:
4947:
4940:
4933:
4885:
4880:
4875:
4870:
4865:
4860:
4855:
4850:
4845:
4825:
4820:
4815:
4810:
4805:
4800:
4795:
4790:
4785:
4765:
4760:
4755:
4750:
4745:
4740:
4735:
4730:
4725:
4705:
4700:
4695:
4690:
4685:
4680:
4675:
4670:
4665:
4645:
4640:
4635:
4630:
4625:
4620:
4615:
4610:
4605:
4585:
4580:
4575:
4570:
4565:
4560:
4555:
4550:
4545:
4517:
4510:
4503:
4496:
4489:
4482:
4421:
4416:
4411:
4406:
4401:
4396:
4391:
4386:
4381:
4368:
4363:
4358:
4353:
4348:
4343:
4338:
4333:
4328:
4315:
4310:
4305:
4300:
4295:
4290:
4285:
4280:
4275:
4262:
4257:
4252:
4247:
4242:
4237:
4232:
4227:
4222:
4209:
4204:
4199:
4194:
4189:
4184:
4179:
4174:
4169:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4095:
4088:
4081:
4074:
4067:
4060:
3976:
3965:
3954:
3943:
3932:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1774:
1769:
1764:
1759:
1754:
1749:
1744:
1682:
1677:
1672:
1667:
1662:
1657:
1652:
1641:
1636:
1631:
1626:
1621:
1616:
1611:
1512:chord lengths of the unit 4-cube
1383:
1373:
1357:
1343:
1312:
1305:
1298:
1186:
1166:
1146:
1126:
1106:
1036:
990:
985:
980:
975:
970:
965:
960:
923:
918:
913:
908:
903:
898:
893:
856:
851:
846:
841:
836:
831:
826:
789:
784:
779:
774:
769:
764:
759:
557:The 5-cell is the 4-dimensional
347:
313:
298:
279:
274:
269:
264:
259:
254:
249:
208:
143:
41:
12012:Multi-shell Polyhedral Clusters
7203:to unit-radius and edge length
5280:
2941:{\displaystyle {\sqrt {1}}=1.0}
1421:
1017:
1014:
1011:
1008:
950:
947:
944:
941:
881:
878:
875:
872:
814:
811:
808:
805:
52:needs additional citations for
12271:. Springer New York. pp.
12074:. Cambridge University Press.
12072:Geometries and Transformations
11875:
9297:{\displaystyle {\bar {T}}_{8}}
9282:
9236:
8427:{\displaystyle {\bar {T}}_{8}}
8412:
8366:
7478:
7465:
7397:
7384:
7312:
7299:
6019:
5998:
5905:
5884:
5812:
5791:
5719:
5698:
5626:
5605:
5508:
5505:
5484:
5467:
5440:
5419:
5411:
5387:
5360:
1561:long diameter of the tesseract
1206:
634:
617:
13:
1:
11981:
10195:order-6 tetrahedral honeycomb
9961:{6,3,3} of hyperbolic space.
3924:
3897:Triangular pyramidal pyramid
3842:
1101:Visualization of 4D rotations
552:
12281:10.1007/978-0-387-92714-5_20
12045:The Geometry of Art and Life
11749:
4911:
4457:
3902:
3881:
3853:
3336:
3334:
3295:
3290:
3288:
3247:
3245:
3243:
3241:
3201:
3150:
3148:
3146:
3144:
3104:
3053:
3051:
3049:
3047:
3007:
2956:
2954:
2952:
2950:
2917:
2824:
2819:
2779:
2774:
2734:
2653:
2648:
2608:
2603:
2563:
2482:
2477:
2468:{\displaystyle 2{\text{𝜂}}}
2449:
2444:
2404:
2311:
2306:
2266:
2261:
2221:
2184:
2179:
2139:
2134:
2094:
2057:
2052:
2043:{\displaystyle 2{\text{𝜂}}}
2024:
2019:
1979:
1928:
1923:
1889:
1884:
1844:
1830:
1525:A 3-cube dissected into six
1338:Projections to 3 dimensions
1334:
1141:Simply rotating in Z-W plane
1121:Simply rotating in X-Y plane
518:in three dimensions and the
168:Knowledge's inclusion policy
7:
12196:(Paper 24) H.S.M. Coxeter,
12189:(Paper 23) H.S.M. Coxeter,
12182:(Paper 22) H.S.M. Coxeter,
11411:{p,3,p} regular honeycombs
10181:It is one of three {3,3,p}
7579:
5286:As a Boerdijk–Coxeter helix
4470:{ }∨{ }∨( )
4022:of the pyramid are made of
4002:, constructed as a regular
3600:{\displaystyle {\sqrt {1}}}
3279:{\displaystyle {\text{𝜼}}}
3263:
3192:{\displaystyle _{3}R^{4}/l}
3154:
3095:{\displaystyle _{2}R^{4}/l}
3057:
2998:{\displaystyle _{1}R^{4}/l}
2960:
2908:{\displaystyle _{0}R^{4}/l}
2870:
2725:{\displaystyle _{2}R^{3}/l}
2687:
2554:{\displaystyle _{1}R^{3}/l}
2516:
2395:{\displaystyle _{0}R^{3}/l}
2357:
2218:
2091:
1976:
1841:
1828:
957:
890:
823:
756:
507:{\displaystyle \alpha _{4}}
10:
13586:
13537:
12964:
12827:grand stellated dodecaplex
12783:great stellated dodecaplex
12396:
11410:
11100:
10202:
9966:
9959:hexagonal tiling honeycomb
9072:
8205:
7680:compound of two tetrahedra
5457:to the origin with radius
539:regular convex 4-polytopes
449:{3,3,3}. It is a 5-vertex
29:
12563:
12559:
12548:
12406:
12402:
12391:
12164:Regular Complex Polytopes
12101:10.1007/978-3-030-88059-0
12025:10.1007/978-3-319-64123-2
12002:10.1007/s10711-011-9647-3
11807:Miyazaki & Ishii 2021
11448:
11426:
11136:
11127:
11117:
11112:
11083:It is self-dual like the
10231:
10225:
10215:
10210:
10095:
10041:
10000:
9995:
9989:
9986:
9979:
9974:
9971:
9809:
9094:
8941:
8224:
4920:
4917:
4901:
4469:
4466:
4463:
4447:
4444:
4018:the hyperplane. The four
1823:
1337:
1002:
935:
866:
799:
421:
399:
387:
373:
361:
341:
331:
321:
306:
288:
242:
232:
227:Convex regular 4-polytope
222:
207:
200:
12553:
12227:The Symmetries of Things
11716:
6037:regular tetrahedral base
3972:Cantitruncated 5-simplex
1468:generated by reflections
1464:characteristic simplexes
1259:orthographic projections
1216:Stereographic projection
1201:Right isoclinic rotation
604:. Its dichoral angle is
593:of 120 regular 5-cells.
561:, the simplest possible
537:, and is one of the six
3994:is a special case of a
1368:side of the pentatope.
1181:Left isoclinic rotation
1046:Geodesics and rotations
12855:great grand dodecaplex
12042:Ghyka, Matila (1977).
12009:Diudea, M. V. (2018).
9298:
9252:
8428:
8382:
7694:constructed from the
7565:of, respectively, the
7537:
7485:
7404:
7319:
7228:
7197:
7163:
7090:
7011:
6921:
6831:
6791:
6742:
6664:
6586:
6508:
6434:
6397:
6363:
6305:
6236:
6167:
6098:
6026:
5912:
5819:
5726:
5633:
5538:
5447:
5349:
5337:
5300:Boerdijk–Coxeter helix
5295:
3918:{3}∨( )∨( )
3832:
3799:
3766:
3733:
3700:
3667:
3634:
3601:
3571:
3538:
3505:
3468:
3435:
3402:
3369:
3326:
3280:
3233:
3193:
3136:
3096:
3039:
2999:
2942:
2909:
2848:
2811:
2766:
2726:
2677:
2640:
2595:
2555:
2506:
2469:
2436:
2396:
2335:
2298:
2253:
2208:
2171:
2126:
2081:
2044:
2011:
1954:
1915:
1876:
1530:
1408:characteristic 5-cells
1365:triangular dipyramidal
1223:
1059:
703:
654:
508:
12333:Der 5-Zeller (5-cell)
11860:} in four dimensions.
11737:characteristic angles
9299:
9253:
8429:
8383:
7538:
7486:
7405:
7320:
7229:
7198:
7164:
7091:
7012:
6922:
6832:
6792:
6743:
6665:
6587:
6509:
6435:
6398:
6364:
6306:
6237:
6168:
6099:
6027:
5913:
5820:
5727:
5634:
5539:
5448:
5367:Cartesian coordinates
5345:
5332:Clifford displacement
5293:
4467:{3}∨( )∨( )
3961:Bitruncated 5-simplex
3833:
3800:
3767:
3734:
3701:
3668:
3635:
3602:
3572:
3539:
3506:
3477:characteristic angles
3469:
3436:
3403:
3370:
3327:
3281:
3234:
3194:
3137:
3097:
3040:
3000:
2943:
2910:
2849:
2812:
2767:
2727:
2678:
2641:
2596:
2556:
2507:
2470:
2437:
2397:
2336:
2299:
2254:
2209:
2172:
2127:
2082:
2045:
2012:
1955:
1916:
1877:
1742:. The regular 5-cell
1723:of polytopes: like a
1524:
1462:Orthoschemes are the
1214:
1053:
701:
655:
524:4-dimensional pyramid
509:
12699:stellated dodecaplex
12250:, Manuscript (1991)
9272:
9226:
8402:
8356:
7496:
7415:
7330:
7241:
7207:
7176:
7101:
7022:
6932:
6844:
6804:
6753:
6675:
6597:
6519:
6447:
6413:
6376:
6316:
6247:
6178:
6109:
6046:
5923:
5830:
5737:
5644:
5551:
5461:
5384:
5365:The simplest set of
4048:Symmetry , order 24
3811:
3778:
3745:
3712:
3679:
3646:
3613:
3587:
3580:characteristic radii
3550:
3517:
3484:
3447:
3414:
3381:
3348:
3300:
3268:
3206:
3159:
3109:
3062:
3012:
2965:
2922:
2875:
2829:
2784:
2739:
2692:
2658:
2613:
2568:
2521:
2487:
2454:
2409:
2362:
2316:
2271:
2226:
2189:
2144:
2099:
2062:
2029:
1984:
1933:
1894:
1849:
1252:triangular bipyramid
1062:The 5-cell has only
684:configuration matrix
668:disphenoidal 30-cell
608:
491:
61:improve this article
13570:Pyramids (geometry)
13565:Regular 4-polytopes
13532:pentagonal polytope
13431:Uniform 10-polytope
12991:Fundamental convex
12385:Regular 4-polytopes
12324:Klitzing, Richard.
11990:Geometriae Dedicata
11908:Kim & Rote 2016
11489:{∞,3,∞}
10183:regular 4-polytopes
7571:rectified penteract
5353:tetrahedron as its
4049:
4035:tetrahedral pyramid
4031:uniform 5-polytopes
3992:tetrahedral pyramid
3950:Truncated 5-simplex
1736:tetrahedral pyramid
1535:tetrahedral pyramid
1412:fundamental domains
1261:
533:is bounded by five
478:tetrahedral pyramid
13401:Uniform 9-polytope
13351:Uniform 8-polytope
13301:Uniform 7-polytope
13258:Uniform 6-polytope
13228:Uniform 5-polytope
13188:Uniform polychoron
13151:Uniform polyhedron
12999:in dimensions 2–10
12643:stellated 120-cell
12510:hecatonicosachoron
12307:Weisstein, Eric W.
12092:Complex Symmetries
12050:Dover Publications
11087:{3,4,3}, having a
10203:{3,3,p} polytopes
9967:{p,3,3} polytopes
9294:
9248:
8424:
8378:
7678:} and the 3D
7590:bitruncated 5-cell
7533:
7481:
7400:
7315:
7224:
7221:
7193:
7190:
7159:
7086:
7007:
6917:
6827:
6818:
6787:
6738:
6660:
6582:
6504:
6430:
6427:
6393:
6391:
6359:
6301:
6232:
6163:
6094:
6022:
6011:
5986:
5971:
5956:
5941:
5908:
5897:
5866:
5815:
5804:
5767:
5722:
5711:
5668:
5629:
5618:
5569:
5534:
5525:
5503:
5443:
5438:
5376:, where 𝜙 is the
5350:
5330:, also known as a
5328:isoclinic rotation
5296:
4047:
4024:triangular pyramid
4006:base in a 3-space
4000:polyhedral pyramid
3828:
3825:
3795:
3792:
3762:
3759:
3729:
3726:
3696:
3693:
3663:
3660:
3630:
3627:
3597:
3567:
3564:
3534:
3531:
3501:
3498:
3479:𝟀, 𝝉, 𝟁), plus
3464:
3461:
3431:
3428:
3398:
3395:
3365:
3362:
3322:
3320:
3276:
3229:
3220:
3189:
3132:
3123:
3092:
3035:
3026:
2995:
2938:
2905:
2844:
2842:
2807:
2797:
2762:
2753:
2722:
2673:
2671:
2636:
2626:
2591:
2582:
2551:
2502:
2500:
2465:
2432:
2423:
2392:
2331:
2329:
2294:
2284:
2249:
2240:
2204:
2202:
2167:
2157:
2122:
2113:
2077:
2075:
2040:
2007:
1998:
1950:
1911:
1872:
1863:
1563:itself, of length
1531:
1257:
1224:
1069:isoclinic rotation
1060:
704:
678:As a configuration
672:bitruncated 5-cell
650:
535:regular tetrahedra
504:
13553:
13552:
13540:Polytope families
12997:uniform polytopes
12959:
12958:
12955:
12954:
12951:
12950:
12946:
12945:
12544:
12543:
12540:
12539:
12535:
12534:
12290:978-0-387-92713-8
12248:Uniform Polytopes
12235:978-1-56881-220-5
12176:978-0-471-01003-6
12150:
12140:Regular Polytopes
12110:978-3-030-88059-0
12081:978-1-107-10340-5
12059:978-0-486-23542-4
12034:978-3-319-64123-2
11796:
11714:
11713:
11406:
11405:
11081:
11080:
10179:
10178:
9940:regular polychora
9936:
9935:
9285:
9239:
9068:
9067:
8415:
8369:
8201:
8200:
7766:
7755:
7744:
7692:uniform polychora
7476:
7441:
7428:
7395:
7366:
7353:
7343:
7310:
7287:
7274:
7264:
7254:
7222:
7220:
7191:
7189:
7172:Scaling these by
7150:
7141:
7132:
7125:
7124:
7077:
7068:
7061:
7060:
7047:
7040:
7039:
6998:
6991:
6990:
6975:
6968:
6967:
6957:
6950:
6949:
6905:
6898:
6897:
6887:
6880:
6879:
6869:
6862:
6861:
6819:
6817:
6714:
6704:
6691:
6636:
6623:
6613:
6558:
6545:
6532:
6480:
6470:
6460:
6428:
6426:
6390:
6386:
6352:
6351:
6294:
6293:
6225:
6224:
6156:
6155:
6087:
6086:
6010:
5985:
5970:
5955:
5940:
5896:
5865:
5803:
5766:
5710:
5667:
5617:
5568:
5526:
5524:
5502:
5437:
5304:triangular tiling
5278:
5277:
4894:
4893:
4430:
4429:
3988:
3987:
3838:
3826:
3824:
3805:
3793:
3791:
3772:
3760:
3758:
3739:
3727:
3725:
3706:
3694:
3692:
3673:
3661:
3659:
3640:
3628:
3626:
3607:
3595:
3577:
3565:
3563:
3544:
3532:
3530:
3511:
3499:
3497:
3474:
3462:
3460:
3441:
3429:
3427:
3408:
3396:
3394:
3375:
3363:
3361:
3340:
3339:
3332:
3319:
3310:
3293:
3286:
3274:
3239:
3221:
3219:
3199:
3142:
3124:
3122:
3102:
3045:
3027:
3025:
3005:
2948:
2930:
2915:
2854:
2841:
2822:
2817:
2805:
2796:
2777:
2772:
2754:
2752:
2732:
2683:
2670:
2651:
2646:
2634:
2625:
2606:
2601:
2583:
2581:
2561:
2512:
2499:
2480:
2475:
2463:
2447:
2442:
2424:
2422:
2402:
2341:
2328:
2309:
2304:
2292:
2283:
2264:
2259:
2241:
2239:
2214:
2201:
2182:
2177:
2165:
2156:
2137:
2132:
2114:
2112:
2087:
2074:
2055:
2050:
2038:
2022:
2017:
1999:
1997:
1960:
1948:
1926:
1921:
1909:
1887:
1882:
1864:
1862:
1474:(also called the
1414:of the different
1397:Irregular 5-cells
1394:
1393:
1333:
1332:
1322:Dihedral symmetry
1196:
1176:
1156:
1136:
1116:
1025:
1024:
431:
430:
196:
195:
188:
137:
136:
129:
111:
16:(Redirected from
13577:
13544:Regular polytope
13105:
13094:
13083:
13042:
12985:
12978:
12971:
12962:
12961:
12935:
12933:
12932:
12929:
12926:
12907:
12905:
12904:
12901:
12898:
12879:
12877:
12876:
12873:
12870:
12851:
12849:
12848:
12845:
12842:
12823:
12821:
12820:
12817:
12814:
12807:
12805:
12804:
12801:
12798:
12779:
12777:
12776:
12773:
12770:
12755:grand dodecaplex
12751:
12749:
12748:
12745:
12742:
12727:great dodecaplex
12723:
12721:
12720:
12717:
12714:
12695:
12693:
12692:
12689:
12686:
12667:
12665:
12664:
12661:
12658:
12566:
12565:
12561:
12560:
12550:
12549:
12495:icositetrachoron
12409:
12408:
12404:
12403:
12393:
12392:
12378:
12371:
12364:
12355:
12354:
12329:
12320:
12319:
12294:
12270:
12219:
12217:
12167:
12148:
12144:
12114:
12085:
12063:
12038:
12005:
11975:
11969:
11963:
11957:
11951:
11945:
11939:
11933:
11927:
11917:
11911:
11905:
11899:
11893:
11887:
11886:
11879:
11873:
11867:
11861:
11851:
11845:
11839:
11830:
11820:
11814:
11804:
11798:
11794:
11788:
11782:
11772:
11766:
11760:
11744:
11729:
11706:
11695:
11684:
11673:
11662:
11651:
11640:
11622:
11611:
11600:
11589:
11578:
11567:
11556:
11544:
11537:
11530:
11523:
11516:
11509:
11502:
11408:
11407:
11398:
11387:
11376:
11365:
11354:
11343:
11332:
11314:
11303:
11292:
11281:
11270:
11259:
11248:
11236:
11229:
11222:
11215:
11208:
11201:
11194:
11098:
11097:
11077:
11076:
11075:
11071:
11070:
11066:
11065:
11061:
11060:
11054:
11053:
11052:
11048:
11047:
11043:
11042:
11038:
11037:
11033:
11032:
11022:
11015:
11014:
11013:
11009:
11008:
11004:
11003:
10999:
10998:
10992:
10991:
10990:
10986:
10985:
10981:
10980:
10976:
10975:
10971:
10970:
10960:
10953:
10952:
10951:
10947:
10946:
10942:
10941:
10937:
10936:
10932:
10931:
10921:
10914:
10913:
10912:
10908:
10907:
10903:
10902:
10896:
10895:
10894:
10890:
10889:
10885:
10884:
10880:
10879:
10875:
10874:
10864:
10857:
10856:
10855:
10851:
10850:
10846:
10845:
10841:
10840:
10836:
10835:
10825:
10818:
10817:
10816:
10812:
10811:
10807:
10806:
10800:
10799:
10798:
10794:
10793:
10789:
10788:
10784:
10783:
10779:
10778:
10768:
10761:
10760:
10759:
10755:
10754:
10750:
10749:
10745:
10744:
10740:
10739:
10729:
10715:
10708:
10701:
10694:
10687:
10680:
10673:
10661:
10660:
10659:
10655:
10654:
10650:
10649:
10645:
10644:
10640:
10639:
10635:
10634:
10628:
10627:
10626:
10622:
10621:
10617:
10616:
10612:
10611:
10607:
10606:
10602:
10601:
10597:
10596:
10584:
10583:
10582:
10578:
10577:
10573:
10572:
10568:
10567:
10563:
10562:
10558:
10557:
10551:
10550:
10549:
10545:
10544:
10540:
10539:
10535:
10534:
10530:
10529:
10525:
10524:
10520:
10519:
10508:
10507:
10506:
10502:
10501:
10497:
10496:
10492:
10491:
10487:
10486:
10482:
10481:
10477:
10476:
10465:
10464:
10463:
10459:
10458:
10454:
10453:
10449:
10448:
10444:
10443:
10437:
10436:
10435:
10431:
10430:
10426:
10425:
10421:
10420:
10416:
10415:
10411:
10410:
10406:
10405:
10394:
10393:
10392:
10388:
10387:
10383:
10382:
10378:
10377:
10373:
10372:
10368:
10367:
10363:
10362:
10351:
10350:
10349:
10345:
10344:
10340:
10339:
10335:
10334:
10330:
10329:
10323:
10322:
10321:
10317:
10316:
10312:
10311:
10307:
10306:
10302:
10301:
10297:
10296:
10292:
10291:
10280:
10279:
10278:
10274:
10273:
10269:
10268:
10264:
10263:
10259:
10258:
10254:
10253:
10249:
10248:
10200:
10199:
10171:
10160:
10149:
10138:
10127:
10116:
10105:
10091:
10084:
10077:
10070:
10063:
10056:
10049:
9964:
9963:
9857:
9850:
9843:
9836:
9829:
9822:
9758:
9757:
9756:
9752:
9751:
9747:
9746:
9742:
9741:
9737:
9736:
9732:
9731:
9727:
9726:
9722:
9721:
9717:
9716:
9712:
9711:
9707:
9706:
9702:
9701:
9697:
9696:
9692:
9691:
9687:
9686:
9682:
9681:
9677:
9676:
9669:
9668:
9667:
9663:
9662:
9658:
9657:
9653:
9652:
9648:
9647:
9643:
9642:
9638:
9637:
9633:
9632:
9628:
9627:
9623:
9622:
9618:
9617:
9613:
9612:
9608:
9607:
9603:
9602:
9598:
9597:
9590:
9589:
9588:
9584:
9583:
9579:
9578:
9574:
9573:
9569:
9568:
9564:
9563:
9559:
9558:
9554:
9553:
9549:
9548:
9544:
9543:
9539:
9538:
9534:
9533:
9529:
9528:
9521:
9520:
9519:
9515:
9514:
9510:
9509:
9505:
9504:
9500:
9499:
9495:
9494:
9490:
9489:
9485:
9484:
9480:
9479:
9475:
9474:
9470:
9469:
9462:
9461:
9460:
9456:
9455:
9451:
9450:
9446:
9445:
9441:
9440:
9436:
9435:
9431:
9430:
9426:
9425:
9421:
9420:
9413:
9412:
9411:
9407:
9406:
9402:
9401:
9397:
9396:
9392:
9391:
9387:
9386:
9382:
9381:
9374:
9373:
9372:
9368:
9367:
9363:
9362:
9358:
9357:
9353:
9352:
9345:
9344:
9343:
9339:
9338:
9334:
9333:
9329:
9328:
9324:
9323:
9303:
9301:
9300:
9295:
9293:
9292:
9287:
9286:
9278:
9257:
9255:
9254:
9249:
9247:
9246:
9241:
9240:
9232:
9070:
9069:
8989:
8982:
8975:
8968:
8961:
8954:
8888:
8887:
8886:
8882:
8881:
8877:
8876:
8872:
8871:
8867:
8866:
8862:
8861:
8857:
8856:
8852:
8851:
8847:
8846:
8842:
8841:
8837:
8836:
8832:
8831:
8827:
8826:
8822:
8821:
8817:
8816:
8812:
8811:
8807:
8806:
8799:
8798:
8797:
8793:
8792:
8788:
8787:
8783:
8782:
8778:
8777:
8773:
8772:
8768:
8767:
8763:
8762:
8758:
8757:
8753:
8752:
8748:
8747:
8743:
8742:
8738:
8737:
8733:
8732:
8728:
8727:
8720:
8719:
8718:
8714:
8713:
8709:
8708:
8704:
8703:
8699:
8698:
8694:
8693:
8689:
8688:
8684:
8683:
8679:
8678:
8674:
8673:
8669:
8668:
8664:
8663:
8659:
8658:
8651:
8650:
8649:
8645:
8644:
8640:
8639:
8635:
8634:
8630:
8629:
8625:
8624:
8620:
8619:
8615:
8614:
8610:
8609:
8605:
8604:
8600:
8599:
8592:
8591:
8590:
8586:
8585:
8581:
8580:
8576:
8575:
8571:
8570:
8566:
8565:
8561:
8560:
8556:
8555:
8551:
8550:
8543:
8542:
8541:
8537:
8536:
8532:
8531:
8527:
8526:
8522:
8521:
8517:
8516:
8512:
8511:
8504:
8503:
8502:
8498:
8497:
8493:
8492:
8488:
8487:
8483:
8482:
8475:
8474:
8473:
8469:
8468:
8464:
8463:
8459:
8458:
8454:
8453:
8433:
8431:
8430:
8425:
8423:
8422:
8417:
8416:
8408:
8387:
8385:
8384:
8379:
8377:
8376:
8371:
8370:
8362:
8203:
8202:
8197:
8190:
8183:
8176:
8169:
8162:
8155:
8148:
8141:
8127:
8126:
8125:
8121:
8120:
8116:
8115:
8111:
8110:
8106:
8105:
8101:
8100:
8096:
8095:
8088:
8087:
8086:
8082:
8081:
8077:
8076:
8072:
8071:
8067:
8066:
8062:
8061:
8057:
8056:
8049:
8048:
8047:
8043:
8042:
8038:
8037:
8033:
8032:
8028:
8027:
8023:
8022:
8018:
8017:
8010:
8009:
8008:
8004:
8003:
7999:
7998:
7994:
7993:
7989:
7988:
7984:
7983:
7979:
7978:
7971:
7970:
7969:
7965:
7964:
7960:
7959:
7955:
7954:
7950:
7949:
7945:
7944:
7940:
7939:
7932:
7931:
7930:
7926:
7925:
7921:
7920:
7916:
7915:
7911:
7910:
7906:
7905:
7901:
7900:
7893:
7892:
7891:
7887:
7886:
7882:
7881:
7877:
7876:
7872:
7871:
7867:
7866:
7862:
7861:
7854:
7853:
7852:
7848:
7847:
7843:
7842:
7838:
7837:
7833:
7832:
7828:
7827:
7823:
7822:
7815:
7814:
7813:
7809:
7808:
7804:
7803:
7799:
7798:
7794:
7793:
7789:
7788:
7784:
7783:
7764:
7753:
7742:
7701:
7700:
7677:
7675:
7674:
7671:
7668:
7654:
7645:
7644:
7643:
7639:
7638:
7634:
7633:
7627:
7626:
7625:
7621:
7620:
7616:
7615:
7609:
7608:
7607:
7603:
7602:
7598:
7597:
7552:
7551:
7542:
7540:
7539:
7534:
7532:
7528:
7490:
7488:
7487:
7482:
7477:
7472:
7464:
7459:
7455:
7442:
7437:
7429:
7424:
7409:
7407:
7406:
7401:
7396:
7391:
7383:
7378:
7374:
7367:
7362:
7354:
7349:
7344:
7339:
7324:
7322:
7321:
7316:
7311:
7306:
7298:
7293:
7289:
7288:
7283:
7275:
7270:
7265:
7260:
7255:
7250:
7233:
7231:
7230:
7225:
7223:
7213:
7211:
7202:
7200:
7199:
7194:
7192:
7182:
7180:
7168:
7166:
7165:
7160:
7158:
7154:
7148:
7139:
7130:
7126:
7117:
7116:
7095:
7093:
7092:
7087:
7085:
7081:
7075:
7066:
7062:
7053:
7052:
7045:
7041:
7035:
7031:
7016:
7014:
7013:
7008:
7006:
7002:
6996:
6992:
6986:
6985:
6977:
6973:
6969:
6963:
6959:
6955:
6951:
6945:
6941:
6926:
6924:
6923:
6918:
6916:
6912:
6903:
6899:
6893:
6889:
6885:
6881:
6875:
6871:
6867:
6863:
6857:
6853:
6836:
6834:
6833:
6828:
6820:
6810:
6808:
6796:
6794:
6793:
6788:
6786:
6782:
6747:
6745:
6744:
6739:
6734:
6729:
6725:
6715:
6710:
6705:
6700:
6692:
6687:
6669:
6667:
6666:
6661:
6656:
6651:
6647:
6637:
6632:
6624:
6619:
6614:
6609:
6591:
6589:
6588:
6583:
6578:
6573:
6569:
6559:
6554:
6546:
6541:
6533:
6528:
6513:
6511:
6510:
6505:
6500:
6495:
6491:
6481:
6476:
6471:
6466:
6461:
6456:
6439:
6437:
6436:
6431:
6429:
6419:
6417:
6402:
6400:
6399:
6394:
6392:
6382:
6381:
6368:
6366:
6365:
6360:
6358:
6354:
6353:
6347:
6343:
6310:
6308:
6307:
6302:
6300:
6296:
6295:
6289:
6288:
6280:
6241:
6239:
6238:
6233:
6231:
6227:
6226:
6220:
6219:
6211:
6172:
6170:
6169:
6164:
6162:
6158:
6157:
6151:
6150:
6142:
6103:
6101:
6100:
6095:
6093:
6089:
6088:
6082:
6081:
6073:
6031:
6029:
6028:
6023:
6012:
6003:
5997:
5992:
5988:
5987:
5978:
5972:
5963:
5957:
5948:
5942:
5933:
5917:
5915:
5914:
5909:
5898:
5889:
5883:
5878:
5874:
5867:
5858:
5824:
5822:
5821:
5816:
5805:
5796:
5790:
5785:
5781:
5768:
5759:
5731:
5729:
5728:
5723:
5712:
5703:
5697:
5692:
5688:
5669:
5660:
5638:
5636:
5635:
5630:
5619:
5610:
5604:
5599:
5595:
5570:
5561:
5543:
5541:
5540:
5535:
5527:
5517:
5515:
5504:
5495:
5483:
5452:
5450:
5449:
5444:
5439:
5430:
5418:
5375:
5374:
5320:Clifford polygon
5308:regular pentagon
5274:
5273:
5272:
5268:
5267:
5263:
5262:
5258:
5257:
5253:
5252:
5248:
5247:
5243:
5242:
5238:
5237:
5233:
5232:
5214:
5213:
5212:
5208:
5207:
5203:
5202:
5198:
5197:
5193:
5192:
5188:
5187:
5183:
5182:
5178:
5177:
5173:
5172:
5154:
5153:
5152:
5148:
5147:
5143:
5142:
5138:
5137:
5133:
5132:
5128:
5127:
5123:
5122:
5118:
5117:
5113:
5112:
5094:
5093:
5092:
5088:
5087:
5083:
5082:
5078:
5077:
5073:
5072:
5068:
5067:
5063:
5062:
5058:
5057:
5053:
5052:
5034:
5033:
5032:
5028:
5027:
5023:
5022:
5018:
5017:
5013:
5012:
5008:
5007:
5003:
5002:
4998:
4997:
4993:
4992:
4965:
4958:
4951:
4944:
4937:
4896:
4895:
4890:
4889:
4888:
4884:
4883:
4879:
4878:
4874:
4873:
4869:
4868:
4864:
4863:
4859:
4858:
4854:
4853:
4849:
4848:
4830:
4829:
4828:
4824:
4823:
4819:
4818:
4814:
4813:
4809:
4808:
4804:
4803:
4799:
4798:
4794:
4793:
4789:
4788:
4770:
4769:
4768:
4764:
4763:
4759:
4758:
4754:
4753:
4749:
4748:
4744:
4743:
4739:
4738:
4734:
4733:
4729:
4728:
4710:
4709:
4708:
4704:
4703:
4699:
4698:
4694:
4693:
4689:
4688:
4684:
4683:
4679:
4678:
4674:
4673:
4669:
4668:
4650:
4649:
4648:
4644:
4643:
4639:
4638:
4634:
4633:
4629:
4628:
4624:
4623:
4619:
4618:
4614:
4613:
4609:
4608:
4590:
4589:
4588:
4584:
4583:
4579:
4578:
4574:
4573:
4569:
4568:
4564:
4563:
4559:
4558:
4554:
4553:
4549:
4548:
4521:
4514:
4507:
4500:
4493:
4486:
4439:
4438:
4434:uniform polytope
4426:
4425:
4424:
4420:
4419:
4415:
4414:
4410:
4409:
4405:
4404:
4400:
4399:
4395:
4394:
4390:
4389:
4385:
4384:
4373:
4372:
4371:
4367:
4366:
4362:
4361:
4357:
4356:
4352:
4351:
4347:
4346:
4342:
4341:
4337:
4336:
4332:
4331:
4320:
4319:
4318:
4314:
4313:
4309:
4308:
4304:
4303:
4299:
4298:
4294:
4293:
4289:
4288:
4284:
4283:
4279:
4278:
4267:
4266:
4265:
4261:
4260:
4256:
4255:
4251:
4250:
4246:
4245:
4241:
4240:
4236:
4235:
4231:
4230:
4226:
4225:
4214:
4213:
4212:
4208:
4207:
4203:
4202:
4198:
4197:
4193:
4192:
4188:
4187:
4183:
4182:
4178:
4177:
4173:
4172:
4161:
4160:
4159:
4155:
4154:
4150:
4149:
4145:
4144:
4140:
4139:
4135:
4134:
4130:
4129:
4125:
4124:
4120:
4119:
4099:
4092:
4085:
4078:
4071:
4064:
4050:
4046:
4042:Schläfli symbols
3980:
3969:
3958:
3947:
3936:
3854:
3837:
3835:
3834:
3829:
3827:
3817:
3815:
3807:
3804:
3802:
3801:
3796:
3794:
3784:
3782:
3774:
3771:
3769:
3768:
3763:
3761:
3751:
3749:
3741:
3738:
3736:
3735:
3730:
3728:
3718:
3716:
3708:
3705:
3703:
3702:
3697:
3695:
3685:
3683:
3675:
3672:
3670:
3669:
3664:
3662:
3652:
3650:
3642:
3639:
3637:
3636:
3631:
3629:
3619:
3617:
3609:
3606:
3604:
3603:
3598:
3596:
3591:
3583:
3576:
3574:
3573:
3568:
3566:
3556:
3554:
3546:
3543:
3541:
3540:
3535:
3533:
3523:
3521:
3513:
3510:
3508:
3507:
3502:
3500:
3490:
3488:
3480:
3473:
3471:
3470:
3465:
3463:
3453:
3451:
3443:
3440:
3438:
3437:
3432:
3430:
3420:
3418:
3410:
3407:
3405:
3404:
3399:
3397:
3387:
3385:
3377:
3374:
3372:
3371:
3366:
3364:
3354:
3352:
3344:
3331:
3329:
3328:
3323:
3321:
3315:
3311:
3308:
3305:
3296:
3291:
3285:
3283:
3282:
3277:
3275:
3272:
3264:
3238:
3236:
3235:
3230:
3222:
3212:
3210:
3202:
3198:
3196:
3195:
3190:
3185:
3180:
3179:
3170:
3169:
3155:
3141:
3139:
3138:
3133:
3125:
3115:
3113:
3105:
3101:
3099:
3098:
3093:
3088:
3083:
3082:
3073:
3072:
3058:
3044:
3042:
3041:
3036:
3028:
3018:
3016:
3008:
3004:
3002:
3001:
2996:
2991:
2986:
2985:
2976:
2975:
2961:
2947:
2945:
2944:
2939:
2931:
2926:
2918:
2914:
2912:
2911:
2906:
2901:
2896:
2895:
2886:
2885:
2871:
2853:
2851:
2850:
2845:
2843:
2834:
2825:
2820:
2816:
2814:
2813:
2808:
2806:
2803:
2798:
2789:
2780:
2775:
2771:
2769:
2768:
2763:
2755:
2745:
2743:
2735:
2731:
2729:
2728:
2723:
2718:
2713:
2712:
2703:
2702:
2688:
2682:
2680:
2679:
2674:
2672:
2663:
2654:
2649:
2645:
2643:
2642:
2637:
2635:
2632:
2627:
2618:
2609:
2604:
2600:
2598:
2597:
2592:
2584:
2574:
2572:
2564:
2560:
2558:
2557:
2552:
2547:
2542:
2541:
2532:
2531:
2517:
2511:
2509:
2508:
2503:
2501:
2492:
2483:
2478:
2474:
2472:
2471:
2466:
2464:
2461:
2450:
2445:
2441:
2439:
2438:
2433:
2425:
2415:
2413:
2405:
2401:
2399:
2398:
2393:
2388:
2383:
2382:
2373:
2372:
2358:
2340:
2338:
2337:
2332:
2330:
2321:
2312:
2307:
2303:
2301:
2300:
2295:
2293:
2290:
2285:
2276:
2267:
2262:
2258:
2256:
2255:
2250:
2242:
2232:
2230:
2222:
2213:
2211:
2210:
2205:
2203:
2194:
2185:
2180:
2176:
2174:
2173:
2168:
2166:
2163:
2158:
2149:
2140:
2135:
2131:
2129:
2128:
2123:
2115:
2105:
2103:
2095:
2086:
2084:
2083:
2078:
2076:
2067:
2058:
2053:
2049:
2047:
2046:
2041:
2039:
2036:
2025:
2020:
2016:
2014:
2013:
2008:
2000:
1990:
1988:
1980:
1959:
1957:
1956:
1951:
1949:
1946:
1929:
1924:
1920:
1918:
1917:
1912:
1910:
1907:
1890:
1885:
1881:
1879:
1878:
1873:
1865:
1855:
1853:
1845:
1821:
1820:
1817:
1816:
1815:
1811:
1810:
1806:
1805:
1801:
1800:
1796:
1795:
1791:
1790:
1786:
1785:
1779:
1778:
1777:
1773:
1772:
1768:
1767:
1763:
1762:
1758:
1757:
1753:
1752:
1748:
1747:
1725:Swiss Army knife
1694:
1693:
1687:
1686:
1685:
1681:
1680:
1676:
1675:
1671:
1670:
1666:
1665:
1661:
1660:
1656:
1655:
1646:
1645:
1644:
1640:
1639:
1635:
1634:
1630:
1629:
1625:
1624:
1620:
1619:
1615:
1614:
1605:
1604:
1598:
1597:
1591:
1590:
1584:
1583:
1569:
1568:
1558:
1557:
1551:
1550:
1544:
1543:
1510:, precisely the
1509:
1508:
1502:
1501:
1495:
1494:
1488:
1487:
1442:is an irregular
1387:
1377:
1361:
1347:
1335:
1316:
1309:
1302:
1262:
1256:
1198:
1197:
1178:
1177:
1158:
1157:
1138:
1137:
1118:
1117:
995:
994:
993:
989:
988:
984:
983:
979:
978:
974:
973:
969:
968:
964:
963:
928:
927:
926:
922:
921:
917:
916:
912:
911:
907:
906:
902:
901:
897:
896:
861:
860:
859:
855:
854:
850:
849:
845:
844:
840:
839:
835:
834:
830:
829:
794:
793:
792:
788:
787:
783:
782:
778:
777:
773:
772:
768:
767:
763:
762:
706:
705:
659:
657:
656:
651:
649:
648:
630:
602:triangular prism
513:
511:
510:
505:
503:
502:
462:hypertetrahedron
451:four-dimensional
351:
317:
302:
284:
283:
282:
278:
277:
273:
272:
268:
267:
263:
262:
258:
257:
253:
252:
212:
198:
197:
191:
184:
180:
177:
171:
147:
146:
139:
132:
125:
121:
118:
112:
110:
69:
45:
37:
32:Pentatope number
21:
13585:
13584:
13580:
13579:
13578:
13576:
13575:
13574:
13555:
13554:
13523:
13516:
13509:
13392:
13385:
13378:
13342:
13335:
13328:
13292:
13285:
13119:Regular polygon
13112:
13103:
13096:
13092:
13085:
13081:
13072:
13063:
13056:
13052:
13040:
13034:
13030:
13018:
13000:
12989:
12960:
12947:
12942:
12930:
12927:
12924:
12923:
12921:
12914:
12911:grand tetraplex
12902:
12899:
12896:
12895:
12893:
12886:
12883:great icosaplex
12874:
12871:
12868:
12867:
12865:
12858:
12846:
12843:
12840:
12839:
12837:
12830:
12818:
12815:
12812:
12811:
12809:
12802:
12799:
12796:
12795:
12793:
12786:
12774:
12771:
12768:
12767:
12765:
12758:
12746:
12743:
12740:
12739:
12737:
12730:
12718:
12715:
12712:
12711:
12709:
12702:
12690:
12687:
12684:
12683:
12681:
12674:
12662:
12659:
12656:
12655:
12653:
12642:
12635:
12628:
12626:
12619:
12612:
12610:
12603:
12601:
12594:
12587:
12580:
12578:
12571:
12555:
12536:
12531:
12516:
12501:
12486:
12471:
12456:
12398:
12387:
12382:
12301:
12291:
12240:
12160:Coxeter, H.S.M.
12135:Coxeter, H.S.M.
12111:
12082:
12060:
12035:
11984:
11979:
11978:
11970:
11966:
11958:
11954:
11946:
11942:
11934:
11930:
11925:
11918:
11914:
11906:
11902:
11894:
11890:
11881:
11880:
11876:
11868:
11864:
11852:
11848:
11840:
11833:
11821:
11817:
11805:
11801:
11789:
11785:
11773:
11769:
11761:
11757:
11752:
11747:
11730:
11723:
11719:
11707:
11696:
11685:
11674:
11663:
11652:
11641:
11632:
11623:
11612:
11601:
11590:
11579:
11568:
11557:
11399:
11388:
11377:
11366:
11355:
11344:
11333:
11324:
11315:
11304:
11293:
11282:
11271:
11260:
11249:
11093:Schläfli symbol
11073:
11068:
11063:
11058:
11056:
11055:
11050:
11045:
11040:
11035:
11030:
11028:
11027:
11023:
11011:
11006:
11001:
10996:
10994:
10993:
10988:
10983:
10978:
10973:
10968:
10966:
10965:
10961:
10949:
10944:
10939:
10934:
10929:
10927:
10926:
10922:
10910:
10905:
10900:
10898:
10897:
10892:
10887:
10882:
10877:
10872:
10870:
10869:
10865:
10853:
10848:
10843:
10838:
10833:
10831:
10830:
10826:
10814:
10809:
10804:
10802:
10801:
10796:
10791:
10786:
10781:
10776:
10774:
10773:
10769:
10757:
10752:
10747:
10742:
10737:
10735:
10734:
10730:
10721:
10657:
10652:
10647:
10642:
10637:
10632:
10630:
10629:
10624:
10619:
10614:
10609:
10604:
10599:
10594:
10592:
10591:
10580:
10575:
10570:
10565:
10560:
10555:
10553:
10552:
10547:
10542:
10537:
10532:
10527:
10522:
10517:
10515:
10514:
10504:
10499:
10494:
10489:
10484:
10479:
10474:
10472:
10471:
10461:
10456:
10451:
10446:
10441:
10439:
10438:
10433:
10428:
10423:
10418:
10413:
10408:
10403:
10401:
10400:
10390:
10385:
10380:
10375:
10370:
10365:
10360:
10358:
10357:
10347:
10342:
10337:
10332:
10327:
10325:
10324:
10319:
10314:
10309:
10304:
10299:
10294:
10289:
10287:
10286:
10276:
10271:
10266:
10261:
10256:
10251:
10246:
10244:
10243:
10172:
10161:
10150:
10139:
10128:
10117:
10106:
10097:
9931:
9923:
9915:
9907:
9899:
9891:
9883:
9875:
9754:
9749:
9744:
9739:
9734:
9729:
9724:
9719:
9714:
9709:
9704:
9699:
9694:
9689:
9684:
9679:
9674:
9672:
9665:
9660:
9655:
9650:
9645:
9640:
9635:
9630:
9625:
9620:
9615:
9610:
9605:
9600:
9595:
9593:
9586:
9581:
9576:
9571:
9566:
9561:
9556:
9551:
9546:
9541:
9536:
9531:
9526:
9524:
9517:
9512:
9507:
9502:
9497:
9492:
9487:
9482:
9477:
9472:
9467:
9465:
9458:
9453:
9448:
9443:
9438:
9433:
9428:
9423:
9418:
9416:
9409:
9404:
9399:
9394:
9389:
9384:
9379:
9377:
9370:
9365:
9360:
9355:
9350:
9348:
9341:
9336:
9331:
9326:
9321:
9319:
9314:
9307:
9288:
9277:
9276:
9275:
9273:
9270:
9269:
9267:
9261:
9242:
9231:
9230:
9229:
9227:
9224:
9223:
9221:
9214:
9206:
9198:
9191:
9187:
9181:
9177:
9171:
9167:
9163:
9155:
9080:
9063:
9055:
9047:
9039:
9031:
9023:
9015:
9007:
8896:
8884:
8879:
8874:
8869:
8864:
8859:
8854:
8849:
8844:
8839:
8834:
8829:
8824:
8819:
8814:
8809:
8804:
8802:
8795:
8790:
8785:
8780:
8775:
8770:
8765:
8760:
8755:
8750:
8745:
8740:
8735:
8730:
8725:
8723:
8716:
8711:
8706:
8701:
8696:
8691:
8686:
8681:
8676:
8671:
8666:
8661:
8656:
8654:
8647:
8642:
8637:
8632:
8627:
8622:
8617:
8612:
8607:
8602:
8597:
8595:
8588:
8583:
8578:
8573:
8568:
8563:
8558:
8553:
8548:
8546:
8539:
8534:
8529:
8524:
8519:
8514:
8509:
8507:
8500:
8495:
8490:
8485:
8480:
8478:
8471:
8466:
8461:
8456:
8451:
8449:
8444:
8437:
8418:
8407:
8406:
8405:
8403:
8400:
8399:
8397:
8391:
8372:
8361:
8360:
8359:
8357:
8354:
8353:
8351:
8344:
8336:
8328:
8321:
8317:
8311:
8307:
8301:
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8285:
8210:
8123:
8118:
8113:
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8103:
8098:
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8015:
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8006:
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7869:
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6279:
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4105:
4055:
3981:
3970:
3959:
3948:
3937:
3928:
3926:
3912:{3,3}∨( )
3886:Regular 5-cell
3877:
3872:
3868:
3864:
3860:
3845:
3814:
3812:
3809:
3808:
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3487:
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3417:
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3412:
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3384:
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3345:
3307:
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3301:
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3209:
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3112:
3110:
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3084:
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3065:
3063:
3060:
3059:
3015:
3013:
3010:
3009:
2987:
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2919:
2897:
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2873:
2872:
2832:
2830:
2827:
2826:
2802:
2787:
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2742:
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2736:
2714:
2708:
2704:
2698:
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2689:
2661:
2659:
2656:
2655:
2631:
2616:
2614:
2611:
2610:
2571:
2569:
2566:
2565:
2543:
2537:
2533:
2527:
2524:
2522:
2519:
2518:
2490:
2488:
2485:
2484:
2460:
2455:
2452:
2451:
2412:
2410:
2407:
2406:
2384:
2378:
2374:
2368:
2365:
2363:
2360:
2359:
2319:
2317:
2314:
2313:
2289:
2274:
2272:
2269:
2268:
2229:
2227:
2224:
2223:
2192:
2190:
2187:
2186:
2162:
2147:
2145:
2142:
2141:
2102:
2100:
2097:
2096:
2065:
2063:
2060:
2059:
2035:
2030:
2027:
2026:
1987:
1985:
1982:
1981:
1945:
1934:
1931:
1930:
1906:
1895:
1892:
1891:
1852:
1850:
1847:
1846:
1813:
1808:
1803:
1798:
1793:
1788:
1783:
1781:
1775:
1770:
1765:
1760:
1755:
1750:
1745:
1743:
1691:
1689:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1651:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1610:
1602:
1600:
1599:edges, and one
1595:
1593:
1588:
1586:
1581:
1579:
1566:
1564:
1555:
1553:
1548:
1546:
1541:
1539:
1506:
1504:
1499:
1497:
1492:
1490:
1485:
1483:
1432:right triangles
1424:
1416:symmetry groups
1401:In the case of
1399:
1388:
1378:
1362:
1348:
1290:
1284:
1278:
1269:
1268:
1249:
1241:
1229:
1209:
1202:
1199:
1187:
1182:
1179:
1167:
1162:
1159:
1147:
1142:
1139:
1127:
1122:
1119:
1107:
1080:Hopf fibrations
1056:double rotation
1048:
1042:
1029:Branko Grünbaum
1006:
991:
986:
981:
976:
971:
966:
961:
959:
939:
924:
919:
914:
909:
904:
899:
894:
892:
870:
857:
852:
847:
842:
837:
832:
827:
825:
803:
790:
785:
780:
775:
770:
765:
760:
758:
747:
741:
735:
729:
723:
680:
644:
640:
626:
609:
606:
605:
569:analogous to a
555:
543:Platonic solids
498:
494:
492:
489:
488:
458:
447:Schläfli symbol
382:
352:
280:
275:
270:
265:
260:
255:
250:
248:
244:Coxeter diagram
234:Schläfli symbol
218:
216:simple rotation
202:
192:
181:
175:
172:
158:Please help by
157:
148:
144:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
13583:
13573:
13572:
13567:
13551:
13550:
13535:
13534:
13525:
13521:
13514:
13507:
13503:
13494:
13477:
13468:
13457:
13456:
13454:
13452:
13447:
13438:
13433:
13427:
13426:
13424:
13422:
13417:
13408:
13403:
13397:
13396:
13394:
13390:
13383:
13376:
13372:
13367:
13358:
13353:
13347:
13346:
13344:
13340:
13333:
13326:
13322:
13317:
13308:
13303:
13297:
13296:
13294:
13290:
13283:
13279:
13274:
13265:
13260:
13254:
13253:
13251:
13249:
13244:
13235:
13230:
13224:
13223:
13214:
13209:
13204:
13195:
13190:
13184:
13183:
13174:
13172:
13167:
13158:
13153:
13147:
13146:
13141:
13136:
13131:
13126:
13121:
13115:
13114:
13110:
13106:
13101:
13090:
13079:
13070:
13061:
13054:
13048:
13038:
13032:
13026:
13020:
13014:
13008:
13002:
13001:
12990:
12988:
12987:
12980:
12973:
12965:
12957:
12956:
12953:
12952:
12949:
12948:
12944:
12943:
12941:
12940:
12937:
12917:
12915:
12913:
12912:
12909:
12889:
12887:
12885:
12884:
12881:
12861:
12859:
12857:
12856:
12853:
12833:
12831:
12829:
12828:
12825:
12789:
12787:
12785:
12784:
12781:
12761:
12759:
12757:
12756:
12753:
12733:
12731:
12729:
12728:
12725:
12705:
12703:
12701:
12700:
12697:
12677:
12675:
12673:
12672:
12669:
12649:
12646:
12645:
12638:
12631:
12622:
12615:
12606:
12597:
12590:
12583:
12574:
12564:
12557:
12556:
12546:
12545:
12542:
12541:
12538:
12537:
12533:
12532:
12530:
12529:
12526:
12525:hexacosichoron
12523:
12519:
12517:
12515:
12514:
12511:
12508:
12504:
12502:
12500:
12499:
12496:
12493:
12489:
12487:
12485:
12484:
12481:
12480:hexadecachoron
12478:
12474:
12472:
12470:
12469:
12466:
12463:
12459:
12457:
12455:
12454:
12451:
12448:
12444:
12441:
12440:
12435:
12430:
12425:
12420:
12415:
12407:
12400:
12399:
12389:
12388:
12381:
12380:
12373:
12366:
12358:
12352:
12351:
12346:
12344:Java3D Applets
12341:
12336:
12330:
12321:
12300:
12299:External links
12297:
12296:
12295:
12289:
12260:
12259:
12258:
12257:, Ph.D. (1966)
12253:N.W. Johnson:
12245:Norman Johnson
12242:
12238:
12223:John H. Conway
12220:
12205:
12204:
12203:
12202:
12201:
12194:
12187:
12168:
12156:
12155:
12154:
12151:
12128:H.S.M. Coxeter
12125:
12115:
12109:
12086:
12080:
12068:Johnson, N. W.
12064:
12058:
12039:
12033:
12006:
11983:
11980:
11977:
11976:
11964:
11952:
11940:
11928:
11923:
11912:
11900:
11888:
11874:
11862:
11846:
11831:
11815:
11799:
11783:
11767:
11765:, p. 249.
11754:
11753:
11751:
11748:
11746:
11745:
11720:
11718:
11715:
11712:
11711:
11700:
11689:
11678:
11667:
11656:
11645:
11634:
11628:
11627:
11616:
11605:
11594:
11583:
11572:
11561:
11550:
11546:
11545:
11538:
11531:
11524:
11517:
11510:
11503:
11496:
11492:
11491:
11485:
11480:
11475:
11470:
11465:
11460:
11455:
11451:
11450:
11447:
11444:
11441:
11438:
11435:
11431:
11430:
11425:
11422:
11417:
11413:
11412:
11404:
11403:
11392:
11381:
11370:
11359:
11348:
11337:
11326:
11320:
11319:
11308:
11297:
11286:
11275:
11264:
11253:
11242:
11238:
11237:
11230:
11223:
11216:
11209:
11202:
11195:
11188:
11184:
11183:
11177:
11172:
11167:
11162:
11157:
11152:
11147:
11139:
11138:
11135:
11132:
11129:
11126:
11122:
11121:
11116:
11111:
11107:
11106:
11105:,3} polytopes
11079:
11078:
11016:
10954:
10915:
10858:
10819:
10762:
10723:
10717:
10716:
10709:
10702:
10695:
10688:
10681:
10674:
10667:
10663:
10662:
10585:
10509:
10466:
10395:
10352:
10281:
10238:
10234:
10233:
10230:
10227:
10224:
10220:
10219:
10214:
10209:
10205:
10204:
10177:
10176:
10165:
10154:
10143:
10132:
10121:
10110:
10099:
10093:
10092:
10085:
10078:
10071:
10064:
10057:
10050:
10043:
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10027:
10022:
10017:
10012:
10007:
10002:
9998:
9997:
9994:
9991:
9988:
9984:
9983:
9978:
9973:
9969:
9968:
9934:
9933:
9929:
9925:
9921:
9917:
9913:
9909:
9905:
9901:
9897:
9893:
9889:
9885:
9881:
9877:
9873:
9869:
9865:
9864:
9861:
9858:
9851:
9844:
9837:
9830:
9823:
9816:
9812:
9811:
9808:
9805:
9802:
9799:
9796:
9793:
9790:
9784:
9783:
9781:
9779:
9777:
9775:
9773:
9770:
9768:
9766:
9760:
9759:
9670:
9591:
9522:
9463:
9414:
9375:
9346:
9317:
9309:
9308:
9305:
9291:
9284:
9281:
9265:
9262:
9259:
9245:
9238:
9235:
9219:
9216:
9212:
9208:
9204:
9200:
9196:
9192:
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9175:
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9169:
9165:
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9158:
9150:
9149:
9144:
9139:
9134:
9129:
9124:
9119:
9114:
9109:
9103:
9102:
9099:
9096:
9093:
9089:
9088:
9075:
9066:
9065:
9061:
9057:
9053:
9049:
9045:
9041:
9037:
9033:
9029:
9025:
9021:
9017:
9013:
9009:
9005:
9001:
8997:
8996:
8993:
8990:
8983:
8976:
8969:
8962:
8955:
8948:
8944:
8943:
8940:
8937:
8934:
8931:
8928:
8925:
8922:
8916:
8915:
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1527:3-orthoschemes
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1244:square pyramid
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1094:Clifford torus
1047:
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1039:to the plane.
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501:
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363:Petrie polygon
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13207:Demitesseract
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12269:
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12267:Shaping Space
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11960:Banchoff 2013
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11181:{3,∞,3}
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10589:{3,3,∞}
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10193:{3,3,5}. The
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10036:{∞,3,3}
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9947:vertex figure
9945:
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7708:
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7699:
7697:
7696:Coxeter group
7693:
7683:
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7661:
7653:
7649:
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7591:
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7586:Coxeter plane
7577:
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5329:
5325:
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5318:which is the
5317:
5313:
5310:which is the
5309:
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4464:{3}∨{ }
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3962:
3957:
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3940:
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3931:
3920:
3917:
3915:{3}∨{ }
3914:
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3491:
3478:
3457:
3454:
3424:
3421:
3391:
3388:
3358:
3355:
3316:
3312:
3309:arc sec
3262:
3259:
3257:
3255:
3253:
3251:
3250:
3226:
3223:
3216:
3213:
3186:
3182:
3176:
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3129:
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3119:
3116:
3089:
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3019:
2992:
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2898:
2892:
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2838:
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2799:
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2759:
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2709:
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2578:
2575:
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2538:
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2528:
2525:
2515:
2496:
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2416:
2389:
2385:
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2344:
2325:
2322:
2286:
2280:
2277:
2246:
2243:
2236:
2233:
2217:
2198:
2195:
2159:
2153:
2150:
2119:
2116:
2109:
2106:
2090:
2071:
2068:
2032:
2004:
2001:
1994:
1991:
1975:
1972:
1970:
1968:
1966:
1964:
1963:
1942:
1939:
1936:
1903:
1900:
1897:
1869:
1866:
1859:
1856:
1840:
1827:
1822:
1819:
1741:
1738:based on the
1737:
1733:
1728:
1726:
1722:
1721:genetic codes
1717:
1713:
1709:
1705:
1700:
1698:
1650:
1607:
1585:edges, three
1577:
1573:
1562:
1536:
1528:
1523:
1519:
1517:
1513:
1481:
1477:
1473:
1469:
1465:
1460:
1458:
1457:3-orthoscheme
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1428:4-orthoscheme
1419:
1417:
1413:
1409:
1404:
1386:
1382:
1376:
1372:
1366:
1360:
1356:
1352:
1346:
1342:
1336:
1329:
1327:
1325:
1323:
1320:
1319:
1315:
1311:
1308:
1304:
1301:
1297:
1294:
1293:
1286:
1280:
1274:
1272:
1271:Coxeter plane
1264:
1263:
1260:
1255:
1253:
1245:
1237:
1233:
1221:
1217:
1213:
1184:
1164:
1144:
1124:
1104:
1102:
1099:
1098:
1097:
1095:
1090:
1088:
1087:
1081:
1077:
1072:
1070:
1065:
1057:
1052:
1043:
1040:
1038:
1034:
1030:
1020:
1000:
997:
953:
933:
930:
887:
884:
863:
820:
817:
796:
752:
749:
743:
737:
731:
725:
722:
717:
714:
711:
708:
707:
700:
696:
694:
690:
685:
675:
673:
669:
664:
661:
645:
641:
637:
631:
627:
623:
620:
614:
611:
603:
599:
594:
592:
588:
584:
580:
576:
572:
568:
564:
560:
550:
548:
544:
540:
536:
532:
527:
525:
521:
517:
499:
495:
486:
485:
479:
475:
471:
467:
463:
459:
452:
448:
444:
440:
436:
426:
424:
423:Uniform index
420:
417:
413:
409:
405:
402:
398:
395:
392:
390:
386:
378:
376:
375:Coxeter group
372:
369:
366:
364:
360:
356:
350:
346:
344:
343:Vertex figure
340:
336:
334:
330:
326:
324:
320:
316:
311:
309:
305:
301:
297:
293:
291:
287:
247:
245:
241:
237:
235:
231:
228:
225:
221:
217:
211:
206:
199:
190:
187:
179:
169:
165:
161:
155:
152:This article
150:
141:
140:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
13527:
13496:
13487:
13479:
13470:
13461:
13441:10-orthoplex
13192:
13177:Dodecahedron
13098:
13087:
13076:
13067:
13058:
13049:
13045:
13035:
13027:
13023:
13015:
13011:
12412:
12313:
12266:
12254:
12247:
12226:
12197:
12190:
12183:
12163:
12139:
12121:
12091:
12071:
12044:
12011:
11993:
11989:
11972:Coxeter 1991
11967:
11955:
11948:Coxeter 1973
11943:
11936:Coxeter 1973
11931:
11920:Coxeter 1973
11915:
11903:
11896:Coxeter 1973
11891:
11877:
11870:Coxeter 1973
11865:
11857:
11854:Coxeter 1973
11849:
11818:
11802:
11791:Coxeter 1973
11786:
11770:
11763:Johnson 2018
11758:
11741:golden ratio
11736:
11733:Coxeter 1973
11457:
11446:Paracompact
11424:Euclidean E
11149:
11143:
11134:Paracompact
11102:
11082:
10240:
10229:Paracompact
10189:{3,3,4} and
10180:
9993:Paracompact
9953:{4,3,3} and
9937:
9807:696,729,600
9106:
9084:
9076:
9011:
8939:696,729,600
8236:
8214:
7709:
7689:
7657:
7583:
7575:
7558:
7545:
7171:
6799:
6405:
6404:
6371:
6039:in 3-space:
6034:
5455:circumcenter
5378:golden ratio
5364:
5351:
5323:
5319:
5297:
5281:Construction
4431:
4034:
4028:
4019:
4015:
3995:
3991:
3989:
3889:Tetrahedral
3846:
3579:
3476:
3341:
1731:
1729:
1715:
1711:
1707:
1703:
1701:
1696:
1608:
1575:
1532:
1479:
1475:
1461:
1446:that is the
1427:
1425:
1422:Orthoschemes
1407:
1400:
1225:
1091:
1084:
1073:
1061:
1041:
1033:Venn diagram
1026:
750:
720:
712:
692:
688:
681:
665:
662:
595:
582:
556:
546:
530:
528:
481:
477:
474:pentahedroid
473:
469:
465:
461:
454:
438:
432:
182:
173:
160:spinning off
153:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
13450:10-demicube
13411:9-orthoplex
13361:8-orthoplex
13311:7-orthoplex
13268:6-orthoplex
13238:5-orthoplex
13193:Pentachoron
13181:Icosahedron
13156:Tetrahedron
12641:great grand
12627:icosahedral
12618:great grand
12570:icosahedral
12483:4-orthoplex
12450:pentachoron
12310:"Pentatope"
11823:Diudea 2018
11709:{3,∞}
11625:{∞,3}
11449:Noncompact
11401:{∞,3}
11317:{3,∞}
11137:Noncompact
11089:palindromic
11025:{3,∞}
10232:Noncompact
10174:{∞,3}
9996:Noncompact
9944:tetrahedral
9101:Hyperbolic
9087:dimensions
8231:Hyperbolic
8217:dimensions
7567:5-orthoplex
6406:unit-radius
5361:Coordinates
4445:, order 12
4218:{ }×{5,3,3}
4165:{ }×{4,3,3}
4112:{ }×{3,3,3}
4044:( )∨{3,3}.
4004:tetrahedron
1609:The 4-cube
1592:edges, two
1448:convex hull
1440:orthoscheme
1351:tetrahedral
1207:Projections
571:tetrahedron
516:tetrahedron
487:(Coxeter's
466:pentachoron
355:tetrahedron
203:(4-simplex)
176:August 2024
117:August 2024
18:Pentachoron
13559:Categories
13436:10-simplex
13420:9-demicube
13370:8-demicube
13320:7-demicube
13277:6-demicube
13247:5-demicube
13161:Octahedron
12513:dodecaplex
12349:pyrochoron
12215:1603.07269
11982:References
11825:, p.
11809:, p.
11777:, p.
11775:Ghyka 1977
9804:2,903,040
9098:Euclidean
8936:2,903,040
8228:Euclidean
7555:hyperplane
4908:, order 1
4905:, order 2
4902:, order 2
4454:, order 4
4451:, order 8
4448:, order 6
4377:t{3,4,3,3}
4324:t{4,3,3,3}
4271:t{3,3,3,3}
4008:hyperplane
3861:Order 120
3843:Isometries
1886:104°30′40″
1734:. It is a
1037:projection
575:hyperplane
567:polychoron
563:4-polytope
553:Properties
443:4-polytope
400:Properties
164:relocating
87:newspapers
13484:orthoplex
13406:9-simplex
13356:8-simplex
13306:7-simplex
13263:6-simplex
13233:5-simplex
13202:Tesseract
12671:icosaplex
12611:stellated
12602:stellated
12579:stellated
12528:tetraplex
12465:tesseract
12453:4-simplex
12315:MathWorld
12118:T. Gosset
11996:: 89–97.
11750:Citations
9951:tesseract
9283:¯
9237:~
8413:¯
8367:~
7735:tr{3,3,3}
7730:2t{3,3,3}
7725:rr{3,3,3}
7505:−
7434:−
7359:−
7280:±
7110:−
7049:−
6980:−
6907:±
6822:≈
6720:−
6697:−
6684:−
6642:−
6629:−
6606:−
6564:−
6551:−
6538:−
6486:−
6283:−
6265:−
6256:−
6214:−
6202:−
6187:−
6145:−
6133:−
6124:−
6076:−
6014:−
6008:ϕ
5983:ϕ
5968:ϕ
5953:ϕ
5938:ϕ
5900:−
5894:ϕ
5869:−
5863:ϕ
5807:−
5801:ϕ
5770:−
5764:ϕ
5714:−
5708:ϕ
5671:−
5665:ϕ
5621:−
5615:ϕ
5572:−
5566:ϕ
5529:≈
5500:ϕ
5491:−
5474:−
5471:ϕ
5435:ϕ
5426:−
5316:pentagram
4899:Symmetry
4442:Symmetry
4010:, and an
3939:5-simplex
3878:Order 10
3869:Order 12
3865:Order 24
3857:Symmetry
3292:37°44′40″
3273:𝜼
3127:≈
3030:≈
2836:π
2804:𝜂
2800:−
2791:π
2776:52°15′20″
2757:≈
2665:π
2633:𝜂
2629:−
2620:π
2605:52°15′20″
2586:≈
2494:π
2462:𝜂
2446:75°29′20″
2427:≈
2323:π
2291:𝜂
2287:−
2278:π
2263:52°15′20″
2244:≈
2196:π
2164:𝜂
2160:−
2151:π
2136:52°15′20″
2117:≈
2069:π
2037:𝜂
2021:75°29′20″
2002:≈
1947:𝟁
1940:−
1937:π
1925:75°29′20″
1908:𝜂
1901:−
1898:π
1867:≈
1837:dihedral
1578:has four
1572:antipodal
1476:tesseract
1403:simplexes
1236:pentagram
1086:isoclines
646:∘
638:≈
621:−
615:
598:self-dual
496:α
470:pentatope
416:isohedral
394:Self-dual
13538:Topics:
13501:demicube
13466:polytope
13460:Uniform
13221:600-cell
13217:120-cell
13170:Demicube
13144:Pentagon
13124:Triangle
12636:600-cell
12629:120-cell
12620:120-cell
12613:120-cell
12604:120-cell
12595:120-cell
12588:120-cell
12581:120-cell
12572:120-cell
12498:octaplex
12438:600-cell
12433:120-cell
12162:(1991),
12137:(1973).
12017:Springer
11443:Compact
11131:Compact
11091:{3,p,3}
10191:600-cell
9955:120-cell
9810:∞
9764:Symmetry
8942:∞
8933:103,680
8897:(order)
8894:Symmetry
8133:Schlegel
7720:r{3,3,3}
7715:t{3,3,3}
7705:Schläfli
7660:hexagram
7580:Compound
5324:isocline
4927:Schlegel
4914:Schläfli
4476:Schlegel
4460:Schläfli
4054:Schlegel
3909:{3,3,3}
3905:Schläfli
3873:Order 6
1410:are the
1232:pentagon
1220:3-sphere
591:compound
587:120-cell
579:triangle
520:triangle
435:geometry
412:isotoxal
408:isogonal
368:pentagon
333:Vertices
76:"5-cell"
13475:simplex
13445:10-cube
13212:24-cell
13198:16-cell
13139:Hexagon
12993:regular
12934:
12922:
12906:
12894:
12878:
12866:
12850:
12838:
12822:
12810:
12806:
12794:
12778:
12766:
12750:
12738:
12722:
12710:
12694:
12682:
12666:
12654:
12522:{3,3,5}
12507:{5,3,3}
12492:{3,4,3}
12477:{3,3,4}
12462:{4,3,3}
12447:{3,3,3}
12428:24-cell
12423:16-cell
11633:figure
11483:{8,3,8}
11478:{7,3,7}
11473:{6,3,6}
11468:{5,3,5}
11463:{4,3,4}
11458:{3,3,3}
11440:Affine
11437:Finite
11325:figure
11175:{3,8,3}
11170:{3,7,3}
11165:{3,6,3}
11160:{3,5,3}
11155:{3,4,3}
11150:{3,3,3}
11128:Finite
11085:24-cell
10722:figure
10512:{3,3,8}
10469:{3,3,7}
10398:{3,3,6}
10355:{3,3,5}
10284:{3,3,4}
10241:{3,3,3}
10226:Finite
10187:16-cell
10030:{8,3,3}
10025:{7,3,3}
10020:{6,3,3}
10015:{5,3,3}
10010:{4,3,3}
10005:{3,3,3}
9990:Finite
9942:with a
9801:51,840
9315:diagram
9313:Coxeter
9154:Coxeter
9095:Finite
9081:figures
8445:diagram
8443:Coxeter
8284:Coxeter
8225:Finite
8211:figures
7775:Coxeter
7768:{3,3,3}
7765:0,1,2,3
7757:{3,3,3}
7746:{3,3,3}
7710:{3,3,3}
7676:
7664:
7569:or the
7548:√
7234:gives:
5371:√
4973:Coxeter
4929:diagram
4529:Coxeter
4478:diagram
4107:Coxeter
4056:diagram
4026:cells.
3929:figure
3925:Example
3891:pyramid
1710:is the
1697:smaller
1690:√
1647:can be
1601:√
1594:√
1587:√
1580:√
1565:√
1554:√
1547:√
1540:√
1505:√
1498:√
1491:√
1484:√
1444:simplex
1246:. The A
1238:. The A
1076:circuit
709:Element
583:regular
559:simplex
484:simplex
312:10 {3}
238:{3,3,3}
101:scholar
13415:9-cube
13365:8-cube
13315:7-cube
13272:6-cube
13242:5-cube
13129:Square
13006:Family
12468:4-cube
12418:8-cell
12413:5-cell
12397:Convex
12287:
12275:–266.
12233:
12229:2008,
12174:
12107:
12078:
12056:
12031:
11631:Vertex
11549:Cells
11495:Image
11416:Space
11323:Vertex
11241:Cells
11187:Image
11110:Space
10720:Vertex
10666:Image
10208:Space
10098:{p,3}
10042:Image
9972:Space
9949:: the
9815:Graph
9092:Space
8947:Graph
8930:1,920
8222:Space
7662:{
7149:
7140:
7131:
7076:
7067:
7046:
6997:
6974:
6956:
6904:
6886:
6868:
5532:1.7888
4014:point
3996:5-cell
3927:Vertex
1716:single
1606:edge.
1480:8-cell
1472:4-cube
1295:Graph
753:-figs
612:arccos
439:5-cell
437:, the
404:convex
201:5-cell
103:
96:
89:
82:
74:
13134:p-gon
12936:,3,3}
12892:{3,3,
12780:,3,5}
12736:{5,3,
12696:,5,3}
12652:{3,5,
12634:grand
12625:great
12609:grand
12600:great
12593:grand
12586:great
12577:small
12210:arXiv
11883:"Pen"
11858:p,q,r
11717:Notes
11698:{3,8}
11687:{3,7}
11676:{3,6}
11665:{3,5}
11654:{3,4}
11643:{3,3}
11614:{8,3}
11603:{7,3}
11592:{6,3}
11581:{5,3}
11570:{4,3}
11559:{3,3}
11454:Name
11434:Form
11390:{8,3}
11379:{7,3}
11368:{6,3}
11357:{5,3}
11346:{4,3}
11335:{3,3}
11306:{3,8}
11295:{3,7}
11284:{3,6}
11273:{3,5}
11262:{3,4}
11251:{3,3}
11125:Form
10963:{3,8}
10924:{3,7}
10867:{3,6}
10828:{3,5}
10771:{3,4}
10732:{3,3}
10237:Name
10223:Form
10163:{8,3}
10152:{7,3}
10141:{6,3}
10130:{5,3}
10119:{4,3}
10108:{3,3}
10096:Cells
10001:Name
9987:Form
9868:Name
9788:Order
9156:group
9000:Name
8920:Order
8286:group
7754:0,1,3
7563:facet
6837:are:
6825:1.265
6403:give
5220:01234
5160:01234
4040:with
4033:have
4029:Many
4020:sides
4016:above
3883:Name
3130:0.408
3033:0.612
2760:0.129
2589:0.224
2430:0.387
2247:0.103
2120:0.183
2005:0.316
1870:1.581
1831:edge
1712:order
1503:, or
1450:of a
1436:cells
1226:The A
1064:digon
999:{3,3}
819:{3,3}
715:-face
682:This
642:75.52
589:is a
480:, or
445:with
323:Edges
308:Faces
296:{3,3}
290:Cells
108:JSTOR
94:books
13492:cube
13165:Cube
12995:and
12554:Star
12285:ISBN
12231:ISBN
12172:ISBN
12105:ISBN
12076:ISBN
12054:ISBN
12029:ISBN
11179:...
11146:,3}
10587:...
9874:−1,1
9798:384
9795:120
9006:−1,2
8927:120
5355:cell
5100:0123
5040:0123
4980:0123
4970:Name
4526:Name
4104:Name
4012:apex
3998:, a
3990:The
3227:0.25
1834:arc
1452:tree
1234:and
1021:( )
954:{ }
864:{ }
797:( )
529:The
389:Dual
223:Type
80:news
13041:(p)
12880:,5}
12864:{3,
12852:,3}
12836:{5,
12808:,5,
12724:,5}
12708:{5,
12277:doi
12273:257
12097:doi
12021:doi
11998:doi
11994:159
11487:...
11142:{3,
11101:{3,
10034:...
9792:12
9304:= E
9258:= E
9083:in
8924:12
8434:= E
8388:= E
8213:in
7743:0,3
5338:Net
4836:123
4776:123
4716:012
4656:012
2936:1.0
2821:90°
2650:90°
2479:90°
2308:60°
2219:𝟁
2181:60°
2092:𝝉
2054:60°
1977:𝟀
1842:𝒍
1478:or
1031:'s
948:10
932:{3}
886:{3}
876:10
464:, '
433:In
162:or
63:by
13561::
13546:•
13542:•
13522:21
13518:•
13515:k1
13511:•
13508:k2
13486:•
13443:•
13413:•
13391:21
13387:•
13384:41
13380:•
13377:42
13363:•
13341:21
13337:•
13334:31
13330:•
13327:32
13313:•
13291:21
13287:•
13284:22
13270:•
13240:•
13219:•
13200:•
13179:•
13163:•
13095:/
13084:/
13074:/
13065:/
13043:/
12312:.
12283:.
12239:n1
12200:,
12193:,
12186:,
12130::
12120::
12103:.
12095:.
12052:.
12048:.
12027:.
12019:.
11992:.
11926:".
11834:^
11827:41
11811:46
11779:68
11724:^
11095:.
9930:61
9922:51
9914:41
9906:31
9898:21
9890:11
9882:01
9863:-
9860:-
9772:]
9268:=
9266:10
9222:=
9188:=D
9178:=A
9164:=A
9147:10
9062:62
9054:52
9046:42
9038:32
9030:22
9022:12
9014:02
8995:-
8992:-
8906:]
8398:=
8396:10
8352:=
8318:=D
8308:=A
8294:=A
8277:10
8209:k2
7698:.
7682:.
7646:.
7628:∩
7610:=
7592:.
7573:.
7559:or
7439:45
7364:40
7285:30
7272:10
7037:10
6947:10
6859:10
5519:16
5357:.
5334:.
4596:12
4536:12
3851::
3822:16
3806:,
3789:60
3773:,
3756:15
3740:,
3723:30
3690:16
3674:,
3641:,
3608:,
3561:60
3545:,
3528:20
3512:,
3495:20
3458:15
3442:,
3425:30
3409:,
3392:10
3217:16
2750:60
2579:20
2420:20
2237:15
2110:30
1995:10
1518:.
1496:,
1489:,
1426:A
1018:5
1015:4
1012:6
1009:4
951:2
945:3
942:3
882:3
879:3
873:2
815:4
812:6
809:4
806:5
674:.
660:.
482:4-
476:,
472:,
468:,
460:,
414:,
410:,
406:,
383:,
327:10
294:5
13530:-
13528:n
13520:k
13513:2
13506:1
13499:-
13497:n
13490:-
13488:n
13482:-
13480:n
13473:-
13471:n
13464:-
13462:n
13389:4
13382:2
13375:1
13339:3
13332:2
13325:1
13289:2
13282:1
13111:n
13109:H
13102:2
13099:G
13091:4
13088:F
13080:8
13077:E
13071:7
13068:E
13062:6
13059:E
13050:n
13046:D
13039:2
13036:I
13028:n
13024:B
13016:n
13012:A
12984:e
12977:t
12970:v
12931:2
12928:/
12925:5
12920:{
12908:}
12903:2
12900:/
12897:5
12875:2
12872:/
12869:5
12847:2
12844:/
12841:5
12824:}
12819:2
12816:/
12813:5
12803:2
12800:/
12797:5
12792:{
12775:2
12772:/
12769:5
12764:{
12752:}
12747:2
12744:/
12741:5
12719:2
12716:/
12713:5
12691:2
12688:/
12685:5
12680:{
12668:}
12663:2
12660:/
12657:5
12377:e
12370:t
12363:v
12328:.
12318:.
12293:.
12279::
12241:)
12218:.
12212::
12149:A
12113:.
12099::
12084:.
12062:.
12037:.
12023::
12004:.
12000::
11962:.
11924:4
11885:.
11844:.
11829:.
11813:.
11797:.
11795:A
11781:.
11731:(
11428:H
11420:S
11144:p
11119:H
11114:S
11103:p
10217:H
10212:S
9981:H
9976:S
9928:2
9920:2
9912:2
9904:2
9896:2
9888:2
9880:2
9872:2
9306:8
9290:8
9280:T
9264:E
9260:8
9244:8
9234:E
9220:9
9218:E
9213:8
9211:E
9205:7
9203:E
9197:6
9195:E
9190:5
9186:5
9184:E
9180:4
9176:4
9174:E
9170:1
9168:A
9166:2
9162:3
9160:E
9142:9
9137:8
9132:7
9127:6
9122:5
9117:4
9112:3
9107:n
9085:n
9079:1
9077:k
9074:2
9060:1
9052:1
9044:1
9036:1
9028:1
9020:1
9012:1
9004:1
8436:8
8420:8
8410:T
8394:E
8390:8
8374:8
8364:E
8350:9
8348:E
8343:8
8341:E
8335:7
8333:E
8327:6
8325:E
8320:5
8316:5
8314:E
8310:4
8306:4
8304:E
8300:1
8298:A
8296:2
8292:3
8290:E
8272:9
8267:8
8262:7
8257:6
8252:5
8247:4
8242:3
8237:n
8215:n
8207:1
7762:t
7751:t
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