9219:
9313:
9272:
6472:
8263:
8616:
8989:
8671:
8561:
8406:
8314:
8212:
6925:
9044:
8934:
8726:
8159:
8063:
7961:
7325:
7027:
6770:
8104:
8012:
7906:
7678:
7376:
7274:
7221:
7170:
135:
8365:
7790:
7733:
7623:
7119:
7078:
6976:
6872:
6821:
6627:
7427:
6678:
6576:
4023:
5763:
2737:
9160:
2764:
4088:
5756:
5824:
607:
3750:
5842:
6523:
7570:
7519:
4095:
3900:
3878:
3803:
3781:
494:
3944:
3847:
3792:
6729:
6421:
6380:
6278:
3933:
3911:
3889:
3867:
3836:
3814:
3770:
3739:
3695:
6329:
6227:
3922:
3825:
3728:
3706:
3684:
3236:
3717:
3673:
8877:
8822:
8457:
1768:
8508:
2303:
2263:
2223:
8767:
2283:
2243:
1834:
2290:
2270:
2250:
2237:
1814:
1774:
9101:
1794:
3578:
3496:
3414:
3326:
2277:
2257:
86:
46:
2230:
1841:
4030:
1187:
7849:
4081:
2758:
2217:
2751:
2744:
2731:
2724:
2717:
2710:
2297:
1801:
1781:
1761:
4016:
1828:
1821:
1808:
1788:
184:
3482:
7468:
3564:
3222:
2779:{p} for a p-gon. Regular polygons are self-dual, so the rectification produces the same polygon. The uniform truncation operation doubles the sides to {2p}. The snub operation, alternating the truncation, restores the original polygon {p}. Thus all uniform polygons are also regular. The following operations can be performed on regular polygons to derive the uniform polygons, which are also regular polygons:
3489:
3400:
3312:
3229:
3571:
3407:
3319:
13780:. They are generalisations of the three-dimensional cube, tetrahedron and octahedron, respectively. There are no regular star polytopes in these dimensions. Most uniform higher-dimensional polytopes are obtained by modifying the regular polytopes, or by taking the Cartesian product of polytopes of lower dimensions.
263:. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the same must also be true within each lower-dimensional face of the polytope. In two dimensions (and for two-dimensional faces of higher-dimensional polytopes) a stronger definition is used: only the
13842:
There are two classes of hyperbolic
Coxeter groups, compact and paracompact. Uniform honeycombs generated by compact groups have finite facets and vertex figures, and exist in 2 through 4 dimensions. Paracompact groups have affine or hyperbolic subgraphs, and infinite facets or vertex figures, and
5814:
The four-dimensional uniform star polytopes have not all been enumerated. The ones that have include the 10 regular star (Schläfli-Hess) 4-polytopes and 57 prisms on the uniform star polyhedra, as well as three infinite families: the prisms on the star antiprisms, the duoprisms formed by
970:
and higher. It can be seen as trirectifying its trirectification. Hexication truncates vertices, edges, faces, cells, 4-faces, and 5-faces, replacing each with new facets. 7-faces are replaced by topologically expanded copies of themselves. (Hexication is derived from Greek
808:
and higher. It can be seen as birectifying its birectification. Sterication truncates vertices, edges, faces, and cells, replacing each with new facets. 5-faces are replaced by topologically expanded copies of themselves. (The term, coined by
Johnson, is derived from Greek
5934:
For a given symmetry simplex, a generating point may be placed on any of the four vertices, 6 edges, 4 faces, or the interior volume. On each of these 15 elements there is a point whose images, reflected in the four mirrors, are the vertices of a uniform 4-polytope.
1085:
and higher. Heptellation truncates vertices, edges, faces, cells, 4-faces, 5-faces, and 6-faces, replacing each with new facets. 8-faces are replaced by topologically expanded copies of themselves. (Heptellation is derived from Greek
5819:
two star polygons, and the duoprisms formed by multiplying an ordinary polygon with a star polygon. There is an unknown number of 4-polytope that do not fit into the above categories; over one thousand have been discovered so far.
753:
and higher. Runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves. (The term, coined by
Johnson, is derived from Latin
5942:
followed by inclusion of one to four subscripts 0,1,2,3. If there is one subscript, the generating point is on a corner of the fundamental region, i.e. a point where three mirrors meet. These corners are notated as
917:
and higher. Pentellation truncates vertices, edges, faces, cells, and 4-faces, replacing each with new facets. 6-faces are replaced by topologically expanded copies of themselves. (Pentellation is derived from Greek
9370:
The following table defines 15 forms of truncation. Each form can have from one to four cell types, located in positions 0,1,2,3 as defined above. The cells are labeled by polyhedral truncation notation.
1266:
Vertex figures for single-ringed
Coxeter diagrams can be constructed from the diagram by removing the ringed node, and ringing neighboring nodes. Such vertex figures are themselves vertex-transitive.
1332:
It is useful to classify the uniform polytopes by dimension. This is equivalent to the number of nodes on the
Coxeter diagram, or the number of hyperplanes in the Wythoffian construction. Because (
5554:
5187:
5050:
5681:
4911:
4798:
4683:
4570:
4454:
5853:
in a small number of mirrors. In a 4-dimensional polytope (or 3-dimensional cubic honeycomb) the fundamental region is bounded by four mirrors. A mirror in 4-space is a three-dimensional
4138:, and 53 semiregular star polyhedra. There are also two infinite sets, the star prisms (one for each star polygon) and star antiprisms (one for each rational number greater than 3/2).
1259:
A smaller number of nonreflectional uniform polytopes have a single vertex figure but are not repeated by simple reflections. Most of these can be represented with operations like
5972:(For the two self-dual 4-polytopes, "dual" means a similar 4-polytope in dual position.) Two or more subscripts mean that the generating point is between the corners indicated.
8743:
1269:
Multiringed polytopes can be constructed by a slightly more complicated construction process, and their topology is not a uniform polytope. For example, the vertex figure of a
9289:
8382:
8080:
7095:
6746:
5569:
5209:
5072:
4933:
2323:, which corresponds to the rational number 5/2. Regular star polygons, {p/q}, can be truncated into semiregular star polygons, t{p/q}=t{2p/q}, but become double-coverings if
5574:
5214:
5077:
12574:
12564:
12554:
12533:
12523:
12503:
12492:
12472:
12462:
12441:
12431:
12421:
12408:
12398:
12388:
12378:
9605:
9595:
9585:
9564:
9554:
9534:
9523:
9503:
9493:
9472:
9462:
9452:
9439:
9429:
9419:
9409:
9299:
9279:
8753:
8733:
8392:
8372:
8090:
8070:
7454:
7434:
7085:
6387:
5478:
5459:
5440:
5411:
5401:
5382:
5362:
5343:
5333:
5322:
5312:
5302:
4382:
4363:
4344:
4315:
4305:
4286:
4266:
4247:
4237:
4226:
4216:
4206:
13752:
13742:
13732:
13721:
13711:
13701:
13690:
13680:
13670:
13659:
13649:
13639:
13630:
13620:
13610:
13600:
13557:
13547:
13521:
13511:
13490:
13480:
13469:
13459:
13449:
13425:
13415:
13405:
13362:
13352:
13326:
13295:
13269:
13259:
13220:
13210:
13142:
13111:
13090:
13080:
13054:
13044:
13010:
13000:
12952:
12931:
12900:
12859:
12810:
12718:
12682:
12646:
12602:
12544:
12513:
12482:
12451:
12331:
12321:
12311:
12300:
12290:
12280:
12269:
12259:
12249:
12238:
12228:
12218:
12209:
12199:
12189:
12179:
12147:
12137:
12116:
12106:
12096:
12085:
12065:
12054:
12034:
12025:
12015:
11995:
11957:
11937:
11926:
11906:
11895:
11885:
11875:
11854:
11844:
11835:
11815:
11805:
11775:
11765:
11755:
11744:
11734:
11713:
11703:
11682:
11672:
11653:
11643:
11633:
11583:
11573:
11552:
11542:
11521:
11511:
11500:
11490:
11480:
11461:
11451:
11441:
11419:
11388:
11368:
11357:
11337:
11306:
11297:
11267:
11233:
11213:
11202:
11171:
11161:
11130:
11111:
11091:
11041:
11010:
11000:
10969:
10958:
10938:
10919:
10899:
10871:
10861:
10840:
10830:
10809:
10778:
10749:
10739:
10679:
10669:
10648:
10617:
10596:
10586:
10557:
10547:
10493:
10462:
10441:
10431:
10410:
10400:
10371:
10361:
10333:
10302:
10271:
10211:
10137:
10106:
10054:
10015:
9945:
9893:
9862:
9823:
9732:
9701:
9670:
9631:
9575:
9544:
9513:
9482:
9330:
9320:
9256:
9246:
9236:
9197:
9187:
9177:
9167:
9138:
9128:
9118:
9108:
9081:
9071:
9061:
9051:
9026:
9016:
9006:
8996:
8971:
8961:
8951:
8941:
8914:
8894:
8884:
8859:
8849:
8829:
8804:
8784:
8774:
8708:
8688:
8678:
8653:
8643:
8623:
8598:
8578:
8568:
8535:
8525:
8515:
8494:
8484:
8474:
8433:
8423:
8413:
8341:
8331:
8321:
8300:
8290:
8280:
8239:
8229:
8219:
8186:
8176:
8131:
8121:
8039:
8029:
7988:
7978:
7933:
7923:
7886:
7856:
7827:
7797:
7770:
7740:
7715:
7685:
7660:
7630:
7597:
7577:
7556:
7536:
7495:
7475:
7444:
7403:
7383:
7362:
7342:
7301:
7281:
7238:
7228:
7207:
7197:
7136:
7126:
7044:
7034:
7013:
7003:
6942:
6932:
6889:
6848:
6787:
6695:
6654:
6593:
6530:
6509:
6428:
6397:
6336:
6315:
6234:
6123:
5708:
5698:
5603:
5593:
5583:
5564:
5498:
5488:
5469:
5449:
5430:
5420:
5391:
5372:
5353:
5243:
5233:
5223:
5204:
5106:
5086:
4957:
4928:
4835:
4825:
4710:
4700:
4607:
4471:
4402:
4392:
4373:
4353:
4334:
4324:
4295:
4276:
4257:
4123:
4113:
4103:
4058:
4048:
4038:
3536:
3454:
3372:
3284:
3194:
3103:
3045:
3035:
2987:
2956:
2946:
2907:
2846:
2677:
2668:
2648:
2620:
2591:
2562:
2553:
2533:
2505:
2476:
2447:
2418:
2204:
2194:
2176:
2157:
2148:
2138:
2120:
2101:
2092:
2082:
2064:
2045:
2036:
2026:
2008:
1989:
1980:
1970:
1952:
1738:
1729:
1719:
1701:
1682:
1673:
1663:
1645:
1626:
1617:
1607:
1589:
1570:
1561:
1551:
1533:
1514:
205:
195:
156:
146:
107:
97:
77:
67:
7105:
6756:
6736:
6407:
6133:
6113:
5980:
The 15 constructive forms by family are summarized below. The self-dual families are listed in one column, and others as two columns with shared entries on the symmetric
4938:
13572:
13536:
13500:
13440:
13377:
13341:
13316:
13305:
13280:
13244:
13235:
13195:
13167:
13157:
13131:
13121:
13100:
13069:
13035:
13025:
12972:
12962:
12941:
12921:
12910:
12885:
12874:
12844:
12835:
12825:
12795:
12769:
12759:
12749:
12738:
12728:
12707:
12697:
12671:
12661:
12637:
12627:
12617:
12127:
12075:
12044:
12005:
11947:
11916:
11864:
11825:
11724:
11693:
11662:
11623:
11593:
11562:
11531:
11471:
11409:
11399:
11378:
11347:
11326:
11316:
11287:
11277:
11223:
11192:
11182:
11151:
11140:
11120:
11101:
11081:
11051:
11031:
11020:
10989:
10979:
10948:
10929:
10909:
10851:
10820:
10799:
10789:
10768:
10758:
10729:
10719:
10689:
10658:
10638:
10627:
10607:
10576:
10567:
10537:
10513:
10503:
10482:
10472:
10451:
10420:
10391:
10381:
10323:
10313:
10292:
10282:
10261:
10251:
10240:
10230:
10220:
10201:
10191:
10181:
10147:
10127:
10116:
10096:
10085:
10075:
10065:
10044:
10034:
10025:
10005:
9995:
9965:
9955:
9934:
9924:
9914:
9903:
9883:
9872:
9852:
9843:
9833:
9813:
9783:
9773:
9763:
9752:
9742:
9721:
9711:
9690:
9680:
9661:
9651:
9641:
9350:
9340:
9226:
8904:
8839:
8794:
8698:
8633:
8588:
8545:
8464:
8443:
8351:
8270:
8249:
8196:
8166:
8141:
8111:
8049:
8019:
7998:
7968:
7943:
7913:
7876:
7866:
7817:
7807:
7760:
7750:
7705:
7695:
7650:
7640:
7607:
7587:
7546:
7526:
7505:
7485:
7413:
7393:
7352:
7332:
7311:
7291:
7258:
7248:
7187:
7177:
7156:
7146:
7064:
7054:
6993:
6983:
6962:
6952:
6909:
6899:
6879:
6858:
6838:
6828:
6807:
6797:
6777:
6715:
6705:
6685:
6664:
6644:
6634:
6613:
6603:
6583:
6560:
6550:
6540:
6499:
6489:
6479:
6458:
6448:
6438:
6366:
6356:
6346:
6305:
6295:
6285:
6264:
6254:
6244:
6213:
6203:
6193:
6183:
6173:
6163:
6153:
6143:
6103:
6093:
6083:
6073:
6063:
6053:
6043:
6033:
5723:
5096:
5067:
4967:
4947:
4815:
4720:
4597:
4587:
4491:
4481:
3556:
3546:
3474:
3464:
3392:
3382:
3304:
3294:
3214:
3204:
3123:
3113:
3002:
2897:
2856:
2697:
2640:
2625:
2611:
2582:
2525:
2496:
2467:
2438:
2199:
2186:
2181:
2167:
2162:
2130:
2125:
2111:
2106:
2074:
2069:
2055:
2050:
2018:
2013:
1999:
1994:
1962:
1748:
1711:
1706:
1692:
1655:
1636:
1599:
1580:
1543:
1524:
235:
225:
215:
176:
166:
117:
2687:
2658:
2630:
2601:
2572:
2543:
2515:
2486:
2457:
2428:
13706:
13685:
13562:
13526:
13516:
13430:
13367:
13331:
13285:
13249:
13225:
13200:
13147:
13059:
13015:
12890:
12864:
12849:
12815:
12800:
12687:
12651:
12607:
9294:
9284:
8748:
8738:
8387:
8377:
8085:
8075:
7449:
7439:
7100:
7090:
6751:
6741:
6402:
6392:
6128:
6118:
5713:
4118:
2992:
2692:
2663:
2635:
2606:
2577:
2548:
2520:
2491:
2462:
2433:
1957:
1743:
13747:
13737:
13716:
13675:
13654:
13644:
13625:
13615:
13605:
13567:
13552:
13531:
13495:
13485:
13464:
13454:
13435:
13420:
13410:
13372:
13357:
13336:
13321:
13300:
13290:
13264:
13254:
13230:
13215:
13205:
13162:
13152:
13126:
13116:
13095:
13085:
13064:
13049:
13030:
13020:
13005:
12967:
12957:
12936:
12926:
12905:
12895:
12869:
12854:
12830:
12820:
12805:
12764:
12754:
12733:
12723:
12702:
12692:
12666:
12656:
12632:
12622:
12612:
12569:
12559:
12549:
12528:
12518:
12508:
12487:
12477:
12467:
12446:
12436:
12426:
12403:
12393:
12383:
12326:
12316:
12295:
12285:
12264:
12254:
12233:
12223:
12204:
12194:
12184:
12142:
12132:
12111:
12101:
12080:
12070:
12049:
12039:
12020:
12010:
12000:
11952:
11942:
11921:
11911:
11890:
11880:
11859:
11849:
11830:
11820:
11810:
11770:
11760:
11739:
11729:
11708:
11698:
11677:
11667:
11648:
11638:
11628:
11588:
11578:
11557:
11547:
11526:
11516:
11495:
11485:
11466:
11456:
11446:
11414:
11404:
11383:
11373:
11352:
11342:
11321:
11311:
11292:
11282:
11272:
11228:
11218:
11197:
11187:
11166:
11156:
11135:
11125:
11106:
11096:
11086:
11046:
11036:
11015:
11005:
10984:
10974:
10953:
10943:
10924:
10914:
10904:
10866:
10856:
10835:
10825:
10804:
10794:
10773:
10763:
10744:
10734:
10724:
10684:
10674:
10653:
10643:
10622:
10612:
10591:
10581:
10562:
10552:
10542:
10508:
10498:
10477:
10467:
10446:
10436:
10415:
10405:
10386:
10376:
10366:
10328:
10318:
10297:
10287:
10266:
10256:
10235:
10225:
10206:
10196:
10186:
10142:
10132:
10111:
10101:
10080:
10070:
10049:
10039:
10020:
10010:
10000:
9960:
9950:
9929:
9919:
9898:
9888:
9867:
9857:
9838:
9828:
9818:
9778:
9768:
9747:
9737:
9716:
9706:
9685:
9675:
9656:
9646:
9636:
9600:
9590:
9580:
9559:
9549:
9539:
9518:
9508:
9498:
9477:
9467:
9457:
9434:
9424:
9414:
9345:
9335:
9325:
9251:
9241:
9231:
9192:
9182:
9172:
9133:
9123:
9113:
9076:
9066:
9056:
9021:
9011:
9001:
8966:
8956:
8946:
8909:
8899:
8889:
8854:
8844:
8834:
8799:
8789:
8779:
8703:
8693:
8683:
8648:
8638:
8628:
8593:
8583:
8573:
8540:
8530:
8520:
8489:
8479:
8469:
8438:
8428:
8418:
8346:
8336:
8326:
8295:
8285:
8275:
8244:
8234:
8224:
8191:
8181:
8171:
8136:
8126:
8116:
8044:
8034:
8024:
7993:
7983:
7973:
7938:
7928:
7918:
7881:
7871:
7861:
7822:
7812:
7802:
7765:
7755:
7745:
7710:
7700:
7690:
7655:
7645:
7635:
7602:
7592:
7582:
7551:
7541:
7531:
7500:
7490:
7480:
7408:
7398:
7388:
7357:
7347:
7337:
7306:
7296:
7286:
7253:
7243:
7233:
7202:
7192:
7182:
7151:
7141:
7131:
7059:
7049:
7039:
7008:
6998:
6988:
6957:
6947:
6937:
6904:
6894:
6884:
6853:
6843:
6833:
6802:
6792:
6782:
6710:
6700:
6690:
6659:
6649:
6639:
6608:
6598:
6588:
6555:
6545:
6535:
6504:
6494:
6484:
6453:
6443:
6433:
6361:
6351:
6341:
6310:
6300:
6290:
6259:
6249:
6239:
6208:
6198:
6188:
6168:
6158:
6148:
6098:
6088:
6078:
6058:
6048:
6038:
5718:
5703:
5598:
5588:
5493:
5483:
5464:
5454:
5435:
5425:
5406:
5396:
5377:
5367:
5348:
5338:
5317:
5307:
5238:
5228:
5101:
5091:
4962:
4952:
4830:
4820:
4715:
4705:
4602:
4592:
4486:
4476:
4397:
4387:
4368:
4358:
4339:
4329:
4310:
4300:
4281:
4271:
4252:
4242:
4221:
4211:
4108:
4053:
4043:
3551:
3541:
3469:
3459:
3387:
3377:
3299:
3289:
3209:
3199:
3118:
3108:
3040:
2997:
2951:
2902:
2851:
2682:
2653:
2596:
2567:
2538:
2510:
2481:
2452:
2423:
2143:
2087:
2031:
1975:
1724:
1687:
1668:
1650:
1631:
1612:
1594:
1575:
1556:
1538:
1519:
230:
220:
210:
200:
171:
161:
151:
112:
102:
72:
481:
and higher. A truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges. (The term, coined by
289:) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows
336:
Equivalently, the
Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension. This approach was first used by
13831:
Related to the subject of finite uniform polytopes are uniform honeycombs in
Euclidean and hyperbolic spaces. Euclidean uniform honeycombs are generated by
4162:, applies to all dimensions; it consists of the letter 't', followed by a series of subscripted numbers corresponding to the ringed nodes of the
441:-polytopes in any combination. The resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them.
3994:
1256:, marking active mirrors by rings, have reflectional symmetry, and can be simply constructed by recursive reflections of the vertex figure.
5857:, but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of the
5869:
14614:
5984:. The final 10th row lists the snub 24-cell constructions. This includes all nonprismatic uniform 4-polytopes, except for the
13954:
13875:
681:. A cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically
14004:
2327:
is even. A truncation can also be made with a reverse orientation polygon t{p/(p-q)}={2p/(p-q)}, for example t{5/3}={10/3}.
1206:, removes alternate vertices from a polytope with only even-sided faces. An alternated omnitruncated polytope is called a
588:. A cantellation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically
14049:
13949:, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
5513:
5146:
5009:
1388:
polygons of twice as many sides, t{p}={2p}. The first few regular polygons (and quasiregular forms) are displayed below:
13909:, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
5641:
4873:
4758:
4643:
4533:
4417:
1213:
The resulting polytopes always can be constructed, and are not generally reflective, and also do not in general have
5788:, 17 prisms on the Platonic and Archimedean solids (excluding the cube-prism, which has already been counted as the
1159:
In addition combinations of truncations can be performed which also generate new uniform polytopes. For example, a
12355:
and inactive nodes must be even-order. Half construction have the vertices of an identically ringed construction.
3076:
In three dimensions, the situation gets more interesting. There are five convex regular polyhedra, known as the
13915:
13881:
13864:
8720:
4159:
326:
1289:
Uniform polytopes have equal edge-lengths, and all vertices are an equal distance from the center, called the
1356:+1)-dimensional. Hence, the tilings of two-dimensional space are grouped with the three-dimensional solids.
13836:
13815:, one can obtain 39 new 6-polytopes, 127 new 7-polytopes and 255 new 8-polytopes. A notable example is the
8057:
13930:
8359:
3937:
17:
9218:
14631:
13922:
13832:
1321:
341:
267:
are considered as uniform, disallowing polygons that alternate between two different lengths of edges.
1252:, the arrangement of edges, faces, cells, etc. around each vertex. Uniform polytopes represented by a
457:
cut cells. Each higher operation also cuts lower ones too, so a cantellation also truncates vertices.
13945:
7421:
5785:
3829:
14072:
9794:
8665:
8308:
5924:
4865:
4135:
678:
585:
357:
14042:
10881:
10523:
9038:
8983:
8871:
8816:
8502:
8451:
7072:
6723:
4996:
4749:
3840:
1191:
575:
12584:
11243:
9154:
9150:
8761:
8610:
8400:
3926:
3882:
1260:
1203:
9312:
9271:
6471:
5931:{4,3,4}, by applying the same operations used to generate the Wythoffian uniform 4-polytopes.
14586:
14579:
14572:
10343:
9385:
The red background shows the truncations of the parent, and blue the truncations of the dual.
9095:
8615:
8555:
8262:
8206:
8006:
7955:
7370:
4636:
3904:
3710:
3688:
3617:
1364:
The only one-dimensional polytope is the line segment. It corresponds to the
Coxeter family A
1270:
472:
188:
50:
14111:
14089:
14077:
9382:
The green background is shown on forms that are equivalent to either the parent or the dual.
8988:
8670:
8560:
8405:
8313:
8211:
6924:
14243:
14190:
11248:
9043:
8933:
8928:
8153:
8149:
7727:
7672:
7564:
7513:
5291:
5001:
4195:
4167:
3858:
3807:
3743:
3664:
3613:
2351:
1397:
1385:
682:
589:
318:
298:
271:
90:
8725:
8158:
8062:
7960:
7324:
7026:
6769:
8:
14598:
14497:
14247:
8103:
8098:
8011:
7905:
7843:
7839:
7677:
7462:
7375:
7273:
7220:
7169:
7021:
6672:
3761:
2356:
1152:
260:
14012:
8364:
7789:
7732:
7622:
7118:
7077:
6975:
6871:
6820:
6626:
134:
14467:
14417:
14367:
14324:
14294:
14254:
14217:
14035:
13902:
13784:
9266:
7900:
7784:
7426:
7268:
7215:
7164:
6866:
6815:
6677:
6575:
5850:
5779:
4151:
3071:
1135:
1082:
1028:
967:
914:
862:
805:
750:
674:
581:
13852:
4155:
4022:
3089:
2776:
2369:
1898:
1462:
1174:
If all truncations are applied at once, the operation can be more generally called an
392:
14606:
13950:
13910:
13871:
8257:
7617:
7319:
7113:
6970:
6764:
5816:
4166:, and followed by the Schläfli symbol of the regular seed polytope. For example, the
3948:
3893:
3609:
3156:
1408:
1309:
1301:
290:
253:
139:
13990:
5807:. Both of these special 4-polytope are composed of subgroups of the vertices of the
14610:
14175:
14164:
14153:
14142:
14133:
14124:
14059:
13805:
13798:
13791:
13787:
9159:
6919:
6570:
5917:
4009:
3957:
2810:
1349:
306:
278:
5762:
14200:
14185:
13812:
12367:
9398:
5988:
5981:
5928:
5828:
5800:
5755:
5634:
5506:
5284:
4524:
4188:
4163:
3989:
3168:
3151:
3144:
3096:
2800:
2736:
2408:
2316:
1942:
1504:
1377:
1345:
1253:
337:
330:
322:
302:
264:
61:
5823:
4087:
2763:
1376:
In two dimensions, there is an infinite family of convex uniform polytopes, the
1296:
Uniform polytopes whose circumradius is equal to the edge length can be used as
685:
copies of themselves. (The composite term combines cantellation and truncation)
14550:
13777:
12160:
5985:
5797:
5138:
4147:
3785:
3360:
3163:
3077:
1413:
1274:
1175:
294:
13907:
Geometrical deduction of semiregular from regular polytopes and space fillings
7569:
7518:
6522:
5841:
4094:
1308:
divides into 6 equilateral triangles and is the vertex figure for the regular
14625:
14567:
14455:
14448:
14441:
14405:
14398:
14391:
14355:
14348:
13816:
10160:
6728:
6420:
6379:
6374:
6277:
3982:
3943:
3932:
3846:
3835:
3796:
3749:
3738:
3732:
1313:
1297:
1249:
606:
592:
copies of themselves. (The term, coined by
Johnson, is derived from the verb
365:
282:
6328:
6226:
3921:
3899:
3877:
3824:
3802:
3780:
3727:
3705:
3683:
14507:
9307:
9213:
5804:
4134:
The uniform star polyhedra include a further 4 regular star polyhedra, the
3910:
3888:
3871:
3866:
3813:
3791:
3769:
3716:
3694:
3672:
3443:
3235:
2362:
2312:
1886:
1876:
1866:
1861:
759:
503:
493:
286:
13929:, Philosophical Transactions of the Royal Society of London, Londne, 1954
8876:
8821:
8507:
8456:
14516:
14477:
14427:
14377:
14334:
14304:
14236:
14222:
8766:
5862:
5858:
5834:
There are 3 right dihedral angles (2 intersecting perpendicular mirrors):
3915:
3754:
3721:
3677:
3525:
3175:
1881:
1871:
1229:
745:
9100:
3577:
3495:
3413:
3325:
1767:
395:
can be used for representing rectified forms, with a single subscript:
14502:
14486:
14436:
14386:
14343:
14313:
14227:
13974:
13967:
13960:
5854:
5792:), and two infinite sets: the prisms on the convex antiprisms, and the
4029:
3818:
3699:
3355:
2319:
greater than 2), but these are non-convex. The simplest example is the
2302:
2289:
2282:
2269:
2262:
2249:
2242:
2236:
2222:
1856:
1833:
1455:
1317:
85:
7848:
4080:
2276:
2256:
1813:
1793:
1773:
14558:
14472:
14422:
14372:
14329:
14299:
14268:
13811:
come into play. By placing rings on a nonzero number of nodes of the
13769:
6323:
5888:
5789:
4146:
The
Wythoffian uniform polyhedra and tilings can be defined by their
4074:
3961:
3851:
3270:
2341:
2336:
2320:
2229:
1851:
1840:
1419:
1221:
1195:
45:
4015:
2757:
2216:
1186:
14532:
14287:
14283:
14210:
6517:
6466:
5906:
5902:
5808:
5793:
2750:
2743:
2730:
2723:
2716:
2709:
2346:
2296:
1891:
1800:
1780:
1760:
1445:
1435:
1425:
1381:
1225:
1031:
and higher. It can be seen as tritruncating its trirectification.
256:
245:
53:
13768:
In five and higher dimensions, there are 3 regular polytopes, the
3608:
In addition to these, there are also 13 semiregular polyhedra, or
1827:
1820:
1807:
1787:
14541:
14511:
14278:
14273:
14264:
14205:
13773:
7467:
6621:
6415:
6272:
5895:
5884:
5796:. There are also 41 convex semiregular 4-polytope, including the
3180:
1450:
1440:
1430:
1402:
1305:
1278:
1233:
478:
57:
3620:
on the Platonic solids, as demonstrated in the following table:
865:
and higher. It can be seen as bitruncating its birectification.
183:
14481:
14431:
14381:
14338:
14308:
14259:
14195:
6221:
5877:
3488:
3481:
1088:
920:
482:
3570:
3563:
3228:
3221:
1316:
divides into 8 regular tetrahedra and 6 square pyramids (half
13895:
5950:: vertex of the parent 4-polytope (center of the dual's cell)
3406:
3399:
3318:
3311:
973:
598:
1384:. Truncated regular polygons become bicolored geometrically
14231:
3960:, one for each regular polygon, and a corresponding set of
3774:
3265:
1277:
polytope (all nodes ringed) will always have an irregular
14019:
13878:(Chapter 3: Wythoff's Construction for Uniform Polytopes)
5962:: center of the parent's face (center of the dual's edge)
5956:: center of the parent's edge (center of the dual's face)
5872:
is generated by one of four symmetry groups, as follows:
13839:. Two affine Coxeter groups can be multiplied together.
5927:
in Euclidean 3-space are analogously generated from the
5912:
group : contains only repeated members of the family.
5849:
Every regular polytope can be seen as the images of a
5653:
5525:
5158:
5021:
4882:
4770:
4655:
4542:
4426:
360:. The zeroth rectification is the original form. The (
5644:
5516:
5149:
5012:
4876:
4761:
4646:
4536:
4420:
437:
Truncation operations that can be applied to regular
13986:, Dissertation, Universität Hamburg, Hamburg (2004)
317:
Nearly every uniform polytope can be generated by a
5549:{\displaystyle s{\begin{Bmatrix}p\\q\end{Bmatrix}}}
5182:{\displaystyle t{\begin{Bmatrix}p\\q\end{Bmatrix}}}
5045:{\displaystyle r{\begin{Bmatrix}p\\q\end{Bmatrix}}}
13946:Kaleidoscopes: Selected Writings of H.S.M. Coxeter
13894:, Ph.D. Dissertation, University of Toronto, 1966
5968:: center of the parent's cell (vertex of the dual)
5676:{\displaystyle s{\begin{Bmatrix}p,q\end{Bmatrix}}}
5675:
5548:
5181:
5044:
4906:{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}
4905:
4793:{\displaystyle t{\begin{Bmatrix}q,p\end{Bmatrix}}}
4792:
4678:{\displaystyle t{\begin{Bmatrix}p,q\end{Bmatrix}}}
4677:
4564:
4448:
270:This is a generalization of the older category of
4565:{\displaystyle {\begin{Bmatrix}q,p\end{Bmatrix}}}
4449:{\displaystyle {\begin{Bmatrix}p,q\end{Bmatrix}}}
1327:
1273:regular polytope (with 2 rings) is a pyramid. An
14623:
1248:Uniform polytopes can be constructed from their
13835:and hyperbolic honeycombs are generated by the
12351:rather than ringed nodes. Branches neighboring
1220:The set of polytopes formed by alternating the
13892:The Theory of Uniform Polytopes and Honeycombs
13580:
13385:
13175:
12980:
12777:
12580:
12414:
12155:
11965:
11783:
11601:
11427:
11241:
11059:
10879:
10697:
10521:
10341:
10155:
9973:
9791:
9611:
9445:
9379:-gonal prism is represented as : {n}×{ }.
5632:
5504:
5328:
5133:
4994:
4862:
4747:
4634:
4518:
4408:
4232:
3019:
2973:
2923:
2872:
2823:
2815:
14043:
14020:uniform, convex polytopes in four dimensions:
13763:
677:and higher. It can be seen as truncating its
584:and higher. It can be seen as rectifying its
5938:The extended Schläfli symbols are made by a
1846:
1392:
5909:{5,3,3}, and their ten regular stellations.
14050:
14036:
347:
9209:
8926:
8553:
8204:
7898:
7615:
7266:
6917:
6568:
6219:
1340:-dimensional spherical space, tilings of
1336:+1)-dimensional polytopes are tilings of
13783:In six, seven and eight dimensions, the
5840:
5822:
1185:
1034:There are also higher hexiruncinations:
388:reduced 5-faces to vertices, and so on.
14615:List of regular polytopes and compounds
13984:Vierdimensionale Archimedische Polytope
5975:
1320:), and it is the vertex figure for the
1228:. In three dimensions, this produces a
602:, meaning to cut with a slanted face.)
432:
14:
14624:
13975:Regular and Semi-Regular Polytopes III
5845:Summary chart of truncation operations
3616:, or by performing operations such as
1232:; in four dimensions, this produces a
14002:
13968:Regular and Semi-Regular Polytopes II
13868:The Beauty of Geometry: Twelve Essays
13826:
12363:
9394:
5861:; thus the mirrors form an irregular
5283:
5278:
4187:
4182:
2787:
1380:, the simplest being the equilateral
1095:There are higher heptellations also:
927:There are also higher pentellations:
688:There are higher cantellations also:
614:There are higher cantellations also:
27:Isogonal polytope with uniform facets
13961:Regular and Semi Regular Polytopes I
868:There are higher sterications also:
816:There are higher sterications also:
765:There are higher runcinations also:
309:to be considered polytopes as well.
13939:, 3rd Edition, Dover New York, 1973
3065:
980:There are higher hexications also:
501:There are higher truncations also:
24:
13843:exist in 2 through 10 dimensions.
9365:
5773:
3956:There is also the infinite set of
364:−1)-th rectification is the
281:. Further, star regular faces and
252:of dimension three or higher is a
25:
14643:
13996:
9214:alternated cantitruncated 16-cell
5836:Edges 1 to 2, 0 to 2, and 1 to 3.
4170:is represented by the notation: t
2775:Regular polygons, represented by
2311:There is also an infinite set of
1371:
325:. Notable exceptions include the
13750:
13745:
13740:
13735:
13730:
13719:
13714:
13709:
13704:
13699:
13688:
13683:
13678:
13673:
13668:
13657:
13652:
13647:
13642:
13637:
13628:
13623:
13618:
13613:
13608:
13603:
13598:
13570:
13565:
13560:
13555:
13550:
13545:
13534:
13529:
13524:
13519:
13514:
13509:
13498:
13493:
13488:
13483:
13478:
13467:
13462:
13457:
13452:
13447:
13438:
13433:
13428:
13423:
13418:
13413:
13408:
13403:
13375:
13370:
13365:
13360:
13355:
13350:
13339:
13334:
13329:
13324:
13319:
13314:
13303:
13298:
13293:
13288:
13283:
13278:
13267:
13262:
13257:
13252:
13247:
13242:
13233:
13228:
13223:
13218:
13213:
13208:
13203:
13198:
13193:
13165:
13160:
13155:
13150:
13145:
13140:
13129:
13124:
13119:
13114:
13109:
13098:
13093:
13088:
13083:
13078:
13067:
13062:
13057:
13052:
13047:
13042:
13033:
13028:
13023:
13018:
13013:
13008:
13003:
12998:
12970:
12965:
12960:
12955:
12950:
12939:
12934:
12929:
12924:
12919:
12908:
12903:
12898:
12893:
12888:
12883:
12872:
12867:
12862:
12857:
12852:
12847:
12842:
12833:
12828:
12823:
12818:
12813:
12808:
12803:
12798:
12793:
12767:
12762:
12757:
12752:
12747:
12736:
12731:
12726:
12721:
12716:
12705:
12700:
12695:
12690:
12685:
12680:
12669:
12664:
12659:
12654:
12649:
12644:
12635:
12630:
12625:
12620:
12615:
12610:
12605:
12600:
12572:
12567:
12562:
12557:
12552:
12547:
12542:
12531:
12526:
12521:
12516:
12511:
12506:
12501:
12490:
12485:
12480:
12475:
12470:
12465:
12460:
12449:
12444:
12439:
12434:
12429:
12424:
12419:
12406:
12401:
12396:
12391:
12386:
12381:
12376:
12329:
12324:
12319:
12314:
12309:
12298:
12293:
12288:
12283:
12278:
12267:
12262:
12257:
12252:
12247:
12236:
12231:
12226:
12221:
12216:
12207:
12202:
12197:
12192:
12187:
12182:
12177:
12145:
12140:
12135:
12130:
12125:
12114:
12109:
12104:
12099:
12094:
12083:
12078:
12073:
12068:
12063:
12052:
12047:
12042:
12037:
12032:
12023:
12018:
12013:
12008:
12003:
11998:
11993:
11955:
11950:
11945:
11940:
11935:
11924:
11919:
11914:
11909:
11904:
11893:
11888:
11883:
11878:
11873:
11862:
11857:
11852:
11847:
11842:
11833:
11828:
11823:
11818:
11813:
11808:
11803:
11773:
11768:
11763:
11758:
11753:
11742:
11737:
11732:
11727:
11722:
11711:
11706:
11701:
11696:
11691:
11680:
11675:
11670:
11665:
11660:
11651:
11646:
11641:
11636:
11631:
11626:
11621:
11591:
11586:
11581:
11576:
11571:
11560:
11555:
11550:
11545:
11540:
11529:
11524:
11519:
11514:
11509:
11498:
11493:
11488:
11483:
11478:
11469:
11464:
11459:
11454:
11449:
11444:
11439:
11417:
11412:
11407:
11402:
11397:
11386:
11381:
11376:
11371:
11366:
11355:
11350:
11345:
11340:
11335:
11324:
11319:
11314:
11309:
11304:
11295:
11290:
11285:
11280:
11275:
11270:
11265:
11231:
11226:
11221:
11216:
11211:
11200:
11195:
11190:
11185:
11180:
11169:
11164:
11159:
11154:
11149:
11138:
11133:
11128:
11123:
11118:
11109:
11104:
11099:
11094:
11089:
11084:
11079:
11049:
11044:
11039:
11034:
11029:
11018:
11013:
11008:
11003:
10998:
10987:
10982:
10977:
10972:
10967:
10956:
10951:
10946:
10941:
10936:
10927:
10922:
10917:
10912:
10907:
10902:
10897:
10869:
10864:
10859:
10854:
10849:
10838:
10833:
10828:
10823:
10818:
10807:
10802:
10797:
10792:
10787:
10776:
10771:
10766:
10761:
10756:
10747:
10742:
10737:
10732:
10727:
10722:
10717:
10687:
10682:
10677:
10672:
10667:
10656:
10651:
10646:
10641:
10636:
10625:
10620:
10615:
10610:
10605:
10594:
10589:
10584:
10579:
10574:
10565:
10560:
10555:
10550:
10545:
10540:
10535:
10511:
10506:
10501:
10496:
10491:
10480:
10475:
10470:
10465:
10460:
10449:
10444:
10439:
10434:
10429:
10418:
10413:
10408:
10403:
10398:
10389:
10384:
10379:
10374:
10369:
10364:
10359:
10331:
10326:
10321:
10316:
10311:
10300:
10295:
10290:
10285:
10280:
10269:
10264:
10259:
10254:
10249:
10238:
10233:
10228:
10223:
10218:
10209:
10204:
10199:
10194:
10189:
10184:
10179:
10145:
10140:
10135:
10130:
10125:
10114:
10109:
10104:
10099:
10094:
10083:
10078:
10073:
10068:
10063:
10052:
10047:
10042:
10037:
10032:
10023:
10018:
10013:
10008:
10003:
9998:
9993:
9963:
9958:
9953:
9948:
9943:
9932:
9927:
9922:
9917:
9912:
9901:
9896:
9891:
9886:
9881:
9870:
9865:
9860:
9855:
9850:
9841:
9836:
9831:
9826:
9821:
9816:
9811:
9781:
9776:
9771:
9766:
9761:
9750:
9745:
9740:
9735:
9730:
9719:
9714:
9709:
9704:
9699:
9688:
9683:
9678:
9673:
9668:
9659:
9654:
9649:
9644:
9639:
9634:
9629:
9603:
9598:
9593:
9588:
9583:
9578:
9573:
9562:
9557:
9552:
9547:
9542:
9537:
9532:
9521:
9516:
9511:
9506:
9501:
9496:
9491:
9480:
9475:
9470:
9465:
9460:
9455:
9450:
9437:
9432:
9427:
9422:
9417:
9412:
9407:
9348:
9343:
9338:
9333:
9328:
9323:
9318:
9311:
9297:
9292:
9287:
9282:
9277:
9270:
9254:
9249:
9244:
9239:
9234:
9229:
9224:
9217:
9195:
9190:
9185:
9180:
9175:
9170:
9165:
9158:
9136:
9131:
9126:
9121:
9116:
9111:
9106:
9099:
9079:
9074:
9069:
9064:
9059:
9054:
9049:
9042:
9024:
9019:
9014:
9009:
9004:
8999:
8994:
8987:
8969:
8964:
8959:
8954:
8949:
8944:
8939:
8932:
8912:
8907:
8902:
8897:
8892:
8887:
8882:
8875:
8857:
8852:
8847:
8842:
8837:
8832:
8827:
8820:
8802:
8797:
8792:
8787:
8782:
8777:
8772:
8765:
8751:
8746:
8741:
8736:
8731:
8724:
8706:
8701:
8696:
8691:
8686:
8681:
8676:
8669:
8651:
8646:
8641:
8636:
8631:
8626:
8621:
8614:
8596:
8591:
8586:
8581:
8576:
8571:
8566:
8559:
8543:
8538:
8533:
8528:
8523:
8518:
8513:
8506:
8492:
8487:
8482:
8477:
8472:
8467:
8462:
8455:
8441:
8436:
8431:
8426:
8421:
8416:
8411:
8404:
8390:
8385:
8380:
8375:
8370:
8363:
8349:
8344:
8339:
8334:
8329:
8324:
8319:
8312:
8298:
8293:
8288:
8283:
8278:
8273:
8268:
8261:
8247:
8242:
8237:
8232:
8227:
8222:
8217:
8210:
8194:
8189:
8184:
8179:
8174:
8169:
8164:
8157:
8139:
8134:
8129:
8124:
8119:
8114:
8109:
8102:
8088:
8083:
8078:
8073:
8068:
8061:
8047:
8042:
8037:
8032:
8027:
8022:
8017:
8010:
7996:
7991:
7986:
7981:
7976:
7971:
7966:
7959:
7941:
7936:
7931:
7926:
7921:
7916:
7911:
7904:
7884:
7879:
7874:
7869:
7864:
7859:
7854:
7847:
7825:
7820:
7815:
7810:
7805:
7800:
7795:
7788:
7768:
7763:
7758:
7753:
7748:
7743:
7738:
7731:
7713:
7708:
7703:
7698:
7693:
7688:
7683:
7676:
7658:
7653:
7648:
7643:
7638:
7633:
7628:
7621:
7605:
7600:
7595:
7590:
7585:
7580:
7575:
7568:
7554:
7549:
7544:
7539:
7534:
7529:
7524:
7517:
7503:
7498:
7493:
7488:
7483:
7478:
7473:
7466:
7452:
7447:
7442:
7437:
7432:
7425:
7411:
7406:
7401:
7396:
7391:
7386:
7381:
7374:
7360:
7355:
7350:
7345:
7340:
7335:
7330:
7323:
7309:
7304:
7299:
7294:
7289:
7284:
7279:
7272:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7219:
7205:
7200:
7195:
7190:
7185:
7180:
7175:
7168:
7154:
7149:
7144:
7139:
7134:
7129:
7124:
7117:
7103:
7098:
7093:
7088:
7083:
7076:
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7025:
7011:
7006:
7001:
6996:
6991:
6986:
6981:
6974:
6960:
6955:
6950:
6945:
6940:
6935:
6930:
6923:
6907:
6902:
6897:
6892:
6887:
6882:
6877:
6870:
6856:
6851:
6846:
6841:
6836:
6831:
6826:
6819:
6805:
6800:
6795:
6790:
6785:
6780:
6775:
6768:
6754:
6749:
6744:
6739:
6734:
6727:
6713:
6708:
6703:
6698:
6693:
6688:
6683:
6676:
6662:
6657:
6652:
6647:
6642:
6637:
6632:
6625:
6611:
6606:
6601:
6596:
6591:
6586:
6581:
6574:
6558:
6553:
6548:
6543:
6538:
6533:
6528:
6521:
6507:
6502:
6497:
6492:
6487:
6482:
6477:
6470:
6456:
6451:
6446:
6441:
6436:
6431:
6426:
6419:
6405:
6400:
6395:
6390:
6385:
6378:
6364:
6359:
6354:
6349:
6344:
6339:
6334:
6327:
6313:
6308:
6303:
6298:
6293:
6288:
6283:
6276:
6262:
6257:
6252:
6247:
6242:
6237:
6232:
6225:
6211:
6206:
6201:
6196:
6191:
6186:
6181:
6171:
6166:
6161:
6156:
6151:
6146:
6141:
6131:
6126:
6121:
6116:
6111:
6101:
6096:
6091:
6086:
6081:
6076:
6071:
6061:
6056:
6051:
6046:
6041:
6036:
6031:
5784:In four dimensions, there are 6
5761:
5754:
5721:
5716:
5711:
5706:
5701:
5696:
5601:
5596:
5591:
5586:
5581:
5572:
5567:
5562:
5496:
5491:
5486:
5481:
5476:
5467:
5462:
5457:
5452:
5447:
5438:
5433:
5428:
5423:
5418:
5409:
5404:
5399:
5394:
5389:
5380:
5375:
5370:
5365:
5360:
5351:
5346:
5341:
5336:
5331:
5320:
5315:
5310:
5305:
5300:
5241:
5236:
5231:
5226:
5221:
5212:
5207:
5202:
5104:
5099:
5094:
5089:
5084:
5075:
5070:
5065:
4965:
4960:
4955:
4950:
4945:
4936:
4931:
4926:
4833:
4828:
4823:
4818:
4813:
4718:
4713:
4708:
4703:
4698:
4605:
4600:
4595:
4590:
4585:
4489:
4484:
4479:
4474:
4469:
4400:
4395:
4390:
4385:
4380:
4371:
4366:
4361:
4356:
4351:
4342:
4337:
4332:
4327:
4322:
4313:
4308:
4303:
4298:
4293:
4284:
4279:
4274:
4269:
4264:
4255:
4250:
4245:
4240:
4235:
4224:
4219:
4214:
4209:
4204:
4141:
4121:
4116:
4111:
4106:
4101:
4093:
4086:
4079:
4056:
4051:
4046:
4041:
4036:
4028:
4021:
4014:
3942:
3931:
3920:
3909:
3898:
3887:
3876:
3865:
3845:
3834:
3823:
3812:
3801:
3790:
3779:
3768:
3748:
3737:
3726:
3715:
3704:
3693:
3682:
3671:
3576:
3569:
3562:
3554:
3549:
3544:
3539:
3534:
3494:
3487:
3480:
3472:
3467:
3462:
3457:
3452:
3412:
3405:
3398:
3390:
3385:
3380:
3375:
3370:
3324:
3317:
3310:
3302:
3297:
3292:
3287:
3282:
3234:
3227:
3220:
3212:
3207:
3202:
3197:
3192:
3121:
3116:
3111:
3106:
3101:
3043:
3038:
3033:
3000:
2995:
2990:
2985:
2954:
2949:
2944:
2905:
2900:
2895:
2854:
2849:
2844:
2762:
2756:
2749:
2742:
2735:
2729:
2722:
2715:
2708:
2695:
2690:
2685:
2680:
2675:
2666:
2661:
2656:
2651:
2646:
2638:
2633:
2628:
2623:
2618:
2609:
2604:
2599:
2594:
2589:
2580:
2575:
2570:
2565:
2560:
2551:
2546:
2541:
2536:
2531:
2523:
2518:
2513:
2508:
2503:
2494:
2489:
2484:
2479:
2474:
2465:
2460:
2455:
2450:
2445:
2436:
2431:
2426:
2421:
2416:
2301:
2295:
2288:
2281:
2275:
2268:
2261:
2255:
2248:
2241:
2235:
2228:
2221:
2215:
2202:
2197:
2192:
2184:
2179:
2174:
2165:
2160:
2155:
2146:
2141:
2136:
2128:
2123:
2118:
2109:
2104:
2099:
2090:
2085:
2080:
2072:
2067:
2062:
2053:
2048:
2043:
2034:
2029:
2024:
2016:
2011:
2006:
1997:
1992:
1987:
1978:
1973:
1968:
1960:
1955:
1950:
1839:
1832:
1826:
1819:
1812:
1806:
1799:
1792:
1786:
1779:
1772:
1766:
1759:
1746:
1741:
1736:
1727:
1722:
1717:
1709:
1704:
1699:
1690:
1685:
1680:
1671:
1666:
1661:
1653:
1648:
1643:
1634:
1629:
1624:
1615:
1610:
1605:
1597:
1592:
1587:
1578:
1573:
1568:
1559:
1554:
1549:
1541:
1536:
1531:
1522:
1517:
1512:
1359:
1243:
605:
492:
233:
228:
223:
218:
213:
208:
203:
198:
193:
182:
174:
169:
164:
159:
154:
149:
144:
133:
115:
110:
105:
100:
95:
84:
75:
70:
65:
44:
13389:Alternated truncated rectified
5991:, which has no Coxeter family.
4154:of the object. An extension of
4066:
4001:
3523:
3441:
3353:
3263:
3173:
2367:
2331:
1896:
1460:
1284:
384:reduces 4-faces to vertices, a
12347:Half constructions exist with
8721:runcicantellated demitesseract
1328:Uniform polytopes by dimension
1202:One special operation, called
1181:
327:great dirhombicosidodecahedron
13:
1:
13858:
12342:
380:reduces cells to vertices, a
376:reduces faces to vertices, a
372:reduces edges to vertices, a
312:
13870:, Dover Publications, 1999,
9308:Alternated truncated 24-cell
8058:cantitruncated demitesseract
5993:
5880:{3,3,3}, which is self-dual;
4099:
4034:
3612:, which can be obtained via
1263:of other uniform polytopes.
329:in three dimensions and the
7:
13973:(Paper 24) H.S.M. Coxeter,
13966:(Paper 23) H.S.M. Coxeter,
13959:(Paper 22) H.S.M. Coxeter,
13846:
8360:omnitruncated demitesseract
1352:are also considered to be (
1304:. For example, the regular
10:
14648:
14604:
14031:
13764:Five and higher dimensions
13584:Alternated omnitruncation
5786:convex regular 4-polytopes
5777:
3084:
3069:
2703:
2210:
1940:
1754:
1502:
1322:alternated cubic honeycomb
342:Conway polyhedron notation
340:, and is the basis of the
131:
123:
42:
13837:hyperbolic Coxeter groups
12373:
12366:
12360:
11969:(or runcitruncated dual)
11605:(or cantitruncated dual)
9404:
9397:
9391:
9149:
8148:
7838:
7422:cantellated demitesseract
6178:
6068:
6020:
6002:
5925:convex uniform honeycombs
5916:(The groups are named in
5694:
5557:
5297:
5290:
5275:
4811:
4696:
4583:
4467:
4201:
4194:
4179:
3025:
2977:
2809:
2806:
2799:
2794:
2784:
2350:
2340:
352:Regular n-polytopes have
31:Convex uniform polytopes
13179:Alternated bitruncation
8666:runcitruncated tesseract
8309:cantitruncated tesseract
4136:Kepler-Poinsot polyhedra
277:, but also includes the
14009:Glossary for Hyperspace
9039:omnitruncated tesseract
8872:runcitruncated 120-cell
8817:runcitruncated 600-cell
8503:cantitruncated 120-cell
8452:cantitruncated 600-cell
7073:truncated demitesseract
6724:rectified demitesseract
5827:Example tetrahedron in
4158:notation, also used by
1192:truncated cuboctahedron
348:Rectification operators
321:, and represented by a
297:and higher dimensional
12984:Alternated truncation
11063:(or cantellated dual)
9155:omnitruncated 600-cell
9151:omnitruncated 120-cell
8762:runcitruncated 24-cell
8611:runcitruncated 16-cell
8401:cantitruncated 24-cell
8258:cantitruncated 16-cell
5846:
5838:
5677:
5550:
5183:
5046:
4907:
4794:
4679:
4566:
4450:
4150:, which specifies the
1281:as its vertex figure.
1199:
13833:affine Coxeter groups
12779:Alternated rectified
9096:omnitruncated 24-cell
8984:omnitruncated 16-cell
8556:runcitruncated 5-cell
8207:cantitruncated 5-cell
8007:bitruncated tesseract
7371:cantellated tesseract
5887:{3,3,4} and its dual
5844:
5826:
5767:Generating triangles
5678:
5551:
5184:
5047:
4908:
4795:
4680:
4567:
4451:
3614:Wythoff constructions
1189:
718:quadricantitruncation
189:Truncated 5-orthoplex
13923:M.S. Longuet-Higgins
13887:, Manuscript (1991)
10701:(or truncated dual)
9977:(or rectified dual)
8929:omnitruncated 5-cell
8154:bitruncated 120-cell
8150:bitruncated 600-cell
7728:runcinated tesseract
7565:cantellated 120-cell
7514:cantellated 600-cell
5976:Constructive summary
5868:Each of the sixteen
5642:
5514:
5147:
5010:
4874:
4759:
4753:(or truncated dual)
4644:
4534:
4418:
4168:truncated octahedron
1217:polytope solutions.
1190:An alternation of a
1153:Uniform 10-polytopes
884:tristericantellation
433:Truncation operators
402:-th rectification =
333:in four dimensions.
319:Wythoff construction
91:Truncated octahedron
14599:pentagonal polytope
14498:Uniform 10-polytope
14058:Fundamental convex
14015:on 4 February 2007.
14003:Olshevsky, George.
13925:and J.C.P. Miller:
12374:Cells by position:
12157:Runcicantitruncated
9405:Cells by position:
8099:bitruncated 24-cell
7956:bitruncated 16-cell
7844:runcinated 120-cell
7840:runcinated 600-cell
7463:cantellated 24-cell
7320:cantellated 16-cell
7022:truncated tesseract
6673:rectified tesseract
5898:{3,4,3}, self-dual;
5870:regular 4-polytopes
1136:Uniform 9-polytopes
1083:Uniform 8-polytopes
1029:Uniform 7-polytopes
968:Uniform 7-polytopes
915:Uniform 6-polytopes
870:bistericantellation
863:Uniform 5-polytopes
806:Uniform 5-polytopes
485:, comes from Latin
386:quintirectification
259:bounded by uniform
32:
14468:Uniform 9-polytope
14418:Uniform 8-polytope
14368:Uniform 7-polytope
14325:Uniform 6-polytope
14295:Uniform 5-polytope
14255:Uniform polychoron
14218:Uniform polyhedron
14066:in dimensions 2–10
14005:"Uniform polytope"
13827:Uniform honeycombs
9267:snub demitesseract
7901:bitruncated 5-cell
7785:runcinated 24-cell
7673:runcinated 16-cell
7269:cantellated 5-cell
7216:truncated 120-cell
7165:truncated 600-cell
6867:rectified 120-cell
6816:rectified 600-cell
5905:{3,3,5}, its dual
5851:fundamental region
5847:
5839:
5780:Uniform 4-polytope
5673:
5667:
5546:
5540:
5179:
5173:
5042:
5036:
4903:
4897:
4790:
4784:
4675:
4669:
4562:
4556:
4446:
4440:
4152:fundamental region
3610:Archimedean solids
3072:Uniform polyhedron
1302:uniform honeycombs
1200:
1171:applied together.
751:Uniform 4-polytope
704:tricantitruncation
644:quadricantellation
382:quadirectification
291:uniform honeycombs
30:
14632:Uniform polytopes
14620:
14619:
14607:Polytope families
14064:uniform polytopes
13955:978-0-471-01003-6
13937:Regular Polytopes
13927:Uniform Polyhedra
13885:Uniform Polytopes
13876:978-0-486-40919-1
13788:simple Lie groups
13761:
13760:
12340:
12339:
9363:
9362:
7618:runcinated 5-cell
7114:truncated 24-cell
6971:truncated 16-cell
6765:rectified 24-cell
6622:rectified 16-cell
5771:
5770:
5749:
5748:
5271:
5270:
4132:
4131:
3954:
3953:
3606:
3605:
3063:
3062:
2773:
2772:
2309:
2308:
1310:triangular tiling
1051:trihexiruncinated
859:Stericantellation
690:bicantitruncation
279:regular polytopes
254:vertex-transitive
242:
241:
140:Truncated 16-cell
16:(Redirected from
14639:
14611:Regular polytope
14172:
14161:
14150:
14109:
14052:
14045:
14038:
14029:
14028:
14025:
14016:
14011:. Archived from
13989:
13935:H.S.M. Coxeter,
13921:H.S.M. Coxeter,
13890:Norman Johnson:
13813:Coxeter diagrams
13755:
13754:
13753:
13749:
13748:
13744:
13743:
13739:
13738:
13734:
13733:
13724:
13723:
13722:
13718:
13717:
13713:
13712:
13708:
13707:
13703:
13702:
13693:
13692:
13691:
13687:
13686:
13682:
13681:
13677:
13676:
13672:
13671:
13662:
13661:
13660:
13656:
13655:
13651:
13650:
13646:
13645:
13641:
13640:
13633:
13632:
13631:
13627:
13626:
13622:
13621:
13617:
13616:
13612:
13611:
13607:
13606:
13602:
13601:
13575:
13574:
13573:
13569:
13568:
13564:
13563:
13559:
13558:
13554:
13553:
13549:
13548:
13539:
13538:
13537:
13533:
13532:
13528:
13527:
13523:
13522:
13518:
13517:
13513:
13512:
13503:
13502:
13501:
13497:
13496:
13492:
13491:
13487:
13486:
13482:
13481:
13472:
13471:
13470:
13466:
13465:
13461:
13460:
13456:
13455:
13451:
13450:
13443:
13442:
13441:
13437:
13436:
13432:
13431:
13427:
13426:
13422:
13421:
13417:
13416:
13412:
13411:
13407:
13406:
13380:
13379:
13378:
13374:
13373:
13369:
13368:
13364:
13363:
13359:
13358:
13354:
13353:
13344:
13343:
13342:
13338:
13337:
13333:
13332:
13328:
13327:
13323:
13322:
13318:
13317:
13308:
13307:
13306:
13302:
13301:
13297:
13296:
13292:
13291:
13287:
13286:
13282:
13281:
13272:
13271:
13270:
13266:
13265:
13261:
13260:
13256:
13255:
13251:
13250:
13246:
13245:
13238:
13237:
13236:
13232:
13231:
13227:
13226:
13222:
13221:
13217:
13216:
13212:
13211:
13207:
13206:
13202:
13201:
13197:
13196:
13170:
13169:
13168:
13164:
13163:
13159:
13158:
13154:
13153:
13149:
13148:
13144:
13143:
13134:
13133:
13132:
13128:
13127:
13123:
13122:
13118:
13117:
13113:
13112:
13103:
13102:
13101:
13097:
13096:
13092:
13091:
13087:
13086:
13082:
13081:
13072:
13071:
13070:
13066:
13065:
13061:
13060:
13056:
13055:
13051:
13050:
13046:
13045:
13038:
13037:
13036:
13032:
13031:
13027:
13026:
13022:
13021:
13017:
13016:
13012:
13011:
13007:
13006:
13002:
13001:
12975:
12974:
12973:
12969:
12968:
12964:
12963:
12959:
12958:
12954:
12953:
12944:
12943:
12942:
12938:
12937:
12933:
12932:
12928:
12927:
12923:
12922:
12913:
12912:
12911:
12907:
12906:
12902:
12901:
12897:
12896:
12892:
12891:
12887:
12886:
12877:
12876:
12875:
12871:
12870:
12866:
12865:
12861:
12860:
12856:
12855:
12851:
12850:
12846:
12845:
12838:
12837:
12836:
12832:
12831:
12827:
12826:
12822:
12821:
12817:
12816:
12812:
12811:
12807:
12806:
12802:
12801:
12797:
12796:
12772:
12771:
12770:
12766:
12765:
12761:
12760:
12756:
12755:
12751:
12750:
12741:
12740:
12739:
12735:
12734:
12730:
12729:
12725:
12724:
12720:
12719:
12710:
12709:
12708:
12704:
12703:
12699:
12698:
12694:
12693:
12689:
12688:
12684:
12683:
12674:
12673:
12672:
12668:
12667:
12663:
12662:
12658:
12657:
12653:
12652:
12648:
12647:
12640:
12639:
12638:
12634:
12633:
12629:
12628:
12624:
12623:
12619:
12618:
12614:
12613:
12609:
12608:
12604:
12603:
12577:
12576:
12575:
12571:
12570:
12566:
12565:
12561:
12560:
12556:
12555:
12551:
12550:
12546:
12545:
12536:
12535:
12534:
12530:
12529:
12525:
12524:
12520:
12519:
12515:
12514:
12510:
12509:
12505:
12504:
12495:
12494:
12493:
12489:
12488:
12484:
12483:
12479:
12478:
12474:
12473:
12469:
12468:
12464:
12463:
12454:
12453:
12452:
12448:
12447:
12443:
12442:
12438:
12437:
12433:
12432:
12428:
12427:
12423:
12422:
12411:
12410:
12409:
12405:
12404:
12400:
12399:
12395:
12394:
12390:
12389:
12385:
12384:
12380:
12379:
12364:Schläfli symbol
12358:
12357:
12334:
12333:
12332:
12328:
12327:
12323:
12322:
12318:
12317:
12313:
12312:
12303:
12302:
12301:
12297:
12296:
12292:
12291:
12287:
12286:
12282:
12281:
12272:
12271:
12270:
12266:
12265:
12261:
12260:
12256:
12255:
12251:
12250:
12241:
12240:
12239:
12235:
12234:
12230:
12229:
12225:
12224:
12220:
12219:
12212:
12211:
12210:
12206:
12205:
12201:
12200:
12196:
12195:
12191:
12190:
12186:
12185:
12181:
12180:
12150:
12149:
12148:
12144:
12143:
12139:
12138:
12134:
12133:
12129:
12128:
12119:
12118:
12117:
12113:
12112:
12108:
12107:
12103:
12102:
12098:
12097:
12088:
12087:
12086:
12082:
12081:
12077:
12076:
12072:
12071:
12067:
12066:
12057:
12056:
12055:
12051:
12050:
12046:
12045:
12041:
12040:
12036:
12035:
12028:
12027:
12026:
12022:
12021:
12017:
12016:
12012:
12011:
12007:
12006:
12002:
12001:
11997:
11996:
11967:Runcicantellated
11960:
11959:
11958:
11954:
11953:
11949:
11948:
11944:
11943:
11939:
11938:
11929:
11928:
11927:
11923:
11922:
11918:
11917:
11913:
11912:
11908:
11907:
11898:
11897:
11896:
11892:
11891:
11887:
11886:
11882:
11881:
11877:
11876:
11867:
11866:
11865:
11861:
11860:
11856:
11855:
11851:
11850:
11846:
11845:
11838:
11837:
11836:
11832:
11831:
11827:
11826:
11822:
11821:
11817:
11816:
11812:
11811:
11807:
11806:
11778:
11777:
11776:
11772:
11771:
11767:
11766:
11762:
11761:
11757:
11756:
11747:
11746:
11745:
11741:
11740:
11736:
11735:
11731:
11730:
11726:
11725:
11716:
11715:
11714:
11710:
11709:
11705:
11704:
11700:
11699:
11695:
11694:
11685:
11684:
11683:
11679:
11678:
11674:
11673:
11669:
11668:
11664:
11663:
11656:
11655:
11654:
11650:
11649:
11645:
11644:
11640:
11639:
11635:
11634:
11630:
11629:
11625:
11624:
11603:Bicantitruncated
11596:
11595:
11594:
11590:
11589:
11585:
11584:
11580:
11579:
11575:
11574:
11565:
11564:
11563:
11559:
11558:
11554:
11553:
11549:
11548:
11544:
11543:
11534:
11533:
11532:
11528:
11527:
11523:
11522:
11518:
11517:
11513:
11512:
11503:
11502:
11501:
11497:
11496:
11492:
11491:
11487:
11486:
11482:
11481:
11474:
11473:
11472:
11468:
11467:
11463:
11462:
11458:
11457:
11453:
11452:
11448:
11447:
11443:
11442:
11422:
11421:
11420:
11416:
11415:
11411:
11410:
11406:
11405:
11401:
11400:
11391:
11390:
11389:
11385:
11384:
11380:
11379:
11375:
11374:
11370:
11369:
11360:
11359:
11358:
11354:
11353:
11349:
11348:
11344:
11343:
11339:
11338:
11329:
11328:
11327:
11323:
11322:
11318:
11317:
11313:
11312:
11308:
11307:
11300:
11299:
11298:
11294:
11293:
11289:
11288:
11284:
11283:
11279:
11278:
11274:
11273:
11269:
11268:
11236:
11235:
11234:
11230:
11229:
11225:
11224:
11220:
11219:
11215:
11214:
11205:
11204:
11203:
11199:
11198:
11194:
11193:
11189:
11188:
11184:
11183:
11174:
11173:
11172:
11168:
11167:
11163:
11162:
11158:
11157:
11153:
11152:
11143:
11142:
11141:
11137:
11136:
11132:
11131:
11127:
11126:
11122:
11121:
11114:
11113:
11112:
11108:
11107:
11103:
11102:
11098:
11097:
11093:
11092:
11088:
11087:
11083:
11082:
11054:
11053:
11052:
11048:
11047:
11043:
11042:
11038:
11037:
11033:
11032:
11023:
11022:
11021:
11017:
11016:
11012:
11011:
11007:
11006:
11002:
11001:
10992:
10991:
10990:
10986:
10985:
10981:
10980:
10976:
10975:
10971:
10970:
10961:
10960:
10959:
10955:
10954:
10950:
10949:
10945:
10944:
10940:
10939:
10932:
10931:
10930:
10926:
10925:
10921:
10920:
10916:
10915:
10911:
10910:
10906:
10905:
10901:
10900:
10874:
10873:
10872:
10868:
10867:
10863:
10862:
10858:
10857:
10853:
10852:
10843:
10842:
10841:
10837:
10836:
10832:
10831:
10827:
10826:
10822:
10821:
10812:
10811:
10810:
10806:
10805:
10801:
10800:
10796:
10795:
10791:
10790:
10781:
10780:
10779:
10775:
10774:
10770:
10769:
10765:
10764:
10760:
10759:
10752:
10751:
10750:
10746:
10745:
10741:
10740:
10736:
10735:
10731:
10730:
10726:
10725:
10721:
10720:
10692:
10691:
10690:
10686:
10685:
10681:
10680:
10676:
10675:
10671:
10670:
10661:
10660:
10659:
10655:
10654:
10650:
10649:
10645:
10644:
10640:
10639:
10630:
10629:
10628:
10624:
10623:
10619:
10618:
10614:
10613:
10609:
10608:
10599:
10598:
10597:
10593:
10592:
10588:
10587:
10583:
10582:
10578:
10577:
10570:
10569:
10568:
10564:
10563:
10559:
10558:
10554:
10553:
10549:
10548:
10544:
10543:
10539:
10538:
10516:
10515:
10514:
10510:
10509:
10505:
10504:
10500:
10499:
10495:
10494:
10485:
10484:
10483:
10479:
10478:
10474:
10473:
10469:
10468:
10464:
10463:
10454:
10453:
10452:
10448:
10447:
10443:
10442:
10438:
10437:
10433:
10432:
10423:
10422:
10421:
10417:
10416:
10412:
10411:
10407:
10406:
10402:
10401:
10394:
10393:
10392:
10388:
10387:
10383:
10382:
10378:
10377:
10373:
10372:
10368:
10367:
10363:
10362:
10336:
10335:
10334:
10330:
10329:
10325:
10324:
10320:
10319:
10315:
10314:
10305:
10304:
10303:
10299:
10298:
10294:
10293:
10289:
10288:
10284:
10283:
10274:
10273:
10272:
10268:
10267:
10263:
10262:
10258:
10257:
10253:
10252:
10243:
10242:
10241:
10237:
10236:
10232:
10231:
10227:
10226:
10222:
10221:
10214:
10213:
10212:
10208:
10207:
10203:
10202:
10198:
10197:
10193:
10192:
10188:
10187:
10183:
10182:
10150:
10149:
10148:
10144:
10143:
10139:
10138:
10134:
10133:
10129:
10128:
10119:
10118:
10117:
10113:
10112:
10108:
10107:
10103:
10102:
10098:
10097:
10088:
10087:
10086:
10082:
10081:
10077:
10076:
10072:
10071:
10067:
10066:
10057:
10056:
10055:
10051:
10050:
10046:
10045:
10041:
10040:
10036:
10035:
10028:
10027:
10026:
10022:
10021:
10017:
10016:
10012:
10011:
10007:
10006:
10002:
10001:
9997:
9996:
9968:
9967:
9966:
9962:
9961:
9957:
9956:
9952:
9951:
9947:
9946:
9937:
9936:
9935:
9931:
9930:
9926:
9925:
9921:
9920:
9916:
9915:
9906:
9905:
9904:
9900:
9899:
9895:
9894:
9890:
9889:
9885:
9884:
9875:
9874:
9873:
9869:
9868:
9864:
9863:
9859:
9858:
9854:
9853:
9846:
9845:
9844:
9840:
9839:
9835:
9834:
9830:
9829:
9825:
9824:
9820:
9819:
9815:
9814:
9786:
9785:
9784:
9780:
9779:
9775:
9774:
9770:
9769:
9765:
9764:
9755:
9754:
9753:
9749:
9748:
9744:
9743:
9739:
9738:
9734:
9733:
9724:
9723:
9722:
9718:
9717:
9713:
9712:
9708:
9707:
9703:
9702:
9693:
9692:
9691:
9687:
9686:
9682:
9681:
9677:
9676:
9672:
9671:
9664:
9663:
9662:
9658:
9657:
9653:
9652:
9648:
9647:
9643:
9642:
9638:
9637:
9633:
9632:
9608:
9607:
9606:
9602:
9601:
9597:
9596:
9592:
9591:
9587:
9586:
9582:
9581:
9577:
9576:
9567:
9566:
9565:
9561:
9560:
9556:
9555:
9551:
9550:
9546:
9545:
9541:
9540:
9536:
9535:
9526:
9525:
9524:
9520:
9519:
9515:
9514:
9510:
9509:
9505:
9504:
9500:
9499:
9495:
9494:
9485:
9484:
9483:
9479:
9478:
9474:
9473:
9469:
9468:
9464:
9463:
9459:
9458:
9454:
9453:
9442:
9441:
9440:
9436:
9435:
9431:
9430:
9426:
9425:
9421:
9420:
9416:
9415:
9411:
9410:
9395:Schläfli symbol
9389:
9388:
9353:
9352:
9351:
9347:
9346:
9342:
9341:
9337:
9336:
9332:
9331:
9327:
9326:
9322:
9321:
9315:
9302:
9301:
9300:
9296:
9295:
9291:
9290:
9286:
9285:
9281:
9280:
9274:
9259:
9258:
9257:
9253:
9252:
9248:
9247:
9243:
9242:
9238:
9237:
9233:
9232:
9228:
9227:
9221:
9200:
9199:
9198:
9194:
9193:
9189:
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9184:
9183:
9179:
9178:
9174:
9173:
9169:
9168:
9162:
9141:
9140:
9139:
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9134:
9130:
9129:
9125:
9124:
9120:
9119:
9115:
9114:
9110:
9109:
9103:
9084:
9083:
9082:
9078:
9077:
9073:
9072:
9068:
9067:
9063:
9062:
9058:
9057:
9053:
9052:
9046:
9029:
9028:
9027:
9023:
9022:
9018:
9017:
9013:
9012:
9008:
9007:
9003:
9002:
8998:
8997:
8991:
8974:
8973:
8972:
8968:
8967:
8963:
8962:
8958:
8957:
8953:
8952:
8948:
8947:
8943:
8942:
8936:
8917:
8916:
8915:
8911:
8910:
8906:
8905:
8901:
8900:
8896:
8895:
8891:
8890:
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8885:
8879:
8862:
8861:
8860:
8856:
8855:
8851:
8850:
8846:
8845:
8841:
8840:
8836:
8835:
8831:
8830:
8824:
8807:
8806:
8805:
8801:
8800:
8796:
8795:
8791:
8790:
8786:
8785:
8781:
8780:
8776:
8775:
8769:
8756:
8755:
8754:
8750:
8749:
8745:
8744:
8740:
8739:
8735:
8734:
8728:
8711:
8710:
8709:
8705:
8704:
8700:
8699:
8695:
8694:
8690:
8689:
8685:
8684:
8680:
8679:
8673:
8656:
8655:
8654:
8650:
8649:
8645:
8644:
8640:
8639:
8635:
8634:
8630:
8629:
8625:
8624:
8618:
8601:
8600:
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8595:
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8584:
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8575:
8574:
8570:
8569:
8563:
8548:
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8546:
8542:
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8537:
8536:
8532:
8531:
8527:
8526:
8522:
8521:
8517:
8516:
8510:
8497:
8496:
8495:
8491:
8490:
8486:
8485:
8481:
8480:
8476:
8475:
8471:
8470:
8466:
8465:
8459:
8446:
8445:
8444:
8440:
8439:
8435:
8434:
8430:
8429:
8425:
8424:
8420:
8419:
8415:
8414:
8408:
8395:
8394:
8393:
8389:
8388:
8384:
8383:
8379:
8378:
8374:
8373:
8367:
8354:
8353:
8352:
8348:
8347:
8343:
8342:
8338:
8337:
8333:
8332:
8328:
8327:
8323:
8322:
8316:
8303:
8302:
8301:
8297:
8296:
8292:
8291:
8287:
8286:
8282:
8281:
8277:
8276:
8272:
8271:
8265:
8252:
8251:
8250:
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8245:
8241:
8240:
8236:
8235:
8231:
8230:
8226:
8225:
8221:
8220:
8214:
8199:
8198:
8197:
8193:
8192:
8188:
8187:
8183:
8182:
8178:
8177:
8173:
8172:
8168:
8167:
8161:
8144:
8143:
8142:
8138:
8137:
8133:
8132:
8128:
8127:
8123:
8122:
8118:
8117:
8113:
8112:
8106:
8093:
8092:
8091:
8087:
8086:
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8081:
8077:
8076:
8072:
8071:
8065:
8052:
8051:
8050:
8046:
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8040:
8036:
8035:
8031:
8030:
8026:
8025:
8021:
8020:
8014:
8001:
8000:
7999:
7995:
7994:
7990:
7989:
7985:
7984:
7980:
7979:
7975:
7974:
7970:
7969:
7963:
7946:
7945:
7944:
7940:
7939:
7935:
7934:
7930:
7929:
7925:
7924:
7920:
7919:
7915:
7914:
7908:
7889:
7888:
7887:
7883:
7882:
7878:
7877:
7873:
7872:
7868:
7867:
7863:
7862:
7858:
7857:
7851:
7830:
7829:
7828:
7824:
7823:
7819:
7818:
7814:
7813:
7809:
7808:
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7803:
7799:
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7773:
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7766:
7762:
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7757:
7756:
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7751:
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7735:
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7701:
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7696:
7692:
7691:
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7680:
7663:
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7656:
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7647:
7646:
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7641:
7637:
7636:
7632:
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7625:
7610:
7609:
7608:
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7599:
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7588:
7584:
7583:
7579:
7578:
7572:
7559:
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7557:
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7548:
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7538:
7537:
7533:
7532:
7528:
7527:
7521:
7508:
7507:
7506:
7502:
7501:
7497:
7496:
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7486:
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7481:
7477:
7476:
7470:
7457:
7456:
7455:
7451:
7450:
7446:
7445:
7441:
7440:
7436:
7435:
7429:
7416:
7415:
7414:
7410:
7409:
7405:
7404:
7400:
7399:
7395:
7394:
7390:
7389:
7385:
7384:
7378:
7365:
7364:
7363:
7359:
7358:
7354:
7353:
7349:
7348:
7344:
7343:
7339:
7338:
7334:
7333:
7327:
7314:
7313:
7312:
7308:
7307:
7303:
7302:
7298:
7297:
7293:
7292:
7288:
7287:
7283:
7282:
7276:
7261:
7260:
7259:
7255:
7254:
7250:
7249:
7245:
7244:
7240:
7239:
7235:
7234:
7230:
7229:
7223:
7210:
7209:
7208:
7204:
7203:
7199:
7198:
7194:
7193:
7189:
7188:
7184:
7183:
7179:
7178:
7172:
7159:
7158:
7157:
7153:
7152:
7148:
7147:
7143:
7142:
7138:
7137:
7133:
7132:
7128:
7127:
7121:
7108:
7107:
7106:
7102:
7101:
7097:
7096:
7092:
7091:
7087:
7086:
7080:
7067:
7066:
7065:
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7060:
7056:
7055:
7051:
7050:
7046:
7045:
7041:
7040:
7036:
7035:
7029:
7016:
7015:
7014:
7010:
7009:
7005:
7004:
7000:
6999:
6995:
6994:
6990:
6989:
6985:
6984:
6978:
6965:
6964:
6963:
6959:
6958:
6954:
6953:
6949:
6948:
6944:
6943:
6939:
6938:
6934:
6933:
6927:
6920:truncated 5-cell
6912:
6911:
6910:
6906:
6905:
6901:
6900:
6896:
6895:
6891:
6890:
6886:
6885:
6881:
6880:
6874:
6861:
6860:
6859:
6855:
6854:
6850:
6849:
6845:
6844:
6840:
6839:
6835:
6834:
6830:
6829:
6823:
6810:
6809:
6808:
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6803:
6799:
6798:
6794:
6793:
6789:
6788:
6784:
6783:
6779:
6778:
6772:
6759:
6758:
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6752:
6748:
6747:
6743:
6742:
6738:
6737:
6731:
6718:
6717:
6716:
6712:
6711:
6707:
6706:
6702:
6701:
6697:
6696:
6692:
6691:
6687:
6686:
6680:
6667:
6666:
6665:
6661:
6660:
6656:
6655:
6651:
6650:
6646:
6645:
6641:
6640:
6636:
6635:
6629:
6616:
6615:
6614:
6610:
6609:
6605:
6604:
6600:
6599:
6595:
6594:
6590:
6589:
6585:
6584:
6578:
6571:rectified 5-cell
6563:
6562:
6561:
6557:
6556:
6552:
6551:
6547:
6546:
6542:
6541:
6537:
6536:
6532:
6531:
6525:
6512:
6511:
6510:
6506:
6505:
6501:
6500:
6496:
6495:
6491:
6490:
6486:
6485:
6481:
6480:
6474:
6461:
6460:
6459:
6455:
6454:
6450:
6449:
6445:
6444:
6440:
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6435:
6434:
6430:
6429:
6423:
6410:
6409:
6408:
6404:
6403:
6399:
6398:
6394:
6393:
6389:
6388:
6382:
6369:
6368:
6367:
6363:
6362:
6358:
6357:
6353:
6352:
6348:
6347:
6343:
6342:
6338:
6337:
6331:
6318:
6317:
6316:
6312:
6311:
6307:
6306:
6302:
6301:
6297:
6296:
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6286:
6280:
6267:
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6265:
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6260:
6256:
6255:
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6250:
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6245:
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6235:
6229:
6216:
6215:
6214:
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6185:
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6176:
6175:
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6155:
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6059:
6055:
6054:
6050:
6049:
6045:
6044:
6040:
6039:
6035:
6034:
5994:
5982:Coxeter diagrams
5918:Coxeter notation
5765:
5758:
5751:
5750:
5726:
5725:
5724:
5720:
5719:
5715:
5714:
5710:
5709:
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5679:
5674:
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5413:
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5364:
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3476:
3475:
3471:
3470:
3466:
3465:
3461:
3460:
3456:
3455:
3416:
3409:
3402:
3395:
3394:
3393:
3389:
3388:
3384:
3383:
3379:
3378:
3374:
3373:
3328:
3321:
3314:
3307:
3306:
3305:
3301:
3300:
3296:
3295:
3291:
3290:
3286:
3285:
3238:
3231:
3224:
3217:
3216:
3215:
3211:
3210:
3206:
3205:
3201:
3200:
3196:
3195:
3126:
3125:
3124:
3120:
3119:
3115:
3114:
3110:
3109:
3105:
3104:
3083:
3082:
3066:Three dimensions
3048:
3047:
3046:
3042:
3041:
3037:
3036:
3005:
3004:
3003:
2999:
2998:
2994:
2993:
2989:
2988:
2959:
2958:
2957:
2953:
2952:
2948:
2947:
2910:
2909:
2908:
2904:
2903:
2899:
2898:
2859:
2858:
2857:
2853:
2852:
2848:
2847:
2782:
2781:
2766:
2760:
2753:
2746:
2739:
2733:
2726:
2719:
2712:
2700:
2699:
2698:
2694:
2693:
2689:
2688:
2684:
2683:
2679:
2678:
2671:
2670:
2669:
2665:
2664:
2660:
2659:
2655:
2654:
2650:
2649:
2643:
2642:
2641:
2637:
2636:
2632:
2631:
2627:
2626:
2622:
2621:
2614:
2613:
2612:
2608:
2607:
2603:
2602:
2598:
2597:
2593:
2592:
2585:
2584:
2583:
2579:
2578:
2574:
2573:
2569:
2568:
2564:
2563:
2556:
2555:
2554:
2550:
2549:
2545:
2544:
2540:
2539:
2535:
2534:
2528:
2527:
2526:
2522:
2521:
2517:
2516:
2512:
2511:
2507:
2506:
2499:
2498:
2497:
2493:
2492:
2488:
2487:
2483:
2482:
2478:
2477:
2470:
2469:
2468:
2464:
2463:
2459:
2458:
2454:
2453:
2449:
2448:
2441:
2440:
2439:
2435:
2434:
2430:
2429:
2425:
2424:
2420:
2419:
2330:
2329:
2305:
2299:
2292:
2285:
2279:
2272:
2265:
2259:
2252:
2245:
2239:
2232:
2225:
2219:
2207:
2206:
2205:
2201:
2200:
2196:
2195:
2189:
2188:
2187:
2183:
2182:
2178:
2177:
2170:
2169:
2168:
2164:
2163:
2159:
2158:
2151:
2150:
2149:
2145:
2144:
2140:
2139:
2133:
2132:
2131:
2127:
2126:
2122:
2121:
2114:
2113:
2112:
2108:
2107:
2103:
2102:
2095:
2094:
2093:
2089:
2088:
2084:
2083:
2077:
2076:
2075:
2071:
2070:
2066:
2065:
2058:
2057:
2056:
2052:
2051:
2047:
2046:
2039:
2038:
2037:
2033:
2032:
2028:
2027:
2021:
2020:
2019:
2015:
2014:
2010:
2009:
2002:
2001:
2000:
1996:
1995:
1991:
1990:
1983:
1982:
1981:
1977:
1976:
1972:
1971:
1965:
1964:
1963:
1959:
1958:
1954:
1953:
1843:
1836:
1830:
1823:
1816:
1810:
1803:
1796:
1790:
1783:
1776:
1770:
1763:
1751:
1750:
1749:
1745:
1744:
1740:
1739:
1732:
1731:
1730:
1726:
1725:
1721:
1720:
1714:
1713:
1712:
1708:
1707:
1703:
1702:
1695:
1694:
1693:
1689:
1688:
1684:
1683:
1676:
1675:
1674:
1670:
1669:
1665:
1664:
1658:
1657:
1656:
1652:
1651:
1647:
1646:
1639:
1638:
1637:
1633:
1632:
1628:
1627:
1620:
1619:
1618:
1614:
1613:
1609:
1608:
1602:
1601:
1600:
1596:
1595:
1591:
1590:
1583:
1582:
1581:
1577:
1576:
1572:
1571:
1564:
1563:
1562:
1558:
1557:
1553:
1552:
1546:
1545:
1544:
1540:
1539:
1535:
1534:
1527:
1526:
1525:
1521:
1520:
1516:
1515:
1391:
1390:
1378:regular polygons
1350:hyperbolic space
1036:bihexiruncinated
609:
547:quintitruncation
533:quadritruncation
496:
378:trirectification
307:hyperbolic space
265:regular polygons
250:uniform polytope
238:
237:
236:
232:
231:
227:
226:
222:
221:
217:
216:
212:
211:
207:
206:
202:
201:
197:
196:
186:
179:
178:
177:
173:
172:
168:
167:
163:
162:
158:
157:
153:
152:
148:
147:
137:
120:
119:
118:
114:
113:
109:
108:
104:
103:
99:
98:
88:
80:
79:
78:
74:
73:
69:
68:
48:
33:
29:
21:
14647:
14646:
14642:
14641:
14640:
14638:
14637:
14636:
14622:
14621:
14590:
14583:
14576:
14459:
14452:
14445:
14409:
14402:
14395:
14359:
14352:
14186:Regular polygon
14179:
14170:
14163:
14159:
14152:
14148:
14139:
14130:
14123:
14119:
14107:
14101:
14097:
14085:
14067:
14056:
14023:
14022:, Marco Möller
13999:
13987:
13861:
13853:Schläfli symbol
13849:
13829:
13820:
13809:
13802:
13795:
13766:
13756:
13751:
13746:
13741:
13736:
13731:
13729:
13725:
13720:
13715:
13710:
13705:
13700:
13698:
13694:
13689:
13684:
13679:
13674:
13669:
13667:
13663:
13658:
13653:
13648:
13643:
13638:
13636:
13629:
13624:
13619:
13614:
13609:
13604:
13599:
13597:
13593:
13583:
13576:
13571:
13566:
13561:
13556:
13551:
13546:
13544:
13540:
13535:
13530:
13525:
13520:
13515:
13510:
13508:
13504:
13499:
13494:
13489:
13484:
13479:
13477:
13473:
13468:
13463:
13458:
13453:
13448:
13446:
13439:
13434:
13429:
13424:
13419:
13414:
13409:
13404:
13402:
13398:
13388:
13381:
13376:
13371:
13366:
13361:
13356:
13351:
13349:
13345:
13340:
13335:
13330:
13325:
13320:
13315:
13313:
13309:
13304:
13299:
13294:
13289:
13284:
13279:
13277:
13273:
13268:
13263:
13258:
13253:
13248:
13243:
13241:
13234:
13229:
13224:
13219:
13214:
13209:
13204:
13199:
13194:
13192:
13188:
13178:
13171:
13166:
13161:
13156:
13151:
13146:
13141:
13139:
13135:
13130:
13125:
13120:
13115:
13110:
13108:
13104:
13099:
13094:
13089:
13084:
13079:
13077:
13073:
13068:
13063:
13058:
13053:
13048:
13043:
13041:
13034:
13029:
13024:
13019:
13014:
13009:
13004:
12999:
12997:
12993:
12983:
12976:
12971:
12966:
12961:
12956:
12951:
12949:
12945:
12940:
12935:
12930:
12925:
12920:
12918:
12914:
12909:
12904:
12899:
12894:
12889:
12884:
12882:
12878:
12873:
12868:
12863:
12858:
12853:
12848:
12843:
12841:
12834:
12829:
12824:
12819:
12814:
12809:
12804:
12799:
12794:
12792:
12788:
12773:
12768:
12763:
12758:
12753:
12748:
12746:
12742:
12737:
12732:
12727:
12722:
12717:
12715:
12711:
12706:
12701:
12696:
12691:
12686:
12681:
12679:
12675:
12670:
12665:
12660:
12655:
12650:
12645:
12643:
12636:
12631:
12626:
12621:
12616:
12611:
12606:
12601:
12599:
12595:
12583:
12573:
12568:
12563:
12558:
12553:
12548:
12543:
12541:
12540:
12532:
12527:
12522:
12517:
12512:
12507:
12502:
12500:
12499:
12491:
12486:
12481:
12476:
12471:
12466:
12461:
12459:
12458:
12450:
12445:
12440:
12435:
12430:
12425:
12420:
12418:
12417:
12407:
12402:
12397:
12392:
12387:
12382:
12377:
12375:
12369:
12345:
12335:
12330:
12325:
12320:
12315:
12310:
12308:
12304:
12299:
12294:
12289:
12284:
12279:
12277:
12273:
12268:
12263:
12258:
12253:
12248:
12246:
12242:
12237:
12232:
12227:
12222:
12217:
12215:
12208:
12203:
12198:
12193:
12188:
12183:
12178:
12176:
12172:
12158:
12151:
12146:
12141:
12136:
12131:
12126:
12124:
12120:
12115:
12110:
12105:
12100:
12095:
12093:
12089:
12084:
12079:
12074:
12069:
12064:
12062:
12058:
12053:
12048:
12043:
12038:
12033:
12031:
12024:
12019:
12014:
12009:
12004:
11999:
11994:
11992:
11988:
11981:
11977:
11975:
11968:
11961:
11956:
11951:
11946:
11941:
11936:
11934:
11930:
11925:
11920:
11915:
11910:
11905:
11903:
11899:
11894:
11889:
11884:
11879:
11874:
11872:
11868:
11863:
11858:
11853:
11848:
11843:
11841:
11834:
11829:
11824:
11819:
11814:
11809:
11804:
11802:
11798:
11791:
11785:Runcitruncated
11779:
11774:
11769:
11764:
11759:
11754:
11752:
11748:
11743:
11738:
11733:
11728:
11723:
11721:
11717:
11712:
11707:
11702:
11697:
11692:
11690:
11686:
11681:
11676:
11671:
11666:
11661:
11659:
11652:
11647:
11642:
11637:
11632:
11627:
11622:
11620:
11616:
11609:
11604:
11597:
11592:
11587:
11582:
11577:
11572:
11570:
11566:
11561:
11556:
11551:
11546:
11541:
11539:
11535:
11530:
11525:
11520:
11515:
11510:
11508:
11504:
11499:
11494:
11489:
11484:
11479:
11477:
11470:
11465:
11460:
11455:
11450:
11445:
11440:
11438:
11429:Cantitruncated
11423:
11418:
11413:
11408:
11403:
11398:
11396:
11392:
11387:
11382:
11377:
11372:
11367:
11365:
11361:
11356:
11351:
11346:
11341:
11336:
11334:
11330:
11325:
11320:
11315:
11310:
11305:
11303:
11296:
11291:
11286:
11281:
11276:
11271:
11266:
11264:
11260:
11246:
11237:
11232:
11227:
11222:
11217:
11212:
11210:
11206:
11201:
11196:
11191:
11186:
11181:
11179:
11175:
11170:
11165:
11160:
11155:
11150:
11148:
11144:
11139:
11134:
11129:
11124:
11119:
11117:
11110:
11105:
11100:
11095:
11090:
11085:
11080:
11078:
11074:
11067:
11062:
11055:
11050:
11045:
11040:
11035:
11030:
11028:
11024:
11019:
11014:
11009:
11004:
10999:
10997:
10993:
10988:
10983:
10978:
10973:
10968:
10966:
10962:
10957:
10952:
10947:
10942:
10937:
10935:
10928:
10923:
10918:
10913:
10908:
10903:
10898:
10896:
10892:
10875:
10870:
10865:
10860:
10855:
10850:
10848:
10844:
10839:
10834:
10829:
10824:
10819:
10817:
10813:
10808:
10803:
10798:
10793:
10788:
10786:
10782:
10777:
10772:
10767:
10762:
10757:
10755:
10748:
10743:
10738:
10733:
10728:
10723:
10718:
10716:
10712:
10705:
10700:
10693:
10688:
10683:
10678:
10673:
10668:
10666:
10662:
10657:
10652:
10647:
10642:
10637:
10635:
10631:
10626:
10621:
10616:
10611:
10606:
10604:
10600:
10595:
10590:
10585:
10580:
10575:
10573:
10566:
10561:
10556:
10551:
10546:
10541:
10536:
10534:
10517:
10512:
10507:
10502:
10497:
10492:
10490:
10486:
10481:
10476:
10471:
10466:
10461:
10459:
10455:
10450:
10445:
10440:
10435:
10430:
10428:
10424:
10419:
10414:
10409:
10404:
10399:
10397:
10390:
10385:
10380:
10375:
10370:
10365:
10360:
10358:
10354:
10337:
10332:
10327:
10322:
10317:
10312:
10310:
10306:
10301:
10296:
10291:
10286:
10281:
10279:
10275:
10270:
10265:
10260:
10255:
10250:
10248:
10244:
10239:
10234:
10229:
10224:
10219:
10217:
10210:
10205:
10200:
10195:
10190:
10185:
10180:
10178:
10174:
10167:
10158:
10151:
10146:
10141:
10136:
10131:
10126:
10124:
10120:
10115:
10110:
10105:
10100:
10095:
10093:
10089:
10084:
10079:
10074:
10069:
10064:
10062:
10058:
10053:
10048:
10043:
10038:
10033:
10031:
10024:
10019:
10014:
10009:
10004:
9999:
9994:
9992:
9988:
9981:
9976:
9969:
9964:
9959:
9954:
9949:
9944:
9942:
9938:
9933:
9928:
9923:
9918:
9913:
9911:
9907:
9902:
9897:
9892:
9887:
9882:
9880:
9876:
9871:
9866:
9861:
9856:
9851:
9849:
9842:
9837:
9832:
9827:
9822:
9817:
9812:
9810:
9806:
9787:
9782:
9777:
9772:
9767:
9762:
9760:
9756:
9751:
9746:
9741:
9736:
9731:
9729:
9725:
9720:
9715:
9710:
9705:
9700:
9698:
9694:
9689:
9684:
9679:
9674:
9669:
9667:
9660:
9655:
9650:
9645:
9640:
9635:
9630:
9628:
9624:
9604:
9599:
9594:
9589:
9584:
9579:
9574:
9572:
9571:
9563:
9558:
9553:
9548:
9543:
9538:
9533:
9531:
9530:
9522:
9517:
9512:
9507:
9502:
9497:
9492:
9490:
9489:
9481:
9476:
9471:
9466:
9461:
9456:
9451:
9449:
9448:
9438:
9433:
9428:
9423:
9418:
9413:
9408:
9406:
9400:
9368:
9366:Truncated forms
9354:
9349:
9344:
9339:
9334:
9329:
9324:
9319:
9317:
9316:
9310:
9303:
9298:
9293:
9288:
9283:
9278:
9276:
9275:
9269:
9260:
9255:
9250:
9245:
9240:
9235:
9230:
9225:
9223:
9222:
9216:
9205:
9201:
9196:
9191:
9186:
9181:
9176:
9171:
9166:
9164:
9163:
9157:
9153:
9146:
9142:
9137:
9132:
9127:
9122:
9117:
9112:
9107:
9105:
9104:
9098:
9089:
9085:
9080:
9075:
9070:
9065:
9060:
9055:
9050:
9048:
9047:
9041:
9034:
9030:
9025:
9020:
9015:
9010:
9005:
9000:
8995:
8993:
8992:
8986:
8979:
8975:
8970:
8965:
8960:
8955:
8950:
8945:
8940:
8938:
8937:
8931:
8922:
8918:
8913:
8908:
8903:
8898:
8893:
8888:
8883:
8881:
8880:
8874:
8867:
8863:
8858:
8853:
8848:
8843:
8838:
8833:
8828:
8826:
8825:
8819:
8812:
8808:
8803:
8798:
8793:
8788:
8783:
8778:
8773:
8771:
8770:
8764:
8757:
8752:
8747:
8742:
8737:
8732:
8730:
8729:
8723:
8716:
8712:
8707:
8702:
8697:
8692:
8687:
8682:
8677:
8675:
8674:
8668:
8661:
8657:
8652:
8647:
8642:
8637:
8632:
8627:
8622:
8620:
8619:
8613:
8606:
8602:
8597:
8592:
8587:
8582:
8577:
8572:
8567:
8565:
8564:
8558:
8549:
8544:
8539:
8534:
8529:
8524:
8519:
8514:
8512:
8511:
8505:
8498:
8493:
8488:
8483:
8478:
8473:
8468:
8463:
8461:
8460:
8454:
8447:
8442:
8437:
8432:
8427:
8422:
8417:
8412:
8410:
8409:
8403:
8396:
8391:
8386:
8381:
8376:
8371:
8369:
8368:
8362:
8355:
8350:
8345:
8340:
8335:
8330:
8325:
8320:
8318:
8317:
8311:
8304:
8299:
8294:
8289:
8284:
8279:
8274:
8269:
8267:
8266:
8260:
8253:
8248:
8243:
8238:
8233:
8228:
8223:
8218:
8216:
8215:
8209:
8200:
8195:
8190:
8185:
8180:
8175:
8170:
8165:
8163:
8162:
8156:
8152:
8145:
8140:
8135:
8130:
8125:
8120:
8115:
8110:
8108:
8107:
8101:
8094:
8089:
8084:
8079:
8074:
8069:
8067:
8066:
8060:
8053:
8048:
8043:
8038:
8033:
8028:
8023:
8018:
8016:
8015:
8009:
8002:
7997:
7992:
7987:
7982:
7977:
7972:
7967:
7965:
7964:
7958:
7951:
7947:
7942:
7937:
7932:
7927:
7922:
7917:
7912:
7910:
7909:
7903:
7894:
7890:
7885:
7880:
7875:
7870:
7865:
7860:
7855:
7853:
7852:
7846:
7842:
7835:
7831:
7826:
7821:
7816:
7811:
7806:
7801:
7796:
7794:
7793:
7787:
7778:
7774:
7769:
7764:
7759:
7754:
7749:
7744:
7739:
7737:
7736:
7730:
7723:
7719:
7714:
7709:
7704:
7699:
7694:
7689:
7684:
7682:
7681:
7675:
7668:
7664:
7659:
7654:
7649:
7644:
7639:
7634:
7629:
7627:
7626:
7620:
7611:
7606:
7601:
7596:
7591:
7586:
7581:
7576:
7574:
7573:
7567:
7560:
7555:
7550:
7545:
7540:
7535:
7530:
7525:
7523:
7522:
7516:
7509:
7504:
7499:
7494:
7489:
7484:
7479:
7474:
7472:
7471:
7465:
7458:
7453:
7448:
7443:
7438:
7433:
7431:
7430:
7424:
7417:
7412:
7407:
7402:
7397:
7392:
7387:
7382:
7380:
7379:
7373:
7366:
7361:
7356:
7351:
7346:
7341:
7336:
7331:
7329:
7328:
7322:
7315:
7310:
7305:
7300:
7295:
7290:
7285:
7280:
7278:
7277:
7271:
7262:
7257:
7252:
7247:
7242:
7237:
7232:
7227:
7225:
7224:
7218:
7211:
7206:
7201:
7196:
7191:
7186:
7181:
7176:
7174:
7173:
7167:
7160:
7155:
7150:
7145:
7140:
7135:
7130:
7125:
7123:
7122:
7116:
7109:
7104:
7099:
7094:
7089:
7084:
7082:
7081:
7075:
7068:
7063:
7058:
7053:
7048:
7043:
7038:
7033:
7031:
7030:
7024:
7017:
7012:
7007:
7002:
6997:
6992:
6987:
6982:
6980:
6979:
6973:
6966:
6961:
6956:
6951:
6946:
6941:
6936:
6931:
6929:
6928:
6922:
6913:
6908:
6903:
6898:
6893:
6888:
6883:
6878:
6876:
6875:
6869:
6862:
6857:
6852:
6847:
6842:
6837:
6832:
6827:
6825:
6824:
6818:
6811:
6806:
6801:
6796:
6791:
6786:
6781:
6776:
6774:
6773:
6767:
6760:
6755:
6750:
6745:
6740:
6735:
6733:
6732:
6726:
6719:
6714:
6709:
6704:
6699:
6694:
6689:
6684:
6682:
6681:
6675:
6668:
6663:
6658:
6653:
6648:
6643:
6638:
6633:
6631:
6630:
6624:
6617:
6612:
6607:
6602:
6597:
6592:
6587:
6582:
6580:
6579:
6573:
6564:
6559:
6554:
6549:
6544:
6539:
6534:
6529:
6527:
6526:
6520:
6513:
6508:
6503:
6498:
6493:
6488:
6483:
6478:
6476:
6475:
6469:
6462:
6457:
6452:
6447:
6442:
6437:
6432:
6427:
6425:
6424:
6418:
6411:
6406:
6401:
6396:
6391:
6386:
6384:
6383:
6377:
6370:
6365:
6360:
6355:
6350:
6345:
6340:
6335:
6333:
6332:
6326:
6319:
6314:
6309:
6304:
6299:
6294:
6289:
6284:
6282:
6281:
6275:
6268:
6263:
6258:
6253:
6248:
6243:
6238:
6233:
6231:
6230:
6224:
6212:
6207:
6202:
6197:
6192:
6187:
6182:
6180:
6179:
6172:
6167:
6162:
6157:
6152:
6147:
6142:
6140:
6139:
6132:
6127:
6122:
6117:
6112:
6110:
6109:
6102:
6097:
6092:
6087:
6082:
6077:
6072:
6070:
6069:
6062:
6057:
6052:
6047:
6042:
6037:
6032:
6030:
6029:
6024:
6018:
6012:
6006:
6000:
5989:grand antiprism
5978:
5929:cubic honeycomb
5835:
5832:
5829:cubic honeycomb
5801:grand antiprism
5782:
5776:
5774:Four dimensions
5766:
5722:
5717:
5712:
5707:
5702:
5697:
5695:
5691:
5666:
5665:
5649:
5648:
5643:
5640:
5639:
5616:
5602:
5597:
5592:
5587:
5582:
5580:
5573:
5568:
5563:
5561:
5539:
5538:
5532:
5531:
5521:
5520:
5515:
5512:
5511:
5497:
5492:
5487:
5482:
5477:
5475:
5468:
5463:
5458:
5453:
5448:
5446:
5439:
5434:
5429:
5424:
5419:
5417:
5410:
5405:
5400:
5395:
5390:
5388:
5381:
5376:
5371:
5366:
5361:
5359:
5352:
5347:
5342:
5337:
5332:
5330:
5321:
5316:
5311:
5306:
5301:
5299:
5293:
5286:
5280:
5242:
5237:
5232:
5227:
5222:
5220:
5213:
5208:
5203:
5201:
5197:
5172:
5171:
5165:
5164:
5154:
5153:
5148:
5145:
5144:
5136:
5105:
5100:
5095:
5090:
5085:
5083:
5076:
5071:
5066:
5064:
5060:
5035:
5034:
5028:
5027:
5017:
5016:
5011:
5008:
5007:
4999:
4966:
4961:
4956:
4951:
4946:
4944:
4937:
4932:
4927:
4925:
4921:
4896:
4895:
4889:
4888:
4878:
4877:
4875:
4872:
4871:
4834:
4829:
4824:
4819:
4814:
4812:
4808:
4783:
4782:
4766:
4765:
4760:
4757:
4756:
4752:
4719:
4714:
4709:
4704:
4699:
4697:
4693:
4668:
4667:
4651:
4650:
4645:
4642:
4641:
4606:
4601:
4596:
4591:
4586:
4584:
4580:
4555:
4554:
4538:
4537:
4535:
4532:
4531:
4521:
4490:
4485:
4480:
4475:
4470:
4468:
4464:
4439:
4438:
4422:
4421:
4419:
4416:
4415:
4401:
4396:
4391:
4386:
4381:
4379:
4372:
4367:
4362:
4357:
4352:
4350:
4343:
4338:
4333:
4328:
4323:
4321:
4314:
4309:
4304:
4299:
4294:
4292:
4285:
4280:
4275:
4270:
4265:
4263:
4256:
4251:
4246:
4241:
4236:
4234:
4225:
4220:
4215:
4210:
4205:
4203:
4197:
4190:
4184:
4173:
4164:Coxeter diagram
4144:
4127:
4122:
4117:
4112:
4107:
4102:
4100:
4071:
4062:
4057:
4052:
4047:
4042:
4037:
4035:
4006:
3997:
3992:
3984:
3947:
3936:
3925:
3914:
3903:
3892:
3881:
3870:
3861:
3850:
3839:
3828:
3817:
3806:
3795:
3784:
3773:
3764:
3753:
3742:
3731:
3720:
3709:
3698:
3687:
3676:
3667:
3653:
3651:
3643:
3638:
3599:
3592:
3584:
3555:
3550:
3545:
3540:
3535:
3533:
3517:
3510:
3502:
3473:
3468:
3463:
3458:
3453:
3451:
3435:
3428:
3420:
3391:
3386:
3381:
3376:
3371:
3369:
3358:
3347:
3340:
3332:
3303:
3298:
3293:
3288:
3283:
3281:
3274:
3268:
3257:
3250:
3242:
3213:
3208:
3203:
3198:
3193:
3191:
3184:
3178:
3159:
3147:
3140:
3135:
3130:
3122:
3117:
3112:
3107:
3102:
3100:
3099:
3092:
3078:Platonic solids
3074:
3068:
3058:
3044:
3039:
3034:
3032:
3015:
3001:
2996:
2991:
2986:
2984:
2969:
2955:
2950:
2945:
2943:
2936:
2919:
2906:
2901:
2896:
2894:
2887:
2877:
2868:
2855:
2850:
2845:
2843:
2836:
2802:
2796:
2791:
2789:
2777:Schläfli symbol
2761:
2734:
2696:
2691:
2686:
2681:
2676:
2674:
2667:
2662:
2657:
2652:
2647:
2645:
2644:
2639:
2634:
2629:
2624:
2619:
2617:
2610:
2605:
2600:
2595:
2590:
2588:
2581:
2576:
2571:
2566:
2561:
2559:
2552:
2547:
2542:
2537:
2532:
2530:
2529:
2524:
2519:
2514:
2509:
2504:
2502:
2495:
2490:
2485:
2480:
2475:
2473:
2466:
2461:
2456:
2451:
2446:
2444:
2437:
2432:
2427:
2422:
2417:
2415:
2410:
2395:
2384:
2317:rational number
2300:
2280:
2260:
2240:
2220:
2203:
2198:
2193:
2191:
2190:
2185:
2180:
2175:
2173:
2166:
2161:
2156:
2154:
2147:
2142:
2137:
2135:
2134:
2129:
2124:
2119:
2117:
2110:
2105:
2100:
2098:
2091:
2086:
2081:
2079:
2078:
2073:
2068:
2063:
2061:
2054:
2049:
2044:
2042:
2035:
2030:
2025:
2023:
2022:
2017:
2012:
2007:
2005:
1998:
1993:
1988:
1986:
1979:
1974:
1969:
1967:
1966:
1961:
1956:
1951:
1949:
1944:
1936:
1928:
1920:
1912:
1904:
1831:
1811:
1791:
1771:
1747:
1742:
1737:
1735:
1728:
1723:
1718:
1716:
1715:
1710:
1705:
1700:
1698:
1691:
1686:
1681:
1679:
1672:
1667:
1662:
1660:
1659:
1654:
1649:
1644:
1642:
1635:
1630:
1625:
1623:
1616:
1611:
1606:
1604:
1603:
1598:
1593:
1588:
1586:
1579:
1574:
1569:
1567:
1560:
1555:
1550:
1548:
1547:
1542:
1537:
1532:
1530:
1523:
1518:
1513:
1511:
1506:
1495:
1487:
1479:
1471:
1417:
1411:
1400:
1374:
1367:
1362:
1330:
1287:
1254:Coxeter diagram
1246:
1184:
1161:runcitruncation
1145:
1124:
1113:
1107:triheptellation
1103:
1075:
1058:
1043:
1017:
1004:
989:
956:
945:
939:tripentellation
935:
907:
890:
876:
851:
838:
824:
794:
783:
773:
741:
724:
710:
696:
671:Cantitruncation
663:
650:
636:
630:tricantellation
622:
568:
553:
539:
525:
511:
489:'to cut off'.)
465:
445:cuts vertices,
435:
420:
416:
412:
407:
393:Schläfli symbol
374:birectification
350:
338:Johannes Kepler
331:grand antiprism
323:Coxeter diagram
315:
293:(2-dimensional
234:
229:
224:
219:
214:
209:
204:
199:
194:
192:
187:
175:
170:
165:
160:
155:
150:
145:
143:
138:
116:
111:
106:
101:
96:
94:
89:
76:
71:
66:
64:
62:Coxeter diagram
49:
28:
23:
22:
15:
12:
11:
5:
14645:
14635:
14634:
14618:
14617:
14602:
14601:
14592:
14588:
14581:
14574:
14570:
14561:
14544:
14535:
14524:
14523:
14521:
14519:
14514:
14505:
14500:
14494:
14493:
14491:
14489:
14484:
14475:
14470:
14464:
14463:
14461:
14457:
14450:
14443:
14439:
14434:
14425:
14420:
14414:
14413:
14411:
14407:
14400:
14393:
14389:
14384:
14375:
14370:
14364:
14363:
14361:
14357:
14350:
14346:
14341:
14332:
14327:
14321:
14320:
14318:
14316:
14311:
14302:
14297:
14291:
14290:
14281:
14276:
14271:
14262:
14257:
14251:
14250:
14241:
14239:
14234:
14225:
14220:
14214:
14213:
14208:
14203:
14198:
14193:
14188:
14182:
14181:
14177:
14173:
14168:
14157:
14146:
14137:
14128:
14121:
14115:
14105:
14099:
14093:
14087:
14081:
14075:
14069:
14068:
14057:
14055:
14054:
14047:
14040:
14032:
14027:
14026:
14017:
13998:
13997:External links
13995:
13994:
13993:
13982:Marco Möller,
13980:
13979:
13978:
13971:
13964:
13942:
13941:
13940:
13933:
13916:H.S.M. Coxeter
13913:
13903:A. Boole Stott
13900:
13899:
13898:
13882:Norman Johnson
13879:
13860:
13857:
13856:
13855:
13848:
13845:
13828:
13825:
13818:
13807:
13800:
13793:
13778:cross-polytope
13765:
13762:
13759:
13758:
13727:
13696:
13665:
13634:
13595:
13591:
13588:
13585:
13579:
13578:
13542:
13506:
13475:
13444:
13400:
13396:
13393:
13390:
13387:Snub rectified
13384:
13383:
13347:
13311:
13275:
13239:
13190:
13186:
13183:
13180:
13174:
13173:
13137:
13106:
13075:
13039:
12995:
12991:
12988:
12985:
12979:
12978:
12947:
12916:
12880:
12839:
12790:
12786:
12783:
12780:
12776:
12775:
12744:
12713:
12677:
12641:
12597:
12593:
12590:
12587:
12579:
12578:
12537:
12496:
12455:
12413:
12412:
12372:
12365:
12362:
12344:
12341:
12338:
12337:
12306:
12275:
12244:
12213:
12174:
12170:
12167:
12164:
12154:
12153:
12122:
12091:
12060:
12029:
11990:
11986:
11983:
11979:
11973:
11970:
11964:
11963:
11932:
11901:
11870:
11839:
11800:
11796:
11793:
11789:
11786:
11782:
11781:
11750:
11719:
11688:
11657:
11618:
11614:
11611:
11606:
11600:
11599:
11568:
11537:
11506:
11475:
11436:
11433:
11430:
11426:
11425:
11394:
11363:
11332:
11301:
11262:
11258:
11255:
11252:
11240:
11239:
11208:
11177:
11146:
11115:
11076:
11072:
11069:
11064:
11058:
11057:
11026:
10995:
10964:
10933:
10894:
10890:
10887:
10884:
10878:
10877:
10846:
10815:
10784:
10753:
10714:
10710:
10707:
10702:
10696:
10695:
10664:
10633:
10602:
10571:
10532:
10529:
10526:
10520:
10519:
10488:
10457:
10426:
10395:
10356:
10352:
10349:
10346:
10340:
10339:
10308:
10277:
10246:
10215:
10176:
10172:
10169:
10164:
10154:
10153:
10122:
10091:
10060:
10029:
9990:
9986:
9983:
9978:
9972:
9971:
9940:
9909:
9878:
9847:
9808:
9804:
9801:
9798:
9790:
9789:
9758:
9727:
9696:
9665:
9626:
9622:
9619:
9616:
9610:
9609:
9568:
9527:
9486:
9444:
9443:
9403:
9396:
9393:
9387:
9386:
9383:
9380:
9367:
9364:
9361:
9360:
9358:
9356:
9305:
9264:
9262:
9211:
9208:
9207:
9203:
9148:
9144:
9093:
9091:
9087:
9036:
9032:
8981:
8977:
8925:
8924:
8920:
8869:
8865:
8814:
8810:
8759:
8718:
8714:
8663:
8659:
8608:
8604:
8552:
8551:
8500:
8449:
8398:
8357:
8306:
8255:
8203:
8202:
8147:
8096:
8055:
8004:
7953:
7949:
7897:
7896:
7892:
7837:
7833:
7782:
7780:
7776:
7725:
7721:
7670:
7666:
7614:
7613:
7562:
7511:
7460:
7419:
7368:
7317:
7265:
7264:
7213:
7162:
7111:
7070:
7019:
6968:
6916:
6915:
6864:
6813:
6762:
6721:
6670:
6619:
6567:
6566:
6515:
6464:
6413:
6372:
6321:
6270:
6218:
6217:
6177:
6137:
6107:
6067:
6026:
6025:
6022:
6019:
6016:
6013:
6010:
6007:
6004:
6001:
5998:
5986:non-Wythoffian
5977:
5974:
5970:
5969:
5963:
5957:
5951:
5914:
5913:
5910:
5899:
5892:
5881:
5798:non-Wythoffian
5778:Main article:
5775:
5772:
5769:
5768:
5759:
5747:
5746:
5744:
5741:
5738:
5735:
5732:
5729:
5727:
5693:
5689:
5686:
5683:
5670:
5664:
5661:
5658:
5655:
5654:
5652:
5647:
5637:
5631:
5630:
5627:
5624:
5621:
5618:
5613:
5610:
5607:
5578:
5559:
5556:
5543:
5537:
5534:
5533:
5530:
5527:
5526:
5524:
5519:
5509:
5507:Snub rectified
5503:
5502:
5473:
5444:
5415:
5386:
5357:
5327:
5326:
5296:
5289:
5282:
5277:
5269:
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5250:
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5218:
5199:
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5192:
5189:
5176:
5170:
5167:
5166:
5163:
5160:
5159:
5157:
5152:
5142:
5135:Cantitruncated
5132:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5081:
5062:
5058:
5055:
5052:
5039:
5033:
5030:
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5026:
5023:
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5015:
5005:
4993:
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4989:
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4894:
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4787:
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4778:
4775:
4772:
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4695:
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4688:
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4672:
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4657:
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4649:
4639:
4633:
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4629:
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4623:
4620:
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4614:
4611:
4582:
4578:
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4529:
4517:
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4513:
4510:
4507:
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4498:
4495:
4466:
4462:
4459:
4456:
4443:
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4425:
4413:
4407:
4406:
4377:
4348:
4319:
4290:
4261:
4231:
4230:
4200:
4193:
4186:
4181:
4171:
4148:Wythoff symbol
4143:
4140:
4130:
4129:
4098:
4091:
4084:
4077:
4072:
4069:
4065:
4064:
4033:
4026:
4019:
4012:
4007:
4004:
4000:
3999:
3987:
3980:
3977:
3974:
3971:
3952:
3951:
3940:
3929:
3918:
3907:
3896:
3885:
3874:
3863:
3855:
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3843:
3832:
3821:
3810:
3799:
3788:
3777:
3766:
3758:
3757:
3746:
3735:
3724:
3713:
3702:
3691:
3680:
3669:
3661:
3660:
3657:
3654:Cantitruncated
3648:
3645:
3640:
3635:
3632:
3629:
3626:
3604:
3603:
3600:
3597:
3594:
3589:
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3581:
3574:
3567:
3560:
3531:
3528:
3522:
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3518:
3515:
3512:
3507:
3504:
3499:
3492:
3485:
3478:
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3440:
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3436:
3433:
3430:
3425:
3422:
3417:
3410:
3403:
3396:
3367:
3364:
3352:
3351:
3348:
3345:
3342:
3337:
3334:
3329:
3322:
3315:
3308:
3279:
3276:
3262:
3261:
3258:
3255:
3252:
3247:
3244:
3239:
3232:
3225:
3218:
3189:
3186:
3172:
3171:
3166:
3161:
3154:
3149:
3142:
3137:
3132:
3131:(transparent)
3127:
3094:
3087:
3070:Main article:
3067:
3064:
3061:
3060:
3055:
3052:
3049:
3030:
3027:
3024:
3018:
3017:
3012:
3009:
3006:
2982:
2979:
2976:
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2922:
2921:
2917:
2914:
2911:
2892:
2889:
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2879:
2871:
2870:
2866:
2863:
2860:
2841:
2838:
2834:
2831:
2828:
2822:
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2818:
2814:
2813:
2808:
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2798:
2793:
2786:
2771:
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2767:
2754:
2747:
2740:
2727:
2720:
2713:
2706:
2702:
2701:
2672:
2615:
2586:
2557:
2500:
2471:
2442:
2413:
2405:
2404:
2397:
2392:
2389:
2386:
2381:
2378:
2375:
2372:
2366:
2365:
2359:
2354:
2349:
2344:
2339:
2334:
2315:(one for each
2307:
2306:
2293:
2286:
2273:
2266:
2253:
2246:
2233:
2226:
2213:
2209:
2208:
2171:
2152:
2115:
2096:
2059:
2040:
2003:
1984:
1947:
1939:
1938:
1933:
1930:
1925:
1922:
1917:
1914:
1909:
1906:
1901:
1895:
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1869:
1864:
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1854:
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1824:
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1804:
1797:
1784:
1777:
1764:
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1753:
1752:
1733:
1696:
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1640:
1621:
1584:
1565:
1528:
1509:
1501:
1500:
1497:
1492:
1489:
1484:
1481:
1476:
1473:
1468:
1465:
1459:
1458:
1453:
1448:
1443:
1438:
1433:
1428:
1423:
1406:
1395:
1373:
1372:Two dimensions
1370:
1365:
1361:
1358:
1329:
1326:
1298:vertex figures
1286:
1283:
1245:
1242:
1183:
1180:
1176:omnitruncation
1157:
1156:
1143:
1139:
1122:
1118:
1117:
1116:
1111:
1101:
1097:biheptellation
1073:
1069:
1068:
1067:
1066:
1065:
1056:
1041:
1025:Hexiruncinated
1015:
1011:
1002:
987:
954:
950:
949:
948:
943:
933:
929:bipentellation
905:
901:
900:
899:
898:
897:
888:
874:
849:
845:
836:
832:tristerication
822:
792:
788:
787:
786:
781:
777:triruncination
771:
739:
735:
734:
733:
732:
731:
722:
708:
694:
661:
657:
648:
634:
620:
616:bicantellation
611:
610:
566:
562:
561:
560:
551:
537:
523:
509:
498:
497:
463:
434:
431:
430:
429:
418:
414:
410:
405:
349:
346:
314:
311:
283:vertex figures
240:
239:
180:
130:
129:
126:
122:
121:
82:
41:
40:
37:
26:
9:
6:
4:
3:
2:
14644:
14633:
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14627:
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14608:
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14509:
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14303:
14301:
14298:
14296:
14293:
14292:
14289:
14285:
14282:
14280:
14277:
14275:
14274:Demitesseract
14272:
14270:
14266:
14263:
14261:
14258:
14256:
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14252:
14249:
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14010:
14006:
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14000:
13992:
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13507:
13476:
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13386:
13348:
13312:
13276:
13240:
13191:
13184:
13181:
13176:
13138:
13107:
13076:
13040:
12996:
12989:
12986:
12981:
12948:
12917:
12881:
12840:
12791:
12784:
12781:
12778:
12745:
12714:
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12642:
12598:
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12588:
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12497:
12456:
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12371:
12359:
12356:
12354:
12350:
12307:
12276:
12245:
12214:
12175:
12168:
12165:
12162:
12161:omnitruncated
12156:
12123:
12092:
12061:
12030:
11991:
11984:
11971:
11966:
11933:
11902:
11871:
11840:
11801:
11794:
11787:
11784:
11751:
11720:
11689:
11658:
11619:
11612:
11607:
11602:
11569:
11538:
11507:
11476:
11437:
11434:
11431:
11428:
11395:
11364:
11333:
11302:
11263:
11256:
11253:
11250:
11245:
11242:
11209:
11178:
11147:
11116:
11077:
11070:
11065:
11061:Bicantellated
11060:
11027:
10996:
10965:
10934:
10895:
10888:
10885:
10883:
10880:
10847:
10816:
10785:
10754:
10715:
10708:
10703:
10698:
10665:
10634:
10603:
10572:
10533:
10530:
10527:
10525:
10522:
10489:
10458:
10427:
10396:
10357:
10350:
10347:
10345:
10342:
10309:
10278:
10247:
10216:
10177:
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10162:
10156:
10123:
10092:
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10030:
9991:
9984:
9979:
9974:
9941:
9910:
9879:
9848:
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9796:
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9759:
9728:
9697:
9666:
9627:
9620:
9617:
9615:
9612:
9569:
9528:
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9357:
9314:
9309:
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9215:
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9210:
9161:
9156:
9152:
9102:
9097:
9094:
9092:
9045:
9040:
9037:
8990:
8985:
8982:
8935:
8930:
8927:
8878:
8873:
8870:
8823:
8818:
8815:
8768:
8763:
8760:
8727:
8722:
8719:
8672:
8667:
8664:
8617:
8612:
8609:
8562:
8557:
8554:
8509:
8504:
8501:
8458:
8453:
8450:
8407:
8402:
8399:
8366:
8361:
8358:
8315:
8310:
8307:
8264:
8259:
8256:
8213:
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8151:
8105:
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8005:
7962:
7957:
7954:
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7902:
7899:
7850:
7845:
7841:
7791:
7786:
7783:
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7726:
7679:
7674:
7671:
7624:
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7512:
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7377:
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7369:
7326:
7321:
7318:
7275:
7270:
7267:
7222:
7217:
7214:
7171:
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7163:
7120:
7115:
7112:
7079:
7074:
7071:
7028:
7023:
7020:
6977:
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6865:
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6814:
6771:
6766:
6763:
6730:
6725:
6722:
6679:
6674:
6671:
6628:
6623:
6620:
6577:
6572:
6569:
6524:
6519:
6516:
6473:
6468:
6465:
6422:
6417:
6414:
6381:
6376:
6375:demitesseract
6373:
6330:
6325:
6322:
6279:
6274:
6271:
6228:
6223:
6220:
6138:
6108:
6028:
6027:
6014:
6008:
5996:
5995:
5992:
5990:
5987:
5983:
5973:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5946:
5945:
5944:
5941:
5936:
5932:
5930:
5926:
5923:Eight of the
5921:
5919:
5911:
5908:
5904:
5900:
5897:
5893:
5890:
5886:
5882:
5879:
5875:
5874:
5873:
5871:
5866:
5864:
5860:
5856:
5852:
5843:
5837:
5830:
5825:
5821:
5818:
5812:
5810:
5806:
5802:
5799:
5795:
5791:
5787:
5781:
5764:
5760:
5757:
5753:
5752:
5745:
5742:
5739:
5736:
5733:
5730:
5728:
5687:
5684:
5668:
5662:
5659:
5656:
5650:
5645:
5638:
5636:
5633:
5628:
5625:
5622:
5619:
5614:
5611:
5608:
5579:
5560:
5541:
5535:
5528:
5522:
5517:
5510:
5508:
5505:
5474:
5445:
5416:
5387:
5358:
5329:
5295:
5288:
5274:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5219:
5200:
5193:
5190:
5174:
5168:
5161:
5155:
5150:
5143:
5140:
5139:Omnitruncated
5134:
5129:
5126:
5123:
5120:
5117:
5114:
5111:
5082:
5063:
5056:
5053:
5037:
5031:
5024:
5018:
5013:
5006:
5003:
4998:
4995:
4990:
4987:
4984:
4981:
4978:
4975:
4972:
4943:
4924:
4917:
4914:
4898:
4892:
4885:
4879:
4870:
4868:
4867:
4863:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4804:
4801:
4785:
4779:
4776:
4773:
4767:
4762:
4755:
4751:
4748:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4689:
4686:
4670:
4664:
4661:
4658:
4652:
4647:
4640:
4638:
4635:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4576:
4573:
4557:
4551:
4548:
4545:
4539:
4530:
4527:
4526:
4519:
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4460:
4457:
4441:
4435:
4432:
4429:
4423:
4414:
4412:
4409:
4378:
4349:
4320:
4291:
4262:
4233:
4199:
4192:
4178:
4175:
4169:
4165:
4161:
4157:
4153:
4149:
4142:Constructions
4139:
4137:
4096:
4092:
4089:
4085:
4082:
4078:
4076:
4073:
4067:
4031:
4027:
4024:
4020:
4017:
4013:
4011:
4008:
4002:
3996:
3991:
3988:
3986:
3981:
3978:
3975:
3972:
3969:
3968:
3965:
3963:
3959:
3950:
3945:
3941:
3939:
3934:
3930:
3928:
3923:
3919:
3917:
3912:
3908:
3906:
3901:
3897:
3895:
3890:
3886:
3884:
3879:
3875:
3873:
3868:
3864:
3860:
3857:
3856:
3853:
3848:
3844:
3842:
3837:
3833:
3831:
3826:
3822:
3820:
3815:
3811:
3809:
3804:
3800:
3798:
3793:
3789:
3787:
3782:
3778:
3776:
3771:
3767:
3763:
3760:
3759:
3756:
3751:
3747:
3745:
3740:
3736:
3734:
3729:
3725:
3723:
3718:
3714:
3712:
3707:
3703:
3701:
3696:
3692:
3690:
3685:
3681:
3679:
3674:
3670:
3666:
3663:
3662:
3658:
3650:Omnitruncated
3649:
3646:
3641:
3636:
3633:
3630:
3627:
3625:
3624:
3621:
3619:
3615:
3611:
3602:Dodecahedron
3601:
3595:
3590:
3587:
3582:
3579:
3575:
3572:
3568:
3565:
3561:
3532:
3529:
3527:
3524:
3519:
3513:
3508:
3505:
3500:
3497:
3493:
3490:
3486:
3483:
3479:
3450:
3447:
3445:
3442:
3437:
3431:
3426:
3423:
3418:
3415:
3411:
3408:
3404:
3401:
3397:
3368:
3365:
3362:
3357:
3354:
3349:
3343:
3338:
3335:
3330:
3327:
3323:
3320:
3316:
3313:
3309:
3280:
3277:
3275:(Hexahedron)
3272:
3267:
3264:
3259:
3253:
3248:
3245:
3240:
3237:
3233:
3230:
3226:
3223:
3219:
3190:
3187:
3182:
3177:
3174:
3170:
3167:
3165:
3162:
3158:
3155:
3153:
3150:
3146:
3143:
3138:
3133:
3128:
3098:
3095:
3091:
3088:
3085:
3081:
3079:
3073:
3056:
3053:
3050:
3031:
3028:
3023:
3020:
3013:
3010:
3007:
2983:
2980:
2974:
2967:
2964:
2961:
2942:
2939:
2932:
2929:
2927:
2924:
2918:
2915:
2912:
2893:
2890:
2883:
2880:
2876:
2873:
2867:
2864:
2861:
2842:
2839:
2832:
2829:
2827:
2824:
2819:
2816:
2812:
2804:
2783:
2780:
2778:
2768:
2765:
2759:
2755:
2752:
2748:
2745:
2741:
2738:
2732:
2728:
2725:
2721:
2718:
2714:
2711:
2707:
2704:
2673:
2616:
2587:
2558:
2501:
2472:
2443:
2414:
2412:
2407:
2406:
2402:
2398:
2393:
2390:
2387:
2382:
2379:
2376:
2373:
2371:
2368:
2364:
2360:
2358:
2355:
2353:
2348:
2345:
2343:
2338:
2335:
2332:
2328:
2326:
2322:
2318:
2314:
2313:star polygons
2304:
2298:
2294:
2291:
2287:
2284:
2278:
2274:
2271:
2267:
2264:
2258:
2254:
2251:
2247:
2244:
2238:
2234:
2231:
2227:
2224:
2218:
2214:
2211:
2172:
2153:
2116:
2097:
2060:
2041:
2004:
1985:
1948:
1946:
1941:
1934:
1931:
1926:
1923:
1918:
1915:
1910:
1907:
1902:
1900:
1897:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1847:
1842:
1838:
1835:
1829:
1825:
1822:
1818:
1815:
1809:
1805:
1802:
1798:
1795:
1789:
1785:
1782:
1778:
1775:
1769:
1765:
1762:
1758:
1755:
1734:
1697:
1678:
1641:
1622:
1585:
1566:
1529:
1510:
1508:
1503:
1498:
1493:
1490:
1485:
1482:
1477:
1474:
1469:
1466:
1464:
1461:
1457:
1454:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1432:
1429:
1427:
1424:
1421:
1415:
1410:
1407:
1404:
1399:
1396:
1393:
1389:
1387:
1383:
1379:
1369:
1360:One dimension
1357:
1355:
1351:
1347:
1344:-dimensional
1343:
1339:
1335:
1325:
1323:
1319:
1315:
1314:cuboctahedron
1311:
1307:
1303:
1299:
1294:
1292:
1282:
1280:
1276:
1275:omnitruncated
1272:
1267:
1264:
1262:
1257:
1255:
1251:
1250:vertex figure
1244:Vertex figure
1241:
1239:
1238:demitesseract
1235:
1231:
1227:
1224:are known as
1223:
1218:
1216:
1211:
1209:
1205:
1197:
1193:
1188:
1179:
1177:
1172:
1170:
1166:
1162:
1154:
1151:- applied to
1150:
1146:
1140:
1137:
1134:- applied to
1133:
1129:
1125:
1119:
1114:
1108:
1104:
1098:
1094:
1093:
1091:
1090:
1084:
1081:- applied to
1080:
1076:
1070:
1063:
1059:
1052:
1048:
1044:
1037:
1033:
1032:
1030:
1027:- applied to
1026:
1022:
1018:
1012:
1009:
1005:
998:
997:trihexication
994:
990:
983:
979:
978:
976:
975:
969:
966:- applied to
965:
961:
957:
951:
946:
940:
936:
930:
926:
925:
923:
922:
916:
913:- applied to
912:
908:
902:
895:
891:
885:
881:
877:
871:
867:
866:
864:
861:- applied to
860:
856:
852:
846:
843:
839:
833:
829:
825:
819:
818:bisterication
815:
814:
812:
807:
804:- applied to
803:
799:
795:
789:
784:
778:
774:
768:
767:biruncination
764:
763:
761:
758:'carpenter's
757:
752:
749:- applied to
748:
747:
742:
736:
729:
725:
719:
715:
711:
705:
701:
697:
691:
687:
686:
684:
680:
679:rectification
676:
673:- applied to
672:
668:
664:
658:
655:
651:
645:
641:
637:
631:
627:
623:
617:
613:
612:
608:
604:
603:
601:
600:
595:
591:
587:
586:rectification
583:
580:- applied to
579:
578:
573:
569:
563:
558:
554:
548:
544:
540:
534:
530:
526:
520:
519:tritruncation
516:
512:
506:
505:
500:
499:
495:
491:
490:
488:
484:
480:
477:- applied to
476:
475:
470:
466:
460:
459:
458:
456:
452:
448:
444:
440:
427:
424:
408:
401:
398:
397:
396:
394:
389:
387:
383:
379:
375:
371:
370:rectification
367:
363:
359:
358:rectification
355:
345:
343:
339:
334:
332:
328:
324:
320:
310:
308:
304:
300:
296:
292:
288:
287:star polygons
284:
280:
276:
274:
268:
266:
262:
258:
255:
251:
247:
190:
185:
181:
141:
136:
132:
127:
124:
92:
87:
83:
63:
59:
55:
52:
47:
43:
38:
35:
34:
19:
14594:
14563:
14554:
14546:
14537:
14528:
14508:10-orthoplex
14244:Dodecahedron
14165:
14154:
14143:
14134:
14125:
14116:
14112:
14102:
14094:
14090:
14082:
14078:
14063:
14013:the original
14008:
13983:
13944:
13936:
13926:
13906:
13891:
13884:
13867:
13841:
13830:
13782:
13767:
12352:
12348:
12346:
10699:Tritruncated
9793:
9613:
9376:
9369:
5979:
5971:
5965:
5959:
5953:
5947:
5939:
5937:
5933:
5922:
5915:
5894:group : the
5876:group : the
5867:
5848:
5833:
5813:
5805:snub 24-cell
5783:
5609:| 2 p q
5249:2 p q |
5112:p q | 2
4973:2 | p q
4864:
4841:2 p | q
4726:2 q | p
4613:p | 2 q
4523:
4497:q | 2 p
4410:
4145:
4133:
3955:
3647:Cantellated
3607:
3520:Icosahedron
3444:Dodecahedron
3075:
3021:
2925:
2874:
2825:
2774:
2400:
2324:
2310:
1887:Enneadecagon
1877:Heptadecagon
1867:Pentadecagon
1862:Tetradecagon
1386:quasiregular
1375:
1363:
1353:
1341:
1337:
1333:
1331:
1295:
1291:circumradius
1290:
1288:
1285:Circumradius
1268:
1265:
1258:
1247:
1237:
1219:
1214:
1212:
1207:
1201:
1173:
1168:
1164:
1160:
1158:
1148:
1141:
1131:
1127:
1120:
1109:
1106:
1099:
1096:
1087:
1079:Heptellation
1078:
1071:
1061:
1054:
1050:
1046:
1039:
1035:
1024:
1020:
1013:
1007:
1000:
996:
992:
985:
982:bihexication
981:
972:
963:
959:
952:
941:
938:
931:
928:
919:
911:Pentellation
910:
903:
893:
886:
883:
879:
872:
869:
858:
854:
847:
841:
834:
831:
827:
820:
817:
810:
801:
797:
790:
779:
776:
769:
766:
755:
744:
737:
727:
720:
717:
713:
706:
703:
699:
692:
689:
670:
666:
659:
653:
646:
643:
639:
632:
629:
625:
618:
615:
597:
593:
577:Cantellation
576:
571:
564:
556:
549:
546:
542:
535:
532:
528:
521:
518:
514:
507:
504:bitruncation
502:
486:
473:
468:
461:
454:
453:cuts faces,
450:
449:cuts edges,
447:cantellation
446:
442:
438:
436:
425:
422:
403:
399:
391:An extended
390:
385:
381:
377:
373:
369:
361:
353:
351:
335:
316:
272:
269:
249:
243:
14517:10-demicube
14478:9-orthoplex
14428:8-orthoplex
14378:7-orthoplex
14335:6-orthoplex
14305:5-orthoplex
14260:Pentachoron
14248:Icosahedron
14223:Tetrahedron
14024:(in German)
13988:(in German)
13785:exceptional
13182:2s{2p,q,2r}
12782:hr{2p,2q,r}
11610:= tr{r,q,p}
11068:= rr{r,q,p}
10882:Cantellated
10524:Bitruncated
10157:Trirectifed
9975:Birectified
5863:tetrahedron
5859:hypersphere
5817:multiplying
4997:Cantellated
4750:Bitruncated
4520:Birectified
3949:(3.3.3.3.5)
3859:Icosahedral
3852:(3.3.3.3.4)
3755:(3.3.3.3.3)
3665:Tetrahedral
3642:Birectified
3639:(tr. dual)
3637:Bitruncated
3526:Icosahedron
3361:3-orthoplex
3350:Octahedron
3176:Tetrahedron
3059:(order 2p)
3016:(order 2p)
2970:(order 4p)
2920:(order 2p)
2869:(order 2p)
1882:Octadecagon
1872:Hexadecagon
1414:2-orthoplex
1312:. Also the
1261:alternation
1230:tetrahedron
1204:alternation
1194:produces a
1182:Alternation
1165:runcination
1155:and higher.
1138:and higher.
1132:Octellation
802:Sterication
746:Runcination
455:sterication
451:runcination
273:semiregular
56:or uniform
18:Sterication
14503:10-simplex
14487:9-demicube
14437:8-demicube
14387:7-demicube
14344:6-demicube
14314:5-demicube
14228:Octahedron
13859:References
13392:sr{p,q,2r}
13189:{2p,q,2r}
12879:hr{2p,2q}
12789:{2p,2q,r}
12585:Alternated
12361:Operation
12343:Half forms
11608:t2r{p,q,r}
11435:tr{p,q,r}
11244:Runcinated
11066:r2r{p,q,r}
10706:= t{r,q,p}
10531:2t{p,q,r}
9982:= r{r,q,p}
9392:Operation
9261:sr{3,3,4}
8550:tr{5,3,3}
8499:tr{3,3,5}
8448:tr{3,4,3}
8356:tr{4,3,3}
8305:tr{3,3,4}
8254:tr{3,3,3}
8201:2t{3,3,5}
8146:2t{3,4,3}
8054:2t{4,3,3}
8003:2t{3,3,4}
7612:rr{5,3,3}
7561:rr{3,3,5}
7510:rr{3,4,3}
7418:rr{4,3,3}
7367:rr{3,3,4}
7316:rr{3,3,3}
5855:hyperplane
5298:Position:
5276:Operation
4202:Position:
4180:Operation
3962:antiprisms
3762:Octahedral
3634:Rectified
3631:Truncated
3618:truncation
3356:Octahedron
3185:(Pyramid)
2785:Operation
2352:Enneagrams
2342:Heptagrams
1857:Tridecagon
1456:Hendecagon
1318:octahedron
1222:hypercubes
1169:truncation
1149:Ennecation
1092:'seven'.)
964:Hexication
813:'solid'.)
474:Truncation
443:Truncation
356:orders of
313:Operations
299:honeycombs
14551:orthoplex
14473:9-simplex
14423:8-simplex
14373:7-simplex
14330:6-simplex
14300:5-simplex
14269:Tesseract
13770:hypercube
13587:os{p,q,r}
13399:{p,q,2r}
12994:{p,2q,r}
12987:s{p,2q,r}
12596:{p,2q,r}
12589:h{p,2q,r}
12305:{ }×{2r}
12274:{2p}×{ }
12121:{ }×{2r}
11900:{2p}×{ }
11432:tr{p,q,r}
10886:rr{p,q,r}
10704:3t{p,q,r}
10528:2t{p,q,r}
10344:Truncated
10168:= {r,q,p}
10166:3r{p,q,r}
9980:2r{p,q,r}
9795:Rectified
9355:s{3,4,3}
7263:t{5,3,3}
7212:t{3,3,5}
7161:t{3,4,3}
7069:t{4,3,3}
7018:t{3,3,4}
6967:t{3,3,3}
6914:r{5,3,3}
6863:r{3,3,5}
6812:r{3,4,3}
6720:r{4,3,3}
6669:r{3,3,4}
6618:r{3,3,3}
6324:tesseract
5889:tesseract
5794:duoprisms
5790:tesseract
4866:Rectified
4637:Truncated
4075:Antiprism
3927:(3.4.5.4)
3894:(3.5.3.5)
3883:(3.10.10)
3830:(3.4.4.4)
3797:(3.4.3.4)
3733:(3.4.3.4)
3700:(3.3.3.3)
3181:3-simplex
3141:(sphere)
2926:Truncated
2875:Rectified
2807:Position
2337:Pentagram
2321:pentagram
1852:Dodecagon
1403:2-simplex
1346:Euclidean
1271:truncated
1226:demicubes
1196:snub cube
924:'five'.)
675:polyhedra
582:polyhedra
303:Euclidean
275:polytopes
51:Truncated
14626:Category
14605:Topics:
14568:demicube
14533:polytope
14527:Uniform
14288:600-cell
14284:120-cell
14237:Demicube
14211:Pentagon
14191:Triangle
13847:See also
13821:polytope
13757:sr{q,r}
13726:{ }×{r}
13695:{p}×{ }
13664:sr{p,q}
13594:{p,q,r}
13582:Omnisnub
13577:s{q,2r}
13541:s{2,2r}
13474:sr{p,q}
13382:s{q,2r}
13274:s{q,2p}
13172:h{2q,r}
13074:s{p,2q}
12977:h{2q,r}
12676:h{p,2q}
12336:tr{q,r}
12243:tr{p,q}
12173:{p,q,r}
12166:o{p,q,r}
12090:{p}×{ }
12059:tr{p,q}
11989:{p,q,r}
11962:rr{q,r}
11931:{ }×{r}
11799:{p,q,r}
11780:tr{q,r}
11718:{p}×{ }
11617:{p,q,r}
11567:{ }×{r}
11505:tr{p,q}
11393:{ }×{r}
11362:{p}×{ }
11261:{p,q,r}
11254:e{p,q,r}
11249:expanded
11238:rr{q,r}
11176:{p}×{ }
11075:{p,q,r}
11025:{ }×{r}
10963:rr{p,q}
10893:{p,q,r}
10713:{p,q,r}
10355:{p,q,r}
10348:t{p,q,r}
10175:{p,q,r}
9989:{p,q,r}
9807:{p,q,r}
9800:r{p,q,r}
9625:{p,q,r}
9304:sr{3,3}
9206:{5,3,3}
9147:{3,4,3}
9090:{3,3,4}
9035:{3,3,4}
8980:{3,3,3}
8923:{5,3,3}
8868:{3,3,5}
8813:{3,4,3}
8758:rr{3,3}
8717:{4,3,3}
8662:{3,3,4}
8607:{3,3,3}
8397:tr{3,3}
8095:2t{3,3}
7952:{3,3,3}
7895:{3,3,5}
7836:{3,4,3}
7779:{4,3,3}
7724:{3,3,4}
7669:{3,3,3}
7459:2r{3,3}
6565:{5,3,3}
6518:120-cell
6514:{3,3,5}
6467:600-cell
6463:{3,4,3}
6371:{4,3,3}
6320:{3,3,4}
6269:{3,3,3}
5907:120-cell
5903:600-cell
5901:group :
5891:{4,3,3};
5883:group :
5809:600-cell
5803:and the
5685:s{p,2q}
5558:sr{p,q}
5279:Schläfli
5191:tr{p,q}
5054:rr{p,q}
5002:expanded
4183:Schläfli
4156:Schläfli
4128:sr{2,p}
4063:tr{2,p}
3998:symbols
3995:Schläfli
3976:Picture
3938:(4.6.10)
3164:Symmetry
3157:Vertices
3136:(solid)
3090:Schläfli
2811:Symmetry
2792:Symbols
2790:Schläfli
2788:Extended
2370:Schläfli
2363:n-agrams
2357:Decagram
2347:Octagram
1899:Schläfli
1892:Icosagon
1463:Schläfli
1446:Enneagon
1436:Heptagon
1426:Pentagon
1398:Triangle
1382:triangle
977:'six'.)
683:expanded
590:expanded
487:truncare
479:polygons
417:, ..., p
257:polytope
246:geometry
54:triangle
14542:simplex
14512:10-cube
14279:24-cell
14265:16-cell
14206:Hexagon
14060:regular
13865:Coxeter
13774:simplex
13592:0,1,2,3
12370:diagram
12368:Coxeter
12171:0,1,2,3
12152:t{r,q}
11982:{r,q,p}
11976:{p,q,r}
11869:t{p,q}
11792:{p,q,r}
11687:t{q,p}
11598:t{q,r}
11145:r{p,q}
11056:r{q,r}
10876:t{r,q}
10694:t{q,r}
10601:t{q,p}
10425:t{p,q}
10152:r{q,r}
9877:r{p,q}
9618:{p,q,r}
9401:diagram
9399:Coxeter
9204:0,1,2,3
9145:0,1,2,3
9088:0,1,2,3
9033:0,1,2,3
8978:0,1,2,3
7110:t{3,3}
6761:r{3,3}
6416:24-cell
6273:16-cell
5896:24-cell
5885:16-cell
5292:Wythoff
5287:diagram
5285:Coxeter
5281:Symbol
5118:{ }×{ }
4915:r{p,q}
4802:t{q,p}
4687:t{p,q}
4196:Wythoff
4191:diagram
4189:Coxeter
4185:Symbol
4174:{3,4}.
4160:Coxeter
3990:Diagram
3979:Tiling
3905:(5.6.6)
3841:(4.6.8)
3808:(4.6.6)
3786:(3.8.8)
3744:(4.6.6)
3711:(3.6.6)
3689:(3.6.6)
3644:(dual)
3628:Parent
3260:(self)
3097:Diagram
2878:(Dual)
2803:diagram
2801:Coxeter
2797:result
2795:Regular
2769:
2411:diagram
2409:Coxeter
2396:t{5/3}
2385:t{4/3}
1945:diagram
1943:Coxeter
1507:diagram
1505:Coxeter
1451:Decagon
1441:Octagon
1431:Hexagon
1306:hexagon
1279:simplex
1234:16-cell
1215:uniform
811:stereos
756:runcina
596:, like
295:tilings
60:, with
58:hexagon
14482:9-cube
14432:8-cube
14382:7-cube
14339:6-cube
14309:5-cube
14196:Square
14073:Family
13991:ONLINE
13953:
13896:ONLINE
13874:
13177:Bisnub
11424:{r,q}
11331:{p,q}
10783:{q,p}
10518:{q,r}
10338:{r,q}
10059:{q,p}
9970:{q,r}
9695:{p,q}
9614:Parent
6412:{3,3}
6222:5-cell
5878:5-cell
5692:{p,q}
5294:symbol
5255:{ }×{}
5198:{p,q}
5061:{p,q}
4922:{p,q}
4809:{p,q}
4694:{p,q}
4581:{p,q}
4574:{q,p}
4465:{p,q}
4458:{p,q}
4411:Parent
4198:symbol
3985:figure
3983:Vertex
3958:prisms
3862:5-3-2
3765:4-3-2
3668:3-3-2
3530:{3,5}
3448:{5,3}
3366:{3,4}
3278:{4,3}
3271:3-cube
3188:{3,3}
3093:{p,q}
2978:h{2p}
2826:Parent
2705:Image
2394:{10/3}
2391:{9/4}
2388:{9/2}
2380:{7/3}
2377:{7/2}
2374:{5/2}
2212:Image
1937:t{10}
1756:Image
1420:2-cube
1409:Square
1115:, etc.
1064:, etc.
1010:, etc.
947:, etc.
896:, etc.
844:, etc.
785:, etc.
730:, etc.
656:, etc.
559:, etc.
483:Kepler
261:facets
14201:p-gon
13397:0,1,2
12353:holes
12349:holes
11987:0,2,3
11797:0,1,3
11615:1,2,3
8921:0,1,3
8866:0,1,3
8811:0,1,3
8715:0,1,3
8660:0,1,3
8605:0,1,3
5831:cell.
5731:s{2p}
5196:0,1,2
4010:Prism
3973:Name
3916:{3,5}
3872:{5,3}
3819:{3,4}
3775:{4,3}
3722:{3,3}
3678:{3,3}
3659:Snub
3511:{3}2
3438:Cube
3152:Edges
3145:Faces
3139:Image
3134:Image
3129:Image
3086:Name
3026:s{p}
2975:Half
2940:{2p}
2383:{8/3}
2333:Name
1932:{19}
1929:t{9}
1924:{17}
1921:t{8}
1916:{15}
1913:t{7}
1908:{13}
1905:t{6}
1848:Name
1499:{11}
1496:t{5}
1488:t{4}
1480:t{3}
1472:t{2}
1394:Name
1236:, or
1163:is a
1089:hepta
1057:2,5,8
1042:1,4,7
1016:0,3,6
921:pente
889:2,4,6
875:1,3,5
850:0,2,4
760:plane
723:3,4,5
709:2,3,4
695:1,2,3
662:0,1,2
599:bevel
301:) of
14559:cube
14232:Cube
14062:and
13951:ISBN
13872:ISBN
13804:and
13776:and
12982:Snub
12582:Half
12159:(or
11247:(or
10161:dual
10159:(or
5635:Snub
5267:{ }
5258:{2q}
5252:{2p}
5137:(or
5130:{ }
5000:(or
4856:{ }
4850:{2q}
4744:{ }
4729:{2p}
4525:dual
4522:(or
4515:{ }
3993:and
3593:{5}
3585:{3}
3503:{5}
3429:{4}
3421:{3}
3341:{3}
3333:{4}
3266:Cube
3251:{3}
3243:{3}
3169:Dual
3160:{q}
3148:{p}
3029:{p}
3022:Snub
2981:{p}
2937:{p}
2930:t{p}
2891:{p}
2888:{p}
2881:r{p}
2840:{p}
2837:{p}
2820:(0)
2817:(1)
1935:{20}
1927:{18}
1919:{16}
1911:{14}
1903:{12}
1494:{10}
1491:{9}
1483:{7}
1475:{5}
1467:{3}
1348:and
1300:for
1208:snub
1167:and
1128:4r4r
1062:3t5r
1047:3t4r
1021:3t3r
1008:3r5r
993:3r4r
960:3r3r
894:2t4r
880:2t3r
855:2t2r
842:2r4r
828:2r3r
798:2r2r
762:'.)
594:cant
421:} =
368:. A
366:dual
305:and
248:, a
14108:(p)
13931:PDF
13911:PDF
13505:--
13346:--
13310:--
13187:1,2
13136:--
13105:--
12992:0,1
12946:--
12915:--
12774:--
12743:--
12712:--
12539:(0)
12498:(1)
12457:(2)
12416:(3)
11978:= e
11749:--
11536:--
11259:0,3
11207:--
11073:1,3
10994:--
10891:0,2
10845:--
10814:--
10711:2,3
10663:--
10632:--
10487:--
10456:--
10353:0,1
10307:--
10276:--
10245:--
10121:--
10090:--
9939:--
9908:--
9788:--
9757:--
9726:--
9570:(0)
9529:(1)
9488:(2)
9447:(3)
9375:An
7950:1,2
7893:0,3
7834:0,3
7777:0,3
7722:0,3
7667:0,3
5920:.)
5743:{3}
5737:{q}
5734:{3}
5690:0,1
5629:--
5620:{q}
5617:{3}
5615:{3}
5612:{p}
5264:{ }
5261:{ }
5124:{ }
5121:{q}
5115:{p}
5059:0,2
4991:--
4988:{ }
4982:{q}
4976:{p}
4859:--
4853:{ }
4847:{ }
4844:{p}
4807:1,2
4741:{ }
4735:{q}
4732:{ }
4692:0,1
4631:--
4625:{ }
4622:{q}
4619:{ }
4503:{ }
4500:{p}
4172:0,1
3588:30
3506:30
3424:12
3336:12
3054:--
3051:--
3011:--
3008:--
2965:{}
2962:{}
2935:0,1
2916:{}
2913:--
2865:--
2862:{}
2830:{p}
2401:p/q
2361:...
1486:{8}
1478:{6}
1470:{4}
1144:0,9
1126:or
1123:0,8
1112:2,9
1102:1,8
1074:0,7
1060:or
1045:or
1019:or
1006:or
1003:2,8
991:or
988:1,7
974:hex
958:or
955:0,6
944:2,7
934:1,6
906:0,5
892:or
878:or
853:or
840:or
837:2,6
826:or
823:1,5
796:or
793:0,4
782:2,5
772:1,4
740:0,3
728:t4r
726:or
714:t3r
712:or
700:t2r
698:or
665:or
654:r4r
652:or
649:3,5
640:r3r
638:or
635:2,4
626:r2r
624:or
621:1,3
570:or
567:0,2
555:or
552:4,5
541:or
538:3,4
527:or
524:2,3
513:or
510:1,2
467:or
464:0,1
419:n-1
413:, p
244:In
128:5D
125:4D
39:3D
36:2D
14628::
14613:•
14609:•
14589:21
14585:•
14582:k1
14578:•
14575:k2
14553:•
14510:•
14480:•
14458:21
14454:•
14451:41
14447:•
14444:42
14430:•
14408:21
14404:•
14401:31
14397:•
14394:32
14380:•
14358:21
14354:•
14351:22
14337:•
14307:•
14286:•
14267:•
14246:•
14230:•
14162:/
14151:/
14141:/
14132:/
14110:/
14007:.
13977:,
13970:,
13963:,
13918::
13905::
13823:.
13819:21
13797:,
13790:,
13772:,
13590:ht
13395:ht
13185:ht
12990:ht
12785:ht
12592:ht
12163:)
11974:3t
11251:)
10163:)
6003:BC
5865:.
5811:.
5740:--
5688:ht
5626:--
5623:--
5141:)
5127:--
5004:)
4985:--
4979:--
4738:--
4628:--
4616:--
4528:)
4512:--
4509:--
4506:--
4005:2p
3970:#
3964:.
3656:)
3591:12
3583:20
3509:20
3501:12
3363:)
3246:6
3080::
3057:]=
2968:]=
2403:}
1422:)
1405:)
1368:.
1324:.
1293:.
1240:.
1210:.
1178:.
1147::
1130::
1105:,
1077::
1053::
1049:,
1038::
1023::
999::
995:,
984::
962::
937:,
909::
882:,
857::
830:,
800::
775:,
743::
716:,
702:,
669::
667:tr
642:,
628:,
574::
572:rr
557:5t
545:,
543:4t
531:,
529:3t
517:,
515:2t
471::
409:{p
344:.
191:,
142:,
93:,
81:.
14597:-
14595:n
14587:k
14580:2
14573:1
14566:-
14564:n
14557:-
14555:n
14549:-
14547:n
14540:-
14538:n
14531:-
14529:n
14456:4
14449:2
14442:1
14406:3
14399:2
14392:1
14356:2
14349:1
14178:n
14176:H
14169:2
14166:G
14158:4
14155:F
14147:8
14144:E
14138:7
14135:E
14129:6
14126:E
14117:n
14113:D
14106:2
14103:I
14095:n
14091:B
14083:n
14079:A
14051:e
14044:t
14037:v
13817:4
13808:8
13806:E
13801:7
13799:E
13794:6
13792:E
12787:1
12594:0
12169:t
11985:t
11980:t
11972:e
11795:t
11790:t
11788:e
11613:t
11257:t
11071:t
10889:t
10709:t
10351:t
10173:3
10171:t
9987:2
9985:t
9805:1
9803:t
9623:0
9621:t
9377:n
9202:t
9143:t
9086:t
9031:t
8976:t
8919:t
8864:t
8809:t
8713:t
8658:t
8603:t
7948:t
7891:t
7832:t
7775:t
7720:t
7665:t
6023:4
6021:H
6017:4
6015:F
6011:4
6009:D
6005:4
5999:4
5997:A
5966:3
5960:2
5954:1
5948:0
5940:t
5669:}
5663:q
5660:,
5657:p
5651:{
5646:s
5542:}
5536:q
5529:p
5523:{
5518:s
5194:t
5175:}
5169:q
5162:p
5156:{
5151:t
5057:t
5038:}
5032:q
5025:p
5019:{
5014:r
4920:1
4918:t
4899:}
4893:q
4886:p
4880:{
4805:t
4786:}
4780:p
4777:,
4774:q
4768:{
4763:t
4690:t
4671:}
4665:q
4662:,
4659:p
4653:{
4648:t
4579:2
4577:t
4558:}
4552:p
4549:,
4546:q
4540:{
4463:0
4461:t
4442:}
4436:q
4433:,
4430:p
4424:{
4070:p
4068:A
4003:P
3652:(
3598:h
3596:I
3516:h
3514:I
3434:h
3432:O
3427:6
3419:8
3359:(
3346:h
3344:O
3339:8
3331:6
3273:)
3269:(
3256:d
3254:T
3249:4
3241:4
3183:)
3179:(
3014:=
2933:t
2886:1
2884:t
2835:0
2833:t
2399:{
2325:q
1418:(
1416:)
1412:(
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1366:1
1354:n
1342:n
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1198:.
1142:t
1121:t
1110:t
1100:t
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1055:t
1040:t
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953:t
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887:t
873:t
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780:t
770:t
738:t
721:t
707:t
693:t
660:t
647:t
633:t
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565:t
550:t
536:t
522:t
508:t
469:t
462:t
439:n
428:.
426:r
423:k
415:2
411:1
406:k
404:t
400:k
362:n
354:n
285:(
20:)
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