4748:
4773:
4782:
4802:
601:
1695:
6492:
6485:
1058:
6499:
5650:
5643:
2322:
5657:
6478:
6464:
5615:
6471:
5636:
5622:
379:
6457:
6450:
5629:
4920:
1300:
5608:
4827:
1942:
1889:
6443:
1247:
982:
5601:
2512:
2422:
1799:
1690:
548:
1811:
2434:
1157:
4704:
4697:
4655:
4648:
4811:
2231:
2224:
2182:
2175:
1644:
1637:
1595:
1588:
1193:
487:
863:
856:
814:
807:
1663:
6535:
6521:
2458:
2446:
1835:
1181:
4641:
268:
6570:
5728:
2168:
1581:
219:
170:
6563:
6556:
5721:
5714:
1169:
800:
70:
6549:
6542:
6528:
5707:
5700:
5693:
5686:
119:
21:
6514:
5679:
5672:
912:
899:
890:
475:
2254:
4 faces. The hexagonal prisms are connected to the truncated tetrahedra via their hexagonal faces, and to the rhombicuboctahedra via 3 of their square faces each, and to the cubes via the other 3 square faces. The truncated tetrahedra are joined to the rhombicuboctahedra via their triangular faces, and the hexagonal prisms via their hexagonal faces.
1823:
499:
971:
2253:
The small rhombicuboctahedral cells are joined via their 6 axial square faces to the cubical cells, and joined via their 12 non-axial square faces to the hexagonal prisms. The cubical cells are joined to the rhombicuboctahedra via 2 opposite faces, and joined to the hexagonal prisms via the remaining
931:
Eight of the cubical cells are connected to the other 24 cubical cells via all 6 square faces. The other 24 cubical cells are connected to the former 8 cells via only two opposite square faces; the remaining 4 faces are connected to the triangular prisms. The triangular prisms are connected to the
2720:
The truncated cuboctahedra cells are joined to the octagonal prisms via their octagonal faces, the truncated octahedra via their hexagonal faces, and the hexagonal prisms via their square faces. The octagonal prisms are joined to the hexagonal prisms and the truncated octahedra via their square
1995:
radially, and filling in the spaces between them with cubes. In the process, the octahedral cells expand into truncated tetrahedra (half of their triangular faces are expanded into hexagons by pulling apart the edges), and the triangular prisms expand into hexagonal prisms (each with its three
2297:
This layout of cells is similar to the layout of the faces of the great rhombicuboctahedron under the projection into 2-dimensional space. Hence, the runcitruncated 16-cell may be thought of as one of the 4-dimensional analogues of the great rhombicuboctahedron. The other analogue is the
2736:, dividing the full group order of a subgroup order by removing one mirror at a time. Edges exist at 4 symmetry positions. Squares exist at 3 positions, hexagons 2 positions, and octagons one. Finally the 4 types of cells exist centered on the 4 corners of the fundamental simplex.
4760:
are shown in blue, with 4 more truncated octahedra on the other side of these prisms also shown in yellow. Cells obscured from 4D viewpoint culled for clarity's sake. Some of the other hexagonal and octagonal prisms may be discerned from this view as well.
2608:
by radially displacing the truncated cuboctahedral cells so that octagonal prisms can be inserted between their octagonal faces. As a result, the triangular prisms expand into hexagonal prisms, and the truncated tetrahedra expand into truncated octahedra.
4564:
The remaining 6 truncated cuboctahedra project to the (non-regular) octagonal faces of the envelope. These are connected to the central truncated cuboctahedron via 6 octagonal prisms, which are the images of the octagonal prism cells, a pair to each
4583:, which is analogous to the layout of faces in the octagon-first projection of the truncated cuboctahedron into 2 dimensions. Thus, the omnitruncated tesseract may be thought of as another analogue of the truncated cuboctahedron in 4 dimensions.
1485:
2108:
2710:
2280:
The 6 cuboidal volumes connecting the axial square faces of the central small rhombicuboctahedron to the center of the octagons correspond with the image of 12 of the cubical cells (each pair of the twelve share the same
740:
1518:
Twelve right-angle triangular prisms connect the inner octagonal prisms. These are the images of 24 of the triangular prism cells. The remaining 8 triangular prisms project onto the triangular faces of the
4574:
Finally, the 8 volumes between the hexagonal faces of the projection envelope and the hexagonal faces of the central truncated cuboctahedron are the images of the 16 truncated octahedra, two cells to each
4731:
1026:
Finally, the 8 tetrahedral volumes connecting the vertices of the central cube to the triangular faces of the envelope are the images of the 16 tetrahedra (again, a pair of cells per image).
1020:
The 12 wedge-shaped volumes connecting the edges of the central cube to the non-axial square faces of the envelope are the images of 24 of the triangular prisms (a pair of cells per image).
1014:
Six cuboidal volumes connect this central cube to the 6 axial square faces of the rhombicuboctahedron. These are the images of 12 of the cubical cells (each pair of cubes share an image).
2290:
The remaining 12 spaces connecting the non-axial square faces of the central small rhombicuboctahedron to the square faces of the envelope are the images of 24 of the hexagonal prisms.
4938:, constructed by removing alternating long rectangles from the octagons, is also not uniform. Like the omnisnub tesseract, it has a highest symmetry construction of order 192, with 8
2277:
Six of the small rhombicuboctahedra project onto the 6 octagonal faces of this envelope, and the other two project to a small rhombicuboctahedron lying at the center of this envelope.
1522:
The 8 remaining volumes lying between the triangular faces of the envelope and the inner truncated cube are the images of the 16 cuboctahedral cells, a pair of cells to each image.
1509:
Six octagonal prisms connect this central truncated cube to the square faces of the envelope. These are the images of 12 of the octagonal prism cells, two cells to each image.
1370:
642:
radially, and filling in the gaps with tetrahedra (vertex figures), cubes (face prisms), and triangular prisms (edge figure prisms). The same process applied to a
4571:
The remaining hexagonal prisms are projected to 12 non-regular hexagonal prism images, lying where a cube's edges would be. Each image corresponds to two cells.
2287:
The 8 volumes connecting the hexagons of the envelope to the triangular faces of the central rhombicuboctahedron are the images of the 16 truncated tetrahedra.
2009:
1038:
in an analogous way to the runcinated tesseract. Hence, the runcinated tesseract may be thought of as the 4-dimensional analogue of the rhombicuboctahedron.
4554:
In the truncated cuboctahedron first parallel projection of the omnitruncated tesseract into 3 dimensions, the images of its cells are laid out as follows:
4730:
1495:
In the truncated cube first parallel projection of the runcitruncated tesseract into 3-dimensional space, the projection image is laid out as follows:
2622:
7389:
660:
6732:
6688:
4747:
6824:
2616:
of the vertices of an omnitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:
6667:
6234:
6166:
6059:
5353:
5168:
6677:, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
6332:
6322:
6312:
6302:
6293:
6283:
6263:
6254:
6244:
6239:
6225:
6215:
6205:
6176:
6171:
6147:
6137:
6118:
6088:
6079:
6064:
6050:
6030:
6011:
6001:
5996:
5982:
5972:
5933:
5928:
5904:
5875:
5860:
5846:
5490:
5480:
5470:
5460:
5451:
5431:
5421:
5402:
5392:
5382:
5363:
5358:
5334:
5324:
5305:
5275:
5256:
5236:
5207:
5197:
5173:
5139:
5090:
4883:
4873:
4863:
4853:
4477:
4467:
4457:
4447:
4363:
4353:
4343:
4333:
4245:
4235:
4225:
4215:
4122:
4112:
4102:
4092:
4012:
4002:
3992:
3982:
3898:
3888:
3878:
3868:
3784:
3774:
3764:
3754:
3670:
3660:
3650:
3640:
3552:
3542:
3532:
3522:
3429:
3419:
3409:
3399:
3321:
3311:
3301:
3291:
3213:
3203:
3193:
3183:
3105:
3095:
3085:
3075:
2986:
2976:
2966:
2956:
2871:
2861:
2851:
2841:
2781:
2771:
2761:
2751:
2400:
2390:
2380:
2370:
1777:
1767:
1747:
1135:
1115:
1105:
453:
423:
309:
299:
289:
279:
260:
250:
230:
211:
191:
181:
160:
111:
81:
32:
5991:
5923:
5855:
6273:
6195:
6186:
6157:
6127:
6108:
6098:
6069:
6040:
6020:
5962:
5952:
5943:
5914:
5894:
5884:
5865:
5836:
5826:
5816:
5441:
5412:
5373:
5344:
5314:
5295:
5285:
5266:
5246:
5227:
5217:
5188:
5178:
5159:
5149:
5129:
5120:
5110:
5100:
1757:
1125:
443:
433:
240:
201:
150:
140:
130:
101:
91:
62:
52:
42:
6742:
6650:
6327:
6317:
6307:
6288:
6278:
6268:
6249:
6220:
6210:
6200:
6181:
6152:
6142:
6132:
6113:
6103:
6093:
6074:
6045:
6035:
6025:
6006:
5977:
5967:
5957:
5938:
5909:
5899:
5889:
5870:
5841:
5831:
5821:
5485:
5475:
5465:
5446:
5436:
5426:
5407:
5397:
5387:
5368:
5339:
5329:
5319:
5300:
5290:
5280:
5261:
5251:
5241:
5222:
5212:
5202:
5183:
5154:
5144:
5134:
5115:
5105:
5095:
4878:
4868:
4858:
4472:
4462:
4452:
4358:
4348:
4338:
4240:
4230:
4220:
4117:
4107:
4097:
4007:
3997:
3987:
3893:
3883:
3873:
3779:
3769:
3759:
3665:
3655:
3645:
3547:
3537:
3527:
3424:
3414:
3404:
3316:
3306:
3296:
3208:
3198:
3188:
3100:
3090:
3080:
2981:
2971:
2961:
2866:
2856:
2846:
2776:
2766:
2756:
2395:
2385:
2375:
1772:
1762:
1752:
1130:
1120:
1110:
448:
438:
428:
304:
294:
284:
255:
245:
235:
206:
196:
186:
155:
145:
135:
106:
96:
86:
57:
47:
37:
6683:, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
7406:
2266:
of the runcitruncated 16-cell under the parallel projection, small rhombicuboctahedron first, into 3-dimensional space:
1996:
original square faces joined, as before, to small rhombicuboctahedra, and its three new square faces joined to cubes).
4801:
600:
5806:
5080:
6760:
6761:
2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 15, 19, 20, and 21
4772:
359:
There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations.
6491:
6484:
4781:
1999:
The vertices of a runcitruncated 16-cell having an edge length of 2 is given by all permutations of the following
1694:
654:
The
Cartesian coordinates of the vertices of the runcinated tesseract with edge length 2 are all permutations of:
1364:
of the vertices of the runcitruncated tesseract having an edge length of 2 is given by all permutations of:
986:
944:
6498:
5649:
5642:
4743:
rendered in blue, and the remaining cells in green. Cells obscured from 4D viewpoint culled for clarity's sake.
1057:
6658:
5656:
5064:
5057:
2725:
2605:
2321:
1926:
1667:
1284:
1011:
The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope.
6477:
6463:
5614:
6817:
6470:
5797:
5635:
5621:
5071:
4736:
4592:
4483:
2581:
2415:
2299:
2271:
2119:
1932:
1532:
1000:
970:
751:
6456:
6449:
5628:
4919:
378:
5790:
5783:
5607:
4846:
4580:
2549:
2284:
The remaining 12 cubical cells project onto the 12 square faces of the great rhombicuboctahedral envelope.
1299:
1290:
7361:
7354:
7347:
5776:
5050:
5036:
1992:
1991:
The runcitruncated 16-cell may be constructed by contracting the small rhombicuboctahedral cells of the
1972:
1804:
1708:
591:
585:
317:
7018:
6965:
6691:
5769:
5043:
4753:
4128:
2733:
2613:
2585:
2427:
2000:
1506:
Two of the truncated cube cells project to a truncated cube in the center of the projection envelope.
1480:{\displaystyle \left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)}
1361:
936:
4911:
filling the gaps at the deleted vertices. It has 272 cells, 944 faces, 864 edges, and 192 vertices.
4826:
7373:
7272:
7022:
5762:
5029:
5022:
4995:
4987:
4939:
1968:
1941:
1792:
1704:
1500:
1346:
1034:
under projection to 2 dimensions. The rhombicuboctahedron is also constructed from the cube or the
1031:
1004:
952:
948:
6623:
6608:
6593:
4994:
has two edge lengths in the ratio of 1 : 2, and occurs as a vertex-faceting of the scaliform
7242:
7192:
7142:
7099:
7069:
7029:
6992:
6810:
6720:
2721:
faces, and the hexagonal prisms are joined to the truncated octahedra via their hexagonal faces.
2577:
2341:
2263:
1964:
1718:
1076:
394:
345:
6340:
5498:
2348:
1725:
1083:
401:
7381:
6728:
6684:
6670:, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
6663:
5755:
5748:
4662:
2472:
2189:
1888:
1854:
1602:
1212:
821:
518:
4849:
of the omnitruncated tesseract, can not be made uniform, but it can be given
Coxeter diagram
7385:
6950:
6939:
6928:
6917:
6908:
6899:
6886:
6864:
6852:
6838:
6834:
6442:
6434:
5592:
4991:
4973:
4969:
4900:
4888:
4793:
3048:
2326:
1945:
1699:
1334:
1303:
1246:
1186:
1062:
917:
881:
627:
604:
480:
383:
5600:
2511:
2421:
1798:
1023:
The 8 triangular faces of the envelope are the images of the remaining 8 triangular prisms.
981:
6975:
6960:
4757:
4740:
4369:
4251:
2593:
2589:
2538:
2451:
2439:
2362:
1980:
1915:
1828:
1739:
1515:
The 6 square faces of the envelope are the images of the remaining 6 truncated cube cells.
1330:
1273:
1174:
1097:
1030:
This layout of cells in projection is analogous to the layout of the faces of the (small)
960:
574:
415:
6773:
1512:
The remaining 12 octagonal prisms are projected to the rectangular faces of the envelope.
1353:
cells outward radially, and inserting octagonal prisms between them. In the process, the
7325:
6640:
2521:
2103:{\displaystyle \left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)}
1689:
1350:
1322:
1150:
557:
547:
2293:
Finally, the last 8 hexagonal prisms project onto the hexagonal faces of the envelope.
1810:
1662:
7400:
7342:
7230:
7223:
7216:
7180:
7173:
7166:
7130:
7123:
6847:
4599:
2823:
2505:
2126:
1898:
1882:
1539:
1326:
1256:
1240:
1162:
956:
758:
541:
324:
2433:
1156:
7282:
4561:
Two of the truncated cuboctahedra project to the center of the projection envelope.
975:
940:
4568:
The 8 hexagonal faces of the envelope are the images of 8 of the hexagonal prisms.
6792:
4558:
The projection envelope is in the shape of a non-uniform truncated cuboctahedron.
7291:
7252:
7202:
7152:
7109:
7079:
7011:
6997:
6795:
4950:
4908:
4896:
2933:
2705:{\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}
1354:
1007:
envelope. The images of its cells are laid out within this envelope as follows:
904:
619:
468:
349:
4810:
1192:
911:
898:
889:
486:
7277:
7261:
7211:
7161:
7118:
7088:
7002:
4904:
4703:
4696:
4654:
4647:
2457:
2445:
1834:
1180:
1035:
1017:
The 18 square faces of the envelope are the images of the other cubical cells.
630:. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.
7333:
7247:
7197:
7147:
7104:
7074:
7043:
5017:
4892:
2230:
2223:
2181:
2174:
1643:
1636:
1594:
1587:
1357:
expand into cuboctahedra, and triangular prisms fill in the remaining gaps.
639:
353:
25:
6789:
6534:
6520:
1168:
862:
855:
813:
806:
6569:
5727:
4640:
267:
7307:
7062:
7058:
6985:
6798:
2729:
1849:
1207:
513:
333:
6562:
6555:
5720:
5713:
2167:
1580:
735:{\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)}
638:
The runcinated tesseract may be constructed by expanding the cells of a
474:
218:
169:
7316:
7286:
7053:
7048:
7039:
6980:
6548:
6541:
6527:
5743:
5706:
5699:
5692:
5685:
4957:
symmetry), 24 rectangular trapezoprisms (topologically equivalent to a
4018:
3790:
3435:
2555:
2482:
2477:
1859:
1822:
1217:
799:
643:
498:
123:
69:
6513:
5678:
5671:
118:
20:
7256:
7206:
7156:
7113:
7083:
7034:
6970:
3904:
3676:
3558:
6778:
x3o3o4x - sidpith, x3o3x4x - proh, x3x3o4x - prit, x3x3x4x - gidpith
6766:
1003:
of the runcinated tesseract into 3-dimensional space has a (small)
2309:
1940:
1677:
1298:
1045:
599:
366:
15:
1670:
with its 128 blue triangular faces and its 192 green quad faces.
7006:
6645:
On the
Regular and Semi-Regular Figures in Space of n Dimensions
4958:
2728:, all incidence counts between elements are shown. The diagonal
1976:
1816:
623:
492:
947:
between them. This dissection can be seen analogous to the 3D
4579:
This layout of cells in projection is similar to that of the
2319:
2312:
1687:
1680:
1055:
1048:
376:
369:
1345:
The runcitruncated tesseract may be constructed from the
2604:
The omnitruncated tesseract can be constructed from the
4756:
cells, highlighted in yellow. Four of the surrounding
4739:
cells, highlighted in yellow. Six of the surrounding
2738:
2625:
2012:
1373:
663:
4720:
6681:Kaleidoscopes: Selected Writings of H.S.M. Coxeter
2704:
2102:
1479:
734:
1499:The projection envelope is a non-uniform (small)
1503:, with 6 square faces and 12 rectangular faces.
6753:The Theory of Uniform Polytopes and Honeycombs
4752:Perspective projection centered on one of the
4735:Perspective projection centered on one of the
6818:
8:
6647:, Messenger of Mathematics, Macmillan, 1900
616:(small) disprismatotesseractihexadecachoron
6825:
6811:
6803:
5005:
4745:
4728:
2690:
2668:
2646:
2624:
2574:great disprismatotesseractihexadecachoron
2085:
2054:
2011:
1462:
1431:
1403:
1372:
717:
662:
166:
17:
6723:, Heidi Burgiel, Chaim Goodman-Strauss,
4918:
4825:
4791:
4779:
4770:
4590:
2117:
1530:
965:
879:
749:
352:(a 3rd order truncation) of the regular
7390:List of regular polytopes and compounds
6788:H4 uniform polytopes with coordinates:
6735:(Chapter 26. pp. 409: Hemicubes: 1
6584:
932:tetrahedra via their triangular faces.
6711:Regular and Semi-Regular Polytopes III
6662:, (3rd edition, 1973), Dover edition,
6704:Regular and Semi-Regular Polytopes II
7:
6697:Regular and Semi Regular Polytopes I
4983:-symmetry wedges) filling the gaps.
4907:(as triangular antiprisms), and 192
4777:Centered on truncated cuboctahedron
2331:centered on truncated cuboctahedron,
2262:The following is the layout of the
6774:"4D uniform polytopes (polychora)"
552:Equilateral-triangular antipodium
14:
4786:Centered on truncated octahedron
2333:truncated octahedral cells shown
6767:http://www.polytope.de/nr17.html
6568:
6561:
6554:
6547:
6540:
6533:
6526:
6519:
6512:
6497:
6490:
6483:
6476:
6469:
6462:
6455:
6448:
6441:
6330:
6325:
6320:
6315:
6310:
6305:
6300:
6291:
6286:
6281:
6276:
6271:
6266:
6261:
6252:
6247:
6242:
6237:
6232:
6223:
6218:
6213:
6208:
6203:
6198:
6193:
6184:
6179:
6174:
6169:
6164:
6155:
6150:
6145:
6140:
6135:
6130:
6125:
6116:
6111:
6106:
6101:
6096:
6091:
6086:
6077:
6072:
6067:
6062:
6057:
6048:
6043:
6038:
6033:
6028:
6023:
6018:
6009:
6004:
5999:
5994:
5989:
5980:
5975:
5970:
5965:
5960:
5955:
5950:
5941:
5936:
5931:
5926:
5921:
5912:
5907:
5902:
5897:
5892:
5887:
5882:
5873:
5868:
5863:
5858:
5853:
5844:
5839:
5834:
5829:
5824:
5819:
5814:
5726:
5719:
5712:
5705:
5698:
5691:
5684:
5677:
5670:
5655:
5648:
5641:
5634:
5627:
5620:
5613:
5606:
5599:
5488:
5483:
5478:
5473:
5468:
5463:
5458:
5449:
5444:
5439:
5434:
5429:
5424:
5419:
5410:
5405:
5400:
5395:
5390:
5385:
5380:
5371:
5366:
5361:
5356:
5351:
5342:
5337:
5332:
5327:
5322:
5317:
5312:
5303:
5298:
5293:
5288:
5283:
5278:
5273:
5264:
5259:
5254:
5249:
5244:
5239:
5234:
5225:
5220:
5215:
5210:
5205:
5200:
5195:
5186:
5181:
5176:
5171:
5166:
5157:
5152:
5147:
5142:
5137:
5132:
5127:
5118:
5113:
5108:
5103:
5098:
5093:
5088:
4881:
4876:
4871:
4866:
4861:
4856:
4851:
4815:Dual to omnitruncated tesseract
4809:
4800:
4780:
4771:
4746:
4729:
4702:
4695:
4653:
4646:
4639:
4475:
4470:
4465:
4460:
4455:
4450:
4445:
4361:
4356:
4351:
4346:
4341:
4336:
4331:
4243:
4238:
4233:
4228:
4223:
4218:
4213:
4120:
4115:
4110:
4105:
4100:
4095:
4090:
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3896:
3891:
3886:
3881:
3876:
3871:
3866:
3782:
3777:
3772:
3767:
3762:
3757:
3752:
3668:
3663:
3658:
3653:
3648:
3643:
3638:
3550:
3545:
3540:
3535:
3530:
3525:
3520:
3427:
3422:
3417:
3412:
3407:
3402:
3397:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3211:
3206:
3201:
3196:
3191:
3186:
3181:
3103:
3098:
3093:
3088:
3083:
3078:
3073:
2984:
2979:
2974:
2969:
2964:
2959:
2954:
2869:
2864:
2859:
2854:
2849:
2844:
2839:
2779:
2774:
2769:
2764:
2759:
2754:
2749:
2732:numbers are derived through the
2510:
2456:
2444:
2432:
2420:
2398:
2393:
2388:
2383:
2378:
2373:
2368:
2320:
2229:
2222:
2180:
2173:
2166:
1887:
1833:
1821:
1809:
1797:
1775:
1770:
1765:
1760:
1755:
1750:
1745:
1693:
1688:
1661:
1642:
1635:
1593:
1586:
1579:
1319:prismatorhombated hexadecachoron
1245:
1191:
1179:
1167:
1155:
1133:
1128:
1123:
1118:
1113:
1108:
1103:
1056:
980:
969:
910:
897:
888:
861:
854:
812:
805:
798:
546:
497:
485:
473:
451:
446:
441:
436:
431:
426:
421:
377:
307:
302:
297:
292:
287:
282:
277:
266:
258:
253:
248:
243:
238:
233:
228:
217:
209:
204:
199:
194:
189:
184:
179:
168:
158:
153:
148:
143:
138:
133:
128:
117:
109:
104:
99:
94:
89:
84:
79:
68:
60:
55:
50:
45:
40:
35:
30:
19:
4529:
4526:
4523:
4520:
4417:
4412:
4409:
4406:
4299:
4296:
4291:
4288:
4182:
4179:
4176:
4171:
4052:
4049:
4046:
4043:
4040:
4037:
3940:
3935:
3932:
3929:
3926:
3923:
3826:
3823:
3818:
3815:
3812:
3809:
3712:
3709:
3706:
3701:
3698:
3695:
3594:
3591:
3588:
3585:
3580:
3577:
3477:
3474:
3471:
3468:
3465:
3460:
3341:
3338:
3335:
3332:
3235:
3230:
3227:
3224:
3127:
3124:
3119:
3116:
3014:
3011:
3008:
3003:
2885:
2544:
2534:
2520:
2504:
2496:
2488:
2464:
2407:
2361:
2347:
2337:
1921:
1911:
1897:
1881:
1873:
1865:
1841:
1784:
1738:
1724:
1714:
1279:
1269:
1255:
1239:
1231:
1223:
1199:
1142:
1096:
1082:
1072:
1068:with cuboctahedral cells shown
580:
570:
556:
540:
532:
524:
505:
460:
414:
400:
390:
2092:
2073:
2061:
2045:
1469:
1450:
1438:
1422:
1410:
1394:
724:
708:
1:
4936:runcic snub rectified 16-cell
2270:The projection envelope is a
1066:centered on a truncated cube,
646:also yields the same figure.
967:
225:(Runcicantellated tesseract)
6709:(Paper 24) H.S.M. Coxeter,
6702:(Paper 23) H.S.M. Coxeter,
6695:(Paper 22) H.S.M. Coxeter,
4436:
4318:
4200:
4081:
3971:
3853:
3743:
3625:
3507:
3388:
3280:
3172:
3064:
2945:
2834:
2516:Chiral scalene tetrahedron
1961:prismatorhombated tesseract
7423:
7379:
6806:
5734:
5008:
4766:Stereographic projections
1957:runcicantellated tesseract
1321:is bounded by 80 cells: 8
176:(Runcicantellated 16-cell)
5002:Related uniform polytopes
4891:, and constructed from 8
4765:
4723:
4185:
4132:
3492:
3439:
3047:
2994:
2816:
2810:
2804:
2547:
2537:
2525:
2509:
2499:
2491:
2366:
2352:
2340:
1924:
1914:
1902:
1886:
1876:
1868:
1743:
1729:
1717:
1282:
1272:
1260:
1244:
1234:
1226:
1101:
1087:
1075:
987:rhombicuboctahedral prism
945:rhombicuboctahedral prism
583:
573:
561:
545:
535:
527:
419:
405:
393:
316:
6725:The Symmetries of Things
4806:Omnitruncated tesseract
4724:Perspective projections
4593:orthographic projections
2606:cantitruncated tesseract
2120:orthographic projections
1668:Stereographic projection
1533:orthographic projections
1315:runcicantellated 16-cell
1311:runcitruncated tesseract
1050:Runcitruncated tesseract
1042:Runcitruncated tesseract
939:can be dissected into 2
752:orthographic projections
174:Runcitruncated tesseract
4986:A variant with regular
4932:bialternatosnub 16-cell
4925:bialternatosnub 16-cell
4915:Bialternatosnub 16-cell
4737:truncated cuboctahedral
2566:omnitruncated tesseract
2314:Omnitruncated tesseract
2306:Omnitruncated tesseract
2300:omnitruncated tesseract
2272:truncated cuboctahedron
1001:orthographic projection
274:(Omnitruncated 16-cell)
272:Omnitruncated tesseract
5009:B4 symmetry polytopes
4927:
4923:Vertex figure for the
4834:
4830:Vertex figure for the
4581:runcitruncated 16-cell
2706:
2582:truncated cuboctahedra
2104:
1953:runcitruncated 16-cell
1948:
1682:Runcitruncated 16-cell
1674:Runcitruncated 16-cell
1481:
1306:
736:
607:
318:Orthogonal projections
223:Runcitruncated 16-cell
4922:
4829:
2707:
2614:Cartesian coordinates
2570:omnitruncated 16-cell
2105:
2001:Cartesian coordinates
1993:cantellated tesseract
1944:
1709:truncated tetrahedron
1482:
1362:Cartesian coordinates
1302:
737:
650:Cartesian coordinates
603:
6748:, Manuscript (1991)
4754:truncated octahedral
2734:Wythoff construction
2726:configuration matrix
2623:
2010:
1973:truncated tetrahedra
1893:Trapezoidal pyramid
1371:
1251:Rectangular pyramid
937:runcinated tesseract
661:
612:runcinated tesseract
371:Runcinated tesseract
363:Runcinated tesseract
338:runcinated tesseract
332:In four-dimensional
76:(Runcinated 16-cell)
74:Runcinated tesseract
7407:Uniform 4-polytopes
7374:pentagonal polytope
7273:Uniform 10-polytope
6833:Fundamental convex
6772:Klitzing, Richard.
6763:, George Olshevsky.
6622:Klitzing, Richard.
6607:Klitzing, Richard.
6594:"x3x3x4x - gidpith"
6592:Klitzing, Richard.
4996:runcic snub 24-cell
4839:full snub tesseract
4822:Full snub tesseract
4796:
4595:
2586:truncated octahedra
2122:
1705:rhombicuboctahedron
1535:
1501:rhombicuboctahedron
1347:truncated tesseract
1032:rhombicuboctahedron
1005:rhombicuboctahedral
949:rhombicuboctahedron
884:
754:
386:with 16 tetrahedra
7243:Uniform 9-polytope
7193:Uniform 8-polytope
7143:Uniform 7-polytope
7100:Uniform 6-polytope
7070:Uniform 5-polytope
7030:Uniform polychoron
6993:Uniform polyhedron
6841:in dimensions 2–10
4940:rhombicuboctahedra
4928:
4843:omnisnub tesseract
4835:
4832:omnisnub tesseract
4792:
4711:Dihedral symmetry
4591:
2702:
2342:Uniform 4-polytope
2238:Dihedral symmetry
2118:
2100:
1969:rhombicuboctahedra
1949:
1719:Uniform 4-polytope
1651:Dihedral symmetry
1531:
1477:
1307:
1077:Uniform 4-polytope
916:Wireframe with 32
903:Wireframe with 16
880:
870:Dihedral symmetry
750:
732:
608:
395:Uniform 4-polytope
346:uniform 4-polytope
342:runcinated 16-cell
7395:
7394:
7382:Polytope families
6839:uniform polytopes
6746:Uniform Polytopes
6733:978-1-56881-220-5
6689:978-0-471-01003-6
6675:Regular Polytopes
6659:Regular Polytopes
6576:
6575:
6427:
6418:
6407:
6396:
6387:
6376:
6365:
6354:
5585:
5576:
5565:
5554:
5545:
5534:
5523:
5512:
4992:triangular prisms
4974:triangular prisms
4970:triangular prisms
4901:square antiprisms
4819:
4818:
4790:
4789:
4719:
4718:
4663:Dihedral symmetry
4547:
4546:
2695:
2680:
2673:
2658:
2651:
2639:
2576:is bounded by 80
2562:
2561:
2246:
2245:
2190:Dihedral symmetry
2090:
2069:
2059:
2041:
2029:
1963:is bounded by 80
1939:
1938:
1700:Schlegel diagrams
1659:
1658:
1603:Dihedral symmetry
1467:
1446:
1436:
1418:
1408:
1390:
1349:by expanding the
1335:triangular prisms
1297:
1296:
992:
991:
924:
923:
918:triangular prisms
882:Schlegel diagrams
878:
877:
822:Dihedral symmetry
722:
704:
692:
680:
628:triangular prisms
598:
597:
330:
329:
7414:
7386:Regular polytope
6947:
6936:
6925:
6884:
6827:
6820:
6813:
6804:
6777:
6673:H.S.M. Coxeter,
6628:
6627:
6619:
6613:
6612:
6604:
6598:
6597:
6589:
6572:
6565:
6558:
6551:
6544:
6537:
6530:
6523:
6516:
6501:
6494:
6487:
6480:
6473:
6466:
6459:
6452:
6445:
6425:
6416:
6405:
6394:
6385:
6374:
6363:
6352:
6335:
6334:
6333:
6329:
6328:
6324:
6323:
6319:
6318:
6314:
6313:
6309:
6308:
6304:
6303:
6296:
6295:
6294:
6290:
6289:
6285:
6284:
6280:
6279:
6275:
6274:
6270:
6269:
6265:
6264:
6257:
6256:
6255:
6251:
6250:
6246:
6245:
6241:
6240:
6236:
6235:
6228:
6227:
6226:
6222:
6221:
6217:
6216:
6212:
6211:
6207:
6206:
6202:
6201:
6197:
6196:
6189:
6188:
6187:
6183:
6182:
6178:
6177:
6173:
6172:
6168:
6167:
6160:
6159:
6158:
6154:
6153:
6149:
6148:
6144:
6143:
6139:
6138:
6134:
6133:
6129:
6128:
6121:
6120:
6119:
6115:
6114:
6110:
6109:
6105:
6104:
6100:
6099:
6095:
6094:
6090:
6089:
6082:
6081:
6080:
6076:
6075:
6071:
6070:
6066:
6065:
6061:
6060:
6053:
6052:
6051:
6047:
6046:
6042:
6041:
6037:
6036:
6032:
6031:
6027:
6026:
6022:
6021:
6014:
6013:
6012:
6008:
6007:
6003:
6002:
5998:
5997:
5993:
5992:
5985:
5984:
5983:
5979:
5978:
5974:
5973:
5969:
5968:
5964:
5963:
5959:
5958:
5954:
5953:
5946:
5945:
5944:
5940:
5939:
5935:
5934:
5930:
5929:
5925:
5924:
5917:
5916:
5915:
5911:
5910:
5906:
5905:
5901:
5900:
5896:
5895:
5891:
5890:
5886:
5885:
5878:
5877:
5876:
5872:
5871:
5867:
5866:
5862:
5861:
5857:
5856:
5849:
5848:
5847:
5843:
5842:
5838:
5837:
5833:
5832:
5828:
5827:
5823:
5822:
5818:
5817:
5730:
5723:
5716:
5709:
5702:
5695:
5688:
5681:
5674:
5659:
5652:
5645:
5638:
5631:
5624:
5617:
5610:
5603:
5583:
5574:
5563:
5552:
5543:
5532:
5521:
5510:
5493:
5492:
5491:
5487:
5486:
5482:
5481:
5477:
5476:
5472:
5471:
5467:
5466:
5462:
5461:
5454:
5453:
5452:
5448:
5447:
5443:
5442:
5438:
5437:
5433:
5432:
5428:
5427:
5423:
5422:
5415:
5414:
5413:
5409:
5408:
5404:
5403:
5399:
5398:
5394:
5393:
5389:
5388:
5384:
5383:
5376:
5375:
5374:
5370:
5369:
5365:
5364:
5360:
5359:
5355:
5354:
5347:
5346:
5345:
5341:
5340:
5336:
5335:
5331:
5330:
5326:
5325:
5321:
5320:
5316:
5315:
5308:
5307:
5306:
5302:
5301:
5297:
5296:
5292:
5291:
5287:
5286:
5282:
5281:
5277:
5276:
5269:
5268:
5267:
5263:
5262:
5258:
5257:
5253:
5252:
5248:
5247:
5243:
5242:
5238:
5237:
5230:
5229:
5228:
5224:
5223:
5219:
5218:
5214:
5213:
5209:
5208:
5204:
5203:
5199:
5198:
5191:
5190:
5189:
5185:
5184:
5180:
5179:
5175:
5174:
5170:
5169:
5162:
5161:
5160:
5156:
5155:
5151:
5150:
5146:
5145:
5141:
5140:
5136:
5135:
5131:
5130:
5123:
5122:
5121:
5117:
5116:
5112:
5111:
5107:
5106:
5102:
5101:
5097:
5096:
5092:
5091:
5006:
4886:
4885:
4884:
4880:
4879:
4875:
4874:
4870:
4869:
4865:
4864:
4860:
4859:
4855:
4854:
4845:, defined as an
4813:
4804:
4797:
4784:
4775:
4758:hexagonal prisms
4750:
4741:octagonal prisms
4733:
4721:
4706:
4699:
4657:
4650:
4643:
4596:
4480:
4479:
4478:
4474:
4473:
4469:
4468:
4464:
4463:
4459:
4458:
4454:
4453:
4449:
4448:
4366:
4365:
4364:
4360:
4359:
4355:
4354:
4350:
4349:
4345:
4344:
4340:
4339:
4335:
4334:
4248:
4247:
4246:
4242:
4241:
4237:
4236:
4232:
4231:
4227:
4226:
4222:
4221:
4217:
4216:
4125:
4124:
4123:
4119:
4118:
4114:
4113:
4109:
4108:
4104:
4103:
4099:
4098:
4094:
4093:
4015:
4014:
4013:
4009:
4008:
4004:
4003:
3999:
3998:
3994:
3993:
3989:
3988:
3984:
3983:
3901:
3900:
3899:
3895:
3894:
3890:
3889:
3885:
3884:
3880:
3879:
3875:
3874:
3870:
3869:
3787:
3786:
3785:
3781:
3780:
3776:
3775:
3771:
3770:
3766:
3765:
3761:
3760:
3756:
3755:
3673:
3672:
3671:
3667:
3666:
3662:
3661:
3657:
3656:
3652:
3651:
3647:
3646:
3642:
3641:
3555:
3554:
3553:
3549:
3548:
3544:
3543:
3539:
3538:
3534:
3533:
3529:
3528:
3524:
3523:
3432:
3431:
3430:
3426:
3425:
3421:
3420:
3416:
3415:
3411:
3410:
3406:
3405:
3401:
3400:
3324:
3323:
3322:
3318:
3317:
3313:
3312:
3308:
3307:
3303:
3302:
3298:
3297:
3293:
3292:
3216:
3215:
3214:
3210:
3209:
3205:
3204:
3200:
3199:
3195:
3194:
3190:
3189:
3185:
3184:
3108:
3107:
3106:
3102:
3101:
3097:
3096:
3092:
3091:
3087:
3086:
3082:
3081:
3077:
3076:
2989:
2988:
2987:
2983:
2982:
2978:
2977:
2973:
2972:
2968:
2967:
2963:
2962:
2958:
2957:
2874:
2873:
2872:
2868:
2867:
2863:
2862:
2858:
2857:
2853:
2852:
2848:
2847:
2843:
2842:
2784:
2783:
2782:
2778:
2777:
2773:
2772:
2768:
2767:
2763:
2762:
2758:
2757:
2753:
2752:
2739:
2711:
2709:
2708:
2703:
2701:
2697:
2696:
2691:
2678:
2674:
2669:
2656:
2652:
2647:
2637:
2594:hexagonal prisms
2590:octagonal prisms
2514:
2460:
2448:
2436:
2424:
2403:
2402:
2401:
2397:
2396:
2392:
2391:
2387:
2386:
2382:
2381:
2377:
2376:
2372:
2371:
2327:Schlegel diagram
2324:
2310:
2233:
2226:
2184:
2177:
2170:
2123:
2109:
2107:
2106:
2101:
2099:
2095:
2091:
2086:
2067:
2060:
2055:
2039:
2027:
1981:hexagonal prisms
1891:
1837:
1825:
1813:
1801:
1780:
1779:
1778:
1774:
1773:
1769:
1768:
1764:
1763:
1759:
1758:
1754:
1753:
1749:
1748:
1697:
1692:
1678:
1665:
1646:
1639:
1597:
1590:
1583:
1536:
1486:
1484:
1483:
1478:
1476:
1472:
1468:
1463:
1444:
1437:
1432:
1416:
1409:
1404:
1388:
1331:octagonal prisms
1249:
1195:
1183:
1171:
1159:
1138:
1137:
1136:
1132:
1131:
1127:
1126:
1122:
1121:
1117:
1116:
1112:
1111:
1107:
1106:
1098:Coxeter diagrams
1063:Schlegel diagram
1060:
1046:
984:
973:
966:
914:
901:
892:
885:
865:
858:
816:
809:
802:
755:
741:
739:
738:
733:
731:
727:
723:
718:
702:
690:
678:
550:
501:
489:
477:
456:
455:
454:
450:
449:
445:
444:
440:
439:
435:
434:
430:
429:
425:
424:
416:Coxeter diagrams
384:Schlegel diagram
381:
367:
312:
311:
310:
306:
305:
301:
300:
296:
295:
291:
290:
286:
285:
281:
280:
270:
263:
262:
261:
257:
256:
252:
251:
247:
246:
242:
241:
237:
236:
232:
231:
221:
214:
213:
212:
208:
207:
203:
202:
198:
197:
193:
192:
188:
187:
183:
182:
172:
163:
162:
161:
157:
156:
152:
151:
147:
146:
142:
141:
137:
136:
132:
131:
121:
114:
113:
112:
108:
107:
103:
102:
98:
97:
93:
92:
88:
87:
83:
82:
72:
65:
64:
63:
59:
58:
54:
53:
49:
48:
44:
43:
39:
38:
34:
33:
23:
16:
7422:
7421:
7417:
7416:
7415:
7413:
7412:
7411:
7397:
7396:
7365:
7358:
7351:
7234:
7227:
7220:
7184:
7177:
7170:
7134:
7127:
6961:Regular polygon
6954:
6945:
6938:
6934:
6927:
6923:
6914:
6905:
6898:
6894:
6882:
6876:
6872:
6860:
6842:
6831:
6785:
6771:
6738:
6637:
6632:
6631:
6621:
6620:
6616:
6606:
6605:
6601:
6591:
6590:
6586:
6581:
6509:
6436:
6428:
6419:
6410:
6408:
6399:
6397:
6388:
6379:
6377:
6368:
6366:
6357:
6355:
6342:
6331:
6326:
6321:
6316:
6311:
6306:
6301:
6299:
6292:
6287:
6282:
6277:
6272:
6267:
6262:
6260:
6253:
6248:
6243:
6238:
6233:
6231:
6229:
6224:
6219:
6214:
6209:
6204:
6199:
6194:
6192:
6185:
6180:
6175:
6170:
6165:
6163:
6161:
6156:
6151:
6146:
6141:
6136:
6131:
6126:
6124:
6117:
6112:
6107:
6102:
6097:
6092:
6087:
6085:
6078:
6073:
6068:
6063:
6058:
6056:
6054:
6049:
6044:
6039:
6034:
6029:
6024:
6019:
6017:
6010:
6005:
6000:
5995:
5990:
5988:
5986:
5981:
5976:
5971:
5966:
5961:
5956:
5951:
5949:
5942:
5937:
5932:
5927:
5922:
5920:
5918:
5913:
5908:
5903:
5898:
5893:
5888:
5883:
5881:
5874:
5869:
5864:
5859:
5854:
5852:
5850:
5845:
5840:
5835:
5830:
5825:
5820:
5815:
5813:
5808:
5799:
5792:
5785:
5778:
5771:
5764:
5757:
5750:
5667:
5594:
5586:
5577:
5568:
5566:
5557:
5555:
5546:
5537:
5535:
5526:
5524:
5515:
5513:
5500:
5489:
5484:
5479:
5474:
5469:
5464:
5459:
5457:
5450:
5445:
5440:
5435:
5430:
5425:
5420:
5418:
5411:
5406:
5401:
5396:
5391:
5386:
5381:
5379:
5372:
5367:
5362:
5357:
5352:
5350:
5348:
5343:
5338:
5333:
5328:
5323:
5318:
5313:
5311:
5304:
5299:
5294:
5289:
5284:
5279:
5274:
5272:
5265:
5260:
5255:
5250:
5245:
5240:
5235:
5233:
5226:
5221:
5216:
5211:
5206:
5201:
5196:
5194:
5187:
5182:
5177:
5172:
5167:
5165:
5163:
5158:
5153:
5148:
5143:
5138:
5133:
5128:
5126:
5119:
5114:
5109:
5104:
5099:
5094:
5089:
5087:
5082:
5073:
5066:
5059:
5052:
5045:
5038:
5031:
5024:
5004:
4981:
4966:
4947:
4917:
4882:
4877:
4872:
4867:
4862:
4857:
4852:
4850:
4824:
4814:
4805:
4785:
4776:
4751:
4734:
4687:
4681:
4631:
4627:
4621:
4617:
4613:
4607:
4589:
4552:
4542:
4538:
4476:
4471:
4466:
4461:
4456:
4451:
4446:
4444:
4441:
4432:
4428:
4424:
4362:
4357:
4352:
4347:
4342:
4337:
4332:
4330:
4327:
4323:
4314:
4310:
4306:
4244:
4239:
4234:
4229:
4224:
4219:
4214:
4212:
4209:
4205:
4196:
4192:
4136:
4121:
4116:
4111:
4106:
4101:
4096:
4091:
4089:
4086:
4077:
4073:
4011:
4006:
4001:
3996:
3991:
3986:
3981:
3979:
3976:
3967:
3963:
3959:
3897:
3892:
3887:
3882:
3877:
3872:
3867:
3865:
3862:
3858:
3849:
3845:
3783:
3778:
3773:
3768:
3763:
3758:
3753:
3751:
3748:
3739:
3735:
3731:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3637:
3634:
3630:
3621:
3617:
3613:
3551:
3546:
3541:
3536:
3531:
3526:
3521:
3519:
3516:
3512:
3503:
3499:
3443:
3428:
3423:
3418:
3413:
3408:
3403:
3398:
3396:
3393:
3384:
3380:
3320:
3315:
3310:
3305:
3300:
3295:
3290:
3288:
3285:
3276:
3272:
3212:
3207:
3202:
3197:
3192:
3187:
3182:
3180:
3177:
3168:
3164:
3104:
3099:
3094:
3089:
3084:
3079:
3074:
3072:
3069:
3060:
3056:
2998:
2985:
2980:
2975:
2970:
2965:
2960:
2955:
2953:
2950:
2941:
2883:
2870:
2865:
2860:
2855:
2850:
2845:
2840:
2838:
2820:
2814:
2808:
2802:
2796:
2780:
2775:
2770:
2765:
2760:
2755:
2750:
2748:
2745:
2718:
2630:
2626:
2621:
2620:
2602:
2529:
2515:
2480:
2475:
2449:
2437:
2425:
2399:
2394:
2389:
2384:
2379:
2374:
2369:
2367:
2363:Coxeter diagram
2356:
2349:Schläfli symbol
2332:
2330:
2325:
2308:
2260:
2251:
2214:
2208:
2158:
2154:
2148:
2144:
2140:
2134:
2116:
2017:
2013:
2008:
2007:
1989:
1906:
1892:
1857:
1852:
1826:
1814:
1802:
1776:
1771:
1766:
1761:
1756:
1751:
1746:
1744:
1740:Coxeter diagram
1733:
1726:Schläfli symbol
1703:
1698:
1676:
1666:
1627:
1621:
1571:
1567:
1561:
1557:
1553:
1547:
1529:
1493:
1378:
1374:
1369:
1368:
1343:
1323:truncated cubes
1264:
1250:
1215:
1210:
1184:
1172:
1160:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1102:
1091:
1084:Schläfli symbol
1067:
1065:
1061:
1044:
999:The cube-first
997:
985:
974:
961:octagonal prism
929:
915:
902:
893:
846:
840:
790:
786:
780:
776:
772:
766:
748:
668:
664:
659:
658:
652:
636:
565:
551:
516:
490:
478:
452:
447:
442:
437:
432:
427:
422:
420:
409:
402:Schläfli symbol
382:
365:
323:
308:
303:
298:
293:
288:
283:
278:
276:
275:
273:
271:
259:
254:
249:
244:
239:
234:
229:
227:
226:
224:
222:
210:
205:
200:
195:
190:
185:
180:
178:
177:
175:
173:
159:
154:
149:
144:
139:
134:
129:
127:
126:
122:
110:
105:
100:
95:
90:
85:
80:
78:
77:
75:
73:
61:
56:
51:
46:
41:
36:
31:
29:
28:
24:
12:
11:
5:
7420:
7418:
7410:
7409:
7399:
7398:
7393:
7392:
7377:
7376:
7367:
7363:
7356:
7349:
7345:
7336:
7319:
7310:
7299:
7298:
7296:
7294:
7289:
7280:
7275:
7269:
7268:
7266:
7264:
7259:
7250:
7245:
7239:
7238:
7236:
7232:
7225:
7218:
7214:
7209:
7200:
7195:
7189:
7188:
7186:
7182:
7175:
7168:
7164:
7159:
7150:
7145:
7139:
7138:
7136:
7132:
7125:
7121:
7116:
7107:
7102:
7096:
7095:
7093:
7091:
7086:
7077:
7072:
7066:
7065:
7056:
7051:
7046:
7037:
7032:
7026:
7025:
7016:
7014:
7009:
7000:
6995:
6989:
6988:
6983:
6978:
6973:
6968:
6963:
6957:
6956:
6952:
6948:
6943:
6932:
6921:
6912:
6903:
6896:
6890:
6880:
6874:
6868:
6862:
6856:
6850:
6844:
6843:
6832:
6830:
6829:
6822:
6815:
6807:
6802:
6801:
6784:
6783:External links
6781:
6780:
6779:
6769:
6764:
6758:
6757:
6756:
6755:, Ph.D. (1966)
6751:N.W. Johnson:
6743:Norman Johnson
6740:
6736:
6721:John H. Conway
6718:
6717:
6716:
6715:
6714:
6707:
6700:
6678:
6671:
6651:H.S.M. Coxeter
6648:
6636:
6633:
6630:
6629:
6614:
6599:
6583:
6582:
6580:
6577:
6574:
6573:
6566:
6559:
6552:
6545:
6538:
6531:
6524:
6517:
6510:
6507:
6503:
6502:
6495:
6488:
6481:
6474:
6467:
6460:
6453:
6446:
6439:
6431:
6430:
6424:
6421:
6415:
6412:
6404:
6401:
6393:
6390:
6384:
6381:
6373:
6370:
6362:
6359:
6351:
6348:
6345:
6337:
6336:
6297:
6258:
6190:
6122:
6083:
6015:
5947:
5879:
5811:
5803:
5802:
5795:
5791:runcitruncated
5788:
5784:cantitruncated
5781:
5774:
5767:
5760:
5753:
5746:
5741:
5737:
5736:
5732:
5731:
5724:
5717:
5710:
5703:
5696:
5689:
5682:
5675:
5668:
5665:
5661:
5660:
5653:
5646:
5639:
5632:
5625:
5618:
5611:
5604:
5597:
5589:
5588:
5582:
5579:
5573:
5570:
5562:
5559:
5551:
5548:
5542:
5539:
5531:
5528:
5520:
5517:
5509:
5506:
5503:
5495:
5494:
5455:
5416:
5377:
5309:
5270:
5231:
5192:
5124:
5085:
5077:
5076:
5069:
5065:runcitruncated
5062:
5058:cantitruncated
5055:
5048:
5041:
5034:
5027:
5020:
5015:
5011:
5010:
5003:
5000:
4979:
4968:symmetry), 32
4964:
4949:symmetry), 16
4945:
4916:
4913:
4823:
4820:
4817:
4816:
4807:
4788:
4787:
4778:
4768:
4767:
4763:
4762:
4744:
4726:
4725:
4717:
4716:
4714:
4712:
4708:
4707:
4700:
4693:
4689:
4688:
4685:
4682:
4679:
4676:
4675:Coxeter plane
4672:
4671:
4669:
4667:
4665:
4659:
4658:
4651:
4644:
4637:
4633:
4632:
4629:
4625:
4622:
4619:
4615:
4611:
4608:
4605:
4602:
4588:
4585:
4577:
4576:
4572:
4569:
4566:
4562:
4559:
4551:
4548:
4545:
4544:
4540:
4536:
4533:
4528:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4498:
4495:
4492:
4489:
4486:
4481:
4442:
4439:
4435:
4434:
4430:
4426:
4422:
4419:
4416:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4367:
4328:
4325:
4321:
4317:
4316:
4312:
4308:
4304:
4301:
4298:
4295:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4249:
4210:
4207:
4203:
4199:
4198:
4194:
4190:
4187:
4184:
4181:
4178:
4175:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4126:
4087:
4084:
4080:
4079:
4075:
4071:
4068:
4065:
4062:
4059:
4056:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4016:
3977:
3974:
3970:
3969:
3965:
3961:
3957:
3954:
3951:
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3945:
3942:
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3934:
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3928:
3925:
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3913:
3910:
3907:
3902:
3863:
3860:
3856:
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3847:
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3840:
3837:
3834:
3831:
3828:
3825:
3822:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3788:
3749:
3746:
3742:
3741:
3737:
3733:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3674:
3635:
3632:
3628:
3624:
3623:
3619:
3615:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3556:
3517:
3514:
3510:
3506:
3505:
3501:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3433:
3394:
3391:
3387:
3386:
3382:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3340:
3337:
3334:
3331:
3328:
3325:
3286:
3283:
3279:
3278:
3274:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3229:
3226:
3223:
3220:
3217:
3178:
3175:
3171:
3170:
3166:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3118:
3115:
3112:
3109:
3070:
3067:
3063:
3062:
3058:
3054:
3051:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3002:
2999:
2996:
2993:
2990:
2951:
2948:
2944:
2943:
2939:
2936:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2884:
2881:
2878:
2875:
2836:
2833:
2832:
2829:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2785:
2746:
2743:
2717:
2714:
2713:
2712:
2700:
2694:
2689:
2686:
2683:
2677:
2672:
2667:
2664:
2661:
2655:
2650:
2645:
2642:
2636:
2633:
2629:
2601:
2598:
2560:
2559:
2546:
2545:Uniform index
2542:
2541:
2536:
2532:
2531:
2530:, , order 384
2527:
2524:
2522:Symmetry group
2518:
2517:
2508:
2502:
2501:
2498:
2494:
2493:
2490:
2486:
2485:
2469:
2466:
2462:
2461:
2412:
2409:
2405:
2404:
2365:
2359:
2358:
2354:
2351:
2345:
2344:
2339:
2335:
2334:
2317:
2316:
2307:
2304:
2295:
2294:
2291:
2288:
2285:
2282:
2278:
2275:
2259:
2256:
2250:
2247:
2244:
2243:
2241:
2239:
2235:
2234:
2227:
2220:
2216:
2215:
2212:
2209:
2206:
2203:
2202:Coxeter plane
2199:
2198:
2196:
2194:
2192:
2186:
2185:
2178:
2171:
2164:
2160:
2159:
2156:
2152:
2149:
2146:
2142:
2138:
2135:
2132:
2129:
2115:
2112:
2111:
2110:
2098:
2094:
2089:
2084:
2081:
2078:
2075:
2072:
2066:
2063:
2058:
2053:
2050:
2047:
2044:
2038:
2035:
2032:
2026:
2023:
2020:
2016:
1988:
1985:
1937:
1936:
1923:
1922:Uniform index
1919:
1918:
1913:
1909:
1908:
1907:, , order 384
1904:
1901:
1899:Symmetry group
1895:
1894:
1885:
1879:
1878:
1875:
1871:
1870:
1867:
1863:
1862:
1846:
1843:
1839:
1838:
1789:
1786:
1782:
1781:
1742:
1736:
1735:
1731:
1728:
1722:
1721:
1716:
1712:
1711:
1685:
1684:
1675:
1672:
1657:
1656:
1654:
1652:
1648:
1647:
1640:
1633:
1629:
1628:
1625:
1622:
1619:
1616:
1615:Coxeter plane
1612:
1611:
1609:
1607:
1605:
1599:
1598:
1591:
1584:
1577:
1573:
1572:
1569:
1565:
1562:
1559:
1555:
1551:
1548:
1545:
1542:
1528:
1525:
1524:
1523:
1520:
1516:
1513:
1510:
1507:
1504:
1492:
1489:
1488:
1487:
1475:
1471:
1466:
1461:
1458:
1455:
1452:
1449:
1443:
1440:
1435:
1430:
1427:
1424:
1421:
1415:
1412:
1407:
1402:
1399:
1396:
1393:
1387:
1384:
1381:
1377:
1351:truncated cube
1342:
1339:
1295:
1294:
1281:
1280:Uniform index
1277:
1276:
1271:
1267:
1266:
1265:, , order 384
1262:
1259:
1257:Symmetry group
1253:
1252:
1243:
1237:
1236:
1233:
1229:
1228:
1225:
1221:
1220:
1204:
1201:
1197:
1196:
1147:
1144:
1140:
1139:
1100:
1094:
1093:
1089:
1086:
1080:
1079:
1074:
1070:
1069:
1053:
1052:
1043:
1040:
1028:
1027:
1024:
1021:
1018:
1015:
1012:
996:
993:
990:
989:
978:
959:and a central
928:
925:
922:
921:
908:
895:
876:
875:
873:
871:
867:
866:
859:
852:
848:
847:
844:
841:
838:
835:
834:Coxeter plane
831:
830:
828:
826:
824:
818:
817:
810:
803:
796:
792:
791:
788:
784:
781:
778:
774:
770:
767:
764:
761:
747:
744:
743:
742:
730:
726:
721:
716:
713:
710:
707:
701:
698:
695:
689:
686:
683:
677:
674:
671:
667:
651:
648:
635:
632:
596:
595:
582:
581:Uniform index
578:
577:
572:
568:
567:
566:, , order 384
563:
560:
558:Symmetry group
554:
553:
544:
538:
537:
534:
530:
529:
526:
522:
521:
510:
507:
503:
502:
465:
462:
458:
457:
418:
412:
411:
407:
404:
398:
397:
392:
388:
387:
374:
373:
364:
361:
344:) is a convex
328:
327:
321:
314:
313:
264:
215:
165:
164:
115:
66:
13:
10:
9:
6:
4:
3:
2:
7419:
7408:
7405:
7404:
7402:
7391:
7387:
7383:
7378:
7375:
7371:
7368:
7366:
7359:
7352:
7346:
7344:
7340:
7337:
7335:
7331:
7327:
7323:
7320:
7318:
7314:
7311:
7309:
7305:
7301:
7300:
7297:
7295:
7293:
7290:
7288:
7284:
7281:
7279:
7276:
7274:
7271:
7270:
7267:
7265:
7263:
7260:
7258:
7254:
7251:
7249:
7246:
7244:
7241:
7240:
7237:
7235:
7228:
7221:
7215:
7213:
7210:
7208:
7204:
7201:
7199:
7196:
7194:
7191:
7190:
7187:
7185:
7178:
7171:
7165:
7163:
7160:
7158:
7154:
7151:
7149:
7146:
7144:
7141:
7140:
7137:
7135:
7128:
7122:
7120:
7117:
7115:
7111:
7108:
7106:
7103:
7101:
7098:
7097:
7094:
7092:
7090:
7087:
7085:
7081:
7078:
7076:
7073:
7071:
7068:
7067:
7064:
7060:
7057:
7055:
7052:
7050:
7049:Demitesseract
7047:
7045:
7041:
7038:
7036:
7033:
7031:
7028:
7027:
7024:
7020:
7017:
7015:
7013:
7010:
7008:
7004:
7001:
6999:
6996:
6994:
6991:
6990:
6987:
6984:
6982:
6979:
6977:
6974:
6972:
6969:
6967:
6964:
6962:
6959:
6958:
6955:
6949:
6946:
6942:
6935:
6931:
6924:
6920:
6915:
6911:
6906:
6902:
6897:
6895:
6893:
6889:
6879:
6875:
6873:
6871:
6867:
6863:
6861:
6859:
6855:
6851:
6849:
6846:
6845:
6840:
6836:
6828:
6823:
6821:
6816:
6814:
6809:
6808:
6805:
6800:
6797:
6794:
6791:
6787:
6786:
6782:
6775:
6770:
6768:
6765:
6762:
6759:
6754:
6750:
6749:
6747:
6744:
6741:
6734:
6730:
6726:
6722:
6719:
6712:
6708:
6705:
6701:
6698:
6694:
6693:
6692:
6690:
6686:
6682:
6679:
6676:
6672:
6669:
6668:0-486-61480-8
6665:
6661:
6660:
6655:
6654:
6652:
6649:
6646:
6642:
6639:
6638:
6634:
6625:
6618:
6615:
6610:
6603:
6600:
6595:
6588:
6585:
6578:
6571:
6567:
6564:
6560:
6557:
6553:
6550:
6546:
6543:
6539:
6536:
6532:
6529:
6525:
6522:
6518:
6515:
6511:
6505:
6504:
6500:
6496:
6493:
6489:
6486:
6482:
6479:
6475:
6472:
6468:
6465:
6461:
6458:
6454:
6451:
6447:
6444:
6440:
6438:
6433:
6432:
6422:
6413:
6402:
6391:
6382:
6371:
6360:
6349:
6346:
6344:
6339:
6338:
6298:
6259:
6191:
6123:
6084:
6016:
5948:
5880:
5812:
5810:
5805:
5804:
5801:
5798:omnitruncated
5796:
5794:
5789:
5787:
5782:
5780:
5775:
5773:
5768:
5766:
5761:
5759:
5754:
5752:
5747:
5745:
5742:
5739:
5738:
5733:
5729:
5725:
5722:
5718:
5715:
5711:
5708:
5704:
5701:
5697:
5694:
5690:
5687:
5683:
5680:
5676:
5673:
5669:
5663:
5662:
5658:
5654:
5651:
5647:
5644:
5640:
5637:
5633:
5630:
5626:
5623:
5619:
5616:
5612:
5609:
5605:
5602:
5598:
5596:
5591:
5590:
5580:
5571:
5560:
5549:
5540:
5529:
5518:
5507:
5504:
5502:
5497:
5496:
5456:
5417:
5378:
5310:
5271:
5232:
5193:
5125:
5086:
5084:
5079:
5078:
5075:
5072:omnitruncated
5070:
5068:
5063:
5061:
5056:
5054:
5049:
5047:
5042:
5040:
5035:
5033:
5028:
5026:
5021:
5019:
5016:
5013:
5012:
5007:
5001:
4999:
4997:
4993:
4989:
4984:
4982:
4975:
4971:
4967:
4960:
4956:
4952:
4948:
4941:
4937:
4933:
4926:
4921:
4914:
4912:
4910:
4906:
4902:
4898:
4894:
4890:
4848:
4844:
4840:
4833:
4828:
4821:
4812:
4808:
4803:
4799:
4798:
4795:
4783:
4774:
4769:
4764:
4759:
4755:
4749:
4742:
4738:
4732:
4727:
4722:
4715:
4713:
4710:
4709:
4705:
4701:
4698:
4694:
4691:
4690:
4683:
4677:
4674:
4673:
4670:
4668:
4666:
4664:
4661:
4660:
4656:
4652:
4649:
4645:
4642:
4638:
4635:
4634:
4623:
4609:
4603:
4601:
4600:Coxeter plane
4598:
4597:
4594:
4586:
4584:
4582:
4573:
4570:
4567:
4563:
4560:
4557:
4556:
4555:
4549:
4534:
4532:
4517:
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4493:
4490:
4487:
4485:
4482:
4443:
4437:
4420:
4415:
4403:
4400:
4397:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4373:
4371:
4368:
4329:
4319:
4302:
4294:
4285:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4253:
4250:
4211:
4201:
4188:
4174:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4130:
4127:
4088:
4082:
4069:
4066:
4063:
4060:
4057:
4055:
4034:
4031:
4028:
4025:
4022:
4020:
4017:
3978:
3972:
3955:
3952:
3949:
3946:
3943:
3938:
3920:
3917:
3914:
3911:
3908:
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3903:
3864:
3854:
3841:
3838:
3835:
3832:
3829:
3821:
3806:
3803:
3800:
3797:
3794:
3792:
3789:
3750:
3744:
3727:
3724:
3721:
3718:
3715:
3704:
3692:
3689:
3686:
3683:
3680:
3678:
3675:
3636:
3626:
3609:
3606:
3603:
3600:
3597:
3583:
3574:
3571:
3568:
3565:
3562:
3560:
3557:
3518:
3508:
3495:
3489:
3486:
3483:
3480:
3463:
3457:
3454:
3451:
3448:
3445:
3437:
3434:
3395:
3389:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3344:
3329:
3326:
3287:
3281:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3233:
3221:
3218:
3179:
3173:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3122:
3113:
3110:
3071:
3065:
3052:
3050:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3006:
3000:
2991:
2952:
2946:
2937:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2888:
2879:
2876:
2837:
2835:
2830:
2828:
2826:
2822:
2798:
2792:
2789:
2786:
2747:
2741:
2740:
2737:
2735:
2731:
2727:
2722:
2715:
2698:
2692:
2687:
2684:
2681:
2675:
2670:
2665:
2662:
2659:
2653:
2648:
2643:
2640:
2634:
2631:
2627:
2619:
2618:
2617:
2615:
2610:
2607:
2599:
2597:
2595:
2591:
2587:
2583:
2579:
2575:
2571:
2567:
2558:
2557:
2552:
2551:
2543:
2540:
2533:
2523:
2519:
2513:
2507:
2506:Vertex figure
2503:
2495:
2487:
2484:
2479:
2474:
2470:
2467:
2463:
2459:
2455:
2454:
2447:
2443:
2442:
2435:
2431:
2430:
2423:
2419:
2418:
2413:
2410:
2406:
2364:
2360:
2350:
2346:
2343:
2336:
2328:
2323:
2318:
2315:
2311:
2305:
2303:
2301:
2292:
2289:
2286:
2283:
2279:
2276:
2273:
2269:
2268:
2267:
2265:
2257:
2255:
2248:
2242:
2240:
2237:
2236:
2232:
2228:
2225:
2221:
2218:
2217:
2210:
2204:
2201:
2200:
2197:
2195:
2193:
2191:
2188:
2187:
2183:
2179:
2176:
2172:
2169:
2165:
2162:
2161:
2150:
2136:
2130:
2128:
2127:Coxeter plane
2125:
2124:
2121:
2113:
2096:
2087:
2082:
2079:
2076:
2070:
2064:
2056:
2051:
2048:
2042:
2036:
2033:
2030:
2024:
2021:
2018:
2014:
2006:
2005:
2004:
2002:
1997:
1994:
1986:
1984:
1982:
1978:
1974:
1970:
1966:
1962:
1958:
1954:
1947:
1943:
1935:
1934:
1929:
1928:
1920:
1917:
1910:
1900:
1896:
1890:
1884:
1883:Vertex figure
1880:
1872:
1864:
1861:
1856:
1851:
1847:
1844:
1840:
1836:
1832:
1831:
1824:
1820:
1819:
1812:
1808:
1807:
1800:
1796:
1795:
1790:
1787:
1783:
1741:
1737:
1727:
1723:
1720:
1713:
1710:
1706:
1701:
1696:
1691:
1686:
1683:
1679:
1673:
1671:
1669:
1664:
1655:
1653:
1650:
1649:
1645:
1641:
1638:
1634:
1631:
1630:
1623:
1617:
1614:
1613:
1610:
1608:
1606:
1604:
1601:
1600:
1596:
1592:
1589:
1585:
1582:
1578:
1575:
1574:
1563:
1549:
1543:
1541:
1540:Coxeter plane
1538:
1537:
1534:
1526:
1521:
1517:
1514:
1511:
1508:
1505:
1502:
1498:
1497:
1496:
1490:
1473:
1464:
1459:
1456:
1453:
1447:
1441:
1433:
1428:
1425:
1419:
1413:
1405:
1400:
1397:
1391:
1385:
1382:
1379:
1375:
1367:
1366:
1365:
1363:
1358:
1356:
1352:
1348:
1340:
1338:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1305:
1301:
1293:
1292:
1287:
1286:
1278:
1275:
1268:
1258:
1254:
1248:
1242:
1241:Vertex figure
1238:
1230:
1222:
1219:
1214:
1209:
1205:
1202:
1198:
1194:
1190:
1189:
1182:
1178:
1177:
1170:
1166:
1165:
1158:
1154:
1153:
1148:
1145:
1141:
1099:
1095:
1085:
1081:
1078:
1071:
1064:
1059:
1054:
1051:
1047:
1041:
1039:
1037:
1033:
1025:
1022:
1019:
1016:
1013:
1010:
1009:
1008:
1006:
1002:
994:
988:
983:
979:
977:
972:
968:
964:
962:
958:
957:square cupola
954:
950:
946:
942:
941:cubic cupolae
938:
933:
926:
919:
913:
909:
906:
900:
896:
891:
887:
886:
883:
874:
872:
869:
868:
864:
860:
857:
853:
850:
849:
842:
836:
833:
832:
829:
827:
825:
823:
820:
819:
815:
811:
808:
804:
801:
797:
794:
793:
782:
768:
762:
760:
759:Coxeter plane
757:
756:
753:
745:
728:
719:
714:
711:
705:
699:
696:
693:
687:
684:
681:
675:
672:
669:
665:
657:
656:
655:
649:
647:
645:
641:
633:
631:
629:
625:
621:
617:
613:
606:
602:
594:
593:
588:
587:
579:
576:
569:
559:
555:
549:
543:
542:Vertex figure
539:
531:
523:
520:
515:
511:
508:
504:
500:
496:
495:
488:
484:
483:
476:
472:
471:
466:
463:
459:
417:
413:
403:
399:
396:
389:
385:
380:
375:
372:
368:
362:
360:
357:
355:
351:
347:
343:
339:
335:
326:
325:Coxeter plane
319:
315:
269:
265:
220:
216:
171:
167:
125:
120:
116:
71:
67:
27:
22:
18:
7369:
7338:
7329:
7321:
7312:
7303:
7283:10-orthoplex
7019:Dodecahedron
6940:
6929:
6918:
6909:
6900:
6891:
6887:
6877:
6869:
6865:
6857:
6853:
6799:t0123{4,3,3}
6752:
6745:
6724:
6710:
6703:
6696:
6680:
6674:
6657:
6644:
6617:
6602:
6587:
4990:and uniform
4985:
4977:
4962:
4954:
4943:
4935:
4931:
4929:
4924:
4842:
4838:
4836:
4831:
4578:
4553:
4530:
4413:
4292:
4172:
4053:
3936:
3819:
3702:
3581:
3461:
3342:
3231:
3120:
3004:
2886:
2824:
2787:
2723:
2719:
2611:
2603:
2600:Construction
2573:
2569:
2565:
2563:
2554:
2548:
2452:
2440:
2428:
2416:
2313:
2296:
2261:
2252:
1998:
1990:
1987:Construction
1960:
1956:
1952:
1950:
1931:
1925:
1829:
1817:
1805:
1793:
1681:
1660:
1494:
1359:
1344:
1341:Construction
1327:cuboctahedra
1318:
1314:
1310:
1308:
1289:
1283:
1187:
1175:
1163:
1151:
1049:
1029:
998:
976:cubic cupola
934:
930:
653:
637:
634:Construction
615:
611:
609:
590:
584:
493:
481:
469:
370:
358:
341:
337:
331:
7292:10-demicube
7253:9-orthoplex
7203:8-orthoplex
7153:7-orthoplex
7110:6-orthoplex
7080:5-orthoplex
7035:Pentachoron
7023:Icosahedron
6998:Tetrahedron
6796:t013{4,3,3}
6793:t013{3,3,4}
5777:bitruncated
5763:cantellated
5051:bitruncated
5037:cantellated
4847:alternation
4550:Projections
2535:Properties
2258:Projections
1912:Properties
1702:centered on
1491:Projections
1270:Properties
995:Projections
571:Properties
350:runcination
7278:10-simplex
7262:9-demicube
7212:8-demicube
7162:7-demicube
7119:6-demicube
7089:5-demicube
7003:Octahedron
6790:t03{4,3,3}
6635:References
6411:tr{3,3,4}
6400:2t{3,3,4}
6380:rr{3,3,4}
5770:runcinated
5569:tr{4,3,3}
5558:2t{4,3,3}
5538:rr{4,3,3}
5044:runcinated
4988:icosahedra
4972:, with 96
4951:icosahedra
4909:tetrahedra
4897:icosahedra
4893:snub cubes
2724:Seen in a
1355:tetrahedra
1036:octahedron
905:tetrahedra
894:Wireframe
620:tetrahedra
348:, being a
7326:orthoplex
7248:9-simplex
7198:8-simplex
7148:7-simplex
7105:6-simplex
7075:5-simplex
7044:Tesseract
6656:Coxeter,
6641:T. Gosset
6624:"s3s3s4x"
6609:"s3s3s4s"
6369:t{3,3,4}
6358:r{3,3,4}
5756:truncated
5749:rectified
5527:t{4,3,3}
5516:r{4,3,3}
5074:tesseract
5067:tesseract
5060:tesseract
5053:tesseract
5046:tesseract
5039:tesseract
5032:tesseract
5030:truncated
5025:tesseract
5023:rectified
5018:tesseract
4961:but with
4905:octahedra
2716:Structure
2592:, and 32
2497:Vertices
2249:Structure
2071:±
2043:±
2031:±
2019:±
1979:, and 32
1874:Vertices
1519:envelope.
1448:±
1420:±
1392:±
1380:±
1333:, and 32
1232:Vertices
955:into two
953:dissected
927:Structure
706:±
694:±
682:±
670:±
640:tesseract
626:, and 32
533:Vertices
354:tesseract
26:Tesseract
7401:Category
7380:Topics:
7343:demicube
7308:polytope
7302:Uniform
7063:600-cell
7059:120-cell
7012:Demicube
6986:Pentagon
6966:Triangle
6435:Schlegel
6429:{3,3,4}
6420:{3,3,4}
6389:{3,3,4}
6347:{3,3,4}
6341:Schläfli
5593:Schlegel
5587:{4,3,3}
5578:{4,3,3}
5547:{4,3,3}
5505:{4,3,3}
5499:Schläfli
4889:symmetry
2730:f-vector
2357:{3,3,4}
1734:{3,3,4}
1092:{4,3,3}
410:{4,3,3}
334:geometry
7317:simplex
7287:10-cube
7054:24-cell
7040:16-cell
6981:Hexagon
6835:regular
6437:diagram
6426:0,1,2,3
6409:{3,3,4}
6398:{3,3,4}
6378:{3,3,4}
6367:{3,3,4}
6356:{3,3,4}
5809:diagram
5807:Coxeter
5800:16-cell
5793:16-cell
5786:16-cell
5779:16-cell
5772:16-cell
5765:16-cell
5758:16-cell
5751:16-cell
5744:16-cell
5735:
5595:diagram
5584:0,1,2,3
5567:{4,3,3}
5556:{4,3,3}
5536:{4,3,3}
5525:{4,3,3}
5514:{4,3,3}
5083:diagram
5081:Coxeter
4484:tr{4,3}
4370:{8}×{ }
4252:{6}×{ }
4129:tr{3,3}
2827:-figure
2355:0,1,2,3
2281:image).
1794:3.4.4.4
1164:3.4.3.4
644:16-cell
618:has 16
124:16-cell
7257:9-cube
7207:8-cube
7157:7-cube
7114:6-cube
7084:5-cube
6971:Square
6848:Family
6731:
6727:2008,
6687:
6666:
6343:symbol
5501:symbol
4953:(with
4942:(with
4887:, and
4692:Graph
4636:Graph
4587:Images
4575:image.
4565:image.
3385:= 192
3277:= 192
3169:= 192
3061:= 192
2942:= 384
2831:Notes
2679:
2657:
2638:
2539:convex
2489:Edges
2465:Faces
2408:Cells
2219:Graph
2163:Graph
2114:Images
2068:
2040:
2028:
1916:convex
1866:Edges
1842:Faces
1785:Cells
1632:Graph
1576:Graph
1527:Images
1445:
1417:
1389:
1274:convex
1224:Edges
1200:Faces
1143:Cells
951:being
943:and a
851:Graph
795:Graph
746:Images
703:
691:
679:
575:convex
525:Edges
506:Faces
461:Cells
6976:p-gon
6579:Notes
6417:0,1,3
6406:0,1,2
5740:Name
5575:0,1,3
5564:0,1,2
5014:Name
4903:, 32
4899:, 24
4895:, 16
4433:= 24
4315:= 32
4197:= 16
4078:= 48
3968:= 96
3850:= 64
3740:= 96
3622:= 96
3504:= 64
3049:3.( )
2934:4.( )
2790:-face
2588:, 24
2584:, 16
2578:cells
2572:, or
2453:4.4.6
2441:4.4.8
2429:4.6.6
2417:4.6.8
2338:Type
2264:cells
1977:cubes
1975:, 24
1971:, 16
1965:cells
1959:, or
1830:4.4.6
1818:4.4.4
1806:3.6.6
1732:0,1,3
1715:Type
1329:, 24
1325:, 16
1317:, or
1188:3.4.4
1176:4.4.8
1152:3.4.4
1090:0,1,3
1073:Type
624:cubes
622:, 32
494:4.4.4
482:3.4.4
470:3.3.3
391:Type
7334:cube
7007:Cube
6837:and
6729:ISBN
6685:ISBN
6664:ISBN
4976:(as
4959:cube
4930:The
4837:The
4543:= 8
4186:( )
3493:{ }
3327:{ }
3219:{ }
3111:{ }
2992:{ }
2877:( )
2612:The
2580:: 8
2564:The
2500:384
2492:768
2476:128
2471:288
2468:464
1967:: 8
1951:The
1877:192
1869:480
1853:240
1845:368
1707:and
1360:The
1309:The
1235:192
1227:480
1211:192
1206:128
1203:368
935:The
610:The
528:192
517:144
509:208
340:(or
336:, a
320:in B
6883:(p)
6395:1,2
6386:0,3
6375:0,2
6364:0,1
5553:1,2
5544:0,3
5533:0,2
5522:0,1
4934:or
4841:or
4794:Net
4628:/ D
4618:/ A
4614:/ D
4019:{8}
3905:{4}
3791:{6}
3677:{4}
3559:{4}
3436:{6}
3343:192
3232:192
3121:192
3005:192
2887:384
2553:21
2483:{8}
2481:48
2478:{6}
2473:{4}
2450:32
2438:24
2426:16
2411:80
2155:/ D
2145:/ A
2141:/ D
1946:Net
1930:20
1860:{6}
1858:64
1855:{4}
1850:{3}
1848:64
1827:32
1815:24
1803:16
1788:80
1568:/ D
1558:/ A
1554:/ D
1304:Net
1288:19
1218:{8}
1216:48
1213:{4}
1208:{3}
1185:32
1173:24
1161:16
1146:80
787:/ D
777:/ A
773:/ D
614:or
605:Net
589:15
536:64
519:{4}
514:{3}
512:64
491:32
479:32
467:16
464:80
408:0,3
7403::
7388:•
7384:•
7364:21
7360:•
7357:k1
7353:•
7350:k2
7328:•
7285:•
7255:•
7233:21
7229:•
7226:41
7222:•
7219:42
7205:•
7183:21
7179:•
7176:31
7172:•
7169:32
7155:•
7133:21
7129:•
7126:22
7112:•
7082:•
7061:•
7042:•
7021:•
7005:•
6937:/
6926:/
6916:/
6907:/
6885:/
6737:n1
6713:,
6706:,
6699:,
6653::
6643::
6230:=
6162:=
6055:=
5987:=
5919:=
5851:=
5349:=
5164:=
4998:.
4965:2d
4539:/B
4515:12
4500:24
4497:24
4494:24
4488:48
4425:/B
4418:*
4414:24
4374:16
4307:/A
4300:*
4293:32
4256:12
4193:/A
4173:16
4148:12
4145:12
4142:12
4139:24
4074:/B
4067:1
4054:48
3960:/A
3953:1
3937:96
3846:/A
3839:1
3820:64
3732:/A
3725:0
3703:96
3614:/A
3607:0
3582:96
3500:/A
3462:64
3381:/A
3374:1
3273:/A
3266:1
3165:/A
3158:1
3057:/A
2596:.
2568:,
2556:22
2550:20
2414:8
2302:.
2003::
1983:.
1955:,
1933:21
1927:19
1791:8
1337:.
1313:,
1291:20
1285:18
1149:8
963:.
920:.
907:.
592:16
586:14
356:.
7372:-
7370:n
7362:k
7355:2
7348:1
7341:-
7339:n
7332:-
7330:n
7324:-
7322:n
7315:-
7313:n
7306:-
7304:n
7231:4
7224:2
7217:1
7181:3
7174:2
7167:1
7131:2
7124:1
6953:n
6951:H
6944:2
6941:G
6933:4
6930:F
6922:8
6919:E
6913:7
6910:E
6904:6
6901:E
6892:n
6888:D
6881:2
6878:I
6870:n
6866:B
6858:n
6854:A
6826:e
6819:t
6812:v
6776:.
6739:)
6626:.
6611:.
6596:.
6508:4
6506:B
6423:t
6414:t
6403:t
6392:t
6383:t
6372:t
6361:t
6353:1
6350:t
5666:4
5664:B
5581:t
5572:t
5561:t
5550:t
5541:t
5530:t
5519:t
5511:1
5508:t
4980:s
4978:C
4963:D
4955:T
4946:h
4944:T
4686:3
4684:A
4680:4
4678:F
4630:3
4626:2
4624:B
4620:2
4616:4
4612:3
4610:B
4606:4
4604:B
4541:3
4537:4
4535:B
4531:8
4527:*
4524:*
4521:*
4518:6
4512:8
4509:0
4506:0
4503:0
4491:0
4440:3
4438:B
4431:1
4429:A
4427:2
4423:4
4421:B
4410:*
4407:*
4404:2
4401:0
4398:0
4395:4
4392:4
4389:0
4386:8
4383:8
4380:0
4377:8
4326:1
4324:A
4322:2
4320:B
4313:1
4311:A
4309:2
4305:4
4303:B
4297:*
4289:*
4286:0
4283:3
4280:0
4277:3
4274:0
4271:2
4268:6
4265:0
4262:6
4259:6
4208:1
4206:A
4204:2
4202:A
4195:3
4191:4
4189:B
4183:*
4180:*
4177:*
4169:0
4166:0
4163:4
4160:0
4157:6
4154:4
4151:0
4135:3
4133:f
4085:3
4083:A
4076:2
4072:4
4070:B
4064:1
4061:0
4058:0
4050:*
4047:*
4044:*
4041:*
4038:*
4035:4
4032:4
4029:0
4026:0
4023:8
3975:2
3973:B
3966:1
3964:A
3962:1
3958:4
3956:B
3950:0
3947:1
3944:0
3941:*
3933:*
3930:*
3927:*
3924:*
3921:2
3918:0
3915:2
3912:0
3909:4
3861:1
3859:A
3857:1
3855:A
3848:2
3844:4
3842:B
3836:0
3833:0
3830:1
3827:*
3824:*
3816:*
3813:*
3810:*
3807:0
3804:3
3801:3
3798:0
3795:6
3747:2
3745:A
3738:1
3736:A
3734:1
3730:4
3728:B
3722:1
3719:1
3716:0
3713:*
3710:*
3707:*
3699:*
3696:*
3693:2
3690:0
3687:0
3684:2
3681:4
3633:1
3631:A
3629:1
3627:A
3620:1
3618:A
3616:1
3612:4
3610:B
3604:1
3601:0
3598:1
3595:*
3592:*
3589:*
3586:*
3578:*
3575:0
3572:2
3569:0
3566:2
3563:4
3515:1
3513:A
3511:1
3509:A
3502:2
3498:4
3496:B
3490:0
3487:0
3484:1
3481:1
3478:*
3475:*
3472:*
3469:*
3466:*
3458:0
3455:0
3452:3
3449:3
3446:6
3442:2
3440:f
3392:2
3390:A
3383:1
3379:4
3377:B
3371:1
3368:1
3365:0
3362:1
3359:1
3356:0
3353:1
3350:0
3347:0
3339:*
3336:*
3333:*
3330:2
3284:1
3282:A
3275:1
3271:4
3269:B
3263:1
3260:0
3257:1
3254:1
3251:0
3248:1
3245:0
3242:1
3239:0
3236:*
3228:*
3225:*
3222:2
3176:1
3174:A
3167:1
3163:4
3161:B
3155:0
3152:1
3149:1
3146:0
3143:1
3140:1
3137:0
3134:0
3131:1
3128:*
3125:*
3117:*
3114:2
3068:1
3066:A
3059:1
3055:4
3053:B
3045:0
3042:1
3039:1
3036:1
3033:0
3030:0
3027:0
3024:1
3021:1
3018:1
3015:*
3012:*
3009:*
3001:2
2997:1
2995:f
2949:1
2947:A
2940:4
2938:B
2930:1
2927:1
2924:1
2921:1
2918:1
2915:1
2912:1
2909:1
2906:1
2903:1
2900:1
2897:1
2894:1
2891:1
2882:0
2880:f
2825:k
2819:3
2817:f
2813:2
2811:f
2807:1
2805:f
2801:0
2799:f
2795:k
2793:f
2788:k
2744:4
2742:B
2699:)
2693:2
2688:3
2685:+
2682:1
2676:,
2671:2
2666:2
2663:+
2660:1
2654:,
2649:2
2644:+
2641:1
2635:,
2632:1
2628:(
2528:4
2526:B
2353:t
2329:,
2274:.
2213:3
2211:A
2207:4
2205:F
2157:3
2153:2
2151:B
2147:2
2143:4
2139:3
2137:B
2133:4
2131:B
2097:)
2093:)
2088:2
2083:2
2080:+
2077:1
2074:(
2065:,
2062:)
2057:2
2052:+
2049:1
2046:(
2037:,
2034:1
2025:,
2022:1
2015:(
1905:4
1903:B
1730:t
1626:3
1624:A
1620:4
1618:F
1570:3
1566:2
1564:B
1560:2
1556:4
1552:3
1550:B
1546:4
1544:B
1474:)
1470:)
1465:2
1460:2
1457:+
1454:1
1451:(
1442:,
1439:)
1434:2
1429:+
1426:1
1423:(
1414:,
1411:)
1406:2
1401:+
1398:1
1395:(
1386:,
1383:1
1376:(
1263:4
1261:B
1088:t
845:3
843:A
839:4
837:F
789:3
785:2
783:B
779:2
775:4
771:3
769:B
765:4
763:B
729:)
725:)
720:2
715:+
712:1
709:(
700:,
697:1
688:,
685:1
676:,
673:1
666:(
564:4
562:B
406:t
322:4
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