9208:
9302:
9261:
6461:
8252:
8605:
8978:
8660:
8550:
8395:
8303:
8201:
6914:
9033:
8923:
8715:
8148:
8052:
7950:
7314:
7016:
6759:
8093:
8001:
7895:
7667:
7365:
7263:
7210:
7159:
124:
8354:
7779:
7722:
7612:
7108:
7067:
6965:
6861:
6810:
6616:
7416:
6667:
6565:
4012:
5752:
2726:
9149:
2753:
4077:
5745:
5813:
596:
3739:
5831:
6512:
7559:
7508:
4084:
3889:
3867:
3792:
3770:
483:
3933:
3836:
3781:
6718:
6410:
6369:
6267:
3922:
3900:
3878:
3856:
3825:
3803:
3759:
3728:
3684:
6318:
6216:
3911:
3814:
3717:
3695:
3673:
3225:
3706:
3662:
8866:
8811:
8446:
1757:
8497:
2292:
2252:
2212:
8756:
2272:
2232:
1823:
2279:
2259:
2239:
2226:
1803:
1763:
9090:
1783:
3567:
3485:
3403:
3315:
2266:
2246:
75:
35:
2219:
1830:
4019:
1176:
7838:
4070:
2747:
2206:
2740:
2733:
2720:
2713:
2706:
2699:
2286:
1790:
1770:
1750:
4005:
1817:
1810:
1797:
1777:
173:
3471:
7457:
3553:
3211:
2768:{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:
3478:
3389:
3301:
3218:
3560:
3396:
3308:
13769:. 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.
252:. 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
13831:
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
5803:
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
959:
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
797:
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
5923:
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.
1074:
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
5808:
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.
742:
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
5931:
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
906:
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
9359:
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.
1255:
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.
1321:
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 (
5543:
5176:
5039:
5670:
4900:
4787:
4672:
4559:
4443:
5842:
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
4127:, 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).
1248:
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
5961:(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.
8732:
1258:
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
9278:
8371:
8069:
7084:
6735:
5558:
5198:
5061:
4922:
2312:, 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
5563:
5203:
5066:
12563:
12553:
12543:
12522:
12512:
12492:
12481:
12461:
12451:
12430:
12420:
12410:
12397:
12387:
12377:
12367:
9594:
9584:
9574:
9553:
9543:
9523:
9512:
9492:
9482:
9461:
9451:
9441:
9428:
9418:
9408:
9398:
9288:
9268:
8742:
8722:
8381:
8361:
8079:
8059:
7443:
7423:
7074:
6376:
5467:
5448:
5429:
5400:
5390:
5371:
5351:
5332:
5322:
5311:
5301:
5291:
4371:
4352:
4333:
4304:
4294:
4275:
4255:
4236:
4226:
4215:
4205:
4195:
13741:
13731:
13721:
13710:
13700:
13690:
13679:
13669:
13659:
13648:
13638:
13628:
13619:
13609:
13599:
13589:
13546:
13536:
13510:
13500:
13479:
13469:
13458:
13448:
13438:
13414:
13404:
13394:
13351:
13341:
13315:
13284:
13258:
13248:
13209:
13199:
13131:
13100:
13079:
13069:
13043:
13033:
12999:
12989:
12941:
12920:
12889:
12848:
12799:
12707:
12671:
12635:
12591:
12533:
12502:
12471:
12440:
12320:
12310:
12300:
12289:
12279:
12269:
12258:
12248:
12238:
12227:
12217:
12207:
12198:
12188:
12178:
12168:
12136:
12126:
12105:
12095:
12085:
12074:
12054:
12043:
12023:
12014:
12004:
11984:
11946:
11926:
11915:
11895:
11884:
11874:
11864:
11843:
11833:
11824:
11804:
11794:
11764:
11754:
11744:
11733:
11723:
11702:
11692:
11671:
11661:
11642:
11632:
11622:
11572:
11562:
11541:
11531:
11510:
11500:
11489:
11479:
11469:
11450:
11440:
11430:
11408:
11377:
11357:
11346:
11326:
11295:
11286:
11256:
11222:
11202:
11191:
11160:
11150:
11119:
11100:
11080:
11030:
10999:
10989:
10958:
10947:
10927:
10908:
10888:
10860:
10850:
10829:
10819:
10798:
10767:
10738:
10728:
10668:
10658:
10637:
10606:
10585:
10575:
10546:
10536:
10482:
10451:
10430:
10420:
10399:
10389:
10360:
10350:
10322:
10291:
10260:
10200:
10126:
10095:
10043:
10004:
9934:
9882:
9851:
9812:
9721:
9690:
9659:
9620:
9564:
9533:
9502:
9471:
9319:
9309:
9245:
9235:
9225:
9186:
9176:
9166:
9156:
9127:
9117:
9107:
9097:
9070:
9060:
9050:
9040:
9015:
9005:
8995:
8985:
8960:
8950:
8940:
8930:
8903:
8883:
8873:
8848:
8838:
8818:
8793:
8773:
8763:
8697:
8677:
8667:
8642:
8632:
8612:
8587:
8567:
8557:
8524:
8514:
8504:
8483:
8473:
8463:
8422:
8412:
8402:
8330:
8320:
8310:
8289:
8279:
8269:
8228:
8218:
8208:
8175:
8165:
8120:
8110:
8028:
8018:
7977:
7967:
7922:
7912:
7875:
7845:
7816:
7786:
7759:
7729:
7704:
7674:
7649:
7619:
7586:
7566:
7545:
7525:
7484:
7464:
7433:
7392:
7372:
7351:
7331:
7290:
7270:
7227:
7217:
7196:
7186:
7125:
7115:
7033:
7023:
7002:
6992:
6931:
6921:
6878:
6837:
6776:
6684:
6643:
6582:
6519:
6498:
6417:
6386:
6325:
6304:
6223:
6112:
5697:
5687:
5592:
5582:
5572:
5553:
5487:
5477:
5458:
5438:
5419:
5409:
5380:
5361:
5342:
5232:
5222:
5212:
5193:
5095:
5075:
4946:
4917:
4824:
4814:
4699:
4689:
4596:
4460:
4391:
4381:
4362:
4342:
4323:
4313:
4284:
4265:
4246:
4112:
4102:
4092:
4047:
4037:
4027:
3525:
3443:
3361:
3273:
3183:
3092:
3034:
3024:
2976:
2945:
2935:
2896:
2835:
2666:
2657:
2637:
2609:
2580:
2551:
2542:
2522:
2494:
2465:
2436:
2407:
2193:
2183:
2165:
2146:
2137:
2127:
2109:
2090:
2081:
2071:
2053:
2034:
2025:
2015:
1997:
1978:
1969:
1959:
1941:
1727:
1718:
1708:
1690:
1671:
1662:
1652:
1634:
1615:
1606:
1596:
1578:
1559:
1550:
1540:
1522:
1503:
194:
184:
145:
135:
96:
86:
66:
56:
7094:
6745:
6725:
6396:
6122:
6102:
5969:
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
4927:
13561:
13525:
13489:
13429:
13366:
13330:
13305:
13294:
13269:
13233:
13224:
13184:
13156:
13146:
13120:
13110:
13089:
13058:
13024:
13014:
12961:
12951:
12930:
12910:
12899:
12874:
12863:
12833:
12824:
12814:
12784:
12758:
12748:
12738:
12727:
12717:
12696:
12686:
12660:
12650:
12626:
12616:
12606:
12116:
12064:
12033:
11994:
11936:
11905:
11853:
11814:
11713:
11682:
11651:
11612:
11582:
11551:
11520:
11460:
11398:
11388:
11367:
11336:
11315:
11305:
11276:
11266:
11212:
11181:
11171:
11140:
11129:
11109:
11090:
11070:
11040:
11020:
11009:
10978:
10968:
10937:
10918:
10898:
10840:
10809:
10788:
10778:
10757:
10747:
10718:
10708:
10678:
10647:
10627:
10616:
10596:
10565:
10556:
10526:
10502:
10492:
10471:
10461:
10440:
10409:
10380:
10370:
10312:
10302:
10281:
10271:
10250:
10240:
10229:
10219:
10209:
10190:
10180:
10170:
10136:
10116:
10105:
10085:
10074:
10064:
10054:
10033:
10023:
10014:
9994:
9984:
9954:
9944:
9923:
9913:
9903:
9892:
9872:
9861:
9841:
9832:
9822:
9802:
9772:
9762:
9752:
9741:
9731:
9710:
9700:
9679:
9669:
9650:
9640:
9630:
9339:
9329:
9215:
8893:
8828:
8783:
8687:
8622:
8577:
8534:
8453:
8432:
8340:
8259:
8238:
8185:
8155:
8130:
8100:
8038:
8008:
7987:
7957:
7932:
7902:
7865:
7855:
7806:
7796:
7749:
7739:
7694:
7684:
7639:
7629:
7596:
7576:
7535:
7515:
7494:
7474:
7402:
7382:
7341:
7321:
7300:
7280:
7247:
7237:
7176:
7166:
7145:
7135:
7053:
7043:
6982:
6972:
6951:
6941:
6898:
6888:
6868:
6847:
6827:
6817:
6796:
6786:
6766:
6704:
6694:
6674:
6653:
6633:
6623:
6602:
6592:
6572:
6549:
6539:
6529:
6488:
6478:
6468:
6447:
6437:
6427:
6355:
6345:
6335:
6294:
6284:
6274:
6253:
6243:
6233:
6202:
6192:
6182:
6172:
6162:
6152:
6142:
6132:
6092:
6082:
6072:
6062:
6052:
6042:
6032:
6022:
5712:
5085:
5056:
4956:
4936:
4804:
4709:
4586:
4576:
4480:
4470:
3545:
3535:
3463:
3453:
3381:
3371:
3293:
3283:
3203:
3193:
3112:
3102:
2991:
2886:
2845:
2686:
2629:
2614:
2600:
2571:
2514:
2485:
2456:
2427:
2188:
2175:
2170:
2156:
2151:
2119:
2114:
2100:
2095:
2063:
2058:
2044:
2039:
2007:
2002:
1988:
1983:
1951:
1737:
1700:
1695:
1681:
1644:
1625:
1588:
1569:
1532:
1513:
224:
214:
204:
165:
155:
106:
2676:
2647:
2619:
2590:
2561:
2532:
2504:
2475:
2446:
2417:
13695:
13674:
13551:
13515:
13505:
13419:
13356:
13320:
13274:
13238:
13214:
13189:
13136:
13048:
13004:
12879:
12853:
12838:
12804:
12789:
12676:
12640:
12596:
9283:
9273:
8737:
8727:
8376:
8366:
8074:
8064:
7438:
7428:
7089:
7079:
6740:
6730:
6391:
6381:
6117:
6107:
5702:
4107:
2981:
2681:
2652:
2624:
2595:
2566:
2537:
2509:
2480:
2451:
2422:
1946:
1732:
13736:
13726:
13705:
13664:
13643:
13633:
13614:
13604:
13594:
13556:
13541:
13520:
13484:
13474:
13453:
13443:
13424:
13409:
13399:
13361:
13346:
13325:
13310:
13289:
13279:
13253:
13243:
13219:
13204:
13194:
13151:
13141:
13115:
13105:
13084:
13074:
13053:
13038:
13019:
13009:
12994:
12956:
12946:
12925:
12915:
12894:
12884:
12858:
12843:
12819:
12809:
12794:
12753:
12743:
12722:
12712:
12691:
12681:
12655:
12645:
12621:
12611:
12601:
12558:
12548:
12538:
12517:
12507:
12497:
12476:
12466:
12456:
12435:
12425:
12415:
12392:
12382:
12372:
12315:
12305:
12284:
12274:
12253:
12243:
12222:
12212:
12193:
12183:
12173:
12131:
12121:
12100:
12090:
12069:
12059:
12038:
12028:
12009:
11999:
11989:
11941:
11931:
11910:
11900:
11879:
11869:
11848:
11838:
11819:
11809:
11799:
11759:
11749:
11728:
11718:
11697:
11687:
11666:
11656:
11637:
11627:
11617:
11577:
11567:
11546:
11536:
11515:
11505:
11484:
11474:
11455:
11445:
11435:
11403:
11393:
11372:
11362:
11341:
11331:
11310:
11300:
11281:
11271:
11261:
11217:
11207:
11186:
11176:
11155:
11145:
11124:
11114:
11095:
11085:
11075:
11035:
11025:
11004:
10994:
10973:
10963:
10942:
10932:
10913:
10903:
10893:
10855:
10845:
10824:
10814:
10793:
10783:
10762:
10752:
10733:
10723:
10713:
10673:
10663:
10642:
10632:
10611:
10601:
10580:
10570:
10551:
10541:
10531:
10497:
10487:
10466:
10456:
10435:
10425:
10404:
10394:
10375:
10365:
10355:
10317:
10307:
10286:
10276:
10255:
10245:
10224:
10214:
10195:
10185:
10175:
10131:
10121:
10100:
10090:
10069:
10059:
10038:
10028:
10009:
9999:
9989:
9949:
9939:
9918:
9908:
9887:
9877:
9856:
9846:
9827:
9817:
9807:
9767:
9757:
9736:
9726:
9705:
9695:
9674:
9664:
9645:
9635:
9625:
9589:
9579:
9569:
9548:
9538:
9528:
9507:
9497:
9487:
9466:
9456:
9446:
9423:
9413:
9403:
9334:
9324:
9314:
9240:
9230:
9220:
9181:
9171:
9161:
9122:
9112:
9102:
9065:
9055:
9045:
9010:
9000:
8990:
8955:
8945:
8935:
8898:
8888:
8878:
8843:
8833:
8823:
8788:
8778:
8768:
8692:
8682:
8672:
8637:
8627:
8617:
8582:
8572:
8562:
8529:
8519:
8509:
8478:
8468:
8458:
8427:
8417:
8407:
8335:
8325:
8315:
8284:
8274:
8264:
8233:
8223:
8213:
8180:
8170:
8160:
8125:
8115:
8105:
8033:
8023:
8013:
7982:
7972:
7962:
7927:
7917:
7907:
7870:
7860:
7850:
7811:
7801:
7791:
7754:
7744:
7734:
7699:
7689:
7679:
7644:
7634:
7624:
7591:
7581:
7571:
7540:
7530:
7520:
7489:
7479:
7469:
7397:
7387:
7377:
7346:
7336:
7326:
7295:
7285:
7275:
7242:
7232:
7222:
7191:
7181:
7171:
7140:
7130:
7120:
7048:
7038:
7028:
6997:
6987:
6977:
6946:
6936:
6926:
6893:
6883:
6873:
6842:
6832:
6822:
6791:
6781:
6771:
6699:
6689:
6679:
6648:
6638:
6628:
6597:
6587:
6577:
6544:
6534:
6524:
6493:
6483:
6473:
6442:
6432:
6422:
6350:
6340:
6330:
6299:
6289:
6279:
6248:
6238:
6228:
6197:
6187:
6177:
6157:
6147:
6137:
6087:
6077:
6067:
6047:
6037:
6027:
5707:
5692:
5587:
5577:
5482:
5472:
5453:
5443:
5424:
5414:
5395:
5385:
5366:
5356:
5337:
5327:
5306:
5296:
5227:
5217:
5090:
5080:
4951:
4941:
4819:
4809:
4704:
4694:
4591:
4581:
4475:
4465:
4386:
4376:
4357:
4347:
4328:
4318:
4299:
4289:
4270:
4260:
4241:
4231:
4210:
4200:
4097:
4042:
4032:
3540:
3530:
3458:
3448:
3376:
3366:
3288:
3278:
3198:
3188:
3107:
3097:
3029:
2986:
2940:
2891:
2840:
2671:
2642:
2585:
2556:
2527:
2499:
2470:
2441:
2412:
2132:
2076:
2020:
1964:
1713:
1676:
1657:
1639:
1620:
1601:
1583:
1564:
1545:
1527:
1508:
219:
209:
199:
189:
160:
150:
140:
101:
91:
61:
470:
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
278:) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows
325:
Equivalently, the
Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension. This approach was first used by
13820:
Related to the subject of finite uniform polytopes are uniform honeycombs in
Euclidean and hyperbolic spaces. Euclidean uniform honeycombs are generated by
4151:, applies to all dimensions; it consists of the letter 't', followed by a series of subscripted numbers corresponding to the ringed nodes of the
430:-polytopes in any combination. The resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them.
3983:
1245:, marking active mirrors by rings, have reflectional symmetry, and can be simply constructed by recursive reflections of the vertex figure.
5846:, but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of the
5858:
14603:
5973:. The final 10th row lists the snub 24-cell constructions. This includes all nonprismatic uniform 4-polytopes, except for the
13943:
13864:
670:. A cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically
13993:
2316:
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}.
1195:, removes alternate vertices from a polytope with only even-sided faces. An alternated omnitruncated polytope is called a
577:. A cantellation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically
14038:
13938:, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
5502:
5135:
4998:
1377:
polygons of twice as many sides, t{p}={2p}. The first few regular polygons (and quasiregular forms) are displayed below:
13898:, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
5630:
4862:
4747:
4632:
4522:
4406:
1202:
The resulting polytopes always can be constructed, and are not generally reflective, and also do not in general have
5777:, 17 prisms on the Platonic and Archimedean solids (excluding the cube-prism, which has already been counted as the
1148:
In addition combinations of truncations can be performed which also generate new uniform polytopes. For example, a
12344:
and inactive nodes must be even-order. Half construction have the vertices of an identically ringed construction.
3065:
In three dimensions, the situation gets more interesting. There are five convex regular polyhedra, known as the
13904:
13870:
13853:
8709:
4148:
315:
1278:
Uniform polytopes have equal edge-lengths, and all vertices are an equal distance from the center, called the
1345:+1)-dimensional. Hence, the tilings of two-dimensional space are grouped with the three-dimensional solids.
13825:
13804:, one can obtain 39 new 6-polytopes, 127 new 7-polytopes and 255 new 8-polytopes. A notable example is the
8046:
13919:
8348:
3926:
9207:
14620:
13911:
13821:
1310:
330:
256:
are considered as uniform, disallowing polygons that alternate between two different lengths of edges.
1241:, the arrangement of edges, faces, cells, etc. around each vertex. Uniform polytopes represented by a
446:
cut cells. Each higher operation also cuts lower ones too, so a cantellation also truncates vertices.
13934:
7410:
5774:
3818:
14061:
9783:
8654:
8297:
5913:
4854:
4124:
667:
574:
346:
14031:
10870:
10512:
9027:
8972:
8860:
8805:
8491:
8440:
7061:
6712:
4985:
4738:
3829:
1180:
564:
12573:
11232:
9143:
9139:
8750:
8599:
8389:
3915:
3871:
1249:
1192:
9301:
9260:
6460:
5920:{4,3,4}, by applying the same operations used to generate the Wythoffian uniform 4-polytopes.
14575:
14568:
14561:
10332:
9374:
The red background shows the truncations of the parent, and blue the truncations of the dual.
9084:
8604:
8544:
8251:
8195:
7995:
7944:
7359:
4625:
3893:
3699:
3677:
3606:
1353:
The only one-dimensional polytope is the line segment. It corresponds to the
Coxeter family A
1259:
461:
177:
39:
14100:
14078:
14066:
9371:
The green background is shown on forms that are equivalent to either the parent or the dual.
8977:
8659:
8549:
8394:
8302:
8200:
6913:
14232:
14179:
11237:
9032:
8922:
8917:
8142:
8138:
7716:
7661:
7553:
7502:
5280:
4990:
4184:
4156:
3847:
3796:
3732:
3653:
3602:
2340:
1386:
1374:
671:
578:
307:
287:
260:
79:
8714:
8147:
8051:
7949:
7313:
7015:
6758:
8:
14587:
14486:
14236:
8092:
8087:
8000:
7894:
7832:
7828:
7666:
7451:
7364:
7262:
7209:
7158:
7010:
6661:
3750:
2345:
1141:
249:
14001:
8353:
7778:
7721:
7611:
7107:
7066:
6964:
6860:
6809:
6615:
123:
14456:
14406:
14356:
14313:
14283:
14243:
14206:
14024:
13891:
13773:
9255:
7889:
7773:
7415:
7257:
7204:
7153:
6855:
6804:
6666:
6564:
5839:
5768:
4140:
3060:
1124:
1071:
1017:
956:
903:
851:
794:
739:
663:
570:
13841:
4144:
4011:
3078:
2765:
2358:
1887:
1451:
1163:
If all truncations are applied at once, the operation can be more generally called an
381:
14595:
13939:
13899:
13860:
8246:
7606:
7308:
7102:
6959:
6753:
5805:
4155:, and followed by the Schläfli symbol of the regular seed polytope. For example, the
3937:
3882:
3598:
3145:
1397:
1298:
1290:
279:
242:
128:
13979:
5796:. Both of these special 4-polytope are composed of subgroups of the vertices of the
14599:
14164:
14153:
14142:
14131:
14122:
14113:
14048:
13794:
13787:
13780:
13776:
9148:
6908:
6559:
5906:
3998:
3946:
2799:
1338:
295:
267:
5751:
14189:
14174:
13801:
12356:
9387:
5977:
5970:
5917:
5817:
5789:
5744:
5623:
5495:
5273:
4513:
4177:
4152:
3978:
3157:
3140:
3133:
3085:
2789:
2725:
2397:
2305:
1931:
1493:
1366:
1334:
1242:
326:
319:
311:
291:
253:
50:
5812:
4076:
2752:
1365:
In two dimensions, there is an infinite family of convex uniform polytopes, the
1285:
Uniform polytopes whose circumradius is equal to the edge length can be used as
674:
copies of themselves. (The composite term combines cantellation and truncation)
14539:
13766:
12149:
5974:
5786:
5127:
4136:
3774:
3349:
3152:
3066:
1402:
1263:
1164:
283:
13896:
Geometrical deduction of semiregular from regular polytopes and space fillings
7558:
7507:
6511:
5830:
4083:
1297:
divides into 6 equilateral triangles and is the vertex figure for the regular
14614:
14556:
14444:
14437:
14430:
14394:
14387:
14380:
14344:
14337:
13805:
10149:
6717:
6409:
6368:
6363:
6266:
3971:
3932:
3921:
3835:
3824:
3785:
3738:
3727:
3721:
1302:
1286:
1238:
595:
581:
copies of themselves. (The term, coined by
Johnson, is derived from the verb
354:
271:
6317:
6215:
3910:
3888:
3866:
3813:
3791:
3769:
3716:
3694:
3672:
14496:
9296:
9202:
5793:
4123:
The uniform star polyhedra include a further 4 regular star polyhedra, the
3899:
3877:
3860:
3855:
3802:
3780:
3758:
3705:
3683:
3661:
3432:
3224:
2351:
2301:
1875:
1865:
1855:
1850:
748:
492:
482:
275:
13918:, Philosophical Transactions of the Royal Society of London, Londne, 1954
8865:
8810:
8496:
8445:
14505:
14466:
14416:
14366:
14323:
14293:
14225:
14211:
8755:
5851:
5847:
5823:
There are 3 right dihedral angles (2 intersecting perpendicular mirrors):
3904:
3743:
3710:
3666:
3514:
3164:
1870:
1860:
1218:
734:
9089:
3566:
3484:
3402:
3314:
1756:
384:
can be used for representing rectified forms, with a single subscript:
14491:
14475:
14425:
14375:
14332:
14302:
14216:
13963:
13956:
13949:
5843:
5781:), and two infinite sets: the prisms on the convex antiprisms, and the
4018:
3807:
3688:
3344:
2308:
greater than 2), but these are non-convex. The simplest example is the
2291:
2278:
2271:
2258:
2251:
2238:
2231:
2225:
2211:
1845:
1822:
1444:
1306:
74:
7837:
4069:
2265:
2245:
1802:
1782:
1762:
14547:
14461:
14411:
14361:
14318:
14288:
14257:
13800:
come into play. By placing rings on a nonzero number of nodes of the
13758:
6312:
5877:
5778:
4135:
The
Wythoffian uniform polyhedra and tilings can be defined by their
4063:
3950:
3840:
3259:
2330:
2325:
2309:
2218:
1840:
1829:
1408:
1210:
1184:
34:
4004:
2746:
2205:
1175:
14521:
14276:
14272:
14199:
6506:
6455:
5895:
5891:
5797:
5782:
2739:
2732:
2719:
2712:
2705:
2698:
2335:
2285:
1880:
1789:
1769:
1749:
1434:
1424:
1414:
1370:
1214:
1020:
and higher. It can be seen as tritruncating its trirectification.
245:
234:
42:
13757:
In five and higher dimensions, there are 3 regular polytopes, the
3597:
In addition to these, there are also 13 semiregular polyhedra, or
1816:
1809:
1796:
1776:
14530:
14500:
14267:
14262:
14253:
14194:
13762:
7456:
6610:
6404:
6261:
5884:
5873:
5785:. There are also 41 convex semiregular 4-polytope, including the
3169:
1439:
1429:
1419:
1391:
1294:
1267:
1222:
467:
46:
3609:
on the Platonic solids, as demonstrated in the following table:
854:
and higher. It can be seen as bitruncating its birectification.
172:
14470:
14420:
14370:
14327:
14297:
14248:
14184:
6210:
5866:
3477:
3470:
1077:
909:
471:
3559:
3552:
3217:
3210:
1305:
divides into 8 regular tetrahedra and 6 square pyramids (half
13884:
5939:: vertex of the parent 4-polytope (center of the dual's cell)
3395:
3388:
3307:
3300:
962:
587:
1373:. Truncated regular polygons become bicolored geometrically
14220:
3949:, one for each regular polygon, and a corresponding set of
3763:
3254:
1266:
polytope (all nodes ringed) will always have an irregular
14008:
13867:(Chapter 3: Wythoff's Construction for Uniform Polytopes)
5951:: center of the parent's face (center of the dual's edge)
5945:: center of the parent's edge (center of the dual's face)
5861:
is generated by one of four symmetry groups, as follows:
13828:. Two affine Coxeter groups can be multiplied together.
5916:
in Euclidean 3-space are analogously generated from the
5901:
group : contains only repeated members of the family.
5838:
Every regular polytope can be seen as the images of a
5642:
5514:
5147:
5010:
4871:
4759:
4644:
4531:
4415:
349:. The zeroth rectification is the original form. The (
5633:
5505:
5138:
5001:
4865:
4750:
4635:
4525:
4409:
426:
Truncation operations that can be applied to regular
13975:, Dissertation, Universität Hamburg, Hamburg (2004)
306:
Nearly every uniform polytope can be generated by a
5538:{\displaystyle s{\begin{Bmatrix}p\\q\end{Bmatrix}}}
5171:{\displaystyle t{\begin{Bmatrix}p\\q\end{Bmatrix}}}
5034:{\displaystyle r{\begin{Bmatrix}p\\q\end{Bmatrix}}}
13935:Kaleidoscopes: Selected Writings of H.S.M. Coxeter
13883:, Ph.D. Dissertation, University of Toronto, 1966
5957:: center of the parent's cell (vertex of the dual)
5665:{\displaystyle s{\begin{Bmatrix}p,q\end{Bmatrix}}}
5664:
5537:
5170:
5033:
4895:{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}
4894:
4782:{\displaystyle t{\begin{Bmatrix}q,p\end{Bmatrix}}}
4781:
4667:{\displaystyle t{\begin{Bmatrix}p,q\end{Bmatrix}}}
4666:
4553:
4437:
259:This is a generalization of the older category of
4554:{\displaystyle {\begin{Bmatrix}q,p\end{Bmatrix}}}
4438:{\displaystyle {\begin{Bmatrix}p,q\end{Bmatrix}}}
1316:
1262:regular polytope (with 2 rings) is a pyramid. An
14612:
1237:Uniform polytopes can be constructed from their
13824:and hyperbolic honeycombs are generated by the
12340:rather than ringed nodes. Branches neighboring
1209:The set of polytopes formed by alternating the
13881:The Theory of Uniform Polytopes and Honeycombs
13569:
13374:
13164:
12969:
12766:
12569:
12403:
12144:
11954:
11772:
11590:
11416:
11230:
11048:
10868:
10686:
10510:
10330:
10144:
9962:
9780:
9600:
9434:
9368:-gonal prism is represented as : {n}×{ }.
5621:
5493:
5317:
5122:
4983:
4851:
4736:
4623:
4507:
4397:
4221:
3008:
2962:
2912:
2861:
2812:
2804:
14032:
14009:uniform, convex polytopes in four dimensions:
13752:
666:and higher. It can be seen as truncating its
573:and higher. It can be seen as rectifying its
5927:The extended Schläfli symbols are made by a
1835:
1381:
5898:{5,3,3}, and their ten regular stellations.
14039:
14025:
336:
9198:
8915:
8542:
8193:
7887:
7604:
7255:
6906:
6557:
6208:
1329:-dimensional spherical space, tilings of
1325:+1)-dimensional polytopes are tilings of
13772:In six, seven and eight dimensions, the
5829:
5811:
1174:
1023:There are also higher hexiruncinations:
377:reduced 5-faces to vertices, and so on.
14604:List of regular polytopes and compounds
13973:Vierdimensionale Archimedische Polytope
5964:
1309:), and it is the vertex figure for the
1217:. In three dimensions, this produces a
591:, meaning to cut with a slanted face.)
421:
14613:
13964:Regular and Semi-Regular Polytopes III
5834:Summary chart of truncation operations
3605:, or by performing operations such as
1221:; in four dimensions, this produces a
13991:
13957:Regular and Semi-Regular Polytopes II
13857:The Beauty of Geometry: Twelve Essays
13815:
12352:
9383:
5850:; thus the mirrors form an irregular
5272:
5267:
4176:
4171:
2776:
1369:, the simplest being the equilateral
1084:There are higher heptellations also:
916:There are also higher pentellations:
677:There are higher cantellations also:
603:There are higher cantellations also:
16:Isogonal polytope with uniform facets
13950:Regular and Semi Regular Polytopes I
857:There are higher sterications also:
805:There are higher sterications also:
754:There are higher runcinations also:
298:to be considered polytopes as well.
13928:, 3rd Edition, Dover New York, 1973
3054:
969:There are higher hexications also:
490:There are higher truncations also:
13:
13832:exist in 2 through 10 dimensions.
9354:
5762:
3945:There is also the infinite set of
353:−1)-th rectification is the
270:. Further, star regular faces and
241:of dimension three or higher is a
14:
14632:
13985:
9203:alternated cantitruncated 16-cell
5825:Edges 1 to 2, 0 to 2, and 1 to 3.
4159:is represented by the notation: t
2764:Regular polygons, represented by
2300:There is also an infinite set of
1360:
314:. Notable exceptions include the
13739:
13734:
13729:
13724:
13719:
13708:
13703:
13698:
13693:
13688:
13677:
13672:
13667:
13662:
13657:
13646:
13641:
13636:
13631:
13626:
13617:
13612:
13607:
13602:
13597:
13592:
13587:
13559:
13554:
13549:
13544:
13539:
13534:
13523:
13518:
13513:
13508:
13503:
13498:
13487:
13482:
13477:
13472:
13467:
13456:
13451:
13446:
13441:
13436:
13427:
13422:
13417:
13412:
13407:
13402:
13397:
13392:
13364:
13359:
13354:
13349:
13344:
13339:
13328:
13323:
13318:
13313:
13308:
13303:
13292:
13287:
13282:
13277:
13272:
13267:
13256:
13251:
13246:
13241:
13236:
13231:
13222:
13217:
13212:
13207:
13202:
13197:
13192:
13187:
13182:
13154:
13149:
13144:
13139:
13134:
13129:
13118:
13113:
13108:
13103:
13098:
13087:
13082:
13077:
13072:
13067:
13056:
13051:
13046:
13041:
13036:
13031:
13022:
13017:
13012:
13007:
13002:
12997:
12992:
12987:
12959:
12954:
12949:
12944:
12939:
12928:
12923:
12918:
12913:
12908:
12897:
12892:
12887:
12882:
12877:
12872:
12861:
12856:
12851:
12846:
12841:
12836:
12831:
12822:
12817:
12812:
12807:
12802:
12797:
12792:
12787:
12782:
12756:
12751:
12746:
12741:
12736:
12725:
12720:
12715:
12710:
12705:
12694:
12689:
12684:
12679:
12674:
12669:
12658:
12653:
12648:
12643:
12638:
12633:
12624:
12619:
12614:
12609:
12604:
12599:
12594:
12589:
12561:
12556:
12551:
12546:
12541:
12536:
12531:
12520:
12515:
12510:
12505:
12500:
12495:
12490:
12479:
12474:
12469:
12464:
12459:
12454:
12449:
12438:
12433:
12428:
12423:
12418:
12413:
12408:
12395:
12390:
12385:
12380:
12375:
12370:
12365:
12318:
12313:
12308:
12303:
12298:
12287:
12282:
12277:
12272:
12267:
12256:
12251:
12246:
12241:
12236:
12225:
12220:
12215:
12210:
12205:
12196:
12191:
12186:
12181:
12176:
12171:
12166:
12134:
12129:
12124:
12119:
12114:
12103:
12098:
12093:
12088:
12083:
12072:
12067:
12062:
12057:
12052:
12041:
12036:
12031:
12026:
12021:
12012:
12007:
12002:
11997:
11992:
11987:
11982:
11944:
11939:
11934:
11929:
11924:
11913:
11908:
11903:
11898:
11893:
11882:
11877:
11872:
11867:
11862:
11851:
11846:
11841:
11836:
11831:
11822:
11817:
11812:
11807:
11802:
11797:
11792:
11762:
11757:
11752:
11747:
11742:
11731:
11726:
11721:
11716:
11711:
11700:
11695:
11690:
11685:
11680:
11669:
11664:
11659:
11654:
11649:
11640:
11635:
11630:
11625:
11620:
11615:
11610:
11580:
11575:
11570:
11565:
11560:
11549:
11544:
11539:
11534:
11529:
11518:
11513:
11508:
11503:
11498:
11487:
11482:
11477:
11472:
11467:
11458:
11453:
11448:
11443:
11438:
11433:
11428:
11406:
11401:
11396:
11391:
11386:
11375:
11370:
11365:
11360:
11355:
11344:
11339:
11334:
11329:
11324:
11313:
11308:
11303:
11298:
11293:
11284:
11279:
11274:
11269:
11264:
11259:
11254:
11220:
11215:
11210:
11205:
11200:
11189:
11184:
11179:
11174:
11169:
11158:
11153:
11148:
11143:
11138:
11127:
11122:
11117:
11112:
11107:
11098:
11093:
11088:
11083:
11078:
11073:
11068:
11038:
11033:
11028:
11023:
11018:
11007:
11002:
10997:
10992:
10987:
10976:
10971:
10966:
10961:
10956:
10945:
10940:
10935:
10930:
10925:
10916:
10911:
10906:
10901:
10896:
10891:
10886:
10858:
10853:
10848:
10843:
10838:
10827:
10822:
10817:
10812:
10807:
10796:
10791:
10786:
10781:
10776:
10765:
10760:
10755:
10750:
10745:
10736:
10731:
10726:
10721:
10716:
10711:
10706:
10676:
10671:
10666:
10661:
10656:
10645:
10640:
10635:
10630:
10625:
10614:
10609:
10604:
10599:
10594:
10583:
10578:
10573:
10568:
10563:
10554:
10549:
10544:
10539:
10534:
10529:
10524:
10500:
10495:
10490:
10485:
10480:
10469:
10464:
10459:
10454:
10449:
10438:
10433:
10428:
10423:
10418:
10407:
10402:
10397:
10392:
10387:
10378:
10373:
10368:
10363:
10358:
10353:
10348:
10320:
10315:
10310:
10305:
10300:
10289:
10284:
10279:
10274:
10269:
10258:
10253:
10248:
10243:
10238:
10227:
10222:
10217:
10212:
10207:
10198:
10193:
10188:
10183:
10178:
10173:
10168:
10134:
10129:
10124:
10119:
10114:
10103:
10098:
10093:
10088:
10083:
10072:
10067:
10062:
10057:
10052:
10041:
10036:
10031:
10026:
10021:
10012:
10007:
10002:
9997:
9992:
9987:
9982:
9952:
9947:
9942:
9937:
9932:
9921:
9916:
9911:
9906:
9901:
9890:
9885:
9880:
9875:
9870:
9859:
9854:
9849:
9844:
9839:
9830:
9825:
9820:
9815:
9810:
9805:
9800:
9770:
9765:
9760:
9755:
9750:
9739:
9734:
9729:
9724:
9719:
9708:
9703:
9698:
9693:
9688:
9677:
9672:
9667:
9662:
9657:
9648:
9643:
9638:
9633:
9628:
9623:
9618:
9592:
9587:
9582:
9577:
9572:
9567:
9562:
9551:
9546:
9541:
9536:
9531:
9526:
9521:
9510:
9505:
9500:
9495:
9490:
9485:
9480:
9469:
9464:
9459:
9454:
9449:
9444:
9439:
9426:
9421:
9416:
9411:
9406:
9401:
9396:
9337:
9332:
9327:
9322:
9317:
9312:
9307:
9300:
9286:
9281:
9276:
9271:
9266:
9259:
9243:
9238:
9233:
9228:
9223:
9218:
9213:
9206:
9184:
9179:
9174:
9169:
9164:
9159:
9154:
9147:
9125:
9120:
9115:
9110:
9105:
9100:
9095:
9088:
9068:
9063:
9058:
9053:
9048:
9043:
9038:
9031:
9013:
9008:
9003:
8998:
8993:
8988:
8983:
8976:
8958:
8953:
8948:
8943:
8938:
8933:
8928:
8921:
8901:
8896:
8891:
8886:
8881:
8876:
8871:
8864:
8846:
8841:
8836:
8831:
8826:
8821:
8816:
8809:
8791:
8786:
8781:
8776:
8771:
8766:
8761:
8754:
8740:
8735:
8730:
8725:
8720:
8713:
8695:
8690:
8685:
8680:
8675:
8670:
8665:
8658:
8640:
8635:
8630:
8625:
8620:
8615:
8610:
8603:
8585:
8580:
8575:
8570:
8565:
8560:
8555:
8548:
8532:
8527:
8522:
8517:
8512:
8507:
8502:
8495:
8481:
8476:
8471:
8466:
8461:
8456:
8451:
8444:
8430:
8425:
8420:
8415:
8410:
8405:
8400:
8393:
8379:
8374:
8369:
8364:
8359:
8352:
8338:
8333:
8328:
8323:
8318:
8313:
8308:
8301:
8287:
8282:
8277:
8272:
8267:
8262:
8257:
8250:
8236:
8231:
8226:
8221:
8216:
8211:
8206:
8199:
8183:
8178:
8173:
8168:
8163:
8158:
8153:
8146:
8128:
8123:
8118:
8113:
8108:
8103:
8098:
8091:
8077:
8072:
8067:
8062:
8057:
8050:
8036:
8031:
8026:
8021:
8016:
8011:
8006:
7999:
7985:
7980:
7975:
7970:
7965:
7960:
7955:
7948:
7930:
7925:
7920:
7915:
7910:
7905:
7900:
7893:
7873:
7868:
7863:
7858:
7853:
7848:
7843:
7836:
7814:
7809:
7804:
7799:
7794:
7789:
7784:
7777:
7757:
7752:
7747:
7742:
7737:
7732:
7727:
7720:
7702:
7697:
7692:
7687:
7682:
7677:
7672:
7665:
7647:
7642:
7637:
7632:
7627:
7622:
7617:
7610:
7594:
7589:
7584:
7579:
7574:
7569:
7564:
7557:
7543:
7538:
7533:
7528:
7523:
7518:
7513:
7506:
7492:
7487:
7482:
7477:
7472:
7467:
7462:
7455:
7441:
7436:
7431:
7426:
7421:
7414:
7400:
7395:
7390:
7385:
7380:
7375:
7370:
7363:
7349:
7344:
7339:
7334:
7329:
7324:
7319:
7312:
7298:
7293:
7288:
7283:
7278:
7273:
7268:
7261:
7245:
7240:
7235:
7230:
7225:
7220:
7215:
7208:
7194:
7189:
7184:
7179:
7174:
7169:
7164:
7157:
7143:
7138:
7133:
7128:
7123:
7118:
7113:
7106:
7092:
7087:
7082:
7077:
7072:
7065:
7051:
7046:
7041:
7036:
7031:
7026:
7021:
7014:
7000:
6995:
6990:
6985:
6980:
6975:
6970:
6963:
6949:
6944:
6939:
6934:
6929:
6924:
6919:
6912:
6896:
6891:
6886:
6881:
6876:
6871:
6866:
6859:
6845:
6840:
6835:
6830:
6825:
6820:
6815:
6808:
6794:
6789:
6784:
6779:
6774:
6769:
6764:
6757:
6743:
6738:
6733:
6728:
6723:
6716:
6702:
6697:
6692:
6687:
6682:
6677:
6672:
6665:
6651:
6646:
6641:
6636:
6631:
6626:
6621:
6614:
6600:
6595:
6590:
6585:
6580:
6575:
6570:
6563:
6547:
6542:
6537:
6532:
6527:
6522:
6517:
6510:
6496:
6491:
6486:
6481:
6476:
6471:
6466:
6459:
6445:
6440:
6435:
6430:
6425:
6420:
6415:
6408:
6394:
6389:
6384:
6379:
6374:
6367:
6353:
6348:
6343:
6338:
6333:
6328:
6323:
6316:
6302:
6297:
6292:
6287:
6282:
6277:
6272:
6265:
6251:
6246:
6241:
6236:
6231:
6226:
6221:
6214:
6200:
6195:
6190:
6185:
6180:
6175:
6170:
6160:
6155:
6150:
6145:
6140:
6135:
6130:
6120:
6115:
6110:
6105:
6100:
6090:
6085:
6080:
6075:
6070:
6065:
6060:
6050:
6045:
6040:
6035:
6030:
6025:
6020:
5773:In four dimensions, there are 6
5750:
5743:
5710:
5705:
5700:
5695:
5690:
5685:
5590:
5585:
5580:
5575:
5570:
5561:
5556:
5551:
5485:
5480:
5475:
5470:
5465:
5456:
5451:
5446:
5441:
5436:
5427:
5422:
5417:
5412:
5407:
5398:
5393:
5388:
5383:
5378:
5369:
5364:
5359:
5354:
5349:
5340:
5335:
5330:
5325:
5320:
5309:
5304:
5299:
5294:
5289:
5230:
5225:
5220:
5215:
5210:
5201:
5196:
5191:
5093:
5088:
5083:
5078:
5073:
5064:
5059:
5054:
4954:
4949:
4944:
4939:
4934:
4925:
4920:
4915:
4822:
4817:
4812:
4807:
4802:
4707:
4702:
4697:
4692:
4687:
4594:
4589:
4584:
4579:
4574:
4478:
4473:
4468:
4463:
4458:
4389:
4384:
4379:
4374:
4369:
4360:
4355:
4350:
4345:
4340:
4331:
4326:
4321:
4316:
4311:
4302:
4297:
4292:
4287:
4282:
4273:
4268:
4263:
4258:
4253:
4244:
4239:
4234:
4229:
4224:
4213:
4208:
4203:
4198:
4193:
4130:
4110:
4105:
4100:
4095:
4090:
4082:
4075:
4068:
4045:
4040:
4035:
4030:
4025:
4017:
4010:
4003:
3931:
3920:
3909:
3898:
3887:
3876:
3865:
3854:
3834:
3823:
3812:
3801:
3790:
3779:
3768:
3757:
3737:
3726:
3715:
3704:
3693:
3682:
3671:
3660:
3565:
3558:
3551:
3543:
3538:
3533:
3528:
3523:
3483:
3476:
3469:
3461:
3456:
3451:
3446:
3441:
3401:
3394:
3387:
3379:
3374:
3369:
3364:
3359:
3313:
3306:
3299:
3291:
3286:
3281:
3276:
3271:
3223:
3216:
3209:
3201:
3196:
3191:
3186:
3181:
3110:
3105:
3100:
3095:
3090:
3032:
3027:
3022:
2989:
2984:
2979:
2974:
2943:
2938:
2933:
2894:
2889:
2884:
2843:
2838:
2833:
2751:
2745:
2738:
2731:
2724:
2718:
2711:
2704:
2697:
2684:
2679:
2674:
2669:
2664:
2655:
2650:
2645:
2640:
2635:
2627:
2622:
2617:
2612:
2607:
2598:
2593:
2588:
2583:
2578:
2569:
2564:
2559:
2554:
2549:
2540:
2535:
2530:
2525:
2520:
2512:
2507:
2502:
2497:
2492:
2483:
2478:
2473:
2468:
2463:
2454:
2449:
2444:
2439:
2434:
2425:
2420:
2415:
2410:
2405:
2290:
2284:
2277:
2270:
2264:
2257:
2250:
2244:
2237:
2230:
2224:
2217:
2210:
2204:
2191:
2186:
2181:
2173:
2168:
2163:
2154:
2149:
2144:
2135:
2130:
2125:
2117:
2112:
2107:
2098:
2093:
2088:
2079:
2074:
2069:
2061:
2056:
2051:
2042:
2037:
2032:
2023:
2018:
2013:
2005:
2000:
1995:
1986:
1981:
1976:
1967:
1962:
1957:
1949:
1944:
1939:
1828:
1821:
1815:
1808:
1801:
1795:
1788:
1781:
1775:
1768:
1761:
1755:
1748:
1735:
1730:
1725:
1716:
1711:
1706:
1698:
1693:
1688:
1679:
1674:
1669:
1660:
1655:
1650:
1642:
1637:
1632:
1623:
1618:
1613:
1604:
1599:
1594:
1586:
1581:
1576:
1567:
1562:
1557:
1548:
1543:
1538:
1530:
1525:
1520:
1511:
1506:
1501:
1348:
1232:
594:
481:
222:
217:
212:
207:
202:
197:
192:
187:
182:
171:
163:
158:
153:
148:
143:
138:
133:
122:
104:
99:
94:
89:
84:
73:
64:
59:
54:
33:
13378:Alternated truncated rectified
5980:, which has no Coxeter family.
4143:of the object. An extension of
4055:
3990:
3512:
3430:
3342:
3252:
3162:
2356:
2320:
1885:
1449:
1273:
373:reduces 4-faces to vertices, a
12336:Half constructions exist with
8710:runcicantellated demitesseract
1317:Uniform polytopes by dimension
1191:One special operation, called
1170:
316:great dirhombicosidodecahedron
1:
13847:
12331:
369:reduces cells to vertices, a
365:reduces faces to vertices, a
361:reduces edges to vertices, a
301:
13859:, Dover Publications, 1999,
9297:Alternated truncated 24-cell
8047:cantitruncated demitesseract
5982:
5869:{3,3,3}, which is self-dual;
4088:
4023:
3601:, which can be obtained via
1252:of other uniform polytopes.
318:in three dimensions and the
7:
13962:(Paper 24) H.S.M. Coxeter,
13955:(Paper 23) H.S.M. Coxeter,
13948:(Paper 22) H.S.M. Coxeter,
13835:
8349:omnitruncated demitesseract
1341:are also considered to be (
1293:. For example, the regular
10:
14637:
14593:
14020:
13753:Five and higher dimensions
13573:Alternated omnitruncation
5775:convex regular 4-polytopes
5766:
3073:
3058:
2692:
2199:
1929:
1743:
1491:
1311:alternated cubic honeycomb
331:Conway polyhedron notation
329:, and is the basis of the
120:
112:
31:
13826:hyperbolic Coxeter groups
12362:
12355:
12349:
11958:(or runcitruncated dual)
11594:(or cantitruncated dual)
9393:
9386:
9380:
9138:
8137:
7827:
7411:cantellated demitesseract
6167:
6057:
6009:
5991:
5914:convex uniform honeycombs
5905:(The groups are named in
5683:
5546:
5286:
5279:
5264:
4800:
4685:
4572:
4456:
4190:
4183:
4168:
3014:
2966:
2798:
2795:
2788:
2783:
2773:
2339:
2329:
341:Regular n-polytopes have
20:Convex uniform polytopes
13168:Alternated bitruncation
8655:runcitruncated tesseract
8298:cantitruncated tesseract
4125:Kepler-Poinsot polyhedra
266:, but also includes the
13998:Glossary for Hyperspace
9028:omnitruncated tesseract
8861:runcitruncated 120-cell
8806:runcitruncated 600-cell
8492:cantitruncated 120-cell
8441:cantitruncated 600-cell
7062:truncated demitesseract
6713:rectified demitesseract
5816:Example tetrahedron in
4147:notation, also used by
1181:truncated cuboctahedron
337:Rectification operators
310:, and represented by a
286:and higher dimensional
12973:Alternated truncation
11052:(or cantellated dual)
9144:omnitruncated 600-cell
9140:omnitruncated 120-cell
8751:runcitruncated 24-cell
8600:runcitruncated 16-cell
8390:cantitruncated 24-cell
8247:cantitruncated 16-cell
5835:
5827:
5666:
5539:
5172:
5035:
4896:
4783:
4668:
4555:
4439:
4139:, which specifies the
1270:as its vertex figure.
1188:
13822:affine Coxeter groups
12768:Alternated rectified
9085:omnitruncated 24-cell
8973:omnitruncated 16-cell
8545:runcitruncated 5-cell
8196:cantitruncated 5-cell
7996:bitruncated tesseract
7360:cantellated tesseract
5876:{3,3,4} and its dual
5833:
5815:
5756:Generating triangles
5667:
5540:
5173:
5036:
4897:
4784:
4669:
4556:
4440:
3603:Wythoff constructions
1178:
707:quadricantitruncation
178:Truncated 5-orthoplex
13912:M.S. Longuet-Higgins
13876:, Manuscript (1991)
10690:(or truncated dual)
9966:(or rectified dual)
8918:omnitruncated 5-cell
8143:bitruncated 120-cell
8139:bitruncated 600-cell
7717:runcinated tesseract
7554:cantellated 120-cell
7503:cantellated 600-cell
5965:Constructive summary
5857:Each of the sixteen
5631:
5503:
5136:
4999:
4863:
4748:
4742:(or truncated dual)
4633:
4523:
4407:
4157:truncated octahedron
1206:polytope solutions.
1179:An alternation of a
1142:Uniform 10-polytopes
873:tristericantellation
422:Truncation operators
391:-th rectification =
322:in four dimensions.
308:Wythoff construction
80:Truncated octahedron
14588:pentagonal polytope
14487:Uniform 10-polytope
14047:Fundamental convex
14004:on 4 February 2007.
13992:Olshevsky, George.
13914:and J.C.P. Miller:
12363:Cells by position:
12146:Runcicantitruncated
9394:Cells by position:
8088:bitruncated 24-cell
7945:bitruncated 16-cell
7833:runcinated 120-cell
7829:runcinated 600-cell
7452:cantellated 24-cell
7309:cantellated 16-cell
7011:truncated tesseract
6662:rectified tesseract
5887:{3,4,3}, self-dual;
5859:regular 4-polytopes
1125:Uniform 9-polytopes
1072:Uniform 8-polytopes
1018:Uniform 7-polytopes
957:Uniform 7-polytopes
904:Uniform 6-polytopes
859:bistericantellation
852:Uniform 5-polytopes
795:Uniform 5-polytopes
474:, comes from Latin
375:quintirectification
248:bounded by uniform
21:
14457:Uniform 9-polytope
14407:Uniform 8-polytope
14357:Uniform 7-polytope
14314:Uniform 6-polytope
14284:Uniform 5-polytope
14244:Uniform polychoron
14207:Uniform polyhedron
14055:in dimensions 2–10
13994:"Uniform polytope"
13816:Uniform honeycombs
9256:snub demitesseract
7890:bitruncated 5-cell
7774:runcinated 24-cell
7662:runcinated 16-cell
7258:cantellated 5-cell
7205:truncated 120-cell
7154:truncated 600-cell
6856:rectified 120-cell
6805:rectified 600-cell
5894:{3,3,5}, its dual
5840:fundamental region
5836:
5828:
5769:Uniform 4-polytope
5662:
5656:
5535:
5529:
5168:
5162:
5031:
5025:
4892:
4886:
4779:
4773:
4664:
4658:
4551:
4545:
4435:
4429:
4141:fundamental region
3599:Archimedean solids
3061:Uniform polyhedron
1291:uniform honeycombs
1189:
1160:applied together.
740:Uniform 4-polytope
693:tricantitruncation
633:quadricantellation
371:quadirectification
280:uniform honeycombs
19:
14621:Uniform polytopes
14609:
14608:
14596:Polytope families
14053:uniform polytopes
13944:978-0-471-01003-6
13926:Regular Polytopes
13916:Uniform Polyhedra
13874:Uniform Polytopes
13865:978-0-486-40919-1
13777:simple Lie groups
13750:
13749:
12329:
12328:
9352:
9351:
7607:runcinated 5-cell
7103:truncated 24-cell
6960:truncated 16-cell
6754:rectified 24-cell
6611:rectified 16-cell
5760:
5759:
5738:
5737:
5260:
5259:
4121:
4120:
3943:
3942:
3595:
3594:
3052:
3051:
2762:
2761:
2298:
2297:
1299:triangular tiling
1040:trihexiruncinated
848:Stericantellation
679:bicantitruncation
268:regular polytopes
243:vertex-transitive
231:
230:
129:Truncated 16-cell
14628:
14600:Regular polytope
14161:
14150:
14139:
14098:
14041:
14034:
14027:
14018:
14017:
14014:
14005:
14000:. Archived from
13978:
13924:H.S.M. Coxeter,
13910:H.S.M. Coxeter,
13879:Norman Johnson:
13802:Coxeter diagrams
13744:
13743:
13742:
13738:
13737:
13733:
13732:
13728:
13727:
13723:
13722:
13713:
13712:
13711:
13707:
13706:
13702:
13701:
13697:
13696:
13692:
13691:
13682:
13681:
13680:
13676:
13675:
13671:
13670:
13666:
13665:
13661:
13660:
13651:
13650:
13649:
13645:
13644:
13640:
13639:
13635:
13634:
13630:
13629:
13622:
13621:
13620:
13616:
13615:
13611:
13610:
13606:
13605:
13601:
13600:
13596:
13595:
13591:
13590:
13564:
13563:
13562:
13558:
13557:
13553:
13552:
13548:
13547:
13543:
13542:
13538:
13537:
13528:
13527:
13526:
13522:
13521:
13517:
13516:
13512:
13511:
13507:
13506:
13502:
13501:
13492:
13491:
13490:
13486:
13485:
13481:
13480:
13476:
13475:
13471:
13470:
13461:
13460:
13459:
13455:
13454:
13450:
13449:
13445:
13444:
13440:
13439:
13432:
13431:
13430:
13426:
13425:
13421:
13420:
13416:
13415:
13411:
13410:
13406:
13405:
13401:
13400:
13396:
13395:
13369:
13368:
13367:
13363:
13362:
13358:
13357:
13353:
13352:
13348:
13347:
13343:
13342:
13333:
13332:
13331:
13327:
13326:
13322:
13321:
13317:
13316:
13312:
13311:
13307:
13306:
13297:
13296:
13295:
13291:
13290:
13286:
13285:
13281:
13280:
13276:
13275:
13271:
13270:
13261:
13260:
13259:
13255:
13254:
13250:
13249:
13245:
13244:
13240:
13239:
13235:
13234:
13227:
13226:
13225:
13221:
13220:
13216:
13215:
13211:
13210:
13206:
13205:
13201:
13200:
13196:
13195:
13191:
13190:
13186:
13185:
13159:
13158:
13157:
13153:
13152:
13148:
13147:
13143:
13142:
13138:
13137:
13133:
13132:
13123:
13122:
13121:
13117:
13116:
13112:
13111:
13107:
13106:
13102:
13101:
13092:
13091:
13090:
13086:
13085:
13081:
13080:
13076:
13075:
13071:
13070:
13061:
13060:
13059:
13055:
13054:
13050:
13049:
13045:
13044:
13040:
13039:
13035:
13034:
13027:
13026:
13025:
13021:
13020:
13016:
13015:
13011:
13010:
13006:
13005:
13001:
13000:
12996:
12995:
12991:
12990:
12964:
12963:
12962:
12958:
12957:
12953:
12952:
12948:
12947:
12943:
12942:
12933:
12932:
12931:
12927:
12926:
12922:
12921:
12917:
12916:
12912:
12911:
12902:
12901:
12900:
12896:
12895:
12891:
12890:
12886:
12885:
12881:
12880:
12876:
12875:
12866:
12865:
12864:
12860:
12859:
12855:
12854:
12850:
12849:
12845:
12844:
12840:
12839:
12835:
12834:
12827:
12826:
12825:
12821:
12820:
12816:
12815:
12811:
12810:
12806:
12805:
12801:
12800:
12796:
12795:
12791:
12790:
12786:
12785:
12761:
12760:
12759:
12755:
12754:
12750:
12749:
12745:
12744:
12740:
12739:
12730:
12729:
12728:
12724:
12723:
12719:
12718:
12714:
12713:
12709:
12708:
12699:
12698:
12697:
12693:
12692:
12688:
12687:
12683:
12682:
12678:
12677:
12673:
12672:
12663:
12662:
12661:
12657:
12656:
12652:
12651:
12647:
12646:
12642:
12641:
12637:
12636:
12629:
12628:
12627:
12623:
12622:
12618:
12617:
12613:
12612:
12608:
12607:
12603:
12602:
12598:
12597:
12593:
12592:
12566:
12565:
12564:
12560:
12559:
12555:
12554:
12550:
12549:
12545:
12544:
12540:
12539:
12535:
12534:
12525:
12524:
12523:
12519:
12518:
12514:
12513:
12509:
12508:
12504:
12503:
12499:
12498:
12494:
12493:
12484:
12483:
12482:
12478:
12477:
12473:
12472:
12468:
12467:
12463:
12462:
12458:
12457:
12453:
12452:
12443:
12442:
12441:
12437:
12436:
12432:
12431:
12427:
12426:
12422:
12421:
12417:
12416:
12412:
12411:
12400:
12399:
12398:
12394:
12393:
12389:
12388:
12384:
12383:
12379:
12378:
12374:
12373:
12369:
12368:
12353:Schläfli symbol
12347:
12346:
12323:
12322:
12321:
12317:
12316:
12312:
12311:
12307:
12306:
12302:
12301:
12292:
12291:
12290:
12286:
12285:
12281:
12280:
12276:
12275:
12271:
12270:
12261:
12260:
12259:
12255:
12254:
12250:
12249:
12245:
12244:
12240:
12239:
12230:
12229:
12228:
12224:
12223:
12219:
12218:
12214:
12213:
12209:
12208:
12201:
12200:
12199:
12195:
12194:
12190:
12189:
12185:
12184:
12180:
12179:
12175:
12174:
12170:
12169:
12139:
12138:
12137:
12133:
12132:
12128:
12127:
12123:
12122:
12118:
12117:
12108:
12107:
12106:
12102:
12101:
12097:
12096:
12092:
12091:
12087:
12086:
12077:
12076:
12075:
12071:
12070:
12066:
12065:
12061:
12060:
12056:
12055:
12046:
12045:
12044:
12040:
12039:
12035:
12034:
12030:
12029:
12025:
12024:
12017:
12016:
12015:
12011:
12010:
12006:
12005:
12001:
12000:
11996:
11995:
11991:
11990:
11986:
11985:
11956:Runcicantellated
11949:
11948:
11947:
11943:
11942:
11938:
11937:
11933:
11932:
11928:
11927:
11918:
11917:
11916:
11912:
11911:
11907:
11906:
11902:
11901:
11897:
11896:
11887:
11886:
11885:
11881:
11880:
11876:
11875:
11871:
11870:
11866:
11865:
11856:
11855:
11854:
11850:
11849:
11845:
11844:
11840:
11839:
11835:
11834:
11827:
11826:
11825:
11821:
11820:
11816:
11815:
11811:
11810:
11806:
11805:
11801:
11800:
11796:
11795:
11767:
11766:
11765:
11761:
11760:
11756:
11755:
11751:
11750:
11746:
11745:
11736:
11735:
11734:
11730:
11729:
11725:
11724:
11720:
11719:
11715:
11714:
11705:
11704:
11703:
11699:
11698:
11694:
11693:
11689:
11688:
11684:
11683:
11674:
11673:
11672:
11668:
11667:
11663:
11662:
11658:
11657:
11653:
11652:
11645:
11644:
11643:
11639:
11638:
11634:
11633:
11629:
11628:
11624:
11623:
11619:
11618:
11614:
11613:
11592:Bicantitruncated
11585:
11584:
11583:
11579:
11578:
11574:
11573:
11569:
11568:
11564:
11563:
11554:
11553:
11552:
11548:
11547:
11543:
11542:
11538:
11537:
11533:
11532:
11523:
11522:
11521:
11517:
11516:
11512:
11511:
11507:
11506:
11502:
11501:
11492:
11491:
11490:
11486:
11485:
11481:
11480:
11476:
11475:
11471:
11470:
11463:
11462:
11461:
11457:
11456:
11452:
11451:
11447:
11446:
11442:
11441:
11437:
11436:
11432:
11431:
11411:
11410:
11409:
11405:
11404:
11400:
11399:
11395:
11394:
11390:
11389:
11380:
11379:
11378:
11374:
11373:
11369:
11368:
11364:
11363:
11359:
11358:
11349:
11348:
11347:
11343:
11342:
11338:
11337:
11333:
11332:
11328:
11327:
11318:
11317:
11316:
11312:
11311:
11307:
11306:
11302:
11301:
11297:
11296:
11289:
11288:
11287:
11283:
11282:
11278:
11277:
11273:
11272:
11268:
11267:
11263:
11262:
11258:
11257:
11225:
11224:
11223:
11219:
11218:
11214:
11213:
11209:
11208:
11204:
11203:
11194:
11193:
11192:
11188:
11187:
11183:
11182:
11178:
11177:
11173:
11172:
11163:
11162:
11161:
11157:
11156:
11152:
11151:
11147:
11146:
11142:
11141:
11132:
11131:
11130:
11126:
11125:
11121:
11120:
11116:
11115:
11111:
11110:
11103:
11102:
11101:
11097:
11096:
11092:
11091:
11087:
11086:
11082:
11081:
11077:
11076:
11072:
11071:
11043:
11042:
11041:
11037:
11036:
11032:
11031:
11027:
11026:
11022:
11021:
11012:
11011:
11010:
11006:
11005:
11001:
11000:
10996:
10995:
10991:
10990:
10981:
10980:
10979:
10975:
10974:
10970:
10969:
10965:
10964:
10960:
10959:
10950:
10949:
10948:
10944:
10943:
10939:
10938:
10934:
10933:
10929:
10928:
10921:
10920:
10919:
10915:
10914:
10910:
10909:
10905:
10904:
10900:
10899:
10895:
10894:
10890:
10889:
10863:
10862:
10861:
10857:
10856:
10852:
10851:
10847:
10846:
10842:
10841:
10832:
10831:
10830:
10826:
10825:
10821:
10820:
10816:
10815:
10811:
10810:
10801:
10800:
10799:
10795:
10794:
10790:
10789:
10785:
10784:
10780:
10779:
10770:
10769:
10768:
10764:
10763:
10759:
10758:
10754:
10753:
10749:
10748:
10741:
10740:
10739:
10735:
10734:
10730:
10729:
10725:
10724:
10720:
10719:
10715:
10714:
10710:
10709:
10681:
10680:
10679:
10675:
10674:
10670:
10669:
10665:
10664:
10660:
10659:
10650:
10649:
10648:
10644:
10643:
10639:
10638:
10634:
10633:
10629:
10628:
10619:
10618:
10617:
10613:
10612:
10608:
10607:
10603:
10602:
10598:
10597:
10588:
10587:
10586:
10582:
10581:
10577:
10576:
10572:
10571:
10567:
10566:
10559:
10558:
10557:
10553:
10552:
10548:
10547:
10543:
10542:
10538:
10537:
10533:
10532:
10528:
10527:
10505:
10504:
10503:
10499:
10498:
10494:
10493:
10489:
10488:
10484:
10483:
10474:
10473:
10472:
10468:
10467:
10463:
10462:
10458:
10457:
10453:
10452:
10443:
10442:
10441:
10437:
10436:
10432:
10431:
10427:
10426:
10422:
10421:
10412:
10411:
10410:
10406:
10405:
10401:
10400:
10396:
10395:
10391:
10390:
10383:
10382:
10381:
10377:
10376:
10372:
10371:
10367:
10366:
10362:
10361:
10357:
10356:
10352:
10351:
10325:
10324:
10323:
10319:
10318:
10314:
10313:
10309:
10308:
10304:
10303:
10294:
10293:
10292:
10288:
10287:
10283:
10282:
10278:
10277:
10273:
10272:
10263:
10262:
10261:
10257:
10256:
10252:
10251:
10247:
10246:
10242:
10241:
10232:
10231:
10230:
10226:
10225:
10221:
10220:
10216:
10215:
10211:
10210:
10203:
10202:
10201:
10197:
10196:
10192:
10191:
10187:
10186:
10182:
10181:
10177:
10176:
10172:
10171:
10139:
10138:
10137:
10133:
10132:
10128:
10127:
10123:
10122:
10118:
10117:
10108:
10107:
10106:
10102:
10101:
10097:
10096:
10092:
10091:
10087:
10086:
10077:
10076:
10075:
10071:
10070:
10066:
10065:
10061:
10060:
10056:
10055:
10046:
10045:
10044:
10040:
10039:
10035:
10034:
10030:
10029:
10025:
10024:
10017:
10016:
10015:
10011:
10010:
10006:
10005:
10001:
10000:
9996:
9995:
9991:
9990:
9986:
9985:
9957:
9956:
9955:
9951:
9950:
9946:
9945:
9941:
9940:
9936:
9935:
9926:
9925:
9924:
9920:
9919:
9915:
9914:
9910:
9909:
9905:
9904:
9895:
9894:
9893:
9889:
9888:
9884:
9883:
9879:
9878:
9874:
9873:
9864:
9863:
9862:
9858:
9857:
9853:
9852:
9848:
9847:
9843:
9842:
9835:
9834:
9833:
9829:
9828:
9824:
9823:
9819:
9818:
9814:
9813:
9809:
9808:
9804:
9803:
9775:
9774:
9773:
9769:
9768:
9764:
9763:
9759:
9758:
9754:
9753:
9744:
9743:
9742:
9738:
9737:
9733:
9732:
9728:
9727:
9723:
9722:
9713:
9712:
9711:
9707:
9706:
9702:
9701:
9697:
9696:
9692:
9691:
9682:
9681:
9680:
9676:
9675:
9671:
9670:
9666:
9665:
9661:
9660:
9653:
9652:
9651:
9647:
9646:
9642:
9641:
9637:
9636:
9632:
9631:
9627:
9626:
9622:
9621:
9597:
9596:
9595:
9591:
9590:
9586:
9585:
9581:
9580:
9576:
9575:
9571:
9570:
9566:
9565:
9556:
9555:
9554:
9550:
9549:
9545:
9544:
9540:
9539:
9535:
9534:
9530:
9529:
9525:
9524:
9515:
9514:
9513:
9509:
9508:
9504:
9503:
9499:
9498:
9494:
9493:
9489:
9488:
9484:
9483:
9474:
9473:
9472:
9468:
9467:
9463:
9462:
9458:
9457:
9453:
9452:
9448:
9447:
9443:
9442:
9431:
9430:
9429:
9425:
9424:
9420:
9419:
9415:
9414:
9410:
9409:
9405:
9404:
9400:
9399:
9384:Schläfli symbol
9378:
9377:
9342:
9341:
9340:
9336:
9335:
9331:
9330:
9326:
9325:
9321:
9320:
9316:
9315:
9311:
9310:
9304:
9291:
9290:
9289:
9285:
9284:
9280:
9279:
9275:
9274:
9270:
9269:
9263:
9248:
9247:
9246:
9242:
9241:
9237:
9236:
9232:
9231:
9227:
9226:
9222:
9221:
9217:
9216:
9210:
9189:
9188:
9187:
9183:
9182:
9178:
9177:
9173:
9172:
9168:
9167:
9163:
9162:
9158:
9157:
9151:
9130:
9129:
9128:
9124:
9123:
9119:
9118:
9114:
9113:
9109:
9108:
9104:
9103:
9099:
9098:
9092:
9073:
9072:
9071:
9067:
9066:
9062:
9061:
9057:
9056:
9052:
9051:
9047:
9046:
9042:
9041:
9035:
9018:
9017:
9016:
9012:
9011:
9007:
9006:
9002:
9001:
8997:
8996:
8992:
8991:
8987:
8986:
8980:
8963:
8962:
8961:
8957:
8956:
8952:
8951:
8947:
8946:
8942:
8941:
8937:
8936:
8932:
8931:
8925:
8906:
8905:
8904:
8900:
8899:
8895:
8894:
8890:
8889:
8885:
8884:
8880:
8879:
8875:
8874:
8868:
8851:
8850:
8849:
8845:
8844:
8840:
8839:
8835:
8834:
8830:
8829:
8825:
8824:
8820:
8819:
8813:
8796:
8795:
8794:
8790:
8789:
8785:
8784:
8780:
8779:
8775:
8774:
8770:
8769:
8765:
8764:
8758:
8745:
8744:
8743:
8739:
8738:
8734:
8733:
8729:
8728:
8724:
8723:
8717:
8700:
8699:
8698:
8694:
8693:
8689:
8688:
8684:
8683:
8679:
8678:
8674:
8673:
8669:
8668:
8662:
8645:
8644:
8643:
8639:
8638:
8634:
8633:
8629:
8628:
8624:
8623:
8619:
8618:
8614:
8613:
8607:
8590:
8589:
8588:
8584:
8583:
8579:
8578:
8574:
8573:
8569:
8568:
8564:
8563:
8559:
8558:
8552:
8537:
8536:
8535:
8531:
8530:
8526:
8525:
8521:
8520:
8516:
8515:
8511:
8510:
8506:
8505:
8499:
8486:
8485:
8484:
8480:
8479:
8475:
8474:
8470:
8469:
8465:
8464:
8460:
8459:
8455:
8454:
8448:
8435:
8434:
8433:
8429:
8428:
8424:
8423:
8419:
8418:
8414:
8413:
8409:
8408:
8404:
8403:
8397:
8384:
8383:
8382:
8378:
8377:
8373:
8372:
8368:
8367:
8363:
8362:
8356:
8343:
8342:
8341:
8337:
8336:
8332:
8331:
8327:
8326:
8322:
8321:
8317:
8316:
8312:
8311:
8305:
8292:
8291:
8290:
8286:
8285:
8281:
8280:
8276:
8275:
8271:
8270:
8266:
8265:
8261:
8260:
8254:
8241:
8240:
8239:
8235:
8234:
8230:
8229:
8225:
8224:
8220:
8219:
8215:
8214:
8210:
8209:
8203:
8188:
8187:
8186:
8182:
8181:
8177:
8176:
8172:
8171:
8167:
8166:
8162:
8161:
8157:
8156:
8150:
8133:
8132:
8131:
8127:
8126:
8122:
8121:
8117:
8116:
8112:
8111:
8107:
8106:
8102:
8101:
8095:
8082:
8081:
8080:
8076:
8075:
8071:
8070:
8066:
8065:
8061:
8060:
8054:
8041:
8040:
8039:
8035:
8034:
8030:
8029:
8025:
8024:
8020:
8019:
8015:
8014:
8010:
8009:
8003:
7990:
7989:
7988:
7984:
7983:
7979:
7978:
7974:
7973:
7969:
7968:
7964:
7963:
7959:
7958:
7952:
7935:
7934:
7933:
7929:
7928:
7924:
7923:
7919:
7918:
7914:
7913:
7909:
7908:
7904:
7903:
7897:
7878:
7877:
7876:
7872:
7871:
7867:
7866:
7862:
7861:
7857:
7856:
7852:
7851:
7847:
7846:
7840:
7819:
7818:
7817:
7813:
7812:
7808:
7807:
7803:
7802:
7798:
7797:
7793:
7792:
7788:
7787:
7781:
7762:
7761:
7760:
7756:
7755:
7751:
7750:
7746:
7745:
7741:
7740:
7736:
7735:
7731:
7730:
7724:
7707:
7706:
7705:
7701:
7700:
7696:
7695:
7691:
7690:
7686:
7685:
7681:
7680:
7676:
7675:
7669:
7652:
7651:
7650:
7646:
7645:
7641:
7640:
7636:
7635:
7631:
7630:
7626:
7625:
7621:
7620:
7614:
7599:
7598:
7597:
7593:
7592:
7588:
7587:
7583:
7582:
7578:
7577:
7573:
7572:
7568:
7567:
7561:
7548:
7547:
7546:
7542:
7541:
7537:
7536:
7532:
7531:
7527:
7526:
7522:
7521:
7517:
7516:
7510:
7497:
7496:
7495:
7491:
7490:
7486:
7485:
7481:
7480:
7476:
7475:
7471:
7470:
7466:
7465:
7459:
7446:
7445:
7444:
7440:
7439:
7435:
7434:
7430:
7429:
7425:
7424:
7418:
7405:
7404:
7403:
7399:
7398:
7394:
7393:
7389:
7388:
7384:
7383:
7379:
7378:
7374:
7373:
7367:
7354:
7353:
7352:
7348:
7347:
7343:
7342:
7338:
7337:
7333:
7332:
7328:
7327:
7323:
7322:
7316:
7303:
7302:
7301:
7297:
7296:
7292:
7291:
7287:
7286:
7282:
7281:
7277:
7276:
7272:
7271:
7265:
7250:
7249:
7248:
7244:
7243:
7239:
7238:
7234:
7233:
7229:
7228:
7224:
7223:
7219:
7218:
7212:
7199:
7198:
7197:
7193:
7192:
7188:
7187:
7183:
7182:
7178:
7177:
7173:
7172:
7168:
7167:
7161:
7148:
7147:
7146:
7142:
7141:
7137:
7136:
7132:
7131:
7127:
7126:
7122:
7121:
7117:
7116:
7110:
7097:
7096:
7095:
7091:
7090:
7086:
7085:
7081:
7080:
7076:
7075:
7069:
7056:
7055:
7054:
7050:
7049:
7045:
7044:
7040:
7039:
7035:
7034:
7030:
7029:
7025:
7024:
7018:
7005:
7004:
7003:
6999:
6998:
6994:
6993:
6989:
6988:
6984:
6983:
6979:
6978:
6974:
6973:
6967:
6954:
6953:
6952:
6948:
6947:
6943:
6942:
6938:
6937:
6933:
6932:
6928:
6927:
6923:
6922:
6916:
6909:truncated 5-cell
6901:
6900:
6899:
6895:
6894:
6890:
6889:
6885:
6884:
6880:
6879:
6875:
6874:
6870:
6869:
6863:
6850:
6849:
6848:
6844:
6843:
6839:
6838:
6834:
6833:
6829:
6828:
6824:
6823:
6819:
6818:
6812:
6799:
6798:
6797:
6793:
6792:
6788:
6787:
6783:
6782:
6778:
6777:
6773:
6772:
6768:
6767:
6761:
6748:
6747:
6746:
6742:
6741:
6737:
6736:
6732:
6731:
6727:
6726:
6720:
6707:
6706:
6705:
6701:
6700:
6696:
6695:
6691:
6690:
6686:
6685:
6681:
6680:
6676:
6675:
6669:
6656:
6655:
6654:
6650:
6649:
6645:
6644:
6640:
6639:
6635:
6634:
6630:
6629:
6625:
6624:
6618:
6605:
6604:
6603:
6599:
6598:
6594:
6593:
6589:
6588:
6584:
6583:
6579:
6578:
6574:
6573:
6567:
6560:rectified 5-cell
6552:
6551:
6550:
6546:
6545:
6541:
6540:
6536:
6535:
6531:
6530:
6526:
6525:
6521:
6520:
6514:
6501:
6500:
6499:
6495:
6494:
6490:
6489:
6485:
6484:
6480:
6479:
6475:
6474:
6470:
6469:
6463:
6450:
6449:
6448:
6444:
6443:
6439:
6438:
6434:
6433:
6429:
6428:
6424:
6423:
6419:
6418:
6412:
6399:
6398:
6397:
6393:
6392:
6388:
6387:
6383:
6382:
6378:
6377:
6371:
6358:
6357:
6356:
6352:
6351:
6347:
6346:
6342:
6341:
6337:
6336:
6332:
6331:
6327:
6326:
6320:
6307:
6306:
6305:
6301:
6300:
6296:
6295:
6291:
6290:
6286:
6285:
6281:
6280:
6276:
6275:
6269:
6256:
6255:
6254:
6250:
6249:
6245:
6244:
6240:
6239:
6235:
6234:
6230:
6229:
6225:
6224:
6218:
6205:
6204:
6203:
6199:
6198:
6194:
6193:
6189:
6188:
6184:
6183:
6179:
6178:
6174:
6173:
6165:
6164:
6163:
6159:
6158:
6154:
6153:
6149:
6148:
6144:
6143:
6139:
6138:
6134:
6133:
6125:
6124:
6123:
6119:
6118:
6114:
6113:
6109:
6108:
6104:
6103:
6095:
6094:
6093:
6089:
6088:
6084:
6083:
6079:
6078:
6074:
6073:
6069:
6068:
6064:
6063:
6055:
6054:
6053:
6049:
6048:
6044:
6043:
6039:
6038:
6034:
6033:
6029:
6028:
6024:
6023:
5983:
5971:Coxeter diagrams
5907:Coxeter notation
5754:
5747:
5740:
5739:
5715:
5714:
5713:
5709:
5708:
5704:
5703:
5699:
5698:
5694:
5693:
5689:
5688:
5671:
5669:
5668:
5663:
5661:
5660:
5595:
5594:
5593:
5589:
5588:
5584:
5583:
5579:
5578:
5574:
5573:
5566:
5565:
5564:
5560:
5559:
5555:
5554:
5544:
5542:
5541:
5536:
5534:
5533:
5490:
5489:
5488:
5484:
5483:
5479:
5478:
5474:
5473:
5469:
5468:
5461:
5460:
5459:
5455:
5454:
5450:
5449:
5445:
5444:
5440:
5439:
5432:
5431:
5430:
5426:
5425:
5421:
5420:
5416:
5415:
5411:
5410:
5403:
5402:
5401:
5397:
5396:
5392:
5391:
5387:
5386:
5382:
5381:
5374:
5373:
5372:
5368:
5367:
5363:
5362:
5358:
5357:
5353:
5352:
5345:
5344:
5343:
5339:
5338:
5334:
5333:
5329:
5328:
5324:
5323:
5314:
5313:
5312:
5308:
5307:
5303:
5302:
5298:
5297:
5293:
5292:
5262:
5261:
5235:
5234:
5233:
5229:
5228:
5224:
5223:
5219:
5218:
5214:
5213:
5206:
5205:
5204:
5200:
5199:
5195:
5194:
5177:
5175:
5174:
5169:
5167:
5166:
5098:
5097:
5096:
5092:
5091:
5087:
5086:
5082:
5081:
5077:
5076:
5069:
5068:
5067:
5063:
5062:
5058:
5057:
5040:
5038:
5037:
5032:
5030:
5029:
4959:
4958:
4957:
4953:
4952:
4948:
4947:
4943:
4942:
4938:
4937:
4930:
4929:
4928:
4924:
4923:
4919:
4918:
4901:
4899:
4898:
4893:
4891:
4890:
4827:
4826:
4825:
4821:
4820:
4816:
4815:
4811:
4810:
4806:
4805:
4788:
4786:
4785:
4780:
4778:
4777:
4712:
4711:
4710:
4706:
4705:
4701:
4700:
4696:
4695:
4691:
4690:
4673:
4671:
4670:
4665:
4663:
4662:
4599:
4598:
4597:
4593:
4592:
4588:
4587:
4583:
4582:
4578:
4577:
4560:
4558:
4557:
4552:
4550:
4549:
4483:
4482:
4481:
4477:
4476:
4472:
4471:
4467:
4466:
4462:
4461:
4444:
4442:
4441:
4436:
4434:
4433:
4394:
4393:
4392:
4388:
4387:
4383:
4382:
4378:
4377:
4373:
4372:
4365:
4364:
4363:
4359:
4358:
4354:
4353:
4349:
4348:
4344:
4343:
4336:
4335:
4334:
4330:
4329:
4325:
4324:
4320:
4319:
4315:
4314:
4307:
4306:
4305:
4301:
4300:
4296:
4295:
4291:
4290:
4286:
4285:
4278:
4277:
4276:
4272:
4271:
4267:
4266:
4262:
4261:
4257:
4256:
4249:
4248:
4247:
4243:
4242:
4238:
4237:
4233:
4232:
4228:
4227:
4218:
4217:
4216:
4212:
4211:
4207:
4206:
4202:
4201:
4197:
4196:
4166:
4165:
4115:
4114:
4113:
4109:
4108:
4104:
4103:
4099:
4098:
4094:
4093:
4086:
4079:
4072:
4050:
4049:
4048:
4044:
4043:
4039:
4038:
4034:
4033:
4029:
4028:
4021:
4014:
4007:
3956:
3955:
3935:
3924:
3913:
3902:
3891:
3880:
3869:
3858:
3838:
3827:
3816:
3805:
3794:
3783:
3772:
3761:
3741:
3730:
3719:
3708:
3697:
3686:
3675:
3664:
3644:
3612:
3611:
3569:
3562:
3555:
3548:
3547:
3546:
3542:
3541:
3537:
3536:
3532:
3531:
3527:
3526:
3487:
3480:
3473:
3466:
3465:
3464:
3460:
3459:
3455:
3454:
3450:
3449:
3445:
3444:
3405:
3398:
3391:
3384:
3383:
3382:
3378:
3377:
3373:
3372:
3368:
3367:
3363:
3362:
3317:
3310:
3303:
3296:
3295:
3294:
3290:
3289:
3285:
3284:
3280:
3279:
3275:
3274:
3227:
3220:
3213:
3206:
3205:
3204:
3200:
3199:
3195:
3194:
3190:
3189:
3185:
3184:
3115:
3114:
3113:
3109:
3108:
3104:
3103:
3099:
3098:
3094:
3093:
3072:
3071:
3055:Three dimensions
3037:
3036:
3035:
3031:
3030:
3026:
3025:
2994:
2993:
2992:
2988:
2987:
2983:
2982:
2978:
2977:
2948:
2947:
2946:
2942:
2941:
2937:
2936:
2899:
2898:
2897:
2893:
2892:
2888:
2887:
2848:
2847:
2846:
2842:
2841:
2837:
2836:
2771:
2770:
2755:
2749:
2742:
2735:
2728:
2722:
2715:
2708:
2701:
2689:
2688:
2687:
2683:
2682:
2678:
2677:
2673:
2672:
2668:
2667:
2660:
2659:
2658:
2654:
2653:
2649:
2648:
2644:
2643:
2639:
2638:
2632:
2631:
2630:
2626:
2625:
2621:
2620:
2616:
2615:
2611:
2610:
2603:
2602:
2601:
2597:
2596:
2592:
2591:
2587:
2586:
2582:
2581:
2574:
2573:
2572:
2568:
2567:
2563:
2562:
2558:
2557:
2553:
2552:
2545:
2544:
2543:
2539:
2538:
2534:
2533:
2529:
2528:
2524:
2523:
2517:
2516:
2515:
2511:
2510:
2506:
2505:
2501:
2500:
2496:
2495:
2488:
2487:
2486:
2482:
2481:
2477:
2476:
2472:
2471:
2467:
2466:
2459:
2458:
2457:
2453:
2452:
2448:
2447:
2443:
2442:
2438:
2437:
2430:
2429:
2428:
2424:
2423:
2419:
2418:
2414:
2413:
2409:
2408:
2319:
2318:
2294:
2288:
2281:
2274:
2268:
2261:
2254:
2248:
2241:
2234:
2228:
2221:
2214:
2208:
2196:
2195:
2194:
2190:
2189:
2185:
2184:
2178:
2177:
2176:
2172:
2171:
2167:
2166:
2159:
2158:
2157:
2153:
2152:
2148:
2147:
2140:
2139:
2138:
2134:
2133:
2129:
2128:
2122:
2121:
2120:
2116:
2115:
2111:
2110:
2103:
2102:
2101:
2097:
2096:
2092:
2091:
2084:
2083:
2082:
2078:
2077:
2073:
2072:
2066:
2065:
2064:
2060:
2059:
2055:
2054:
2047:
2046:
2045:
2041:
2040:
2036:
2035:
2028:
2027:
2026:
2022:
2021:
2017:
2016:
2010:
2009:
2008:
2004:
2003:
1999:
1998:
1991:
1990:
1989:
1985:
1984:
1980:
1979:
1972:
1971:
1970:
1966:
1965:
1961:
1960:
1954:
1953:
1952:
1948:
1947:
1943:
1942:
1832:
1825:
1819:
1812:
1805:
1799:
1792:
1785:
1779:
1772:
1765:
1759:
1752:
1740:
1739:
1738:
1734:
1733:
1729:
1728:
1721:
1720:
1719:
1715:
1714:
1710:
1709:
1703:
1702:
1701:
1697:
1696:
1692:
1691:
1684:
1683:
1682:
1678:
1677:
1673:
1672:
1665:
1664:
1663:
1659:
1658:
1654:
1653:
1647:
1646:
1645:
1641:
1640:
1636:
1635:
1628:
1627:
1626:
1622:
1621:
1617:
1616:
1609:
1608:
1607:
1603:
1602:
1598:
1597:
1591:
1590:
1589:
1585:
1584:
1580:
1579:
1572:
1571:
1570:
1566:
1565:
1561:
1560:
1553:
1552:
1551:
1547:
1546:
1542:
1541:
1535:
1534:
1533:
1529:
1528:
1524:
1523:
1516:
1515:
1514:
1510:
1509:
1505:
1504:
1380:
1379:
1367:regular polygons
1339:hyperbolic space
1025:bihexiruncinated
598:
536:quintitruncation
522:quadritruncation
485:
367:trirectification
296:hyperbolic space
254:regular polygons
239:uniform polytope
227:
226:
225:
221:
220:
216:
215:
211:
210:
206:
205:
201:
200:
196:
195:
191:
190:
186:
185:
175:
168:
167:
166:
162:
161:
157:
156:
152:
151:
147:
146:
142:
141:
137:
136:
126:
109:
108:
107:
103:
102:
98:
97:
93:
92:
88:
87:
77:
69:
68:
67:
63:
62:
58:
57:
37:
22:
18:
14636:
14635:
14631:
14630:
14629:
14627:
14626:
14625:
14611:
14610:
14579:
14572:
14565:
14448:
14441:
14434:
14398:
14391:
14384:
14348:
14341:
14175:Regular polygon
14168:
14159:
14152:
14148:
14141:
14137:
14128:
14119:
14112:
14108:
14096:
14090:
14086:
14074:
14056:
14045:
14012:
14011:, Marco Möller
13988:
13976:
13850:
13842:Schläfli symbol
13838:
13818:
13809:
13798:
13791:
13784:
13755:
13745:
13740:
13735:
13730:
13725:
13720:
13718:
13714:
13709:
13704:
13699:
13694:
13689:
13687:
13683:
13678:
13673:
13668:
13663:
13658:
13656:
13652:
13647:
13642:
13637:
13632:
13627:
13625:
13618:
13613:
13608:
13603:
13598:
13593:
13588:
13586:
13582:
13572:
13565:
13560:
13555:
13550:
13545:
13540:
13535:
13533:
13529:
13524:
13519:
13514:
13509:
13504:
13499:
13497:
13493:
13488:
13483:
13478:
13473:
13468:
13466:
13462:
13457:
13452:
13447:
13442:
13437:
13435:
13428:
13423:
13418:
13413:
13408:
13403:
13398:
13393:
13391:
13387:
13377:
13370:
13365:
13360:
13355:
13350:
13345:
13340:
13338:
13334:
13329:
13324:
13319:
13314:
13309:
13304:
13302:
13298:
13293:
13288:
13283:
13278:
13273:
13268:
13266:
13262:
13257:
13252:
13247:
13242:
13237:
13232:
13230:
13223:
13218:
13213:
13208:
13203:
13198:
13193:
13188:
13183:
13181:
13177:
13167:
13160:
13155:
13150:
13145:
13140:
13135:
13130:
13128:
13124:
13119:
13114:
13109:
13104:
13099:
13097:
13093:
13088:
13083:
13078:
13073:
13068:
13066:
13062:
13057:
13052:
13047:
13042:
13037:
13032:
13030:
13023:
13018:
13013:
13008:
13003:
12998:
12993:
12988:
12986:
12982:
12972:
12965:
12960:
12955:
12950:
12945:
12940:
12938:
12934:
12929:
12924:
12919:
12914:
12909:
12907:
12903:
12898:
12893:
12888:
12883:
12878:
12873:
12871:
12867:
12862:
12857:
12852:
12847:
12842:
12837:
12832:
12830:
12823:
12818:
12813:
12808:
12803:
12798:
12793:
12788:
12783:
12781:
12777:
12762:
12757:
12752:
12747:
12742:
12737:
12735:
12731:
12726:
12721:
12716:
12711:
12706:
12704:
12700:
12695:
12690:
12685:
12680:
12675:
12670:
12668:
12664:
12659:
12654:
12649:
12644:
12639:
12634:
12632:
12625:
12620:
12615:
12610:
12605:
12600:
12595:
12590:
12588:
12584:
12572:
12562:
12557:
12552:
12547:
12542:
12537:
12532:
12530:
12529:
12521:
12516:
12511:
12506:
12501:
12496:
12491:
12489:
12488:
12480:
12475:
12470:
12465:
12460:
12455:
12450:
12448:
12447:
12439:
12434:
12429:
12424:
12419:
12414:
12409:
12407:
12406:
12396:
12391:
12386:
12381:
12376:
12371:
12366:
12364:
12358:
12334:
12324:
12319:
12314:
12309:
12304:
12299:
12297:
12293:
12288:
12283:
12278:
12273:
12268:
12266:
12262:
12257:
12252:
12247:
12242:
12237:
12235:
12231:
12226:
12221:
12216:
12211:
12206:
12204:
12197:
12192:
12187:
12182:
12177:
12172:
12167:
12165:
12161:
12147:
12140:
12135:
12130:
12125:
12120:
12115:
12113:
12109:
12104:
12099:
12094:
12089:
12084:
12082:
12078:
12073:
12068:
12063:
12058:
12053:
12051:
12047:
12042:
12037:
12032:
12027:
12022:
12020:
12013:
12008:
12003:
11998:
11993:
11988:
11983:
11981:
11977:
11970:
11966:
11964:
11957:
11950:
11945:
11940:
11935:
11930:
11925:
11923:
11919:
11914:
11909:
11904:
11899:
11894:
11892:
11888:
11883:
11878:
11873:
11868:
11863:
11861:
11857:
11852:
11847:
11842:
11837:
11832:
11830:
11823:
11818:
11813:
11808:
11803:
11798:
11793:
11791:
11787:
11780:
11774:Runcitruncated
11768:
11763:
11758:
11753:
11748:
11743:
11741:
11737:
11732:
11727:
11722:
11717:
11712:
11710:
11706:
11701:
11696:
11691:
11686:
11681:
11679:
11675:
11670:
11665:
11660:
11655:
11650:
11648:
11641:
11636:
11631:
11626:
11621:
11616:
11611:
11609:
11605:
11598:
11593:
11586:
11581:
11576:
11571:
11566:
11561:
11559:
11555:
11550:
11545:
11540:
11535:
11530:
11528:
11524:
11519:
11514:
11509:
11504:
11499:
11497:
11493:
11488:
11483:
11478:
11473:
11468:
11466:
11459:
11454:
11449:
11444:
11439:
11434:
11429:
11427:
11418:Cantitruncated
11412:
11407:
11402:
11397:
11392:
11387:
11385:
11381:
11376:
11371:
11366:
11361:
11356:
11354:
11350:
11345:
11340:
11335:
11330:
11325:
11323:
11319:
11314:
11309:
11304:
11299:
11294:
11292:
11285:
11280:
11275:
11270:
11265:
11260:
11255:
11253:
11249:
11235:
11226:
11221:
11216:
11211:
11206:
11201:
11199:
11195:
11190:
11185:
11180:
11175:
11170:
11168:
11164:
11159:
11154:
11149:
11144:
11139:
11137:
11133:
11128:
11123:
11118:
11113:
11108:
11106:
11099:
11094:
11089:
11084:
11079:
11074:
11069:
11067:
11063:
11056:
11051:
11044:
11039:
11034:
11029:
11024:
11019:
11017:
11013:
11008:
11003:
10998:
10993:
10988:
10986:
10982:
10977:
10972:
10967:
10962:
10957:
10955:
10951:
10946:
10941:
10936:
10931:
10926:
10924:
10917:
10912:
10907:
10902:
10897:
10892:
10887:
10885:
10881:
10864:
10859:
10854:
10849:
10844:
10839:
10837:
10833:
10828:
10823:
10818:
10813:
10808:
10806:
10802:
10797:
10792:
10787:
10782:
10777:
10775:
10771:
10766:
10761:
10756:
10751:
10746:
10744:
10737:
10732:
10727:
10722:
10717:
10712:
10707:
10705:
10701:
10694:
10689:
10682:
10677:
10672:
10667:
10662:
10657:
10655:
10651:
10646:
10641:
10636:
10631:
10626:
10624:
10620:
10615:
10610:
10605:
10600:
10595:
10593:
10589:
10584:
10579:
10574:
10569:
10564:
10562:
10555:
10550:
10545:
10540:
10535:
10530:
10525:
10523:
10506:
10501:
10496:
10491:
10486:
10481:
10479:
10475:
10470:
10465:
10460:
10455:
10450:
10448:
10444:
10439:
10434:
10429:
10424:
10419:
10417:
10413:
10408:
10403:
10398:
10393:
10388:
10386:
10379:
10374:
10369:
10364:
10359:
10354:
10349:
10347:
10343:
10326:
10321:
10316:
10311:
10306:
10301:
10299:
10295:
10290:
10285:
10280:
10275:
10270:
10268:
10264:
10259:
10254:
10249:
10244:
10239:
10237:
10233:
10228:
10223:
10218:
10213:
10208:
10206:
10199:
10194:
10189:
10184:
10179:
10174:
10169:
10167:
10163:
10156:
10147:
10140:
10135:
10130:
10125:
10120:
10115:
10113:
10109:
10104:
10099:
10094:
10089:
10084:
10082:
10078:
10073:
10068:
10063:
10058:
10053:
10051:
10047:
10042:
10037:
10032:
10027:
10022:
10020:
10013:
10008:
10003:
9998:
9993:
9988:
9983:
9981:
9977:
9970:
9965:
9958:
9953:
9948:
9943:
9938:
9933:
9931:
9927:
9922:
9917:
9912:
9907:
9902:
9900:
9896:
9891:
9886:
9881:
9876:
9871:
9869:
9865:
9860:
9855:
9850:
9845:
9840:
9838:
9831:
9826:
9821:
9816:
9811:
9806:
9801:
9799:
9795:
9776:
9771:
9766:
9761:
9756:
9751:
9749:
9745:
9740:
9735:
9730:
9725:
9720:
9718:
9714:
9709:
9704:
9699:
9694:
9689:
9687:
9683:
9678:
9673:
9668:
9663:
9658:
9656:
9649:
9644:
9639:
9634:
9629:
9624:
9619:
9617:
9613:
9593:
9588:
9583:
9578:
9573:
9568:
9563:
9561:
9560:
9552:
9547:
9542:
9537:
9532:
9527:
9522:
9520:
9519:
9511:
9506:
9501:
9496:
9491:
9486:
9481:
9479:
9478:
9470:
9465:
9460:
9455:
9450:
9445:
9440:
9438:
9437:
9427:
9422:
9417:
9412:
9407:
9402:
9397:
9395:
9389:
9357:
9355:Truncated forms
9343:
9338:
9333:
9328:
9323:
9318:
9313:
9308:
9306:
9305:
9299:
9292:
9287:
9282:
9277:
9272:
9267:
9265:
9264:
9258:
9249:
9244:
9239:
9234:
9229:
9224:
9219:
9214:
9212:
9211:
9205:
9194:
9190:
9185:
9180:
9175:
9170:
9165:
9160:
9155:
9153:
9152:
9146:
9142:
9135:
9131:
9126:
9121:
9116:
9111:
9106:
9101:
9096:
9094:
9093:
9087:
9078:
9074:
9069:
9064:
9059:
9054:
9049:
9044:
9039:
9037:
9036:
9030:
9023:
9019:
9014:
9009:
9004:
8999:
8994:
8989:
8984:
8982:
8981:
8975:
8968:
8964:
8959:
8954:
8949:
8944:
8939:
8934:
8929:
8927:
8926:
8920:
8911:
8907:
8902:
8897:
8892:
8887:
8882:
8877:
8872:
8870:
8869:
8863:
8856:
8852:
8847:
8842:
8837:
8832:
8827:
8822:
8817:
8815:
8814:
8808:
8801:
8797:
8792:
8787:
8782:
8777:
8772:
8767:
8762:
8760:
8759:
8753:
8746:
8741:
8736:
8731:
8726:
8721:
8719:
8718:
8712:
8705:
8701:
8696:
8691:
8686:
8681:
8676:
8671:
8666:
8664:
8663:
8657:
8650:
8646:
8641:
8636:
8631:
8626:
8621:
8616:
8611:
8609:
8608:
8602:
8595:
8591:
8586:
8581:
8576:
8571:
8566:
8561:
8556:
8554:
8553:
8547:
8538:
8533:
8528:
8523:
8518:
8513:
8508:
8503:
8501:
8500:
8494:
8487:
8482:
8477:
8472:
8467:
8462:
8457:
8452:
8450:
8449:
8443:
8436:
8431:
8426:
8421:
8416:
8411:
8406:
8401:
8399:
8398:
8392:
8385:
8380:
8375:
8370:
8365:
8360:
8358:
8357:
8351:
8344:
8339:
8334:
8329:
8324:
8319:
8314:
8309:
8307:
8306:
8300:
8293:
8288:
8283:
8278:
8273:
8268:
8263:
8258:
8256:
8255:
8249:
8242:
8237:
8232:
8227:
8222:
8217:
8212:
8207:
8205:
8204:
8198:
8189:
8184:
8179:
8174:
8169:
8164:
8159:
8154:
8152:
8151:
8145:
8141:
8134:
8129:
8124:
8119:
8114:
8109:
8104:
8099:
8097:
8096:
8090:
8083:
8078:
8073:
8068:
8063:
8058:
8056:
8055:
8049:
8042:
8037:
8032:
8027:
8022:
8017:
8012:
8007:
8005:
8004:
7998:
7991:
7986:
7981:
7976:
7971:
7966:
7961:
7956:
7954:
7953:
7947:
7940:
7936:
7931:
7926:
7921:
7916:
7911:
7906:
7901:
7899:
7898:
7892:
7883:
7879:
7874:
7869:
7864:
7859:
7854:
7849:
7844:
7842:
7841:
7835:
7831:
7824:
7820:
7815:
7810:
7805:
7800:
7795:
7790:
7785:
7783:
7782:
7776:
7767:
7763:
7758:
7753:
7748:
7743:
7738:
7733:
7728:
7726:
7725:
7719:
7712:
7708:
7703:
7698:
7693:
7688:
7683:
7678:
7673:
7671:
7670:
7664:
7657:
7653:
7648:
7643:
7638:
7633:
7628:
7623:
7618:
7616:
7615:
7609:
7600:
7595:
7590:
7585:
7580:
7575:
7570:
7565:
7563:
7562:
7556:
7549:
7544:
7539:
7534:
7529:
7524:
7519:
7514:
7512:
7511:
7505:
7498:
7493:
7488:
7483:
7478:
7473:
7468:
7463:
7461:
7460:
7454:
7447:
7442:
7437:
7432:
7427:
7422:
7420:
7419:
7413:
7406:
7401:
7396:
7391:
7386:
7381:
7376:
7371:
7369:
7368:
7362:
7355:
7350:
7345:
7340:
7335:
7330:
7325:
7320:
7318:
7317:
7311:
7304:
7299:
7294:
7289:
7284:
7279:
7274:
7269:
7267:
7266:
7260:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7216:
7214:
7213:
7207:
7200:
7195:
7190:
7185:
7180:
7175:
7170:
7165:
7163:
7162:
7156:
7149:
7144:
7139:
7134:
7129:
7124:
7119:
7114:
7112:
7111:
7105:
7098:
7093:
7088:
7083:
7078:
7073:
7071:
7070:
7064:
7057:
7052:
7047:
7042:
7037:
7032:
7027:
7022:
7020:
7019:
7013:
7006:
7001:
6996:
6991:
6986:
6981:
6976:
6971:
6969:
6968:
6962:
6955:
6950:
6945:
6940:
6935:
6930:
6925:
6920:
6918:
6917:
6911:
6902:
6897:
6892:
6887:
6882:
6877:
6872:
6867:
6865:
6864:
6858:
6851:
6846:
6841:
6836:
6831:
6826:
6821:
6816:
6814:
6813:
6807:
6800:
6795:
6790:
6785:
6780:
6775:
6770:
6765:
6763:
6762:
6756:
6749:
6744:
6739:
6734:
6729:
6724:
6722:
6721:
6715:
6708:
6703:
6698:
6693:
6688:
6683:
6678:
6673:
6671:
6670:
6664:
6657:
6652:
6647:
6642:
6637:
6632:
6627:
6622:
6620:
6619:
6613:
6606:
6601:
6596:
6591:
6586:
6581:
6576:
6571:
6569:
6568:
6562:
6553:
6548:
6543:
6538:
6533:
6528:
6523:
6518:
6516:
6515:
6509:
6502:
6497:
6492:
6487:
6482:
6477:
6472:
6467:
6465:
6464:
6458:
6451:
6446:
6441:
6436:
6431:
6426:
6421:
6416:
6414:
6413:
6407:
6400:
6395:
6390:
6385:
6380:
6375:
6373:
6372:
6366:
6359:
6354:
6349:
6344:
6339:
6334:
6329:
6324:
6322:
6321:
6315:
6308:
6303:
6298:
6293:
6288:
6283:
6278:
6273:
6271:
6270:
6264:
6257:
6252:
6247:
6242:
6237:
6232:
6227:
6222:
6220:
6219:
6213:
6201:
6196:
6191:
6186:
6181:
6176:
6171:
6169:
6168:
6161:
6156:
6151:
6146:
6141:
6136:
6131:
6129:
6128:
6121:
6116:
6111:
6106:
6101:
6099:
6098:
6091:
6086:
6081:
6076:
6071:
6066:
6061:
6059:
6058:
6051:
6046:
6041:
6036:
6031:
6026:
6021:
6019:
6018:
6013:
6007:
6001:
5995:
5989:
5978:grand antiprism
5967:
5918:cubic honeycomb
5824:
5821:
5818:cubic honeycomb
5790:grand antiprism
5771:
5765:
5763:Four dimensions
5755:
5711:
5706:
5701:
5696:
5691:
5686:
5684:
5680:
5655:
5654:
5638:
5637:
5632:
5629:
5628:
5605:
5591:
5586:
5581:
5576:
5571:
5569:
5562:
5557:
5552:
5550:
5528:
5527:
5521:
5520:
5510:
5509:
5504:
5501:
5500:
5486:
5481:
5476:
5471:
5466:
5464:
5457:
5452:
5447:
5442:
5437:
5435:
5428:
5423:
5418:
5413:
5408:
5406:
5399:
5394:
5389:
5384:
5379:
5377:
5370:
5365:
5360:
5355:
5350:
5348:
5341:
5336:
5331:
5326:
5321:
5319:
5310:
5305:
5300:
5295:
5290:
5288:
5282:
5275:
5269:
5231:
5226:
5221:
5216:
5211:
5209:
5202:
5197:
5192:
5190:
5186:
5161:
5160:
5154:
5153:
5143:
5142:
5137:
5134:
5133:
5125:
5094:
5089:
5084:
5079:
5074:
5072:
5065:
5060:
5055:
5053:
5049:
5024:
5023:
5017:
5016:
5006:
5005:
5000:
4997:
4996:
4988:
4955:
4950:
4945:
4940:
4935:
4933:
4926:
4921:
4916:
4914:
4910:
4885:
4884:
4878:
4877:
4867:
4866:
4864:
4861:
4860:
4823:
4818:
4813:
4808:
4803:
4801:
4797:
4772:
4771:
4755:
4754:
4749:
4746:
4745:
4741:
4708:
4703:
4698:
4693:
4688:
4686:
4682:
4657:
4656:
4640:
4639:
4634:
4631:
4630:
4595:
4590:
4585:
4580:
4575:
4573:
4569:
4544:
4543:
4527:
4526:
4524:
4521:
4520:
4510:
4479:
4474:
4469:
4464:
4459:
4457:
4453:
4428:
4427:
4411:
4410:
4408:
4405:
4404:
4390:
4385:
4380:
4375:
4370:
4368:
4361:
4356:
4351:
4346:
4341:
4339:
4332:
4327:
4322:
4317:
4312:
4310:
4303:
4298:
4293:
4288:
4283:
4281:
4274:
4269:
4264:
4259:
4254:
4252:
4245:
4240:
4235:
4230:
4225:
4223:
4214:
4209:
4204:
4199:
4194:
4192:
4186:
4179:
4173:
4162:
4153:Coxeter diagram
4133:
4116:
4111:
4106:
4101:
4096:
4091:
4089:
4060:
4051:
4046:
4041:
4036:
4031:
4026:
4024:
3995:
3986:
3981:
3973:
3936:
3925:
3914:
3903:
3892:
3881:
3870:
3859:
3850:
3839:
3828:
3817:
3806:
3795:
3784:
3773:
3762:
3753:
3742:
3731:
3720:
3709:
3698:
3687:
3676:
3665:
3656:
3642:
3640:
3632:
3627:
3588:
3581:
3573:
3544:
3539:
3534:
3529:
3524:
3522:
3506:
3499:
3491:
3462:
3457:
3452:
3447:
3442:
3440:
3424:
3417:
3409:
3380:
3375:
3370:
3365:
3360:
3358:
3347:
3336:
3329:
3321:
3292:
3287:
3282:
3277:
3272:
3270:
3263:
3257:
3246:
3239:
3231:
3202:
3197:
3192:
3187:
3182:
3180:
3173:
3167:
3148:
3136:
3129:
3124:
3119:
3111:
3106:
3101:
3096:
3091:
3089:
3088:
3081:
3067:Platonic solids
3063:
3057:
3047:
3033:
3028:
3023:
3021:
3004:
2990:
2985:
2980:
2975:
2973:
2958:
2944:
2939:
2934:
2932:
2925:
2908:
2895:
2890:
2885:
2883:
2876:
2866:
2857:
2844:
2839:
2834:
2832:
2825:
2791:
2785:
2780:
2778:
2766:Schläfli symbol
2750:
2723:
2685:
2680:
2675:
2670:
2665:
2663:
2656:
2651:
2646:
2641:
2636:
2634:
2633:
2628:
2623:
2618:
2613:
2608:
2606:
2599:
2594:
2589:
2584:
2579:
2577:
2570:
2565:
2560:
2555:
2550:
2548:
2541:
2536:
2531:
2526:
2521:
2519:
2518:
2513:
2508:
2503:
2498:
2493:
2491:
2484:
2479:
2474:
2469:
2464:
2462:
2455:
2450:
2445:
2440:
2435:
2433:
2426:
2421:
2416:
2411:
2406:
2404:
2399:
2384:
2373:
2306:rational number
2289:
2269:
2249:
2229:
2209:
2192:
2187:
2182:
2180:
2179:
2174:
2169:
2164:
2162:
2155:
2150:
2145:
2143:
2136:
2131:
2126:
2124:
2123:
2118:
2113:
2108:
2106:
2099:
2094:
2089:
2087:
2080:
2075:
2070:
2068:
2067:
2062:
2057:
2052:
2050:
2043:
2038:
2033:
2031:
2024:
2019:
2014:
2012:
2011:
2006:
2001:
1996:
1994:
1987:
1982:
1977:
1975:
1968:
1963:
1958:
1956:
1955:
1950:
1945:
1940:
1938:
1933:
1925:
1917:
1909:
1901:
1893:
1820:
1800:
1780:
1760:
1736:
1731:
1726:
1724:
1717:
1712:
1707:
1705:
1704:
1699:
1694:
1689:
1687:
1680:
1675:
1670:
1668:
1661:
1656:
1651:
1649:
1648:
1643:
1638:
1633:
1631:
1624:
1619:
1614:
1612:
1605:
1600:
1595:
1593:
1592:
1587:
1582:
1577:
1575:
1568:
1563:
1558:
1556:
1549:
1544:
1539:
1537:
1536:
1531:
1526:
1521:
1519:
1512:
1507:
1502:
1500:
1495:
1484:
1476:
1468:
1460:
1406:
1400:
1389:
1363:
1356:
1351:
1319:
1276:
1243:Coxeter diagram
1235:
1173:
1150:runcitruncation
1134:
1113:
1102:
1096:triheptellation
1092:
1064:
1047:
1032:
1006:
993:
978:
945:
934:
928:tripentellation
924:
896:
879:
865:
840:
827:
813:
783:
772:
762:
730:
713:
699:
685:
660:Cantitruncation
652:
639:
625:
619:tricantellation
611:
557:
542:
528:
514:
500:
478:'to cut off'.)
454:
434:cuts vertices,
424:
409:
405:
401:
396:
382:Schläfli symbol
363:birectification
339:
327:Johannes Kepler
320:grand antiprism
312:Coxeter diagram
304:
282:(2-dimensional
223:
218:
213:
208:
203:
198:
193:
188:
183:
181:
176:
164:
159:
154:
149:
144:
139:
134:
132:
127:
105:
100:
95:
90:
85:
83:
78:
65:
60:
55:
53:
51:Coxeter diagram
38:
17:
12:
11:
5:
14634:
14624:
14623:
14607:
14606:
14591:
14590:
14581:
14577:
14570:
14563:
14559:
14550:
14533:
14524:
14513:
14512:
14510:
14508:
14503:
14494:
14489:
14483:
14482:
14480:
14478:
14473:
14464:
14459:
14453:
14452:
14450:
14446:
14439:
14432:
14428:
14423:
14414:
14409:
14403:
14402:
14400:
14396:
14389:
14382:
14378:
14373:
14364:
14359:
14353:
14352:
14350:
14346:
14339:
14335:
14330:
14321:
14316:
14310:
14309:
14307:
14305:
14300:
14291:
14286:
14280:
14279:
14270:
14265:
14260:
14251:
14246:
14240:
14239:
14230:
14228:
14223:
14214:
14209:
14203:
14202:
14197:
14192:
14187:
14182:
14177:
14171:
14170:
14166:
14162:
14157:
14146:
14135:
14126:
14117:
14110:
14104:
14094:
14088:
14082:
14076:
14070:
14064:
14058:
14057:
14046:
14044:
14043:
14036:
14029:
14021:
14016:
14015:
14006:
13987:
13986:External links
13984:
13983:
13982:
13971:Marco Möller,
13969:
13968:
13967:
13960:
13953:
13931:
13930:
13929:
13922:
13905:H.S.M. Coxeter
13902:
13892:A. Boole Stott
13889:
13888:
13887:
13871:Norman Johnson
13868:
13849:
13846:
13845:
13844:
13837:
13834:
13817:
13814:
13807:
13796:
13789:
13782:
13767:cross-polytope
13754:
13751:
13748:
13747:
13716:
13685:
13654:
13623:
13584:
13580:
13577:
13574:
13568:
13567:
13531:
13495:
13464:
13433:
13389:
13385:
13382:
13379:
13376:Snub rectified
13373:
13372:
13336:
13300:
13264:
13228:
13179:
13175:
13172:
13169:
13163:
13162:
13126:
13095:
13064:
13028:
12984:
12980:
12977:
12974:
12968:
12967:
12936:
12905:
12869:
12828:
12779:
12775:
12772:
12769:
12765:
12764:
12733:
12702:
12666:
12630:
12586:
12582:
12579:
12576:
12568:
12567:
12526:
12485:
12444:
12402:
12401:
12361:
12354:
12351:
12333:
12330:
12327:
12326:
12295:
12264:
12233:
12202:
12163:
12159:
12156:
12153:
12143:
12142:
12111:
12080:
12049:
12018:
11979:
11975:
11972:
11968:
11962:
11959:
11953:
11952:
11921:
11890:
11859:
11828:
11789:
11785:
11782:
11778:
11775:
11771:
11770:
11739:
11708:
11677:
11646:
11607:
11603:
11600:
11595:
11589:
11588:
11557:
11526:
11495:
11464:
11425:
11422:
11419:
11415:
11414:
11383:
11352:
11321:
11290:
11251:
11247:
11244:
11241:
11229:
11228:
11197:
11166:
11135:
11104:
11065:
11061:
11058:
11053:
11047:
11046:
11015:
10984:
10953:
10922:
10883:
10879:
10876:
10873:
10867:
10866:
10835:
10804:
10773:
10742:
10703:
10699:
10696:
10691:
10685:
10684:
10653:
10622:
10591:
10560:
10521:
10518:
10515:
10509:
10508:
10477:
10446:
10415:
10384:
10345:
10341:
10338:
10335:
10329:
10328:
10297:
10266:
10235:
10204:
10165:
10161:
10158:
10153:
10143:
10142:
10111:
10080:
10049:
10018:
9979:
9975:
9972:
9967:
9961:
9960:
9929:
9898:
9867:
9836:
9797:
9793:
9790:
9787:
9779:
9778:
9747:
9716:
9685:
9654:
9615:
9611:
9608:
9605:
9599:
9598:
9557:
9516:
9475:
9433:
9432:
9392:
9385:
9382:
9376:
9375:
9372:
9369:
9356:
9353:
9350:
9349:
9347:
9345:
9294:
9253:
9251:
9200:
9197:
9196:
9192:
9137:
9133:
9082:
9080:
9076:
9025:
9021:
8970:
8966:
8914:
8913:
8909:
8858:
8854:
8803:
8799:
8748:
8707:
8703:
8652:
8648:
8597:
8593:
8541:
8540:
8489:
8438:
8387:
8346:
8295:
8244:
8192:
8191:
8136:
8085:
8044:
7993:
7942:
7938:
7886:
7885:
7881:
7826:
7822:
7771:
7769:
7765:
7714:
7710:
7659:
7655:
7603:
7602:
7551:
7500:
7449:
7408:
7357:
7306:
7254:
7253:
7202:
7151:
7100:
7059:
7008:
6957:
6905:
6904:
6853:
6802:
6751:
6710:
6659:
6608:
6556:
6555:
6504:
6453:
6402:
6361:
6310:
6259:
6207:
6206:
6166:
6126:
6096:
6056:
6015:
6014:
6011:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5975:non-Wythoffian
5966:
5963:
5959:
5958:
5952:
5946:
5940:
5903:
5902:
5899:
5888:
5881:
5870:
5787:non-Wythoffian
5767:Main article:
5764:
5761:
5758:
5757:
5748:
5736:
5735:
5733:
5730:
5727:
5724:
5721:
5718:
5716:
5682:
5678:
5675:
5672:
5659:
5653:
5650:
5647:
5644:
5643:
5641:
5636:
5626:
5620:
5619:
5616:
5613:
5610:
5607:
5602:
5599:
5596:
5567:
5548:
5545:
5532:
5526:
5523:
5522:
5519:
5516:
5515:
5513:
5508:
5498:
5496:Snub rectified
5492:
5491:
5462:
5433:
5404:
5375:
5346:
5316:
5315:
5285:
5278:
5271:
5266:
5258:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5236:
5207:
5188:
5184:
5181:
5178:
5165:
5159:
5156:
5155:
5152:
5149:
5148:
5146:
5141:
5131:
5124:Cantitruncated
5121:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5070:
5051:
5047:
5044:
5041:
5028:
5022:
5019:
5018:
5015:
5012:
5011:
5009:
5004:
4994:
4982:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4931:
4912:
4908:
4905:
4902:
4889:
4883:
4880:
4879:
4876:
4873:
4872:
4870:
4858:
4850:
4849:
4846:
4843:
4840:
4837:
4834:
4831:
4828:
4799:
4795:
4792:
4789:
4776:
4770:
4767:
4764:
4761:
4760:
4758:
4753:
4743:
4735:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4684:
4680:
4677:
4674:
4661:
4655:
4652:
4649:
4646:
4645:
4643:
4638:
4628:
4622:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4571:
4567:
4564:
4561:
4548:
4542:
4539:
4536:
4533:
4532:
4530:
4518:
4506:
4505:
4502:
4499:
4496:
4493:
4490:
4487:
4484:
4455:
4451:
4448:
4445:
4432:
4426:
4423:
4420:
4417:
4416:
4414:
4402:
4396:
4395:
4366:
4337:
4308:
4279:
4250:
4220:
4219:
4189:
4182:
4175:
4170:
4160:
4137:Wythoff symbol
4132:
4129:
4119:
4118:
4087:
4080:
4073:
4066:
4061:
4058:
4054:
4053:
4022:
4015:
4008:
4001:
3996:
3993:
3989:
3988:
3976:
3969:
3966:
3963:
3960:
3941:
3940:
3929:
3918:
3907:
3896:
3885:
3874:
3863:
3852:
3844:
3843:
3832:
3821:
3810:
3799:
3788:
3777:
3766:
3755:
3747:
3746:
3735:
3724:
3713:
3702:
3691:
3680:
3669:
3658:
3650:
3649:
3646:
3643:Cantitruncated
3637:
3634:
3629:
3624:
3621:
3618:
3615:
3593:
3592:
3589:
3586:
3583:
3578:
3575:
3570:
3563:
3556:
3549:
3520:
3517:
3511:
3510:
3507:
3504:
3501:
3496:
3493:
3488:
3481:
3474:
3467:
3438:
3435:
3429:
3428:
3425:
3422:
3419:
3414:
3411:
3406:
3399:
3392:
3385:
3356:
3353:
3341:
3340:
3337:
3334:
3331:
3326:
3323:
3318:
3311:
3304:
3297:
3268:
3265:
3251:
3250:
3247:
3244:
3241:
3236:
3233:
3228:
3221:
3214:
3207:
3178:
3175:
3161:
3160:
3155:
3150:
3143:
3138:
3131:
3126:
3121:
3120:(transparent)
3116:
3083:
3076:
3059:Main article:
3056:
3053:
3050:
3049:
3044:
3041:
3038:
3019:
3016:
3013:
3007:
3006:
3001:
2998:
2995:
2971:
2968:
2965:
2961:
2960:
2955:
2952:
2949:
2930:
2927:
2923:
2920:
2917:
2911:
2910:
2906:
2903:
2900:
2881:
2878:
2874:
2871:
2868:
2860:
2859:
2855:
2852:
2849:
2830:
2827:
2823:
2820:
2817:
2811:
2810:
2807:
2803:
2802:
2797:
2794:
2787:
2782:
2775:
2760:
2759:
2756:
2743:
2736:
2729:
2716:
2709:
2702:
2695:
2691:
2690:
2661:
2604:
2575:
2546:
2489:
2460:
2431:
2402:
2394:
2393:
2386:
2381:
2378:
2375:
2370:
2367:
2364:
2361:
2355:
2354:
2348:
2343:
2338:
2333:
2328:
2323:
2304:(one for each
2296:
2295:
2282:
2275:
2262:
2255:
2242:
2235:
2222:
2215:
2202:
2198:
2197:
2160:
2141:
2104:
2085:
2048:
2029:
1992:
1973:
1936:
1928:
1927:
1922:
1919:
1914:
1911:
1906:
1903:
1898:
1895:
1890:
1884:
1883:
1878:
1873:
1868:
1863:
1858:
1853:
1848:
1843:
1838:
1834:
1833:
1826:
1813:
1806:
1793:
1786:
1773:
1766:
1753:
1746:
1742:
1741:
1722:
1685:
1666:
1629:
1610:
1573:
1554:
1517:
1498:
1490:
1489:
1486:
1481:
1478:
1473:
1470:
1465:
1462:
1457:
1454:
1448:
1447:
1442:
1437:
1432:
1427:
1422:
1417:
1412:
1395:
1384:
1362:
1361:Two dimensions
1359:
1354:
1350:
1347:
1318:
1315:
1287:vertex figures
1275:
1272:
1234:
1231:
1172:
1169:
1165:omnitruncation
1146:
1145:
1132:
1128:
1111:
1107:
1106:
1105:
1100:
1090:
1086:biheptellation
1062:
1058:
1057:
1056:
1055:
1054:
1045:
1030:
1014:Hexiruncinated
1004:
1000:
991:
976:
943:
939:
938:
937:
932:
922:
918:bipentellation
894:
890:
889:
888:
887:
886:
877:
863:
838:
834:
825:
821:tristerication
811:
781:
777:
776:
775:
770:
766:triruncination
760:
728:
724:
723:
722:
721:
720:
711:
697:
683:
650:
646:
637:
623:
609:
605:bicantellation
600:
599:
555:
551:
550:
549:
540:
526:
512:
498:
487:
486:
452:
423:
420:
419:
418:
407:
403:
399:
394:
338:
335:
303:
300:
272:vertex figures
229:
228:
169:
119:
118:
115:
111:
110:
71:
30:
29:
26:
15:
9:
6:
4:
3:
2:
14633:
14622:
14619:
14618:
14616:
14605:
14601:
14597:
14592:
14589:
14585:
14582:
14580:
14573:
14566:
14560:
14558:
14554:
14551:
14549:
14545:
14541:
14537:
14534:
14532:
14528:
14525:
14523:
14519:
14515:
14514:
14511:
14509:
14507:
14504:
14502:
14498:
14495:
14493:
14490:
14488:
14485:
14484:
14481:
14479:
14477:
14474:
14472:
14468:
14465:
14463:
14460:
14458:
14455:
14454:
14451:
14449:
14442:
14435:
14429:
14427:
14424:
14422:
14418:
14415:
14413:
14410:
14408:
14405:
14404:
14401:
14399:
14392:
14385:
14379:
14377:
14374:
14372:
14368:
14365:
14363:
14360:
14358:
14355:
14354:
14351:
14349:
14342:
14336:
14334:
14331:
14329:
14325:
14322:
14320:
14317:
14315:
14312:
14311:
14308:
14306:
14304:
14301:
14299:
14295:
14292:
14290:
14287:
14285:
14282:
14281:
14278:
14274:
14271:
14269:
14266:
14264:
14263:Demitesseract
14261:
14259:
14255:
14252:
14250:
14247:
14245:
14242:
14241:
14238:
14234:
14231:
14229:
14227:
14224:
14222:
14218:
14215:
14213:
14210:
14208:
14205:
14204:
14201:
14198:
14196:
14193:
14191:
14188:
14186:
14183:
14181:
14178:
14176:
14173:
14172:
14169:
14163:
14160:
14156:
14149:
14145:
14138:
14134:
14129:
14125:
14120:
14116:
14111:
14109:
14107:
14103:
14093:
14089:
14087:
14085:
14081:
14077:
14075:
14073:
14069:
14065:
14063:
14060:
14059:
14054:
14050:
14042:
14037:
14035:
14030:
14028:
14023:
14022:
14019:
14010:
14007:
14003:
13999:
13995:
13990:
13989:
13981:
13974:
13970:
13965:
13961:
13958:
13954:
13951:
13947:
13946:
13945:
13941:
13937:
13936:
13932:
13927:
13923:
13921:
13917:
13913:
13909:
13908:
13906:
13903:
13901:
13897:
13893:
13890:
13886:
13882:
13878:
13877:
13875:
13872:
13869:
13866:
13862:
13858:
13855:
13852:
13851:
13843:
13840:
13839:
13833:
13829:
13827:
13823:
13813:
13811:
13803:
13799:
13792:
13785:
13778:
13775:
13770:
13768:
13764:
13760:
13717:
13686:
13655:
13624:
13585:
13578:
13575:
13570:
13532:
13496:
13465:
13434:
13390:
13383:
13380:
13375:
13337:
13301:
13265:
13229:
13180:
13173:
13170:
13165:
13127:
13096:
13065:
13029:
12985:
12978:
12975:
12970:
12937:
12906:
12870:
12829:
12780:
12773:
12770:
12767:
12734:
12703:
12667:
12631:
12587:
12580:
12577:
12575:
12570:
12527:
12486:
12445:
12404:
12360:
12348:
12345:
12343:
12339:
12296:
12265:
12234:
12203:
12164:
12157:
12154:
12151:
12150:omnitruncated
12145:
12112:
12081:
12050:
12019:
11980:
11973:
11960:
11955:
11922:
11891:
11860:
11829:
11790:
11783:
11776:
11773:
11740:
11709:
11678:
11647:
11608:
11601:
11596:
11591:
11558:
11527:
11496:
11465:
11426:
11423:
11420:
11417:
11384:
11353:
11322:
11291:
11252:
11245:
11242:
11239:
11234:
11231:
11198:
11167:
11136:
11105:
11066:
11059:
11054:
11050:Bicantellated
11049:
11016:
10985:
10954:
10923:
10884:
10877:
10874:
10872:
10869:
10836:
10805:
10774:
10743:
10704:
10697:
10692:
10687:
10654:
10623:
10592:
10561:
10522:
10519:
10516:
10514:
10511:
10478:
10447:
10416:
10385:
10346:
10339:
10336:
10334:
10331:
10298:
10267:
10236:
10205:
10166:
10159:
10154:
10151:
10145:
10112:
10081:
10050:
10019:
9980:
9973:
9968:
9963:
9930:
9899:
9868:
9837:
9798:
9791:
9788:
9786:
9785:
9781:
9748:
9717:
9686:
9655:
9616:
9609:
9606:
9604:
9601:
9558:
9517:
9476:
9435:
9391:
9379:
9373:
9370:
9367:
9363:
9362:
9361:
9348:
9346:
9303:
9298:
9295:
9262:
9257:
9254:
9252:
9209:
9204:
9201:
9199:
9150:
9145:
9141:
9091:
9086:
9083:
9081:
9034:
9029:
9026:
8979:
8974:
8971:
8924:
8919:
8916:
8867:
8862:
8859:
8812:
8807:
8804:
8757:
8752:
8749:
8716:
8711:
8708:
8661:
8656:
8653:
8606:
8601:
8598:
8551:
8546:
8543:
8498:
8493:
8490:
8447:
8442:
8439:
8396:
8391:
8388:
8355:
8350:
8347:
8304:
8299:
8296:
8253:
8248:
8245:
8202:
8197:
8194:
8149:
8144:
8140:
8094:
8089:
8086:
8053:
8048:
8045:
8002:
7997:
7994:
7951:
7946:
7943:
7896:
7891:
7888:
7839:
7834:
7830:
7780:
7775:
7772:
7770:
7723:
7718:
7715:
7668:
7663:
7660:
7613:
7608:
7605:
7560:
7555:
7552:
7509:
7504:
7501:
7458:
7453:
7450:
7417:
7412:
7409:
7366:
7361:
7358:
7315:
7310:
7307:
7264:
7259:
7256:
7211:
7206:
7203:
7160:
7155:
7152:
7109:
7104:
7101:
7068:
7063:
7060:
7017:
7012:
7009:
6966:
6961:
6958:
6915:
6910:
6907:
6862:
6857:
6854:
6811:
6806:
6803:
6760:
6755:
6752:
6719:
6714:
6711:
6668:
6663:
6660:
6617:
6612:
6609:
6566:
6561:
6558:
6513:
6508:
6505:
6462:
6457:
6454:
6411:
6406:
6403:
6370:
6365:
6364:demitesseract
6362:
6319:
6314:
6311:
6268:
6263:
6260:
6217:
6212:
6209:
6127:
6097:
6017:
6016:
6003:
5997:
5985:
5984:
5981:
5979:
5976:
5972:
5962:
5956:
5953:
5950:
5947:
5944:
5941:
5938:
5935:
5934:
5933:
5930:
5925:
5921:
5919:
5915:
5912:Eight of the
5910:
5908:
5900:
5897:
5893:
5889:
5886:
5882:
5879:
5875:
5871:
5868:
5864:
5863:
5862:
5860:
5855:
5853:
5849:
5845:
5841:
5832:
5826:
5819:
5814:
5810:
5807:
5801:
5799:
5795:
5791:
5788:
5784:
5780:
5776:
5770:
5753:
5749:
5746:
5742:
5741:
5734:
5731:
5728:
5725:
5722:
5719:
5717:
5676:
5673:
5657:
5651:
5648:
5645:
5639:
5634:
5627:
5625:
5622:
5617:
5614:
5611:
5608:
5603:
5600:
5597:
5568:
5549:
5530:
5524:
5517:
5511:
5506:
5499:
5497:
5494:
5463:
5434:
5405:
5376:
5347:
5318:
5284:
5277:
5263:
5255:
5252:
5249:
5246:
5243:
5240:
5237:
5208:
5189:
5182:
5179:
5163:
5157:
5150:
5144:
5139:
5132:
5129:
5128:Omnitruncated
5123:
5118:
5115:
5112:
5109:
5106:
5103:
5100:
5071:
5052:
5045:
5042:
5026:
5020:
5013:
5007:
5002:
4995:
4992:
4987:
4984:
4979:
4976:
4973:
4970:
4967:
4964:
4961:
4932:
4913:
4906:
4903:
4887:
4881:
4874:
4868:
4859:
4857:
4856:
4852:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4793:
4790:
4774:
4768:
4765:
4762:
4756:
4751:
4744:
4740:
4737:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4678:
4675:
4659:
4653:
4650:
4647:
4641:
4636:
4629:
4627:
4624:
4619:
4616:
4613:
4610:
4607:
4604:
4601:
4565:
4562:
4546:
4540:
4537:
4534:
4528:
4519:
4516:
4515:
4508:
4503:
4500:
4497:
4494:
4491:
4488:
4485:
4449:
4446:
4430:
4424:
4421:
4418:
4412:
4403:
4401:
4398:
4367:
4338:
4309:
4280:
4251:
4222:
4188:
4181:
4167:
4164:
4158:
4154:
4150:
4146:
4142:
4138:
4131:Constructions
4128:
4126:
4085:
4081:
4078:
4074:
4071:
4067:
4065:
4062:
4056:
4020:
4016:
4013:
4009:
4006:
4002:
4000:
3997:
3991:
3985:
3980:
3977:
3975:
3970:
3967:
3964:
3961:
3958:
3957:
3954:
3952:
3948:
3939:
3934:
3930:
3928:
3923:
3919:
3917:
3912:
3908:
3906:
3901:
3897:
3895:
3890:
3886:
3884:
3879:
3875:
3873:
3868:
3864:
3862:
3857:
3853:
3849:
3846:
3845:
3842:
3837:
3833:
3831:
3826:
3822:
3820:
3815:
3811:
3809:
3804:
3800:
3798:
3793:
3789:
3787:
3782:
3778:
3776:
3771:
3767:
3765:
3760:
3756:
3752:
3749:
3748:
3745:
3740:
3736:
3734:
3729:
3725:
3723:
3718:
3714:
3712:
3707:
3703:
3701:
3696:
3692:
3690:
3685:
3681:
3679:
3674:
3670:
3668:
3663:
3659:
3655:
3652:
3651:
3647:
3639:Omnitruncated
3638:
3635:
3630:
3625:
3622:
3619:
3616:
3614:
3613:
3610:
3608:
3604:
3600:
3591:Dodecahedron
3590:
3584:
3579:
3576:
3571:
3568:
3564:
3561:
3557:
3554:
3550:
3521:
3518:
3516:
3513:
3508:
3502:
3497:
3494:
3489:
3486:
3482:
3479:
3475:
3472:
3468:
3439:
3436:
3434:
3431:
3426:
3420:
3415:
3412:
3407:
3404:
3400:
3397:
3393:
3390:
3386:
3357:
3354:
3351:
3346:
3343:
3338:
3332:
3327:
3324:
3319:
3316:
3312:
3309:
3305:
3302:
3298:
3269:
3266:
3264:(Hexahedron)
3261:
3256:
3253:
3248:
3242:
3237:
3234:
3229:
3226:
3222:
3219:
3215:
3212:
3208:
3179:
3176:
3171:
3166:
3163:
3159:
3156:
3154:
3151:
3147:
3144:
3142:
3139:
3135:
3132:
3127:
3122:
3117:
3087:
3084:
3080:
3077:
3074:
3070:
3068:
3062:
3045:
3042:
3039:
3020:
3017:
3012:
3009:
3002:
2999:
2996:
2972:
2969:
2963:
2956:
2953:
2950:
2931:
2928:
2921:
2918:
2916:
2913:
2907:
2904:
2901:
2882:
2879:
2872:
2869:
2865:
2862:
2856:
2853:
2850:
2831:
2828:
2821:
2818:
2816:
2813:
2808:
2805:
2801:
2793:
2772:
2769:
2767:
2757:
2754:
2748:
2744:
2741:
2737:
2734:
2730:
2727:
2721:
2717:
2714:
2710:
2707:
2703:
2700:
2696:
2693:
2662:
2605:
2576:
2547:
2490:
2461:
2432:
2403:
2401:
2396:
2395:
2391:
2387:
2382:
2379:
2376:
2371:
2368:
2365:
2362:
2360:
2357:
2353:
2349:
2347:
2344:
2342:
2337:
2334:
2332:
2327:
2324:
2321:
2317:
2315:
2311:
2307:
2303:
2302:star polygons
2293:
2287:
2283:
2280:
2276:
2273:
2267:
2263:
2260:
2256:
2253:
2247:
2243:
2240:
2236:
2233:
2227:
2223:
2220:
2216:
2213:
2207:
2203:
2200:
2161:
2142:
2105:
2086:
2049:
2030:
1993:
1974:
1937:
1935:
1930:
1923:
1920:
1915:
1912:
1907:
1904:
1899:
1896:
1891:
1889:
1886:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1836:
1831:
1827:
1824:
1818:
1814:
1811:
1807:
1804:
1798:
1794:
1791:
1787:
1784:
1778:
1774:
1771:
1767:
1764:
1758:
1754:
1751:
1747:
1744:
1723:
1686:
1667:
1630:
1611:
1574:
1555:
1518:
1499:
1497:
1492:
1487:
1482:
1479:
1474:
1471:
1466:
1463:
1458:
1455:
1453:
1450:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1410:
1404:
1399:
1396:
1393:
1388:
1385:
1382:
1378:
1376:
1372:
1368:
1358:
1349:One dimension
1346:
1344:
1340:
1336:
1333:-dimensional
1332:
1328:
1324:
1314:
1312:
1308:
1304:
1303:cuboctahedron
1300:
1296:
1292:
1288:
1283:
1281:
1271:
1269:
1265:
1264:omnitruncated
1261:
1256:
1253:
1251:
1246:
1244:
1240:
1239:vertex figure
1233:Vertex figure
1230:
1228:
1227:demitesseract
1224:
1220:
1216:
1213:are known as
1212:
1207:
1205:
1200:
1198:
1194:
1186:
1182:
1177:
1168:
1166:
1161:
1159:
1155:
1151:
1143:
1140:- applied to
1139:
1135:
1129:
1126:
1123:- applied to
1122:
1118:
1114:
1108:
1103:
1097:
1093:
1087:
1083:
1082:
1080:
1079:
1073:
1070:- applied to
1069:
1065:
1059:
1052:
1048:
1041:
1037:
1033:
1026:
1022:
1021:
1019:
1016:- applied to
1015:
1011:
1007:
1001:
998:
994:
987:
986:trihexication
983:
979:
972:
968:
967:
965:
964:
958:
955:- applied to
954:
950:
946:
940:
935:
929:
925:
919:
915:
914:
912:
911:
905:
902:- applied to
901:
897:
891:
884:
880:
874:
870:
866:
860:
856:
855:
853:
850:- applied to
849:
845:
841:
835:
832:
828:
822:
818:
814:
808:
807:bisterication
804:
803:
801:
796:
793:- applied to
792:
788:
784:
778:
773:
767:
763:
757:
756:biruncination
753:
752:
750:
747:'carpenter's
746:
741:
738:- applied to
737:
736:
731:
725:
718:
714:
708:
704:
700:
694:
690:
686:
680:
676:
675:
673:
669:
668:rectification
665:
662:- applied to
661:
657:
653:
647:
644:
640:
634:
630:
626:
620:
616:
612:
606:
602:
601:
597:
593:
592:
590:
589:
584:
580:
576:
575:rectification
572:
569:- applied to
568:
567:
562:
558:
552:
547:
543:
537:
533:
529:
523:
519:
515:
509:
508:tritruncation
505:
501:
495:
494:
489:
488:
484:
480:
479:
477:
473:
469:
466:- applied to
465:
464:
459:
455:
449:
448:
447:
445:
441:
437:
433:
429:
416:
413:
397:
390:
387:
386:
385:
383:
378:
376:
372:
368:
364:
360:
359:rectification
356:
352:
348:
347:rectification
344:
334:
332:
328:
323:
321:
317:
313:
309:
299:
297:
293:
289:
285:
281:
277:
276:star polygons
273:
269:
265:
263:
257:
255:
251:
247:
244:
240:
236:
179:
174:
170:
130:
125:
121:
116:
113:
81:
76:
72:
52:
48:
44:
41:
36:
32:
27:
24:
23:
14583:
14552:
14543:
14535:
14526:
14517:
14497:10-orthoplex
14233:Dodecahedron
14154:
14143:
14132:
14123:
14114:
14105:
14101:
14091:
14083:
14079:
14071:
14067:
14052:
14002:the original
13997:
13972:
13933:
13925:
13915:
13895:
13880:
13873:
13856:
13830:
13819:
13771:
13756:
12341:
12337:
12335:
10688:Tritruncated
9782:
9602:
9365:
9358:
5968:
5960:
5954:
5948:
5942:
5936:
5928:
5926:
5922:
5911:
5904:
5883:group : the
5865:group : the
5856:
5837:
5822:
5802:
5794:snub 24-cell
5772:
5598:| 2 p q
5238:2 p q |
5101:p q | 2
4962:2 | p q
4853:
4830:2 p | q
4715:2 q | p
4602:p | 2 q
4512:
4486:q | 2 p
4399:
4134:
4122:
3944:
3636:Cantellated
3596:
3509:Icosahedron
3433:Dodecahedron
3064:
3010:
2914:
2863:
2814:
2763:
2389:
2313:
2299:
1876:Enneadecagon
1866:Heptadecagon
1856:Pentadecagon
1851:Tetradecagon
1375:quasiregular
1364:
1352:
1342:
1330:
1326:
1322:
1320:
1284:
1280:circumradius
1279:
1277:
1274:Circumradius
1257:
1254:
1247:
1236:
1226:
1208:
1203:
1201:
1196:
1190:
1162:
1157:
1153:
1149:
1147:
1137:
1130:
1120:
1116:
1109:
1098:
1095:
1088:
1085:
1076:
1068:Heptellation
1067:
1060:
1050:
1043:
1039:
1035:
1028:
1024:
1013:
1009:
1002:
996:
989:
985:
981:
974:
971:bihexication
970:
961:
952:
948:
941:
930:
927:
920:
917:
908:
900:Pentellation
899:
892:
882:
875:
872:
868:
861:
858:
847:
843:
836:
830:
823:
820:
816:
809:
806:
799:
790:
786:
779:
768:
765:
758:
755:
744:
733:
726:
716:
709:
706:
702:
695:
692:
688:
681:
678:
659:
655:
648:
642:
635:
632:
628:
621:
618:
614:
607:
604:
586:
582:
566:Cantellation
565:
560:
553:
545:
538:
535:
531:
524:
521:
517:
510:
507:
503:
496:
493:bitruncation
491:
475:
462:
457:
450:
443:
442:cuts faces,
439:
438:cuts edges,
436:cantellation
435:
431:
427:
425:
414:
411:
392:
388:
380:An extended
379:
374:
370:
366:
362:
358:
350:
342:
340:
324:
305:
261:
258:
238:
232:
14506:10-demicube
14467:9-orthoplex
14417:8-orthoplex
14367:7-orthoplex
14324:6-orthoplex
14294:5-orthoplex
14249:Pentachoron
14237:Icosahedron
14212:Tetrahedron
14013:(in German)
13977:(in German)
13774:exceptional
13171:2s{2p,q,2r}
12771:hr{2p,2q,r}
11599:= tr{r,q,p}
11057:= rr{r,q,p}
10871:Cantellated
10513:Bitruncated
10146:Trirectifed
9964:Birectified
5852:tetrahedron
5848:hypersphere
5806:multiplying
4986:Cantellated
4739:Bitruncated
4509:Birectified
3938:(3.3.3.3.5)
3848:Icosahedral
3841:(3.3.3.3.4)
3744:(3.3.3.3.3)
3654:Tetrahedral
3631:Birectified
3628:(tr. dual)
3626:Bitruncated
3515:Icosahedron
3350:3-orthoplex
3339:Octahedron
3165:Tetrahedron
3048:(order 2p)
3005:(order 2p)
2959:(order 4p)
2909:(order 2p)
2858:(order 2p)
1871:Octadecagon
1861:Hexadecagon
1403:2-orthoplex
1301:. Also the
1250:alternation
1219:tetrahedron
1193:alternation
1183:produces a
1171:Alternation
1154:runcination
1144:and higher.
1127:and higher.
1121:Octellation
791:Sterication
735:Runcination
444:sterication
440:runcination
262:semiregular
45:or uniform
14492:10-simplex
14476:9-demicube
14426:8-demicube
14376:7-demicube
14333:6-demicube
14303:5-demicube
14217:Octahedron
13848:References
13381:sr{p,q,2r}
13178:{2p,q,2r}
12868:hr{2p,2q}
12778:{2p,2q,r}
12574:Alternated
12350:Operation
12332:Half forms
11597:t2r{p,q,r}
11424:tr{p,q,r}
11233:Runcinated
11055:r2r{p,q,r}
10695:= t{r,q,p}
10520:2t{p,q,r}
9971:= r{r,q,p}
9381:Operation
9250:sr{3,3,4}
8539:tr{5,3,3}
8488:tr{3,3,5}
8437:tr{3,4,3}
8345:tr{4,3,3}
8294:tr{3,3,4}
8243:tr{3,3,3}
8190:2t{3,3,5}
8135:2t{3,4,3}
8043:2t{4,3,3}
7992:2t{3,3,4}
7601:rr{5,3,3}
7550:rr{3,3,5}
7499:rr{3,4,3}
7407:rr{4,3,3}
7356:rr{3,3,4}
7305:rr{3,3,3}
5844:hyperplane
5287:Position:
5265:Operation
4191:Position:
4169:Operation
3951:antiprisms
3751:Octahedral
3623:Rectified
3620:Truncated
3607:truncation
3345:Octahedron
3174:(Pyramid)
2774:Operation
2341:Enneagrams
2331:Heptagrams
1846:Tridecagon
1445:Hendecagon
1307:octahedron
1211:hypercubes
1158:truncation
1138:Ennecation
1081:'seven'.)
953:Hexication
802:'solid'.)
463:Truncation
432:Truncation
345:orders of
302:Operations
288:honeycombs
14540:orthoplex
14462:9-simplex
14412:8-simplex
14362:7-simplex
14319:6-simplex
14289:5-simplex
14258:Tesseract
13759:hypercube
13576:os{p,q,r}
13388:{p,q,2r}
12983:{p,2q,r}
12976:s{p,2q,r}
12585:{p,2q,r}
12578:h{p,2q,r}
12294:{ }×{2r}
12263:{2p}×{ }
12110:{ }×{2r}
11889:{2p}×{ }
11421:tr{p,q,r}
10875:rr{p,q,r}
10693:3t{p,q,r}
10517:2t{p,q,r}
10333:Truncated
10157:= {r,q,p}
10155:3r{p,q,r}
9969:2r{p,q,r}
9784:Rectified
9344:s{3,4,3}
7252:t{5,3,3}
7201:t{3,3,5}
7150:t{3,4,3}
7058:t{4,3,3}
7007:t{3,3,4}
6956:t{3,3,3}
6903:r{5,3,3}
6852:r{3,3,5}
6801:r{3,4,3}
6709:r{4,3,3}
6658:r{3,3,4}
6607:r{3,3,3}
6313:tesseract
5878:tesseract
5783:duoprisms
5779:tesseract
4855:Rectified
4626:Truncated
4064:Antiprism
3916:(3.4.5.4)
3883:(3.5.3.5)
3872:(3.10.10)
3819:(3.4.4.4)
3786:(3.4.3.4)
3722:(3.4.3.4)
3689:(3.3.3.3)
3170:3-simplex
3130:(sphere)
2915:Truncated
2864:Rectified
2796:Position
2326:Pentagram
2310:pentagram
1841:Dodecagon
1392:2-simplex
1335:Euclidean
1260:truncated
1215:demicubes
1185:snub cube
913:'five'.)
664:polyhedra
571:polyhedra
292:Euclidean
264:polytopes
40:Truncated
14615:Category
14594:Topics:
14557:demicube
14522:polytope
14516:Uniform
14277:600-cell
14273:120-cell
14226:Demicube
14200:Pentagon
14180:Triangle
13836:See also
13810:polytope
13746:sr{q,r}
13715:{ }×{r}
13684:{p}×{ }
13653:sr{p,q}
13583:{p,q,r}
13571:Omnisnub
13566:s{q,2r}
13530:s{2,2r}
13463:sr{p,q}
13371:s{q,2r}
13263:s{q,2p}
13161:h{2q,r}
13063:s{p,2q}
12966:h{2q,r}
12665:h{p,2q}
12325:tr{q,r}
12232:tr{p,q}
12162:{p,q,r}
12155:o{p,q,r}
12079:{p}×{ }
12048:tr{p,q}
11978:{p,q,r}
11951:rr{q,r}
11920:{ }×{r}
11788:{p,q,r}
11769:tr{q,r}
11707:{p}×{ }
11606:{p,q,r}
11556:{ }×{r}
11494:tr{p,q}
11382:{ }×{r}
11351:{p}×{ }
11250:{p,q,r}
11243:e{p,q,r}
11238:expanded
11227:rr{q,r}
11165:{p}×{ }
11064:{p,q,r}
11014:{ }×{r}
10952:rr{p,q}
10882:{p,q,r}
10702:{p,q,r}
10344:{p,q,r}
10337:t{p,q,r}
10164:{p,q,r}
9978:{p,q,r}
9796:{p,q,r}
9789:r{p,q,r}
9614:{p,q,r}
9293:sr{3,3}
9195:{5,3,3}
9136:{3,4,3}
9079:{3,3,4}
9024:{3,3,4}
8969:{3,3,3}
8912:{5,3,3}
8857:{3,3,5}
8802:{3,4,3}
8747:rr{3,3}
8706:{4,3,3}
8651:{3,3,4}
8596:{3,3,3}
8386:tr{3,3}
8084:2t{3,3}
7941:{3,3,3}
7884:{3,3,5}
7825:{3,4,3}
7768:{4,3,3}
7713:{3,3,4}
7658:{3,3,3}
7448:2r{3,3}
6554:{5,3,3}
6507:120-cell
6503:{3,3,5}
6456:600-cell
6452:{3,4,3}
6360:{4,3,3}
6309:{3,3,4}
6258:{3,3,3}
5896:120-cell
5892:600-cell
5890:group :
5880:{4,3,3};
5872:group :
5798:600-cell
5792:and the
5674:s{p,2q}
5547:sr{p,q}
5268:Schläfli
5180:tr{p,q}
5043:rr{p,q}
4991:expanded
4172:Schläfli
4145:Schläfli
4117:sr{2,p}
4052:tr{2,p}
3987:symbols
3984:Schläfli
3965:Picture
3927:(4.6.10)
3153:Symmetry
3146:Vertices
3125:(solid)
3079:Schläfli
2800:Symmetry
2781:Symbols
2779:Schläfli
2777:Extended
2359:Schläfli
2352:n-agrams
2346:Decagram
2336:Octagram
1888:Schläfli
1881:Icosagon
1452:Schläfli
1435:Enneagon
1425:Heptagon
1415:Pentagon
1387:Triangle
1371:triangle
966:'six'.)
672:expanded
579:expanded
476:truncare
468:polygons
406:, ..., p
246:polytope
235:geometry
43:triangle
14531:simplex
14501:10-cube
14268:24-cell
14254:16-cell
14195:Hexagon
14049:regular
13854:Coxeter
13763:simplex
13581:0,1,2,3
12359:diagram
12357:Coxeter
12160:0,1,2,3
12141:t{r,q}
11971:{r,q,p}
11965:{p,q,r}
11858:t{p,q}
11781:{p,q,r}
11676:t{q,p}
11587:t{q,r}
11134:r{p,q}
11045:r{q,r}
10865:t{r,q}
10683:t{q,r}
10590:t{q,p}
10414:t{p,q}
10141:r{q,r}
9866:r{p,q}
9607:{p,q,r}
9390:diagram
9388:Coxeter
9193:0,1,2,3
9134:0,1,2,3
9077:0,1,2,3
9022:0,1,2,3
8967:0,1,2,3
7099:t{3,3}
6750:r{3,3}
6405:24-cell
6262:16-cell
5885:24-cell
5874:16-cell
5281:Wythoff
5276:diagram
5274:Coxeter
5270:Symbol
5107:{ }×{ }
4904:r{p,q}
4791:t{q,p}
4676:t{p,q}
4185:Wythoff
4180:diagram
4178:Coxeter
4174:Symbol
4163:{3,4}.
4149:Coxeter
3979:Diagram
3968:Tiling
3894:(5.6.6)
3830:(4.6.8)
3797:(4.6.6)
3775:(3.8.8)
3733:(4.6.6)
3700:(3.6.6)
3678:(3.6.6)
3633:(dual)
3617:Parent
3249:(self)
3086:Diagram
2867:(Dual)
2792:diagram
2790:Coxeter
2786:result
2784:Regular
2758:
2400:diagram
2398:Coxeter
2385:t{5/3}
2374:t{4/3}
1934:diagram
1932:Coxeter
1496:diagram
1494:Coxeter
1440:Decagon
1430:Octagon
1420:Hexagon
1295:hexagon
1268:simplex
1223:16-cell
1204:uniform
800:stereos
745:runcina
585:, like
284:tilings
49:, with
47:hexagon
14471:9-cube
14421:8-cube
14371:7-cube
14328:6-cube
14298:5-cube
14185:Square
14062:Family
13980:ONLINE
13942:
13885:ONLINE
13863:
13166:Bisnub
11413:{r,q}
11320:{p,q}
10772:{q,p}
10507:{q,r}
10327:{r,q}
10048:{q,p}
9959:{q,r}
9684:{p,q}
9603:Parent
6401:{3,3}
6211:5-cell
5867:5-cell
5681:{p,q}
5283:symbol
5244:{ }×{}
5187:{p,q}
5050:{p,q}
4911:{p,q}
4798:{p,q}
4683:{p,q}
4570:{p,q}
4563:{q,p}
4454:{p,q}
4447:{p,q}
4400:Parent
4187:symbol
3974:figure
3972:Vertex
3947:prisms
3851:5-3-2
3754:4-3-2
3657:3-3-2
3519:{3,5}
3437:{5,3}
3355:{3,4}
3267:{4,3}
3260:3-cube
3177:{3,3}
3082:{p,q}
2967:h{2p}
2815:Parent
2694:Image
2383:{10/3}
2380:{9/4}
2377:{9/2}
2369:{7/3}
2366:{7/2}
2363:{5/2}
2201:Image
1926:t{10}
1745:Image
1409:2-cube
1398:Square
1104:, etc.
1053:, etc.
999:, etc.
936:, etc.
885:, etc.
833:, etc.
774:, etc.
719:, etc.
645:, etc.
548:, etc.
472:Kepler
250:facets
14190:p-gon
13386:0,1,2
12342:holes
12338:holes
11976:0,2,3
11786:0,1,3
11604:1,2,3
8910:0,1,3
8855:0,1,3
8800:0,1,3
8704:0,1,3
8649:0,1,3
8594:0,1,3
5820:cell.
5720:s{2p}
5185:0,1,2
3999:Prism
3962:Name
3905:{3,5}
3861:{5,3}
3808:{3,4}
3764:{4,3}
3711:{3,3}
3667:{3,3}
3648:Snub
3500:{3}2
3427:Cube
3141:Edges
3134:Faces
3128:Image
3123:Image
3118:Image
3075:Name
3015:s{p}
2964:Half
2929:{2p}
2372:{8/3}
2322:Name
1921:{19}
1918:t{9}
1913:{17}
1910:t{8}
1905:{15}
1902:t{7}
1897:{13}
1894:t{6}
1837:Name
1488:{11}
1485:t{5}
1477:t{4}
1469:t{3}
1461:t{2}
1383:Name
1225:, or
1152:is a
1078:hepta
1046:2,5,8
1031:1,4,7
1005:0,3,6
910:pente
878:2,4,6
864:1,3,5
839:0,2,4
749:plane
712:3,4,5
698:2,3,4
684:1,2,3
651:0,1,2
588:bevel
290:) of
14548:cube
14221:Cube
14051:and
13940:ISBN
13861:ISBN
13793:and
13765:and
12971:Snub
12571:Half
12148:(or
11236:(or
10150:dual
10148:(or
5624:Snub
5256:{ }
5247:{2q}
5241:{2p}
5126:(or
5119:{ }
4989:(or
4845:{ }
4839:{2q}
4733:{ }
4718:{2p}
4514:dual
4511:(or
4504:{ }
3982:and
3582:{5}
3574:{3}
3492:{5}
3418:{4}
3410:{3}
3330:{3}
3322:{4}
3255:Cube
3240:{3}
3232:{3}
3158:Dual
3149:{q}
3137:{p}
3018:{p}
3011:Snub
2970:{p}
2926:{p}
2919:t{p}
2880:{p}
2877:{p}
2870:r{p}
2829:{p}
2826:{p}
2809:(0)
2806:(1)
1924:{20}
1916:{18}
1908:{16}
1900:{14}
1892:{12}
1483:{10}
1480:{9}
1472:{7}
1464:{5}
1456:{3}
1337:and
1289:for
1197:snub
1156:and
1117:4r4r
1051:3t5r
1036:3t4r
1010:3t3r
997:3r5r
982:3r4r
949:3r3r
883:2t4r
869:2t3r
844:2t2r
831:2r4r
817:2r3r
787:2r2r
751:'.)
583:cant
410:} =
357:. A
355:dual
294:and
237:, a
14097:(p)
13920:PDF
13900:PDF
13494:--
13335:--
13299:--
13176:1,2
13125:--
13094:--
12981:0,1
12935:--
12904:--
12763:--
12732:--
12701:--
12528:(0)
12487:(1)
12446:(2)
12405:(3)
11967:= e
11738:--
11525:--
11248:0,3
11196:--
11062:1,3
10983:--
10880:0,2
10834:--
10803:--
10700:2,3
10652:--
10621:--
10476:--
10445:--
10342:0,1
10296:--
10265:--
10234:--
10110:--
10079:--
9928:--
9897:--
9777:--
9746:--
9715:--
9559:(0)
9518:(1)
9477:(2)
9436:(3)
9364:An
7939:1,2
7882:0,3
7823:0,3
7766:0,3
7711:0,3
7656:0,3
5909:.)
5732:{3}
5726:{q}
5723:{3}
5679:0,1
5618:--
5609:{q}
5606:{3}
5604:{3}
5601:{p}
5253:{ }
5250:{ }
5113:{ }
5110:{q}
5104:{p}
5048:0,2
4980:--
4977:{ }
4971:{q}
4965:{p}
4848:--
4842:{ }
4836:{ }
4833:{p}
4796:1,2
4730:{ }
4724:{q}
4721:{ }
4681:0,1
4620:--
4614:{ }
4611:{q}
4608:{ }
4492:{ }
4489:{p}
4161:0,1
3577:30
3495:30
3413:12
3325:12
3043:--
3040:--
3000:--
2997:--
2954:{}
2951:{}
2924:0,1
2905:{}
2902:--
2854:--
2851:{}
2819:{p}
2390:p/q
2350:...
1475:{8}
1467:{6}
1459:{4}
1133:0,9
1115:or
1112:0,8
1101:2,9
1091:1,8
1063:0,7
1049:or
1034:or
1008:or
995:or
992:2,8
980:or
977:1,7
963:hex
947:or
944:0,6
933:2,7
923:1,6
895:0,5
881:or
867:or
842:or
829:or
826:2,6
815:or
812:1,5
785:or
782:0,4
771:2,5
761:1,4
729:0,3
717:t4r
715:or
703:t3r
701:or
689:t2r
687:or
654:or
643:r4r
641:or
638:3,5
629:r3r
627:or
624:2,4
615:r2r
613:or
610:1,3
559:or
556:0,2
544:or
541:4,5
530:or
527:3,4
516:or
513:2,3
502:or
499:1,2
456:or
453:0,1
408:n-1
402:, p
233:In
117:5D
114:4D
28:3D
25:2D
14617::
14602:•
14598:•
14578:21
14574:•
14571:k1
14567:•
14564:k2
14542:•
14499:•
14469:•
14447:21
14443:•
14440:41
14436:•
14433:42
14419:•
14397:21
14393:•
14390:31
14386:•
14383:32
14369:•
14347:21
14343:•
14340:22
14326:•
14296:•
14275:•
14256:•
14235:•
14219:•
14151:/
14140:/
14130:/
14121:/
14099:/
13996:.
13966:,
13959:,
13952:,
13907::
13894::
13812:.
13808:21
13786:,
13779:,
13761:,
13579:ht
13384:ht
13174:ht
12979:ht
12774:ht
12581:ht
12152:)
11963:3t
11240:)
10152:)
5992:BC
5854:.
5800:.
5729:--
5677:ht
5615:--
5612:--
5130:)
5116:--
4993:)
4974:--
4968:--
4727:--
4617:--
4605:--
4517:)
4501:--
4498:--
4495:--
3994:2p
3959:#
3953:.
3645:)
3580:12
3572:20
3498:20
3490:12
3352:)
3235:6
3069::
3046:]=
2957:]=
2392:}
1411:)
1394:)
1357:.
1313:.
1282:.
1229:.
1199:.
1167:.
1136::
1119::
1094:,
1066::
1042::
1038:,
1027::
1012::
988::
984:,
973::
951::
926:,
898::
871:,
846::
819:,
789::
764:,
732::
705:,
691:,
658::
656:tr
631:,
617:,
563::
561:rr
546:5t
534:,
532:4t
520:,
518:3t
506:,
504:2t
460::
398:{p
333:.
180:,
131:,
82:,
70:.
14586:-
14584:n
14576:k
14569:2
14562:1
14555:-
14553:n
14546:-
14544:n
14538:-
14536:n
14529:-
14527:n
14520:-
14518:n
14445:4
14438:2
14431:1
14395:3
14388:2
14381:1
14345:2
14338:1
14167:n
14165:H
14158:2
14155:G
14147:4
14144:F
14136:8
14133:E
14127:7
14124:E
14118:6
14115:E
14106:n
14102:D
14095:2
14092:I
14084:n
14080:B
14072:n
14068:A
14040:e
14033:t
14026:v
13806:4
13797:8
13795:E
13790:7
13788:E
13783:6
13781:E
12776:1
12583:0
12158:t
11974:t
11969:t
11961:e
11784:t
11779:t
11777:e
11602:t
11246:t
11060:t
10878:t
10698:t
10340:t
10162:3
10160:t
9976:2
9974:t
9794:1
9792:t
9612:0
9610:t
9366:n
9191:t
9132:t
9075:t
9020:t
8965:t
8908:t
8853:t
8798:t
8702:t
8647:t
8592:t
7937:t
7880:t
7821:t
7764:t
7709:t
7654:t
6012:4
6010:H
6006:4
6004:F
6000:4
5998:D
5994:4
5988:4
5986:A
5955:3
5949:2
5943:1
5937:0
5929:t
5658:}
5652:q
5649:,
5646:p
5640:{
5635:s
5531:}
5525:q
5518:p
5512:{
5507:s
5183:t
5164:}
5158:q
5151:p
5145:{
5140:t
5046:t
5027:}
5021:q
5014:p
5008:{
5003:r
4909:1
4907:t
4888:}
4882:q
4875:p
4869:{
4794:t
4775:}
4769:p
4766:,
4763:q
4757:{
4752:t
4679:t
4660:}
4654:q
4651:,
4648:p
4642:{
4637:t
4568:2
4566:t
4547:}
4541:p
4538:,
4535:q
4529:{
4452:0
4450:t
4431:}
4425:q
4422:,
4419:p
4413:{
4059:p
4057:A
3992:P
3641:(
3587:h
3585:I
3505:h
3503:I
3423:h
3421:O
3416:6
3408:8
3348:(
3335:h
3333:O
3328:8
3320:6
3262:)
3258:(
3245:d
3243:T
3238:4
3230:4
3172:)
3168:(
3003:=
2922:t
2875:1
2873:t
2824:0
2822:t
2388:{
2314:q
1407:(
1405:)
1401:(
1390:(
1355:1
1343:n
1331:n
1327:n
1323:n
1187:.
1131:t
1110:t
1099:t
1089:t
1061:t
1044:t
1029:t
1003:t
990:t
975:t
942:t
931:t
921:t
893:t
876:t
862:t
837:t
824:t
810:t
780:t
769:t
759:t
727:t
710:t
696:t
682:t
649:t
636:t
622:t
608:t
554:t
539:t
525:t
511:t
497:t
458:t
451:t
428:n
417:.
415:r
412:k
404:2
400:1
395:k
393:t
389:k
351:n
343:n
274:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
