6202:
5587:
11044:
10242:
9338:
8515:
7887:
36:
7345:
7219:
11273:
721:
10198:
6216:
5469:
5343:
3226:
10581:
3311:
3255:
11388:
3340:
3068:
728:
697:
11708:
8866:
6989:
3097:
136:
9883:
9294:
8135:
7102:
5851:
4062:
3946:
3147:
3012:
12108:
6652:
6645:
11497:
10690:
10588:
10364:
9678:
8975:
8642:
6103:
5226:
4421:
4309:
3331:
3237:
3217:
3176:
3079:
3059:
1923:
8012:
6751:
6744:
11613:
11395:
11171:
7460:
7088:
6975:
5580:
5212:
5100:
3322:
3302:
2073:
2051:
1976:
1954:
12000:
7453:
6297:
2117:
2020:
4991:
1965:
10466:
9563:
8744:
7331:
6862:
6758:
6659:
6089:
5455:
4886:
4787:
4428:
4295:
3819:
3246:
3167:
3088:
2106:
2084:
2062:
2040:
2009:
1987:
1943:
1912:
1868:
8236:
7681:
145:
7467:
7205:
6876:
5978:
5837:
5329:
4167:
4048:
3939:
3932:
2095:
1998:
1901:
1879:
1857:
6465:
6475:
4601:
3430:
11504:
10697:
10371:
9876:
9788:
9685:
9461:
8982:
8649:
8128:
5711:
3826:
3709:
3613:
3606:
3158:
3138:
1890:
1846:
11620:
11178:
9570:
8751:
8019:
7338:
6470:
6096:
5462:
4596:
4302:
3003:
11940:
7212:
7095:
6982:
6209:
5985:
5844:
5336:
5219:
5114:
5107:
4174:
4055:
3425:
8231:
4591:
6869:
6765:
6666:
5992:
5718:
5005:
4998:
4893:
4794:
4181:
3716:
10574:
11322:
11123:
10515:
10316:
7387:
7031:
11715:
11280:
11083:
11000:
8873:
7446:
7439:
7324:
7317:
6673:
4900:
4414:
4407:
3833:
3284:
12233:
98:
11381:
6075:
4288:
11449:
11444:
11433:
11344:
11333:
11227:
11216:
10961:
10642:
10637:
10626:
10537:
10526:
10420:
10409:
10159:
9739:
9734:
9723:
9634:
9623:
9612:
9517:
9506:
9412:
9255:
9059:
8927:
8922:
8911:
8815:
8804:
8793:
8698:
8687:
8594:
8432:
8187:
8182:
8090:
8079:
8068:
7970:
7533:
7521:
7516:
7505:
7409:
7398:
7283:
7272:
7261:
7157:
7144:
7044:
6931:
6918:
6814:
6711:
6377:
6365:
6360:
6349:
5929:
5912:
5782:
5664:
5653:
5533:
5522:
5511:
5407:
5396:
5385:
5281:
5268:
5169:
5156:
5056:
5043:
4942:
4839:
4501:
4489:
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4362:
4351:
4245:
4234:
4223:
4117:
4104:
4001:
3988:
3888:
3875:
3769:
3658:
10119:
9963:
9890:
9215:
8471:
8142:
7930:
7198:
7191:
6968:
6961:
6772:
6304:
5725:
5594:
4435:
3723:
3620:
3208:
3194:
3129:
3115:
2994:
2974:
2956:
3420:
5964:
5823:
4160:
4041:
3925:
11701:
10921:
10473:
8859:
8554:
7081:
7074:
6855:
6848:
3293:
3273:
3041:
107:
11266:
5697:
3812:
3702:
3599:
8392:
6311:
3050:
3030:
6082:
5448:
4281:
11490:
10683:
10357:
9781:
9671:
9454:
8968:
8635:
7847:
2983:
2965:
2945:
6195:
5971:
5830:
5322:
5205:
5093:
4153:
4034:
3918:
127:
11606:
11164:
10459:
9556:
8737:
5573:
5704:
4984:
4879:
4780:
3805:
3695:
3592:
5441:
116:
6188:
5315:
5198:
5086:
9795:
9468:
8026:
6290:
6283:
5566:
4801:
4977:
4872:
4773:
7676:
380:
with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, and the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct
337:
The
Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others continues to afflict all the work on this topic (including that of the present author). It arises from the fact that the traditional usage of the term "regular polyhedra" was, and is,
12228:
The alternated cantitruncated. All the original faces end up with half as many sides, and the squares degenerate into edges. Since the omnitruncated forms have 3 faces/vertex, new triangles are formed. Usually these alternated faceting forms are slightly deformed thereafter in order to end again as
363:
polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional
367:
There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is
479:(1415 – 1492) rediscovered the five truncations of the Platonic solids—truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron—and included illustrations and calculations of their metric properties in his book
11885:
The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual. Polyhedra are named by the number of sides of the two regular forms: {p,q} and {q,p}, like cuboctahedron for r{4,3} between a cube and octahedron.
11995:
Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated polyhedron. The polyhedron has its original faces doubled in sides, and contains the faces of the
11945:
The birectified (dual) is a further truncation so that the original faces are reduced to points. New faces are formed under each parent vertex. The number of edges is unchanged and are rotated 90 degrees. A birectification can be seen as the dual.
338:
contrary to syntax and to logic: the words seem to imply that we are dealing, among the objects we call "polyhedra", with those special ones that deserve to be called "regular". But at each stage— Euclid, Kepler, Poinsot, Hess, Brückner,
2134:(The sphere is not cut, only the tiling is cut.) (On a sphere, an edge is the arc of the great circle, the shortest way, between its two vertices. Hence, a digon whose vertices are not polar-opposite is flat: it looks like an edge.)
12162:
The truncation and cantellation operations are applied together to create an omnitruncated form which has the parent's faces doubled in sides, the dual's faces doubled in sides, and squares where the original edges existed.
391:
Double covers. Some non-orientable polyhedra have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces, edges and vertices. They are usually not counted as uniform
12103:
with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. A cantellated polyhedron is named as a rhombi-r{p,q}, like rhombicuboctahedron for
3366:
The tetrahedral symmetry is represented by a fundamental triangle with one vertex with two mirrors, and two vertices with three mirrors, represented by the symbol (3 3 2). It can also be represented by the
7591:
of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polyhedra, the hosohedra and dihedra which exist as tilings on the sphere.
388:
Degenerate compounds. Some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges.
677:
In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its
2784:
2737:
2690:
1683:
1636:
1589:
2600:
2554:
2510:
1499:
1453:
1409:
2643:
2464:
1542:
1363:
656:. Most of them were shown during the International Stereoscopic Union Congress held in 1993, at the Congress Theatre, Eastbourne, England; and again in 2005 at the Kursaal of Besançon, France.
587:(1878) discovered 2, Albert Badoureau (1881) discovered 36 more, and Pitsch (1881) independently discovered 18, of which 3 had not previously been discovered. Together these gave 41 polyhedra.
385:
Hidden faces. Some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra.
398:
Double edges. Skilling's figure has the property that it has double edges (as in the degenerate uniform polyhedra) but its faces cannot be written as a union of two uniform polyhedra.
395:
Double faces. There are several polyhedra with doubled faces produced by
Wythoff's construction. Most authors do not allow doubled faces and remove them as part of the construction.
641:
independently proved the completeness and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility (the
12950:
12898:
12808:
12728:
12733:
12723:
12613:
12538:
12955:
12903:
12813:
12803:
12761:
12751:
5888:
5762:
12973:
12893:
12618:
12543:
1304:
1257:
1210:
1105:
12963:
12945:
12821:
6140:
6029:
5631:
1309:
1262:
1215:
12937:
12917:
12878:
12848:
12785:
12655:
12573:
12462:
11512:
6111:
6000:
5859:
5733:
12908:
12888:
12766:
12746:
12738:
12718:
12695:
12623:
12608:
12598:
12533:
12500:
12445:
12425:
12416:
12386:
12376:
12346:
12327:
12312:
12302:
12259:
12249:
12222:
12212:
12202:
12156:
12146:
12136:
12093:
12073:
12044:
12034:
11979:
11969:
11932:
11869:
11810:
11743:
11733:
11648:
11638:
11628:
11560:
11540:
11532:
11423:
11413:
11403:
11308:
11298:
11288:
11206:
11196:
11103:
11093:
11030:
10931:
10725:
10715:
10705:
10616:
10606:
10596:
10501:
10491:
10481:
10399:
10389:
10296:
10286:
10228:
10129:
9918:
9908:
9823:
9813:
9803:
9713:
9703:
9693:
9598:
9588:
9578:
9496:
9486:
9392:
9382:
9324:
9225:
9010:
9000:
8990:
8901:
8891:
8881:
8779:
8769:
8759:
8677:
8667:
8574:
8564:
8501:
8402:
8170:
8160:
8150:
8054:
8044:
8034:
7950:
7940:
7857:
7495:
7485:
7475:
7373:
7363:
7353:
7247:
7227:
7130:
7120:
7007:
6997:
6894:
6800:
6681:
6339:
6329:
6319:
6244:
6234:
6224:
6150:
6145:
6131:
6121:
6034:
6010:
5898:
5893:
5879:
5767:
5641:
5636:
5622:
5612:
5497:
5487:
5477:
5371:
5351:
5254:
5244:
5132:
5122:
5023:
4928:
4809:
4463:
4453:
4443:
4337:
4327:
4317:
4209:
4189:
4090:
4080:
3964:
3954:
3851:
3751:
3628:
2410:
2400:
2390:
2381:
2371:
2361:
2352:
2332:
2323:
2294:
2284:
2255:
2226:
2216:
2187:
1299:
1291:
1281:
1271:
1252:
1244:
1234:
1224:
1197:
1177:
1168:
1139:
1129:
1100:
1082:
1053:
1043:
1014:
12831:
1110:
316:, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a
12968:
12927:
12863:
12826:
12795:
12775:
12756:
12710:
12680:
12670:
12640:
12588:
12548:
12525:
12515:
12487:
12477:
12435:
12401:
12361:
12274:
12083:
12024:
11989:
11922:
11912:
11879:
11859:
11830:
11820:
11723:
11550:
11522:
11186:
11113:
11020:
11010:
10951:
10941:
10886:
10876:
10857:
10847:
10828:
10818:
10379:
10306:
10218:
10208:
10149:
10139:
10084:
10074:
10055:
10045:
10026:
10016:
9898:
9476:
9402:
9314:
9304:
9245:
9235:
9180:
9170:
9151:
9141:
9122:
9112:
8657:
8584:
8491:
8481:
8422:
8412:
8357:
8347:
8328:
8318:
8299:
8289:
7960:
7877:
7867:
7802:
7792:
7773:
7763:
7744:
7734:
7631:
7621:
7611:
7237:
7110:
7017:
6904:
6884:
6790:
6780:
6701:
6691:
6615:
6605:
6586:
6576:
6557:
6547:
6450:
6440:
6430:
6039:
6020:
5869:
5772:
5753:
5743:
5602:
5361:
5234:
5142:
5033:
5013:
4918:
4908:
4829:
4819:
4743:
4733:
4714:
4699:
4680:
4670:
4574:
4564:
4554:
4199:
4070:
3974:
3861:
3841:
3741:
3731:
3648:
3638:
3562:
3552:
3533:
3523:
3509:
3499:
3403:
3393:
3383:
2342:
2313:
2303:
2274:
2265:
2245:
2236:
2207:
2197:
1205:
1187:
1158:
1148:
1119:
1092:
1072:
1063:
1034:
1024:
12868:
12853:
12700:
12685:
12660:
12645:
12578:
12505:
12467:
12406:
12391:
12366:
12351:
12317:
12264:
11738:
11643:
11418:
10720:
10611:
9913:
9818:
9708:
9005:
8896:
8165:
8155:
2405:
12932:
12922:
12873:
12858:
12790:
12780:
12705:
12690:
12665:
12650:
12593:
12583:
12520:
12510:
12482:
12472:
12440:
12430:
12411:
12396:
12371:
12356:
12322:
12307:
12269:
12254:
12217:
12207:
12151:
12141:
12088:
12078:
12039:
12029:
11984:
11974:
11927:
11917:
11874:
11864:
11825:
11815:
11728:
11633:
11555:
11545:
11527:
11517:
11408:
11303:
11293:
11201:
11191:
11108:
11098:
11025:
11015:
10946:
10936:
10881:
10871:
10852:
10833:
10823:
10710:
10601:
10496:
10486:
10394:
10384:
10301:
10291:
10223:
10213:
10144:
10134:
10079:
10069:
10050:
10031:
10021:
9903:
9808:
9698:
9593:
9583:
9491:
9481:
9397:
9387:
9319:
9309:
9240:
9230:
9175:
9165:
9146:
9127:
9117:
8995:
8886:
8774:
8764:
8672:
8662:
8579:
8569:
8496:
8486:
8417:
8407:
8352:
8342:
8323:
8304:
8294:
8049:
8039:
7955:
7945:
7872:
7862:
7797:
7787:
7768:
7749:
7739:
7626:
7616:
7490:
7480:
7368:
7358:
7242:
7232:
7125:
7115:
7012:
7002:
6899:
6889:
6795:
6785:
6696:
6686:
6610:
6600:
6581:
6562:
6552:
6445:
6435:
6334:
6324:
6239:
6229:
6126:
6116:
6015:
6005:
5874:
5864:
5748:
5738:
5617:
5607:
5492:
5482:
5366:
5356:
5249:
5239:
5137:
5127:
5028:
5018:
4923:
4913:
4824:
4814:
4738:
4728:
4709:
4704:
4685:
4675:
4569:
4559:
4458:
4448:
4332:
4322:
4204:
4194:
4085:
4075:
3969:
3959:
3856:
3846:
3746:
3736:
3643:
3633:
3557:
3547:
3528:
3504:
3398:
3388:
2395:
2376:
2366:
2347:
2337:
2318:
2308:
2289:
2279:
2260:
2250:
2231:
2221:
2202:
2192:
1286:
1276:
1239:
1229:
1192:
1182:
1163:
1153:
1134:
1124:
1087:
1077:
1058:
1048:
1029:
1019:
10765:
13354:
6411:
of the sphere generates 7 uniform polyhedra, and a 1 more by alternation. Only one is repeated from the tetrahedral and octahedral symmetry table above.
13129:
298:
and 39 semiregular. There are also many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John
Skilling called the
13292:
4535:
of the sphere generates 7 uniform polyhedra, and a 7 more by alternation. Six of these forms are repeated from the tetrahedral symmetry table above.
752:
Within the
Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The
6414:
The icosahedral symmetry is represented by a fundamental triangle (5 3 2) counting the mirrors at each vertex. It can also be represented by the
659:
In 1993, Zvi Har'El (1949–2008) produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called
4538:
The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the
7595:
The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the
13025:
13020:
13567:
13015:
900:
Below the convex uniform polyhedra are indexed 1–18 for the nonprismatic forms as they are presented in the tables by symmetry form.
727:
14166:
368:
non-degenerate, then we get the so-called degenerate uniform polyhedra. These require a more general definition of polyhedra.
13516:
13405:
13376:
13228:
13601:
13052:
13037:
2743:
2696:
2649:
1642:
1595:
1548:
626:
2560:
2516:
2470:
1459:
1415:
1369:
13075:
2606:
2427:
1505:
1326:
749:
In more detail the convex uniform polyhedron are given below by their
Wythoff construction within each symmetry group.
262:
There are two infinite classes of uniform polyhedra, together with 75 other polyhedra. They are 2 infinite classes of
642:
481:
299:
79:
57:
17:
295:
50:
13360:
783:
376:
gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional
13388:(1994), "Polyhedra with Hollow Faces", in Tibor Bisztriczky; Peter McMullen; Rolf Schneider; et al. (eds.),
13272:
10798:
9996:
9092:
8269:
7714:
6527:
4650:
3479:
964:
705:
632:
591:
13284:
631:, which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by
182:
342:... —the writers failed to define what are the "polyhedra" among which they are finding the "regular" ones.
7582:
545:
541:
537:
520:
287:
178:
7310:
2110:
12050:
A bitruncation can be seen as the truncation of the dual. A bitruncated cube is a truncated octahedron.
14183:
13428:
Philosophical
Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences
13276:
13141:
Dr. Zvi Har’El (December 14, 1949 – February 2, 2008) and
International Jules Verne Studies - A Tribute
599:
426:
13057:
13010:
6458:
2002:
13487:
Sopov, S. P. (1970). "A proof of the completeness on the list of elementary homogeneous polyhedra".
13624:
12118:
11841:
893:, the first having only two faces, and the second only two vertices. The truncation of the regular
44:
7658:, and on the sphere as an equator line on the longitude, and n equally-spaced lines of longitude.
13594:
13390:
Proceedings of the NATO Advanced Study
Institute on Polytopes: Abstract, Convex and Computational
13267:
12998:
12055:
5434:
2013:
691:
279:
244:
190:
186:
13561:
764:. Many polyhedra are repeated from different construction sources, and are colored differently.
606:
independently discovered eleven of these. Lesavre and
Mercier rediscovered five of them in 1947.
13032:
13003:
12493:
7184:
6954:
4582:
2099:
2055:
867:
291:
283:
248:
61:
14138:
14131:
14124:
13085:
11951:
10189:
7067:
5815:
4026:
3911:
2077:
1883:
1861:
566:
493:
476:
224:
13663:
13641:
13629:
13140:
12229:
uniform polyhedra. The possibility of the latter variation depends on the degree of freedom.
6201:
5586:
13795:
13742:
13496:
13471:
13435:
13337:
13301:
12061:
11687:
11073:
10991:
10566:
8845:
6408:
6067:
5191:
4273:
3360:
2905:
2031:
1980:
1916:
1837:
1806:
842:
830:
768:
743:
603:
470:
8:
14150:
14049:
13799:
13161:
13118:
11373:
10110:
9954:
9050:
8462:
8222:
6180:
5308:
4532:
3411:
1934:
836:
498:
162:
13439:
13305:
11043:
10241:
9337:
8514:
7886:
652:
drew all the uniform polyhedra and their duals in 3D with a Turbo Pascal program called
14019:
13969:
13919:
13876:
13846:
13806:
13587:
13537:
13475:
13459:
13341:
13325:
12993:
10912:
10271:
9367:
8544:
7838:
7344:
7218:
555:
232:
13572:
13385:
13211:
11272:
2420:
1319:
14158:
13534:
13512:
13479:
13451:
13401:
13372:
13345:
13317:
11258:
9862:
9285:
8383:
7588:
7432:
6841:
3264:
3185:
3106:
3021:
2936:
2121:
2066:
874:
848:
559:
516:
506:
466:
377:
360:
275:
252:
216:
174:
6215:
5468:
5342:
14162:
13727:
13716:
13705:
13694:
13685:
13676:
13615:
13611:
13504:
13443:
13393:
13364:
13309:
13237:
13170:
13042:
11482:
10675:
10349:
9663:
8960:
8627:
7833:
7667:
931:
709:
622:
430:
327:, which also applies to shapes in higher-dimensional (or lower-dimensional) space.
324:
263:
236:
158:
154:
10197:
3225:
720:
13752:
13737:
13492:
13467:
13333:
13280:
12181:
11894:
11783:
11598:
11156:
10793:
10580:
10451:
9991:
9548:
9206:
9087:
8729:
8264:
7920:
7709:
7604:
6988:
6522:
6423:
4645:
4547:
3474:
3376:
3363:
of the sphere generates 5 uniform polyhedra, and a 6th form by a snub operation.
3310:
3254:
2179:
1006:
696:
595:
533:
512:
364:
Euclidean space; these are allowed to be non-convex and intersecting each other.
309:
256:
240:
212:
208:
13397:
13260:
11387:
7101:
5850:
4061:
3945:
3339:
3067:
251:(if neither edge- nor face-transitive). The faces and vertices don't need to be
14102:
13414:
13062:
13047:
11707:
10587:
8865:
6102:
5225:
5079:
4308:
3330:
3096:
2874:
1958:
1773:
678:
418:
414:
305:
271:
166:
135:
13556:
13368:
13242:
13223:
11394:
9882:
9293:
8134:
8011:
7459:
3146:
3011:
700:
The great dirhombicosidodecahedron, the only non-Wythoffian uniform polyhedron
14177:
14119:
14007:
14000:
13993:
13957:
13950:
13943:
13907:
13900:
13455:
13321:
13204:
12107:
11496:
10786:
10689:
10465:
10363:
9984:
9677:
9562:
9080:
8974:
8743:
8641:
8257:
7702:
7596:
7452:
7330:
6651:
6644:
6515:
6415:
6296:
6088:
5956:
5454:
4969:
4638:
4539:
4427:
4420:
4294:
4145:
3467:
3368:
3245:
3236:
3216:
3175:
3166:
3087:
3078:
3058:
2116:
2105:
2019:
2008:
1969:
1922:
1911:
1905:
570:
551:
452:
317:
313:
170:
13193:
13189:
11612:
11170:
8235:
7680:
7466:
7204:
7087:
6974:
6875:
6750:
6743:
5977:
5836:
5579:
5328:
5211:
5099:
4166:
4047:
3938:
3931:
3321:
3301:
2094:
2072:
2050:
1997:
1975:
1953:
1900:
1878:
1856:
903:
For the infinite set of prismatic forms, they are indexed in four families:
14059:
13551:
13447:
13313:
13153:
11999:
11503:
10696:
10370:
9875:
9684:
8981:
8648:
8127:
7554:
6861:
6757:
6658:
6638:
6474:
6464:
5710:
4990:
4885:
4786:
4600:
3825:
3818:
3708:
3612:
3605:
3429:
2083:
2061:
2044:
2039:
1986:
1964:
1942:
1889:
1867:
1845:
649:
488:
11619:
11177:
9569:
8750:
8018:
7337:
6095:
5461:
4301:
3002:
771:, so images of both are given. The spherical tilings including the set of
144:
14068:
14029:
13979:
13929:
13886:
13856:
13788:
13774:
9868:
9787:
9460:
8120:
7211:
7094:
6981:
6737:
6469:
6208:
5984:
5843:
5689:
5558:
5335:
5218:
5113:
5106:
4595:
4399:
4173:
4054:
3687:
3585:
3157:
3137:
2088:
1927:
1894:
1850:
671:
584:
8230:
6868:
6764:
6665:
5991:
5717:
5004:
4997:
4892:
4793:
4590:
4180:
3715:
3424:
14054:
14038:
13988:
13938:
13895:
13865:
13779:
13175:
13091:
12988:
11939:
11693:
11279:
11082:
10999:
10573:
8851:
7445:
7438:
7323:
7316:
6672:
4865:
4413:
4406:
3797:
3315:
3283:
3230:
3151:
3072:
2987:
1991:
1872:
907:
889:
772:
757:
753:
736:
462:
12676:
Only possible in uniform tilings (infinite polyhedra), alternation of
12382:
Only possible in uniform tilings (infinite polyhedra), alternation of
11380:
10118:
9962:
9889:
9214:
9058:
8470:
8141:
7929:
7197:
7190:
6967:
6960:
6771:
6303:
5724:
5593:
4434:
3722:
3619:
3207:
3193:
3128:
3114:
2993:
2973:
2955:
381:
realizations. Some of the ways they can be degenerate are as follows:
323:
The concept of uniform polyhedron is a special case of the concept of
14110:
14024:
13974:
13924:
13881:
13851:
13820:
13542:
13463:
13329:
11714:
11700:
11326:
11321:
11127:
11122:
10920:
10514:
10472:
10315:
8872:
8858:
8553:
7655:
7386:
7080:
7073:
7030:
6854:
6847:
6276:
4899:
3832:
3419:
3292:
3272:
3040:
2024:
943:
438:
267:
228:
13130:
Edmond Bonan, "Polyèdres
Eastbourne 1993", Stéréo-Club Français 1993
12232:
11448:
11443:
11432:
11343:
11332:
11265:
11226:
11215:
10960:
10641:
10636:
10625:
10536:
10525:
10419:
10408:
10158:
9738:
9733:
9722:
9633:
9622:
9611:
9516:
9505:
9411:
9254:
8926:
8921:
8910:
8814:
8803:
8792:
8697:
8686:
8593:
8431:
8391:
8186:
8181:
8089:
8078:
8067:
7969:
7532:
7520:
7515:
7504:
7408:
7397:
7282:
7271:
7260:
7156:
7143:
7043:
6930:
6917:
6813:
6710:
6376:
6364:
6359:
6348:
6310:
6074:
5928:
5911:
5781:
5663:
5652:
5532:
5521:
5510:
5406:
5395:
5384:
5280:
5267:
5168:
5155:
5055:
5042:
4941:
4838:
4500:
4488:
4483:
4472:
4372:
4361:
4350:
4287:
4244:
4233:
4222:
4116:
4103:
4000:
3987:
3887:
3874:
3768:
3657:
3049:
3029:
223:
mapping any vertex onto any other. It follows that all vertices are
14084:
13839:
13835:
13762:
13185:
11489:
11454:
10682:
10647:
10630:
10413:
10356:
10163:
9780:
9744:
9670:
9453:
8967:
8932:
8915:
8691:
8634:
8436:
8192:
7846:
7537:
7526:
7509:
7287:
7265:
7148:
7048:
6935:
6922:
6818:
6715:
6381:
6370:
5963:
5937:
5822:
5790:
5668:
5657:
5411:
5173:
5060:
4946:
4505:
4494:
4477:
4249:
4227:
4159:
4108:
4040:
4005:
3924:
3892:
3879:
3773:
3662:
3277:
3198:
3119:
3034:
2982:
2964:
2949:
2944:
919:
883:
776:
220:
200:
106:
97:
11605:
11163:
10458:
9555:
8736:
5696:
3811:
3701:
3598:
870:
operations applied to the polyhedra with an even number of sides.
767:
The Wythoff construction applies equally to uniform polyhedra and
14093:
14063:
13830:
13825:
13816:
13757:
11437:
11220:
10965:
10519:
10320:
9616:
9416:
8797:
8598:
7413:
7391:
7161:
7035:
6081:
5920:
5537:
5515:
5447:
5285:
5160:
4377:
4355:
4280:
4121:
3992:
10764:
There are 24 fundamental triangles, visible in the faces of the
9953:
There are 20 fundamental triangles, visible in the faces of the
9049:
There are 16 fundamental triangles, visible in the faces of the
8221:
There are 12 fundamental triangles, visible in the faces of the
6194:
5970:
5829:
5572:
5321:
5204:
5092:
4152:
4033:
3917:
126:
14033:
13983:
13933:
13890:
13860:
13811:
13747:
13224:"Closed-Form Expressions for Uniform Polyhedra and Their Duals"
12884:
Only possible in uniform tilings (infinite polyhedra), same as
11348:
11337:
11231:
10541:
10530:
10424:
9794:
9727:
9638:
9627:
9521:
9510:
9467:
9259:
8819:
8808:
8702:
8094:
8083:
8072:
8025:
7974:
7666:
There are 8 fundamental triangles, visible in the faces of the
7402:
7276:
6353:
6289:
6282:
5703:
5526:
5400:
5389:
5272:
5047:
4983:
4878:
4843:
4800:
4779:
4366:
4238:
3804:
3694:
3591:
519:, in 1619. He also identified the infinite families of uniform
434:
13426:
Skilling, J. (1975). "The complete set of uniform polyhedra".
5440:
473:(c. 290 – c. 350 AD) mentioned Archimedes listed 13 polyhedra.
115:
11048:
10246:
9342:
8519:
7891:
7675:
6187:
5565:
5314:
5197:
5085:
456:
422:
13271:
13097:
7670:(Octahedron) and alternately colored triangles on a sphere:
4976:
4871:
4772:
609:
372:
gave a rather complicated definition of a polyhedron, while
356:
13783:
13532:
9773:
9446:
8004:
4766:
1947:
732:
704:
The 57 nonprismatic nonconvex forms, with exception of the
13181:
13086:
https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA40
515:(1571–1630) was the first to publish the complete list of
485:. He also discussed the cuboctahedron in a different book.
417:
date back to the classical Greeks and were studied by the
756:
is a regular polyhedron, and a triangular antiprism. The
12099:
In addition to vertex truncation, each original edge is
4585:, and in the alternately colored triangles on a sphere:
3414:, and in the alternately colored triangles on a sphere:
881:
classes which become degenerate as polyhedra : the
6461:, and in the alternately colored triangles on a sphere:
782:
These symmetry groups are formed from the reflectional
715:
670:
Also in 1993, R. Mäder ported this Kaleido solution to
2755:
2708:
2661:
2615:
2572:
2525:
2482:
2436:
1654:
1607:
1560:
1514:
1471:
1424:
1381:
1335:
877:, the spherical Wythoff construction process adds two
594:
discovered the remaining twelve in collaboration with
445:
247:(if also edge-transitive but not face-transitive), or
6457:
There are 120 triangles, visible in the faces of the
2746:
2699:
2652:
2609:
2563:
2519:
2473:
2430:
1645:
1598:
1551:
1508:
1462:
1418:
1372:
1329:
866:
The remaining nonreflective forms are constructed by
577:
4581:
There are 48 triangles, visible in the faces of the
3410:
There are 24 triangles, visible in the faces of the
2779:{\displaystyle s{\begin{Bmatrix}p\\2\end{Bmatrix}}}
2732:{\displaystyle t{\begin{Bmatrix}p\\2\end{Bmatrix}}}
2685:{\displaystyle r{\begin{Bmatrix}p\\2\end{Bmatrix}}}
1678:{\displaystyle s{\begin{Bmatrix}p\\q\end{Bmatrix}}}
1631:{\displaystyle t{\begin{Bmatrix}p\\q\end{Bmatrix}}}
1584:{\displaystyle r{\begin{Bmatrix}p\\q\end{Bmatrix}}}
618:
proved their conjecture that the list was complete.
6398:
3350:
2778:
2731:
2684:
2637:
2595:{\displaystyle t{\begin{Bmatrix}2,p\end{Bmatrix}}}
2594:
2549:{\displaystyle {\begin{Bmatrix}p\\2\end{Bmatrix}}}
2548:
2505:{\displaystyle t{\begin{Bmatrix}p,2\end{Bmatrix}}}
2504:
2458:
1677:
1630:
1583:
1536:
1494:{\displaystyle t{\begin{Bmatrix}q,p\end{Bmatrix}}}
1493:
1448:{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}
1447:
1404:{\displaystyle t{\begin{Bmatrix}p,q\end{Bmatrix}}}
1403:
1357:
441:discovered the regular dodecahedron before 500 BC.
13293:Philosophical Transactions of the Royal Society A
11769:
4522:
2638:{\displaystyle {\begin{Bmatrix}2,p\end{Bmatrix}}}
2459:{\displaystyle {\begin{Bmatrix}p,2\end{Bmatrix}}}
1537:{\displaystyle {\begin{Bmatrix}q,p\end{Bmatrix}}}
1358:{\displaystyle {\begin{Bmatrix}p,q\end{Bmatrix}}}
469:. His original book on the subject was lost, but
14175:
13264:Vielecke und vielflache. Theorie und geschichte.
10755:
9944:
9040:
8212:
13413:
10768:and alternately colored triangles on a sphere.
9957:and alternately colored triangles on a sphere:
9053:and alternately colored triangles on a sphere:
8225:and alternately colored triangles on a sphere:
7638:Below are the first five dihedral symmetries: D
708:, are compiled by Wythoff constructions within
373:
7427:
7301:
7175:
7062:
6949:
6832:
6733:
6633:
6271:
6173:
6062:
5951:
5808:
5682:
5551:
5425:
5299:
5187:
5074:
4960:
4861:
4761:
4391:
4263:
4135:
4019:
3906:
3787:
3680:
3580:
786:, each represented by a fundamental triangle (
286:— the non-convex star polyhedra as in 4
13595:
13026:List of uniform polyhedra by Schwarz triangle
742:The convex uniform polyhedra can be named by
13573:Uniform Polyhedron -- from Wolfram MathWorld
13098:Coxeter, Longuet-Higgins & Miller (1954)
7661:
7570:, I2(p) family (Dph dihedral symmetry)": -->
610:Coxeter, Longuet-Higgins & Miller (1954)
357:Coxeter, Longuet-Higgins & Miller (1954)
255:, so many of the uniform polyhedra are also
13021:List of uniform polyhedra by Wythoff symbol
465:(287 BC – 212 BC) discovered all of the 13
407:
13602:
13588:
13151:
13016:List of uniform polyhedra by vertex figure
769:uniform tilings on the surface of a sphere
674:with a slightly different indexing system.
13503:
13241:
13174:
685:
527:
80:Learn how and when to remove this message
13425:
13384:
13154:"Uniform Solution for Uniform Polyhedra"
695:
638:
612:published the list of uniform polyhedra.
369:
348:
43:This article includes a list of general
14167:List of regular polytopes and compounds
565:The set of four was proven complete by
14:
14176:
13552:Uniform Solution for Uniform Polyhedra
13352:
13221:
13081:
2130:And a sampling of dihedral symmetries:
665:Uniform Solution for Uniform Polyhedra
153:Some examples of uniform polyhedrons:
27:Isogonal polyhedron with regular faces
13533:
13486:
13229:Discrete & Computational Geometry
615:
554:(1809) discovered the other two, the
536:(1619) discovered two of the regular
505:for its 26 faces, which was drawn by
13489:Ukrainskiui Geometricheskiui Sbornik
716:Convex forms by Wythoff construction
29:
13266:. Leipzig, Germany: Teubner, 1900.
13053:Uniform tilings in hyperbolic plane
13038:List of Wenninger polyhedron models
446:Nonregular uniform convex polyhedra
24:
11938:
726:
719:
578:Other 53 nonregular star polyhedra
49:it lacks sufficient corresponding
25:
14195:
13526:
13119:Piero della Francesca's Polyhedra
11836:Any regular polyhedron or tiling
10448:Omnitruncated pentagonal dihedron
956:
643:great disnub dirhombidodecahedron
598:(1930–1932) but did not publish.
482:De quinque corporibus regularibus
300:great disnub dirhombidodecahedron
270:, the convex polyhedrons as in 5
12971:
12966:
12961:
12953:
12948:
12943:
12935:
12930:
12925:
12920:
12915:
12906:
12901:
12896:
12891:
12886:
12876:
12871:
12866:
12861:
12856:
12851:
12846:
12829:
12824:
12819:
12811:
12806:
12801:
12793:
12788:
12783:
12778:
12773:
12764:
12759:
12754:
12749:
12744:
12736:
12731:
12726:
12721:
12716:
12708:
12703:
12698:
12693:
12688:
12683:
12678:
12668:
12663:
12658:
12653:
12648:
12643:
12638:
12621:
12616:
12611:
12606:
12596:
12591:
12586:
12581:
12576:
12571:
12546:
12541:
12536:
12531:
12523:
12518:
12513:
12508:
12503:
12498:
12485:
12480:
12475:
12470:
12465:
12460:
12443:
12438:
12433:
12428:
12423:
12414:
12409:
12404:
12399:
12394:
12389:
12384:
12374:
12369:
12364:
12359:
12354:
12349:
12344:
12325:
12320:
12315:
12310:
12305:
12300:
12272:
12267:
12262:
12257:
12252:
12247:
12231:
12220:
12215:
12210:
12205:
12200:
12154:
12149:
12144:
12139:
12134:
12106:
12091:
12086:
12081:
12076:
12071:
12042:
12037:
12032:
12027:
12022:
11998:
11987:
11982:
11977:
11972:
11967:
11930:
11925:
11920:
11915:
11910:
11877:
11872:
11867:
11862:
11857:
11828:
11823:
11818:
11813:
11808:
11741:
11736:
11731:
11726:
11721:
11713:
11706:
11699:
11646:
11641:
11636:
11631:
11626:
11618:
11611:
11604:
11558:
11553:
11548:
11543:
11538:
11530:
11525:
11520:
11515:
11510:
11502:
11495:
11488:
11447:
11442:
11431:
11421:
11416:
11411:
11406:
11401:
11393:
11386:
11379:
11342:
11331:
11320:
11306:
11301:
11296:
11291:
11286:
11278:
11271:
11264:
11255:Omnitruncated hexagonal dihedron
11225:
11214:
11204:
11199:
11194:
11189:
11184:
11176:
11169:
11162:
11121:
11111:
11106:
11101:
11096:
11091:
11081:
11042:
11028:
11023:
11018:
11013:
11008:
10998:
10959:
10949:
10944:
10939:
10934:
10929:
10919:
10884:
10879:
10874:
10869:
10855:
10850:
10845:
10831:
10826:
10821:
10816:
10723:
10718:
10713:
10708:
10703:
10695:
10688:
10681:
10640:
10635:
10624:
10614:
10609:
10604:
10599:
10594:
10586:
10579:
10572:
10535:
10524:
10513:
10499:
10494:
10489:
10484:
10479:
10471:
10464:
10457:
10418:
10407:
10397:
10392:
10387:
10382:
10377:
10369:
10362:
10355:
10314:
10304:
10299:
10294:
10289:
10284:
10240:
10226:
10221:
10216:
10211:
10206:
10196:
10157:
10147:
10142:
10137:
10132:
10127:
10117:
10082:
10077:
10072:
10067:
10053:
10048:
10043:
10029:
10024:
10019:
10014:
9961:
9916:
9911:
9906:
9901:
9896:
9888:
9881:
9874:
9821:
9816:
9811:
9806:
9801:
9793:
9786:
9779:
9737:
9732:
9721:
9711:
9706:
9701:
9696:
9691:
9683:
9676:
9669:
9632:
9621:
9610:
9596:
9591:
9586:
9581:
9576:
9568:
9561:
9554:
9515:
9504:
9494:
9489:
9484:
9479:
9474:
9466:
9459:
9452:
9410:
9400:
9395:
9390:
9385:
9380:
9336:
9322:
9317:
9312:
9307:
9302:
9292:
9253:
9243:
9238:
9233:
9228:
9223:
9213:
9178:
9173:
9168:
9163:
9149:
9144:
9139:
9125:
9120:
9115:
9110:
9057:
9008:
9003:
8998:
8993:
8988:
8980:
8973:
8966:
8925:
8920:
8909:
8899:
8894:
8889:
8884:
8879:
8871:
8864:
8857:
8813:
8802:
8791:
8777:
8772:
8767:
8762:
8757:
8749:
8742:
8735:
8696:
8685:
8675:
8670:
8665:
8660:
8655:
8647:
8640:
8633:
8592:
8582:
8577:
8572:
8567:
8562:
8552:
8513:
8499:
8494:
8489:
8484:
8479:
8469:
8430:
8420:
8415:
8410:
8405:
8400:
8390:
8355:
8350:
8345:
8340:
8326:
8321:
8316:
8302:
8297:
8292:
8287:
8234:
8229:
8185:
8180:
8168:
8163:
8158:
8153:
8148:
8140:
8133:
8126:
8088:
8077:
8066:
8052:
8047:
8042:
8037:
8032:
8024:
8017:
8010:
7968:
7958:
7953:
7948:
7943:
7938:
7928:
7885:
7875:
7870:
7865:
7860:
7855:
7845:
7800:
7795:
7790:
7785:
7771:
7766:
7761:
7747:
7742:
7737:
7732:
7679:
7674:
7629:
7624:
7619:
7614:
7609:
7603:(p) or , as well as a prismatic
7531:
7519:
7514:
7503:
7493:
7488:
7483:
7478:
7473:
7465:
7458:
7451:
7444:
7437:
7407:
7396:
7385:
7371:
7366:
7361:
7356:
7351:
7343:
7336:
7329:
7322:
7315:
7281:
7270:
7259:
7245:
7240:
7235:
7230:
7225:
7217:
7210:
7203:
7196:
7189:
7155:
7142:
7128:
7123:
7118:
7113:
7108:
7100:
7093:
7086:
7079:
7072:
7042:
7029:
7015:
7010:
7005:
7000:
6995:
6987:
6980:
6973:
6966:
6959:
6929:
6916:
6902:
6897:
6892:
6887:
6882:
6874:
6867:
6860:
6853:
6846:
6812:
6798:
6793:
6788:
6783:
6778:
6770:
6763:
6756:
6749:
6742:
6709:
6699:
6694:
6689:
6684:
6679:
6671:
6664:
6657:
6650:
6643:
6613:
6608:
6603:
6598:
6584:
6579:
6574:
6560:
6555:
6550:
6545:
6473:
6468:
6463:
6448:
6443:
6438:
6433:
6428:
6375:
6363:
6358:
6347:
6337:
6332:
6327:
6322:
6317:
6309:
6302:
6295:
6288:
6281:
6242:
6237:
6232:
6227:
6222:
6214:
6207:
6200:
6193:
6186:
6148:
6143:
6138:
6129:
6124:
6119:
6114:
6109:
6101:
6094:
6087:
6080:
6073:
6037:
6032:
6027:
6018:
6013:
6008:
6003:
5998:
5990:
5983:
5976:
5969:
5962:
5927:
5910:
5896:
5891:
5886:
5877:
5872:
5867:
5862:
5857:
5849:
5842:
5835:
5828:
5821:
5780:
5770:
5765:
5760:
5751:
5746:
5741:
5736:
5731:
5723:
5716:
5709:
5702:
5695:
5662:
5651:
5639:
5634:
5629:
5620:
5615:
5610:
5605:
5600:
5592:
5585:
5578:
5571:
5564:
5531:
5520:
5509:
5495:
5490:
5485:
5480:
5475:
5467:
5460:
5453:
5446:
5439:
5405:
5394:
5383:
5369:
5364:
5359:
5354:
5349:
5341:
5334:
5327:
5320:
5313:
5279:
5266:
5252:
5247:
5242:
5237:
5232:
5224:
5217:
5210:
5203:
5196:
5167:
5154:
5140:
5135:
5130:
5125:
5120:
5112:
5105:
5098:
5091:
5084:
5054:
5041:
5031:
5026:
5021:
5016:
5011:
5003:
4996:
4989:
4982:
4975:
4940:
4926:
4921:
4916:
4911:
4906:
4898:
4891:
4884:
4877:
4870:
4837:
4827:
4822:
4817:
4812:
4807:
4799:
4792:
4785:
4778:
4771:
4741:
4736:
4731:
4726:
4712:
4707:
4702:
4697:
4683:
4678:
4673:
4668:
4599:
4594:
4589:
4572:
4567:
4562:
4557:
4552:
4499:
4487:
4482:
4471:
4461:
4456:
4451:
4446:
4441:
4433:
4426:
4419:
4412:
4405:
4371:
4360:
4349:
4335:
4330:
4325:
4320:
4315:
4307:
4300:
4293:
4286:
4279:
4243:
4232:
4221:
4207:
4202:
4197:
4192:
4187:
4179:
4172:
4165:
4158:
4151:
4115:
4102:
4088:
4083:
4078:
4073:
4068:
4060:
4053:
4046:
4039:
4032:
3999:
3986:
3972:
3967:
3962:
3957:
3952:
3944:
3937:
3930:
3923:
3916:
3886:
3873:
3859:
3854:
3849:
3844:
3839:
3831:
3824:
3817:
3810:
3803:
3767:
3749:
3744:
3739:
3734:
3729:
3721:
3714:
3707:
3700:
3693:
3656:
3646:
3641:
3636:
3631:
3626:
3618:
3611:
3604:
3597:
3590:
3560:
3555:
3550:
3545:
3531:
3526:
3521:
3507:
3502:
3497:
3428:
3423:
3418:
3401:
3396:
3391:
3386:
3381:
3338:
3329:
3320:
3309:
3300:
3291:
3282:
3271:
3253:
3244:
3235:
3224:
3215:
3206:
3192:
3174:
3165:
3156:
3145:
3136:
3127:
3113:
3095:
3086:
3077:
3066:
3057:
3048:
3039:
3028:
3010:
3001:
2992:
2981:
2972:
2963:
2954:
2943:
2408:
2403:
2398:
2393:
2388:
2379:
2374:
2369:
2364:
2359:
2350:
2345:
2340:
2335:
2330:
2321:
2316:
2311:
2306:
2301:
2292:
2287:
2282:
2277:
2272:
2263:
2258:
2253:
2248:
2243:
2234:
2229:
2224:
2219:
2214:
2205:
2200:
2195:
2190:
2185:
2115:
2104:
2093:
2082:
2071:
2060:
2049:
2038:
2018:
2007:
1996:
1985:
1974:
1963:
1952:
1941:
1921:
1910:
1899:
1888:
1877:
1866:
1855:
1844:
1307:
1302:
1297:
1289:
1284:
1279:
1274:
1269:
1260:
1255:
1250:
1242:
1237:
1232:
1227:
1222:
1213:
1208:
1203:
1195:
1190:
1185:
1180:
1175:
1166:
1161:
1156:
1151:
1146:
1137:
1132:
1127:
1122:
1117:
1108:
1103:
1098:
1090:
1085:
1080:
1075:
1070:
1061:
1056:
1051:
1046:
1041:
1032:
1027:
1022:
1017:
1012:
873:Along with the prisms and their
784:point groups in three dimensions
779:which are degenerate polyhedra.
746:operations on the regular form.
491:plagiarized Francesca's work in
143:
134:
125:
114:
105:
96:
34:
13356:Multi-shell Polyhedral Clusters
13273:Coxeter, Harold Scott MacDonald
13210:Mathematica J. 3, 48-57, 1993.
11675:
11587:
11471:
11362:
11247:
11145:
11062:
10983:
10904:
10672:Cantic snub pentagonal dihedron
10664:
10555:
10440:
10346:Truncated pentagonal hosohedron
10338:
10260:
10181:
10102:
9850:
9761:
9652:
9537:
9434:
9356:
9277:
9198:
8949:
8833:
8726:Omnitruncated trigonal dihedron
8718:
8616:
8533:
8454:
8375:
8108:
7992:
7909:
7820:
359:define uniform polyhedra to be
183:Kepler–Poinsot polyhedron
13511:. Cambridge University Press.
13215:
13198:
13145:
13134:
13123:
13112:
13103:
12337:Alternated cantellation (hrr)
11770:Wythoff construction operators
11595:Cantic snub hexagonal dihedron
11153:Truncated hexagonal hosohedron
8001:Omnitruncated digonal dihedron
706:great dirhombicosidodecahedron
13:
1:
13254:
10268:Truncated pentagonal dihedron
9545:Omnitruncated square dihedron
8957:Cantic snub trigonal dihedron
8624:Truncated trigonal hosohedron
7654:, represented the faces of a
663:and summarized it in a paper
374:McMullen & Schulte (2002)
330:
312:(isohedral) and have regular
13576:Has a visual chart of all 75
13421:, Cambridge University Press
13392:, Springer, pp. 43–70,
11719:
11683:
11624:
11594:
11508:
11478:
11399:
11369:
11284:
11254:
11182:
11152:
11089:
11070:Truncated hexagonal dihedron
11069:
11006:
10990:
10927:
10911:
10701:
10671:
10592:
10562:
10477:
10447:
10375:
10345:
10282:
10267:
10204:
10188:
10125:
10109:
9894:
9858:
9799:
9769:
9689:
9659:
9574:
9544:
9472:
9442:
9378:
9363:
9300:
9284:
9221:
9205:
8986:
8956:
8877:
8841:
8755:
8725:
8653:
8623:
8560:
8540:
8477:
8461:
8398:
8382:
8146:
8116:
8030:
8000:
7936:
7916:
7853:
7832:
7583:Prismatic uniform polyhedron
7471:
7431:
7349:
7305:
7223:
7179:
7106:
7066:
6993:
6953:
6880:
6836:
6776:
6736:
6677:
6637:
6315:
6275:
6220:
6107:
5996:
5855:
5811:
5729:
5685:
5598:
5554:
5473:
5429:
5347:
5303:
5230:
5190:
5118:
5078:
5009:
4963:
4904:
4864:
4805:
4765:
4439:
4395:
4313:
4267:
4185:
4139:
4066:
4022:
3950:
3910:
3837:
3791:
3727:
3683:
3624:
3584:
3262:
3183:
3104:
3019:
2934:
2136:
2029:
1932:
1835:
960:
546:great stellated dodecahedron
542:small stellated dodecahedron
179:small stellated dodecahedron
7:
13398:10.1007/978-94-011-0924-6_3
12982:
9770:Cantic snub square dihedron
9443:Truncated square hosohedron
8541:Truncated trigonal dihedron
7311:Truncated icosidodecahedron
928:(only as spherical tilings)
916:(only as spherical tilings)
10:
14200:
14156:
13583:
13568:Uniform polyhedron gallery
13419:Abstract Regular Polytopes
12675:
12492:
7917:Truncated digonal dihedron
7580:
7306:Omnitruncated dodecahedron
4270:Truncated tetratetrahedron
1004:
962:
689:
402:
347:(Branko GrĂĽnbaum
13369:10.1007/978-3-319-64123-2
13243:10.1007/s00454-001-0078-2
13222:Messer, Peter W. (2002).
13058:Pseudo-uniform polyhedron
13011:List of uniform polyhedra
12174:
11794:
11684:Snub hexagonal hosohedron
11479:Cantic hexagonal dihedron
10807:
10804:
10792:
10785:
10782:
10779:
10776:
10773:
10005:
10002:
9990:
9983:
9980:
9977:
9974:
9971:
9364:Truncated square dihedron
9101:
9098:
9086:
9079:
9076:
9073:
9070:
9067:
8278:
8275:
8263:
8256:
8253:
8250:
8247:
8244:
7723:
7720:
7708:
7701:
7698:
7695:
7692:
7689:
7662:(2 2 2) Dihedral symmetry
7646:. The dihedral symmetry D
7308:Omnitruncated icosahedron
6536:
6533:
6521:
6514:
6511:
6508:
6498:
6488:
6485:
6482:
6459:disdyakis triacontahedron
4659:
4656:
4644:
4637:
4634:
4631:
4623:
4615:
4612:
4609:
4268:Omnitruncated tetrahedron
3488:
3485:
3473:
3466:
3463:
3460:
3452:
3444:
3441:
3438:
2417:
1316:
308:to uniform polyhedra are
13417:; Schulte, Egon (2002),
13068:
12177:
11797:
10805:Face counts by position
10563:Snub pentagonal dihedron
10003:Face counts by position
9099:Face counts by position
8276:Face counts by position
7721:Face counts by position
7180:Cantellated dodecahedron
6534:Face counts by position
5432:Omnitruncated octahedron
4657:Face counts by position
3486:Face counts by position
667:, counting figures 1-80.
538:Kepler–Poinsot polyhedra
408:Regular convex polyhedra
288:Kepler–Poinsot polyhedra
13109:Regular Polytopes, p.13
12999:Quasiregular polyhedron
12170:Alternation operations
11370:Snub hexagonal dihedron
7182:Cantellated icosahedron
5435:Truncated cuboctahedron
4140:Cantellated tetrahedron
4023:Bitruncated tetrahedron
3684:Birectified tetrahedron
731:Example forms from the
692:Uniform star polyhedron
191:uniform star polyhedron
187:snub dodecadodecahedron
64:more precise citations.
13448:10.1098/rsta.1975.0022
13353:Diudea, M. V. (2018),
13314:10.1098/rsta.1954.0003
13277:Longuet-Higgins, M. S.
13033:List of Johnson solids
13004:Semiregular polyhedron
12280:Alternated truncation
12009:(also truncated dual)
11943:
9859:Snub square hosohedron
8842:Snub trigonal dihedron
7433:Snub icosidodecahedron
7185:Rhombicosidodecahedron
6955:Truncated dodecahedron
6837:Rectified dodecahedron
6177:Cantic snub octahedron
5306:Cantellated octahedron
4583:disdyakis dodecahedron
4142:Rhombitetratetrahedron
2780:
2733:
2686:
2639:
2596:
2550:
2506:
2460:
1679:
1632:
1585:
1538:
1495:
1449:
1405:
1359:
810: > 1 and
739:
724:
701:
686:Uniform star polyhedra
528:Regular star polyhedra
429:(c. 417 BC – 369 BC),
344:
292:uniform star polyhedra
13557:The Uniform Polyhedra
11942:
10766:dodecagonal bipyramid
10190:Pentagonal hosohedron
8117:Snub digonal dihedron
7555:(p 2 2) Prismatic , I
7068:Truncated icosahedron
6839:Rectified icosahedron
5816:Truncated tetrahedron
4396:Snub tetratetrahedron
4027:truncated tetrahedron
3912:Truncated tetrahedron
3792:Rectified tetrahedron
2781:
2734:
2687:
2640:
2597:
2551:
2507:
2461:
1680:
1633:
1586:
1539:
1496:
1450:
1406:
1360:
940:(truncated hosohedra)
859:= 3,4,5,... – order 4
762:rectified tetrahedron
730:
723:
699:
583:Of the remaining 53,
569:in 1813 and named by
567:Augustin-Louis Cauchy
521:prisms and antiprisms
494:De divina proportione
477:Piero della Francesca
433:(c. 420–380 BC), and
335:
302:, Skilling's figure.
13538:"Uniform Polyhedron"
13152:Har'el, Zvi (1993).
12631:Half rectified (hr)
12193:Snub rectified (sr)
11688:Triangular antiprism
11074:dodecagonal dihedron
10992:Hexagonal hosohedron
10567:Pentagonal antiprism
9660:Snub square dihedron
8846:Triangular antiprism
6409:icosahedral symmetry
6403:icosahedral symmetry
6068:Truncated octahedron
5192:Truncated octahedron
4966:Rectified octahedron
4274:truncated octahedron
3361:tetrahedral symmetry
3355:tetrahedral symmetry
2916:p. 2.p. 2
2744:
2697:
2650:
2607:
2561:
2517:
2471:
2428:
1643:
1596:
1549:
1506:
1460:
1416:
1370:
1327:
897:creates the prisms.
843:Icosahedral symmetry
831:Tetrahedral symmetry
744:Wythoff construction
604:H.C. Longuet-Higgins
600:M.S. Longuet-Higgins
497:in 1509, adding the
471:Pappus of Alexandria
14151:pentagonal polytope
14050:Uniform 10-polytope
13610:Fundamental convex
13440:1975RSPTA.278..111S
13306:1954RSPTA.246..401C
13285:"Uniform polyhedra"
13162:Geometriae Dedicata
12171:
12115:Cantitruncated (tr)
11374:Hexagonal antiprism
10111:Pentagonal dihedron
9955:decagonal bipyramid
9051:octagonal bipyramid
8463:Trigonal hosohedron
8223:hexagonal bipyramid
6181:Rhombicuboctahedron
5309:Rhombicuboctahedron
4533:octahedral symmetry
4527:octahedral symmetry
3412:tetrakis hexahedron
845:(5 3 2) – order 120
837:Octahedral symmetry
806: > 1,
802: > 1,
625:published his book
499:rhombicuboctahedron
425:(c. 424 – 348 BC),
219:—there is an
163:regular tetrahedron
14020:Uniform 9-polytope
13970:Uniform 8-polytope
13920:Uniform 7-polytope
13877:Uniform 6-polytope
13847:Uniform 5-polytope
13807:Uniform polychoron
13770:Uniform polyhedron
13618:in dimensions 2–10
13535:Weisstein, Eric W.
13208:Uniform Polyhedra.
13176:10.1007/BF01263494
12994:Regular polyhedron
12169:
11944:
10913:Hexagonal dihedron
10272:decagonal dihedron
9368:octagonal dihedron
8545:hexagonal dihedron
7839:digonal hosohedron
7566:dihedral symmetry)
6422:or , as well as a
6277:Snub cuboctahedron
5430:Omnitruncated cube
4546:or , as well as a
3375:or , as well as a
2776:
2770:
2729:
2723:
2682:
2676:
2635:
2629:
2592:
2586:
2546:
2540:
2502:
2496:
2456:
2450:
1675:
1669:
1628:
1622:
1581:
1575:
1534:
1528:
1491:
1485:
1445:
1439:
1401:
1395:
1355:
1349:
839:(4 3 2) – order 48
833:(3 3 2) – order 24
740:
725:
702:
556:great dodecahedron
517:Archimedean solids
467:Archimedean solids
437:(fl. 300 BC). The
276:Archimedean solids
205:uniform polyhedron
14184:Uniform polyhedra
14172:
14171:
14159:Polytope families
13616:uniform polytopes
13564:Uniform Polyhedra
13562:Virtual Polyhedra
13518:978-0-521-09859-5
13509:Polyhedron Models
13505:Wenninger, Magnus
13434:(1278): 111–135.
13407:978-94-010-4398-4
13378:978-3-319-64123-2
12980:
12979:
12167:
12166:
11767:
11766:
11259:Dodecagonal prism
10760:dihedral symmetry
10753:
10752:
9949:dihedral symmetry
9942:
9941:
9863:Digonal antiprism
9286:square hosohedron
9045:dihedral symmetry
9038:
9037:
8384:Trigonal dihedron
8217:dihedral symmetry
8210:
8209:
7589:dihedral symmetry
7552:
7551:
6842:Icosidodecahedron
6396:
6395:
4520:
4519:
3348:
3347:
2170:
2127:
2126:
997:
875:dihedral symmetry
849:Dihedral symmetry
710:Schwarz triangles
628:Polyhedron models
560:great icosahedron
507:Leonardo da Vinci
378:abstract polytope
361:vertex-transitive
217:vertex-transitive
175:Archimedean solid
90:
89:
82:
18:Uniform polyhedra
16:(Redirected from
14191:
14163:Regular polytope
13724:
13713:
13702:
13661:
13604:
13597:
13590:
13581:
13580:
13548:
13547:
13522:
13500:
13483:
13422:
13410:
13381:
13349:
13300:(916): 401–450.
13289:
13281:Miller, J. C. P.
13248:
13247:
13245:
13219:
13213:
13202:
13196:
13186:Kaleido software
13180:
13178:
13158:
13149:
13143:
13138:
13132:
13127:
13121:
13116:
13110:
13107:
13101:
13095:
13089:
13079:
13043:Polyhedron model
12976:
12975:
12974:
12970:
12969:
12965:
12964:
12958:
12957:
12956:
12952:
12951:
12947:
12946:
12940:
12939:
12938:
12934:
12933:
12929:
12928:
12924:
12923:
12919:
12918:
12911:
12910:
12909:
12905:
12904:
12900:
12899:
12895:
12894:
12890:
12889:
12881:
12880:
12879:
12875:
12874:
12870:
12869:
12865:
12864:
12860:
12859:
12855:
12854:
12850:
12849:
12834:
12833:
12832:
12828:
12827:
12823:
12822:
12816:
12815:
12814:
12810:
12809:
12805:
12804:
12798:
12797:
12796:
12792:
12791:
12787:
12786:
12782:
12781:
12777:
12776:
12769:
12768:
12767:
12763:
12762:
12758:
12757:
12753:
12752:
12748:
12747:
12741:
12740:
12739:
12735:
12734:
12730:
12729:
12725:
12724:
12720:
12719:
12713:
12712:
12711:
12707:
12706:
12702:
12701:
12697:
12696:
12692:
12691:
12687:
12686:
12682:
12681:
12673:
12672:
12671:
12667:
12666:
12662:
12661:
12657:
12656:
12652:
12651:
12647:
12646:
12642:
12641:
12626:
12625:
12624:
12620:
12619:
12615:
12614:
12610:
12609:
12601:
12600:
12599:
12595:
12594:
12590:
12589:
12585:
12584:
12580:
12579:
12575:
12574:
12551:
12550:
12549:
12545:
12544:
12540:
12539:
12535:
12534:
12528:
12527:
12526:
12522:
12521:
12517:
12516:
12512:
12511:
12507:
12506:
12502:
12501:
12490:
12489:
12488:
12484:
12483:
12479:
12478:
12474:
12473:
12469:
12468:
12464:
12463:
12448:
12447:
12446:
12442:
12441:
12437:
12436:
12432:
12431:
12427:
12426:
12419:
12418:
12417:
12413:
12412:
12408:
12407:
12403:
12402:
12398:
12397:
12393:
12392:
12388:
12387:
12379:
12378:
12377:
12373:
12372:
12368:
12367:
12363:
12362:
12358:
12357:
12353:
12352:
12348:
12347:
12330:
12329:
12328:
12324:
12323:
12319:
12318:
12314:
12313:
12309:
12308:
12304:
12303:
12277:
12276:
12275:
12271:
12270:
12266:
12265:
12261:
12260:
12256:
12255:
12251:
12250:
12235:
12225:
12224:
12223:
12219:
12218:
12214:
12213:
12209:
12208:
12204:
12203:
12172:
12168:
12159:
12158:
12157:
12153:
12152:
12148:
12147:
12143:
12142:
12138:
12137:
12110:
12096:
12095:
12094:
12090:
12089:
12085:
12084:
12080:
12079:
12075:
12074:
12047:
12046:
12045:
12041:
12040:
12036:
12035:
12031:
12030:
12026:
12025:
12007:Bitruncated (2t)
12002:
11992:
11991:
11990:
11986:
11985:
11981:
11980:
11976:
11975:
11971:
11970:
11935:
11934:
11933:
11929:
11928:
11924:
11923:
11919:
11918:
11914:
11913:
11891:Birectified (2r)
11882:
11881:
11880:
11876:
11875:
11871:
11870:
11866:
11865:
11861:
11860:
11833:
11832:
11831:
11827:
11826:
11822:
11821:
11817:
11816:
11812:
11811:
11774:
11773:
11746:
11745:
11744:
11740:
11739:
11735:
11734:
11730:
11729:
11725:
11724:
11717:
11710:
11703:
11651:
11650:
11649:
11645:
11644:
11640:
11639:
11635:
11634:
11630:
11629:
11622:
11615:
11608:
11563:
11562:
11561:
11557:
11556:
11552:
11551:
11547:
11546:
11542:
11541:
11535:
11534:
11533:
11529:
11528:
11524:
11523:
11519:
11518:
11514:
11513:
11506:
11499:
11492:
11483:Triangular prism
11451:
11446:
11435:
11426:
11425:
11424:
11420:
11419:
11415:
11414:
11410:
11409:
11405:
11404:
11397:
11390:
11383:
11346:
11335:
11324:
11311:
11310:
11309:
11305:
11304:
11300:
11299:
11295:
11294:
11290:
11289:
11282:
11275:
11268:
11229:
11218:
11209:
11208:
11207:
11203:
11202:
11198:
11197:
11193:
11192:
11188:
11187:
11180:
11173:
11166:
11125:
11116:
11115:
11114:
11110:
11109:
11105:
11104:
11100:
11099:
11095:
11094:
11085:
11046:
11033:
11032:
11031:
11027:
11026:
11022:
11021:
11017:
11016:
11012:
11011:
11002:
10963:
10954:
10953:
10952:
10948:
10947:
10943:
10942:
10938:
10937:
10933:
10932:
10923:
10889:
10888:
10887:
10883:
10882:
10878:
10877:
10873:
10872:
10860:
10859:
10858:
10854:
10853:
10849:
10848:
10836:
10835:
10834:
10830:
10829:
10825:
10824:
10820:
10819:
10771:
10770:
10728:
10727:
10726:
10722:
10721:
10717:
10716:
10712:
10711:
10707:
10706:
10699:
10692:
10685:
10676:Pentagonal prism
10644:
10639:
10628:
10619:
10618:
10617:
10613:
10612:
10608:
10607:
10603:
10602:
10598:
10597:
10590:
10583:
10576:
10539:
10528:
10517:
10504:
10503:
10502:
10498:
10497:
10493:
10492:
10488:
10487:
10483:
10482:
10475:
10468:
10461:
10422:
10411:
10402:
10401:
10400:
10396:
10395:
10391:
10390:
10386:
10385:
10381:
10380:
10373:
10366:
10359:
10350:pentagonal prism
10318:
10309:
10308:
10307:
10303:
10302:
10298:
10297:
10293:
10292:
10288:
10287:
10244:
10231:
10230:
10229:
10225:
10224:
10220:
10219:
10215:
10214:
10210:
10209:
10200:
10161:
10152:
10151:
10150:
10146:
10145:
10141:
10140:
10136:
10135:
10131:
10130:
10121:
10087:
10086:
10085:
10081:
10080:
10076:
10075:
10071:
10070:
10058:
10057:
10056:
10052:
10051:
10047:
10046:
10034:
10033:
10032:
10028:
10027:
10023:
10022:
10018:
10017:
9969:
9968:
9965:
9921:
9920:
9919:
9915:
9914:
9910:
9909:
9905:
9904:
9900:
9899:
9892:
9885:
9878:
9826:
9825:
9824:
9820:
9819:
9815:
9814:
9810:
9809:
9805:
9804:
9797:
9790:
9783:
9741:
9736:
9725:
9716:
9715:
9714:
9710:
9709:
9705:
9704:
9700:
9699:
9695:
9694:
9687:
9680:
9673:
9664:Square antiprism
9636:
9625:
9614:
9601:
9600:
9599:
9595:
9594:
9590:
9589:
9585:
9584:
9580:
9579:
9572:
9565:
9558:
9519:
9508:
9499:
9498:
9497:
9493:
9492:
9488:
9487:
9483:
9482:
9478:
9477:
9470:
9463:
9456:
9414:
9405:
9404:
9403:
9399:
9398:
9394:
9393:
9389:
9388:
9384:
9383:
9340:
9327:
9326:
9325:
9321:
9320:
9316:
9315:
9311:
9310:
9306:
9305:
9296:
9257:
9248:
9247:
9246:
9242:
9241:
9237:
9236:
9232:
9231:
9227:
9226:
9217:
9183:
9182:
9181:
9177:
9176:
9172:
9171:
9167:
9166:
9154:
9153:
9152:
9148:
9147:
9143:
9142:
9130:
9129:
9128:
9124:
9123:
9119:
9118:
9114:
9113:
9065:
9064:
9061:
9013:
9012:
9011:
9007:
9006:
9002:
9001:
8997:
8996:
8992:
8991:
8984:
8977:
8970:
8961:Triangular prism
8929:
8924:
8913:
8904:
8903:
8902:
8898:
8897:
8893:
8892:
8888:
8887:
8883:
8882:
8875:
8868:
8861:
8817:
8806:
8795:
8782:
8781:
8780:
8776:
8775:
8771:
8770:
8766:
8765:
8761:
8760:
8753:
8746:
8739:
8700:
8689:
8680:
8679:
8678:
8674:
8673:
8669:
8668:
8664:
8663:
8659:
8658:
8651:
8644:
8637:
8628:Triangular prism
8596:
8587:
8586:
8585:
8581:
8580:
8576:
8575:
8571:
8570:
8566:
8565:
8556:
8517:
8504:
8503:
8502:
8498:
8497:
8493:
8492:
8488:
8487:
8483:
8482:
8473:
8434:
8425:
8424:
8423:
8419:
8418:
8414:
8413:
8409:
8408:
8404:
8403:
8394:
8360:
8359:
8358:
8354:
8353:
8349:
8348:
8344:
8343:
8331:
8330:
8329:
8325:
8324:
8320:
8319:
8307:
8306:
8305:
8301:
8300:
8296:
8295:
8291:
8290:
8242:
8241:
8238:
8233:
8189:
8184:
8173:
8172:
8171:
8167:
8166:
8162:
8161:
8157:
8156:
8152:
8151:
8144:
8137:
8130:
8092:
8081:
8070:
8057:
8056:
8055:
8051:
8050:
8046:
8045:
8041:
8040:
8036:
8035:
8028:
8021:
8014:
7972:
7963:
7962:
7961:
7957:
7956:
7952:
7951:
7947:
7946:
7942:
7941:
7932:
7889:
7880:
7879:
7878:
7874:
7873:
7869:
7868:
7864:
7863:
7859:
7858:
7849:
7834:Digonal dihedron
7805:
7804:
7803:
7799:
7798:
7794:
7793:
7789:
7788:
7776:
7775:
7774:
7770:
7769:
7765:
7764:
7752:
7751:
7750:
7746:
7745:
7741:
7740:
7736:
7735:
7687:
7686:
7683:
7678:
7668:square bipyramid
7634:
7633:
7632:
7628:
7627:
7623:
7622:
7618:
7617:
7613:
7612:
7578:
7577:
7573:
7535:
7523:
7518:
7507:
7498:
7497:
7496:
7492:
7491:
7487:
7486:
7482:
7481:
7477:
7476:
7469:
7462:
7455:
7448:
7441:
7411:
7400:
7389:
7376:
7375:
7374:
7370:
7369:
7365:
7364:
7360:
7359:
7355:
7354:
7347:
7340:
7333:
7326:
7319:
7285:
7274:
7263:
7250:
7249:
7248:
7244:
7243:
7239:
7238:
7234:
7233:
7229:
7228:
7221:
7214:
7207:
7200:
7193:
7159:
7146:
7133:
7132:
7131:
7127:
7126:
7122:
7121:
7117:
7116:
7112:
7111:
7104:
7097:
7090:
7083:
7076:
7046:
7033:
7020:
7019:
7018:
7014:
7013:
7009:
7008:
7004:
7003:
6999:
6998:
6991:
6984:
6977:
6970:
6963:
6933:
6920:
6907:
6906:
6905:
6901:
6900:
6896:
6895:
6891:
6890:
6886:
6885:
6878:
6871:
6864:
6857:
6850:
6816:
6803:
6802:
6801:
6797:
6796:
6792:
6791:
6787:
6786:
6782:
6781:
6774:
6767:
6760:
6753:
6746:
6713:
6704:
6703:
6702:
6698:
6697:
6693:
6692:
6688:
6687:
6683:
6682:
6675:
6668:
6661:
6654:
6647:
6618:
6617:
6616:
6612:
6611:
6607:
6606:
6602:
6601:
6589:
6588:
6587:
6583:
6582:
6578:
6577:
6565:
6564:
6563:
6559:
6558:
6554:
6553:
6549:
6548:
6480:
6479:
6477:
6472:
6467:
6453:
6452:
6451:
6447:
6446:
6442:
6441:
6437:
6436:
6432:
6431:
6379:
6367:
6362:
6351:
6342:
6341:
6340:
6336:
6335:
6331:
6330:
6326:
6325:
6321:
6320:
6313:
6306:
6299:
6292:
6285:
6247:
6246:
6245:
6241:
6240:
6236:
6235:
6231:
6230:
6226:
6225:
6218:
6211:
6204:
6197:
6190:
6153:
6152:
6151:
6147:
6146:
6142:
6141:
6134:
6133:
6132:
6128:
6127:
6123:
6122:
6118:
6117:
6113:
6112:
6105:
6098:
6091:
6084:
6077:
6042:
6041:
6040:
6036:
6035:
6031:
6030:
6023:
6022:
6021:
6017:
6016:
6012:
6011:
6007:
6006:
6002:
6001:
5994:
5987:
5980:
5973:
5966:
5931:
5914:
5901:
5900:
5899:
5895:
5894:
5890:
5889:
5882:
5881:
5880:
5876:
5875:
5871:
5870:
5866:
5865:
5861:
5860:
5853:
5846:
5839:
5832:
5825:
5784:
5775:
5774:
5773:
5769:
5768:
5764:
5763:
5756:
5755:
5754:
5750:
5749:
5745:
5744:
5740:
5739:
5735:
5734:
5727:
5720:
5713:
5706:
5699:
5666:
5655:
5644:
5643:
5642:
5638:
5637:
5633:
5632:
5625:
5624:
5623:
5619:
5618:
5614:
5613:
5609:
5608:
5604:
5603:
5596:
5589:
5582:
5575:
5568:
5535:
5524:
5513:
5500:
5499:
5498:
5494:
5493:
5489:
5488:
5484:
5483:
5479:
5478:
5471:
5464:
5457:
5450:
5443:
5409:
5398:
5387:
5374:
5373:
5372:
5368:
5367:
5363:
5362:
5358:
5357:
5353:
5352:
5345:
5338:
5331:
5324:
5317:
5304:Cantellated cube
5283:
5270:
5257:
5256:
5255:
5251:
5250:
5246:
5245:
5241:
5240:
5236:
5235:
5228:
5221:
5214:
5207:
5200:
5171:
5158:
5145:
5144:
5143:
5139:
5138:
5134:
5133:
5129:
5128:
5124:
5123:
5116:
5109:
5102:
5095:
5088:
5058:
5045:
5036:
5035:
5034:
5030:
5029:
5025:
5024:
5020:
5019:
5015:
5014:
5007:
5000:
4993:
4986:
4979:
4944:
4931:
4930:
4929:
4925:
4924:
4920:
4919:
4915:
4914:
4910:
4909:
4902:
4895:
4888:
4881:
4874:
4841:
4832:
4831:
4830:
4826:
4825:
4821:
4820:
4816:
4815:
4811:
4810:
4803:
4796:
4789:
4782:
4775:
4746:
4745:
4744:
4740:
4739:
4735:
4734:
4730:
4729:
4717:
4716:
4715:
4711:
4710:
4706:
4705:
4701:
4700:
4688:
4687:
4686:
4682:
4681:
4677:
4676:
4672:
4671:
4607:
4606:
4603:
4598:
4593:
4577:
4576:
4575:
4571:
4570:
4566:
4565:
4561:
4560:
4556:
4555:
4503:
4491:
4486:
4475:
4466:
4465:
4464:
4460:
4459:
4455:
4454:
4450:
4449:
4445:
4444:
4437:
4430:
4423:
4416:
4409:
4375:
4364:
4353:
4340:
4339:
4338:
4334:
4333:
4329:
4328:
4324:
4323:
4319:
4318:
4311:
4304:
4297:
4290:
4283:
4247:
4236:
4225:
4212:
4211:
4210:
4206:
4205:
4201:
4200:
4196:
4195:
4191:
4190:
4183:
4176:
4169:
4162:
4155:
4119:
4106:
4093:
4092:
4091:
4087:
4086:
4082:
4081:
4077:
4076:
4072:
4071:
4064:
4057:
4050:
4043:
4036:
4003:
3990:
3977:
3976:
3975:
3971:
3970:
3966:
3965:
3961:
3960:
3956:
3955:
3948:
3941:
3934:
3927:
3920:
3890:
3877:
3864:
3863:
3862:
3858:
3857:
3853:
3852:
3848:
3847:
3843:
3842:
3835:
3828:
3821:
3814:
3807:
3794:Tetratetrahedron
3771:
3754:
3753:
3752:
3748:
3747:
3743:
3742:
3738:
3737:
3733:
3732:
3725:
3718:
3711:
3704:
3697:
3660:
3651:
3650:
3649:
3645:
3644:
3640:
3639:
3635:
3634:
3630:
3629:
3622:
3615:
3608:
3601:
3594:
3565:
3564:
3563:
3559:
3558:
3554:
3553:
3549:
3548:
3536:
3535:
3534:
3530:
3529:
3525:
3524:
3512:
3511:
3510:
3506:
3505:
3501:
3500:
3436:
3435:
3432:
3427:
3422:
3406:
3405:
3404:
3400:
3399:
3395:
3394:
3390:
3389:
3385:
3384:
3342:
3333:
3324:
3313:
3304:
3295:
3286:
3275:
3257:
3248:
3239:
3228:
3219:
3210:
3196:
3178:
3169:
3160:
3149:
3140:
3131:
3117:
3099:
3090:
3081:
3070:
3061:
3052:
3043:
3032:
3014:
3005:
2996:
2985:
2976:
2967:
2958:
2947:
2785:
2783:
2782:
2777:
2775:
2774:
2738:
2736:
2735:
2730:
2728:
2727:
2691:
2689:
2688:
2683:
2681:
2680:
2644:
2642:
2641:
2636:
2634:
2633:
2601:
2599:
2598:
2593:
2591:
2590:
2555:
2553:
2552:
2547:
2545:
2544:
2511:
2509:
2508:
2503:
2501:
2500:
2465:
2463:
2462:
2457:
2455:
2454:
2413:
2412:
2411:
2407:
2406:
2402:
2401:
2397:
2396:
2392:
2391:
2384:
2383:
2382:
2378:
2377:
2373:
2372:
2368:
2367:
2363:
2362:
2355:
2354:
2353:
2349:
2348:
2344:
2343:
2339:
2338:
2334:
2333:
2326:
2325:
2324:
2320:
2319:
2315:
2314:
2310:
2309:
2305:
2304:
2297:
2296:
2295:
2291:
2290:
2286:
2285:
2281:
2280:
2276:
2275:
2268:
2267:
2266:
2262:
2261:
2257:
2256:
2252:
2251:
2247:
2246:
2239:
2238:
2237:
2233:
2232:
2228:
2227:
2223:
2222:
2218:
2217:
2210:
2209:
2208:
2204:
2203:
2199:
2198:
2194:
2193:
2189:
2188:
2168:
2137:
2119:
2108:
2097:
2086:
2075:
2064:
2053:
2042:
2022:
2011:
2000:
1989:
1978:
1967:
1956:
1945:
1925:
1914:
1903:
1892:
1881:
1870:
1859:
1848:
1832:3.3.p. 3.q
1684:
1682:
1681:
1676:
1674:
1673:
1637:
1635:
1634:
1629:
1627:
1626:
1590:
1588:
1587:
1582:
1580:
1579:
1543:
1541:
1540:
1535:
1533:
1532:
1500:
1498:
1497:
1492:
1490:
1489:
1454:
1452:
1451:
1446:
1444:
1443:
1410:
1408:
1407:
1402:
1400:
1399:
1364:
1362:
1361:
1356:
1354:
1353:
1312:
1311:
1310:
1306:
1305:
1301:
1300:
1294:
1293:
1292:
1288:
1287:
1283:
1282:
1278:
1277:
1273:
1272:
1265:
1264:
1263:
1259:
1258:
1254:
1253:
1247:
1246:
1245:
1241:
1240:
1236:
1235:
1231:
1230:
1226:
1225:
1218:
1217:
1216:
1212:
1211:
1207:
1206:
1200:
1199:
1198:
1194:
1193:
1189:
1188:
1184:
1183:
1179:
1178:
1171:
1170:
1169:
1165:
1164:
1160:
1159:
1155:
1154:
1150:
1149:
1142:
1141:
1140:
1136:
1135:
1131:
1130:
1126:
1125:
1121:
1120:
1113:
1112:
1111:
1107:
1106:
1102:
1101:
1095:
1094:
1093:
1089:
1088:
1084:
1083:
1079:
1078:
1074:
1073:
1066:
1065:
1064:
1060:
1059:
1055:
1054:
1050:
1049:
1045:
1044:
1037:
1036:
1035:
1031:
1030:
1026:
1025:
1021:
1020:
1016:
1015:
995:
961:
825:
623:Magnus Wenninger
501:, calling it an
431:Timaeus of Locri
352:
341:
325:uniform polytope
209:regular polygons
159:square antiprism
155:triangular prism
147:
138:
129:
118:
109:
100:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
14199:
14198:
14194:
14193:
14192:
14190:
14189:
14188:
14174:
14173:
14142:
14135:
14128:
14011:
14004:
13997:
13961:
13954:
13947:
13911:
13904:
13738:Regular polygon
13731:
13722:
13715:
13711:
13704:
13700:
13691:
13682:
13675:
13671:
13659:
13653:
13649:
13637:
13619:
13608:
13529:
13519:
13415:McMullen, Peter
13408:
13379:
13287:
13257:
13252:
13251:
13220:
13216:
13203:
13199:
13156:
13150:
13146:
13139:
13135:
13128:
13124:
13117:
13113:
13108:
13104:
13096:
13092:
13080:
13076:
13071:
12985:
12972:
12967:
12962:
12960:
12954:
12949:
12944:
12942:
12936:
12931:
12926:
12921:
12916:
12914:
12912:
12907:
12902:
12897:
12892:
12887:
12885:
12877:
12872:
12867:
12862:
12857:
12852:
12847:
12845:
12830:
12825:
12820:
12818:
12812:
12807:
12802:
12800:
12794:
12789:
12784:
12779:
12774:
12772:
12770:
12765:
12760:
12755:
12750:
12745:
12743:
12737:
12732:
12727:
12722:
12717:
12715:
12709:
12704:
12699:
12694:
12689:
12684:
12679:
12677:
12669:
12664:
12659:
12654:
12649:
12644:
12639:
12637:
12622:
12617:
12612:
12607:
12605:
12597:
12592:
12587:
12582:
12577:
12572:
12570:
12566:
12559:
12547:
12542:
12537:
12532:
12530:
12524:
12519:
12514:
12509:
12504:
12499:
12497:
12486:
12481:
12476:
12471:
12466:
12461:
12459:
12444:
12439:
12434:
12429:
12424:
12422:
12420:
12415:
12410:
12405:
12400:
12395:
12390:
12385:
12383:
12375:
12370:
12365:
12360:
12355:
12350:
12345:
12343:
12326:
12321:
12316:
12311:
12306:
12301:
12299:
12295:
12288:
12273:
12268:
12263:
12258:
12253:
12248:
12246:
12230:
12221:
12216:
12211:
12206:
12201:
12199:
12183:
12155:
12150:
12145:
12140:
12135:
12133:
12129:
12125:
12116:
12105:
12092:
12087:
12082:
12077:
12072:
12070:
12059:
12043:
12038:
12033:
12028:
12023:
12021:
12017:
12013:
12008:
11997:
11988:
11983:
11978:
11973:
11968:
11966:
11962:
11958:
11931:
11926:
11921:
11916:
11911:
11909:
11905:
11901:
11892:
11878:
11873:
11868:
11863:
11858:
11856:
11852:
11848:
11829:
11824:
11819:
11814:
11809:
11807:
11803:
11799:
11785:
11772:
11748:s{2,6}=sr{2,3}
11747:
11742:
11737:
11732:
11727:
11722:
11720:
11691:
11685:
11681:
11680:
11656:
11652:
11647:
11642:
11637:
11632:
11627:
11625:
11599:Hexagonal prism
11596:
11592:
11568:
11564:
11559:
11554:
11549:
11544:
11539:
11537:
11531:
11526:
11521:
11516:
11511:
11509:
11480:
11476:
11452:
11436:
11427:
11422:
11417:
11412:
11407:
11402:
11400:
11371:
11367:
11347:
11336:
11325:
11316:
11312:
11307:
11302:
11297:
11292:
11287:
11285:
11256:
11252:
11230:
11219:
11210:
11205:
11200:
11195:
11190:
11185:
11183:
11157:hexagonal prism
11154:
11150:
11126:
11117:
11112:
11107:
11102:
11097:
11092:
11090:
11071:
11067:
11047:
11034:
11029:
11024:
11019:
11014:
11009:
11007:
10988:
10964:
10955:
10950:
10945:
10940:
10935:
10930:
10928:
10909:
10891:
10890:
10885:
10880:
10875:
10870:
10868:
10867:
10862:
10861:
10856:
10851:
10846:
10844:
10843:
10838:
10837:
10832:
10827:
10822:
10817:
10815:
10814:
10808:Element counts
10801:
10796:
10788:
10762:
10759:
10733:
10729:
10724:
10719:
10714:
10709:
10704:
10702:
10673:
10669:
10645:
10629:
10620:
10615:
10610:
10605:
10600:
10595:
10593:
10564:
10560:
10540:
10529:
10518:
10509:
10505:
10500:
10495:
10490:
10485:
10480:
10478:
10452:Decagonal prism
10449:
10445:
10423:
10412:
10403:
10398:
10393:
10388:
10383:
10378:
10376:
10347:
10343:
10319:
10310:
10305:
10300:
10295:
10290:
10285:
10283:
10269:
10265:
10245:
10232:
10227:
10222:
10217:
10212:
10207:
10205:
10186:
10162:
10153:
10148:
10143:
10138:
10133:
10128:
10126:
10107:
10089:
10088:
10083:
10078:
10073:
10068:
10066:
10065:
10060:
10059:
10054:
10049:
10044:
10042:
10041:
10036:
10035:
10030:
10025:
10020:
10015:
10013:
10012:
10006:Element counts
9999:
9994:
9986:
9951:
9948:
9923:s{2,4}=sr{2,2}
9922:
9917:
9912:
9907:
9902:
9897:
9895:
9866:
9860:
9856:
9855:
9831:
9827:
9822:
9817:
9812:
9807:
9802:
9800:
9771:
9767:
9766:
9742:
9726:
9717:
9712:
9707:
9702:
9697:
9692:
9690:
9661:
9657:
9637:
9626:
9615:
9606:
9602:
9597:
9592:
9587:
9582:
9577:
9575:
9549:Octagonal prism
9546:
9542:
9520:
9509:
9500:
9495:
9490:
9485:
9480:
9475:
9473:
9444:
9440:
9439:
9415:
9406:
9401:
9396:
9391:
9386:
9381:
9379:
9365:
9361:
9341:
9328:
9323:
9318:
9313:
9308:
9303:
9301:
9282:
9258:
9249:
9244:
9239:
9234:
9229:
9224:
9222:
9207:square dihedron
9203:
9185:
9184:
9179:
9174:
9169:
9164:
9162:
9161:
9156:
9155:
9150:
9145:
9140:
9138:
9137:
9132:
9131:
9126:
9121:
9116:
9111:
9109:
9108:
9102:Element counts
9095:
9090:
9082:
9047:
9044:
9018:
9014:
9009:
9004:
8999:
8994:
8989:
8987:
8958:
8954:
8930:
8914:
8905:
8900:
8895:
8890:
8885:
8880:
8878:
8849:
8843:
8839:
8838:
8818:
8807:
8796:
8787:
8783:
8778:
8773:
8768:
8763:
8758:
8756:
8730:Hexagonal prism
8727:
8723:
8701:
8690:
8681:
8676:
8671:
8666:
8661:
8656:
8654:
8625:
8621:
8597:
8588:
8583:
8578:
8573:
8568:
8563:
8561:
8542:
8538:
8518:
8505:
8500:
8495:
8490:
8485:
8480:
8478:
8459:
8435:
8426:
8421:
8416:
8411:
8406:
8401:
8399:
8380:
8362:
8361:
8356:
8351:
8346:
8341:
8339:
8338:
8333:
8332:
8327:
8322:
8317:
8315:
8314:
8309:
8308:
8303:
8298:
8293:
8288:
8286:
8285:
8279:Element counts
8272:
8267:
8259:
8219:
8216:
8190:
8174:
8169:
8164:
8159:
8154:
8149:
8147:
8118:
8114:
8113:
8093:
8082:
8071:
8062:
8058:
8053:
8048:
8043:
8038:
8033:
8031:
8002:
7998:
7997:
7973:
7964:
7959:
7954:
7949:
7944:
7939:
7937:
7921:square dihedron
7918:
7914:
7890:
7881:
7876:
7871:
7866:
7861:
7856:
7854:
7837:
7830:
7826:
7825:
7807:
7806:
7801:
7796:
7791:
7786:
7784:
7783:
7778:
7777:
7772:
7767:
7762:
7760:
7759:
7754:
7753:
7748:
7743:
7738:
7733:
7731:
7730:
7724:Element counts
7717:
7712:
7704:
7664:
7649:
7645:
7641:
7630:
7625:
7620:
7615:
7610:
7608:
7605:Coxeter diagram
7602:
7585:
7579:
7575:
7571:
7569:
7568:
7565:
7558:
7536:
7524:
7508:
7499:
7494:
7489:
7484:
7479:
7474:
7472:
7412:
7401:
7390:
7381:
7377:
7372:
7367:
7362:
7357:
7352:
7350:
7309:
7307:
7286:
7275:
7264:
7255:
7251:
7246:
7241:
7236:
7231:
7226:
7224:
7183:
7181:
7160:
7147:
7138:
7134:
7129:
7124:
7119:
7114:
7109:
7107:
7047:
7034:
7025:
7021:
7016:
7011:
7006:
7001:
6996:
6994:
6934:
6921:
6912:
6908:
6903:
6898:
6893:
6888:
6883:
6881:
6840:
6838:
6817:
6804:
6799:
6794:
6789:
6784:
6779:
6777:
6714:
6705:
6700:
6695:
6690:
6685:
6680:
6678:
6620:
6619:
6614:
6609:
6604:
6599:
6597:
6596:
6591:
6590:
6585:
6580:
6575:
6573:
6572:
6567:
6566:
6561:
6556:
6551:
6546:
6544:
6543:
6537:Element counts
6530:
6525:
6517:
6506:
6504:
6500:
6496:
6494:
6490:
6462:
6449:
6444:
6439:
6434:
6429:
6427:
6424:Coxeter diagram
6421:
6405:
6402:
6380:
6368:
6352:
6343:
6338:
6333:
6328:
6323:
6318:
6316:
6252:
6248:
6243:
6238:
6233:
6228:
6223:
6221:
6178:
6154:
6149:
6144:
6139:
6137:
6135:
6130:
6125:
6120:
6115:
6110:
6108:
6043:
6038:
6033:
6028:
6026:
6024:
6019:
6014:
6009:
6004:
5999:
5997:
5936:
5932:
5919:
5915:
5906:
5902:
5897:
5892:
5887:
5885:
5883:
5878:
5873:
5868:
5863:
5858:
5856:
5813:
5789:
5785:
5776:
5771:
5766:
5761:
5759:
5757:
5752:
5747:
5742:
5737:
5732:
5730:
5687:
5667:
5656:
5646:s{3,4}=sr{3,3}
5645:
5640:
5635:
5630:
5628:
5626:
5621:
5616:
5611:
5606:
5601:
5599:
5556:
5555:Snub octahedron
5536:
5525:
5514:
5505:
5501:
5496:
5491:
5486:
5481:
5476:
5474:
5433:
5431:
5410:
5399:
5388:
5379:
5375:
5370:
5365:
5360:
5355:
5350:
5348:
5307:
5305:
5284:
5271:
5262:
5258:
5253:
5248:
5243:
5238:
5233:
5231:
5172:
5159:
5150:
5146:
5141:
5136:
5131:
5126:
5121:
5119:
5059:
5046:
5037:
5032:
5027:
5022:
5017:
5012:
5010:
4967:
4965:
4945:
4932:
4927:
4922:
4917:
4912:
4907:
4905:
4842:
4833:
4828:
4823:
4818:
4813:
4808:
4806:
4748:
4747:
4742:
4737:
4732:
4727:
4725:
4724:
4719:
4718:
4713:
4708:
4703:
4698:
4696:
4695:
4690:
4689:
4684:
4679:
4674:
4669:
4667:
4666:
4660:Element counts
4653:
4648:
4640:
4629:
4625:
4621:
4617:
4573:
4568:
4563:
4558:
4553:
4551:
4548:Coxeter diagram
4545:
4529:
4526:
4504:
4492:
4476:
4467:
4462:
4457:
4452:
4447:
4442:
4440:
4397:
4376:
4365:
4354:
4345:
4341:
4336:
4331:
4326:
4321:
4316:
4314:
4271:
4269:
4248:
4237:
4226:
4217:
4213:
4208:
4203:
4198:
4193:
4188:
4186:
4143:
4141:
4120:
4107:
4098:
4094:
4089:
4084:
4079:
4074:
4069:
4067:
4024:
4004:
3991:
3982:
3978:
3973:
3968:
3963:
3958:
3953:
3951:
3891:
3878:
3869:
3865:
3860:
3855:
3850:
3845:
3840:
3838:
3795:
3793:
3772:
3759:
3755:
3750:
3745:
3740:
3735:
3730:
3728:
3685:
3661:
3652:
3647:
3642:
3637:
3632:
3627:
3625:
3567:
3566:
3561:
3556:
3551:
3546:
3544:
3543:
3538:
3537:
3532:
3527:
3522:
3520:
3519:
3514:
3513:
3508:
3503:
3498:
3496:
3495:
3489:Element counts
3482:
3477:
3469:
3458:
3454:
3450:
3446:
3402:
3397:
3392:
3387:
3382:
3380:
3377:Coxeter diagram
3374:
3357:
3354:
3343:
3334:
3325:
3314:
3305:
3296:
3287:
3276:
3267:
3258:
3249:
3240:
3229:
3220:
3211:
3197:
3188:
3179:
3170:
3161:
3150:
3141:
3132:
3118:
3109:
3100:
3091:
3082:
3071:
3062:
3053:
3044:
3033:
3024:
3015:
3006:
2997:
2986:
2977:
2968:
2959:
2948:
2939:
2868:
2861:
2854:
2847:
2840:
2833:
2826:
2819:
2769:
2768:
2762:
2761:
2751:
2750:
2745:
2742:
2741:
2722:
2721:
2715:
2714:
2704:
2703:
2698:
2695:
2694:
2675:
2674:
2668:
2667:
2657:
2656:
2651:
2648:
2647:
2628:
2627:
2611:
2610:
2608:
2605:
2604:
2585:
2584:
2568:
2567:
2562:
2559:
2558:
2539:
2538:
2532:
2531:
2521:
2520:
2518:
2515:
2514:
2495:
2494:
2478:
2477:
2472:
2469:
2468:
2449:
2448:
2432:
2431:
2429:
2426:
2425:
2421:Schläfli symbol
2419:
2409:
2404:
2399:
2394:
2389:
2387:
2380:
2375:
2370:
2365:
2360:
2358:
2351:
2346:
2341:
2336:
2331:
2329:
2322:
2317:
2312:
2307:
2302:
2300:
2293:
2288:
2283:
2278:
2273:
2271:
2264:
2259:
2254:
2249:
2244:
2242:
2235:
2230:
2225:
2220:
2215:
2213:
2206:
2201:
2196:
2191:
2186:
2184:
2180:Coxeter diagram
2166:
2158:
2153:
2120:
2109:
2098:
2087:
2076:
2065:
2054:
2043:
2034:
2023:
2012:
2001:
1990:
1979:
1968:
1957:
1946:
1937:
1926:
1915:
1904:
1893:
1882:
1871:
1860:
1849:
1840:
1776:
1767:
1760:
1753:
1746:
1739:
1732:
1725:
1718:
1668:
1667:
1661:
1660:
1650:
1649:
1644:
1641:
1640:
1621:
1620:
1614:
1613:
1603:
1602:
1597:
1594:
1593:
1574:
1573:
1567:
1566:
1556:
1555:
1550:
1547:
1546:
1527:
1526:
1510:
1509:
1507:
1504:
1503:
1484:
1483:
1467:
1466:
1461:
1458:
1457:
1438:
1437:
1431:
1430:
1420:
1419:
1417:
1414:
1413:
1394:
1393:
1377:
1376:
1371:
1368:
1367:
1348:
1347:
1331:
1330:
1328:
1325:
1324:
1320:Schläfli symbol
1318:
1308:
1303:
1298:
1296:
1295:
1290:
1285:
1280:
1275:
1270:
1268:
1261:
1256:
1251:
1249:
1248:
1243:
1238:
1233:
1228:
1223:
1221:
1214:
1209:
1204:
1202:
1201:
1196:
1191:
1186:
1181:
1176:
1174:
1167:
1162:
1157:
1152:
1147:
1145:
1138:
1133:
1128:
1123:
1118:
1116:
1109:
1104:
1099:
1097:
1096:
1091:
1086:
1081:
1076:
1071:
1069:
1062:
1057:
1052:
1047:
1042:
1040:
1033:
1028:
1023:
1018:
1013:
1011:
1007:Coxeter diagram
993:
985:
980:
959:
951:
939:
927:
915:
811:
718:
694:
688:
639:Skilling (1975)
596:J. C. P. Miller
580:
530:
513:Johannes Kepler
503:icosihexahedron
448:
415:Platonic solids
410:
405:
370:GrĂĽnbaum (1994)
354:
346:
339:
333:
310:face-transitive
272:Platonic solids
241:edge-transitive
197:
196:
195:
194:
150:
149:
148:
140:
139:
131:
130:
121:
120:
119:
111:
110:
102:
101:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
14197:
14187:
14186:
14170:
14169:
14154:
14153:
14144:
14140:
14133:
14126:
14122:
14113:
14096:
14087:
14076:
14075:
14073:
14071:
14066:
14057:
14052:
14046:
14045:
14043:
14041:
14036:
14027:
14022:
14016:
14015:
14013:
14009:
14002:
13995:
13991:
13986:
13977:
13972:
13966:
13965:
13963:
13959:
13952:
13945:
13941:
13936:
13927:
13922:
13916:
13915:
13913:
13909:
13902:
13898:
13893:
13884:
13879:
13873:
13872:
13870:
13868:
13863:
13854:
13849:
13843:
13842:
13833:
13828:
13823:
13814:
13809:
13803:
13802:
13793:
13791:
13786:
13777:
13772:
13766:
13765:
13760:
13755:
13750:
13745:
13740:
13734:
13733:
13729:
13725:
13720:
13709:
13698:
13689:
13680:
13673:
13667:
13657:
13651:
13645:
13639:
13633:
13627:
13621:
13620:
13609:
13607:
13606:
13599:
13592:
13584:
13579:
13578:
13570:
13565:
13559:
13554:
13549:
13528:
13527:External links
13525:
13524:
13523:
13517:
13501:
13491:(8): 139–156.
13484:
13423:
13411:
13406:
13382:
13377:
13350:
13269:
13256:
13253:
13250:
13249:
13236:(3): 353–375.
13214:
13197:
13144:
13133:
13122:
13111:
13102:
13090:
13073:
13072:
13070:
13067:
13066:
13065:
13063:List of shapes
13060:
13055:
13050:
13048:Uniform tiling
13045:
13040:
13035:
13030:
13029:
13028:
13023:
13018:
13008:
13007:
13006:
13001:
12996:
12984:
12981:
12978:
12977:
12882:
12843:
12840:
12836:
12835:
12674:
12635:
12632:
12628:
12627:
12602:
12568:
12564:
12561:
12557:
12553:
12552:
12491:
12457:
12454:
12450:
12449:
12380:
12341:
12338:
12334:
12333:
12331:
12297:
12293:
12290:
12286:
12285:Cantic snub (s
12282:
12281:
12278:
12244:
12241:
12237:
12236:
12226:
12197:
12194:
12190:
12189:
12186:
12179:
12176:
12165:
12164:
12160:
12131:
12127:
12122:
12112:
12111:
12097:
12068:
12065:
12052:
12051:
12048:
12019:
12015:
12010:
12004:
12003:
11993:
11964:
11960:
11955:
11948:
11947:
11936:
11907:
11903:
11898:
11888:
11887:
11883:
11854:
11850:
11845:
11838:
11837:
11834:
11805:
11801:
11796:
11792:
11791:
11788:
11781:
11778:
11771:
11768:
11765:
11764:
11761:
11758:
11755:
11753:
11751:
11749:
11718:
11711:
11704:
11697:
11682:
11678:
11674:
11673:
11670:
11667:
11664:
11662:
11660:
11658:
11654:
11623:
11616:
11609:
11602:
11593:
11590:
11586:
11585:
11582:
11579:
11576:
11574:
11572:
11570:
11566:
11507:
11500:
11493:
11486:
11477:
11474:
11470:
11469:
11466:
11463:
11460:
11457:
11440:
11429:
11398:
11391:
11384:
11377:
11368:
11365:
11361:
11360:
11357:
11354:
11351:
11340:
11329:
11318:
11317:{2,6}=tr{2,6}
11314:
11283:
11276:
11269:
11262:
11253:
11250:
11246:
11245:
11242:
11239:
11236:
11234:
11223:
11212:
11181:
11174:
11167:
11160:
11151:
11148:
11144:
11143:
11140:
11137:
11134:
11132:
11130:
11119:
11088:
11086:
11079:
11077:
11068:
11065:
11061:
11060:
11057:
11054:
11051:
11040:
11038:
11036:
11005:
11003:
10996:
10994:
10989:
10986:
10982:
10981:
10978:
10975:
10972:
10970:
10968:
10957:
10926:
10924:
10917:
10915:
10910:
10907:
10903:
10902:
10899:
10896:
10893:
10864:
10840:
10810:
10809:
10806:
10803:
10791:
10784:
10781:
10778:
10775:
10761:
10757:
10754:
10751:
10750:
10747:
10744:
10741:
10739:
10737:
10735:
10731:
10700:
10693:
10686:
10679:
10670:
10667:
10663:
10662:
10659:
10656:
10653:
10650:
10633:
10622:
10591:
10584:
10577:
10570:
10561:
10558:
10554:
10553:
10550:
10547:
10544:
10533:
10522:
10511:
10510:{2,5}=tr{2,5}
10507:
10476:
10469:
10462:
10455:
10446:
10443:
10439:
10438:
10435:
10432:
10429:
10427:
10416:
10405:
10374:
10367:
10360:
10353:
10344:
10341:
10337:
10336:
10333:
10330:
10327:
10325:
10323:
10312:
10281:
10279:
10277:
10275:
10266:
10263:
10259:
10258:
10255:
10252:
10249:
10238:
10236:
10234:
10203:
10201:
10194:
10192:
10187:
10184:
10180:
10179:
10176:
10173:
10170:
10168:
10166:
10155:
10124:
10122:
10115:
10113:
10108:
10105:
10101:
10100:
10097:
10094:
10091:
10062:
10038:
10008:
10007:
10004:
10001:
9989:
9982:
9979:
9976:
9973:
9967:
9966:
9950:
9946:
9943:
9940:
9939:
9936:
9933:
9930:
9928:
9926:
9924:
9893:
9886:
9879:
9872:
9857:
9853:
9849:
9848:
9845:
9842:
9839:
9837:
9835:
9833:
9829:
9798:
9791:
9784:
9777:
9768:
9764:
9760:
9759:
9756:
9753:
9750:
9747:
9730:
9719:
9688:
9681:
9674:
9667:
9658:
9655:
9651:
9650:
9647:
9644:
9641:
9630:
9619:
9608:
9607:{2,4}=tr{2,4}
9604:
9573:
9566:
9559:
9552:
9543:
9540:
9536:
9535:
9532:
9529:
9526:
9524:
9513:
9502:
9471:
9464:
9457:
9450:
9441:
9437:
9433:
9432:
9429:
9426:
9423:
9421:
9419:
9408:
9377:
9375:
9373:
9371:
9362:
9359:
9355:
9354:
9351:
9348:
9345:
9334:
9332:
9330:
9299:
9297:
9290:
9288:
9283:
9280:
9276:
9275:
9272:
9269:
9266:
9264:
9262:
9251:
9220:
9218:
9211:
9209:
9204:
9201:
9197:
9196:
9193:
9190:
9187:
9158:
9134:
9104:
9103:
9100:
9097:
9085:
9078:
9075:
9072:
9069:
9063:
9062:
9046:
9042:
9039:
9036:
9035:
9032:
9029:
9026:
9024:
9022:
9020:
9016:
8985:
8978:
8971:
8964:
8955:
8952:
8948:
8947:
8944:
8941:
8938:
8935:
8918:
8907:
8876:
8869:
8862:
8855:
8840:
8836:
8832:
8831:
8828:
8825:
8822:
8811:
8800:
8789:
8788:{2,3}=tr{2,3}
8785:
8754:
8747:
8740:
8733:
8724:
8721:
8717:
8716:
8713:
8710:
8707:
8705:
8694:
8683:
8652:
8645:
8638:
8631:
8622:
8619:
8615:
8614:
8611:
8608:
8605:
8603:
8601:
8590:
8559:
8557:
8550:
8548:
8539:
8536:
8532:
8531:
8528:
8525:
8522:
8511:
8509:
8507:
8476:
8474:
8467:
8465:
8460:
8457:
8453:
8452:
8449:
8446:
8443:
8441:
8439:
8428:
8397:
8395:
8388:
8386:
8381:
8378:
8374:
8373:
8370:
8367:
8364:
8335:
8311:
8281:
8280:
8277:
8274:
8262:
8255:
8252:
8249:
8246:
8240:
8239:
8218:
8214:
8211:
8208:
8207:
8204:
8201:
8198:
8195:
8178:
8176:
8145:
8138:
8131:
8124:
8115:
8111:
8107:
8106:
8103:
8100:
8097:
8086:
8075:
8064:
8063:{2,2}=tr{2,2}
8060:
8029:
8022:
8015:
8008:
7999:
7995:
7991:
7990:
7987:
7984:
7981:
7979:
7977:
7966:
7935:
7933:
7926:
7924:
7915:
7912:
7908:
7907:
7904:
7901:
7898:
7896:
7894:
7883:
7852:
7850:
7843:
7841:
7831:
7828:
7823:
7819:
7818:
7815:
7812:
7809:
7780:
7756:
7726:
7725:
7722:
7719:
7707:
7700:
7697:
7694:
7691:
7685:
7684:
7663:
7660:
7647:
7643:
7639:
7600:
7581:Main article:
7567:
7560:
7556:
7553:
7550:
7549:
7546:
7543:
7540:
7529:
7512:
7501:
7470:
7463:
7456:
7449:
7442:
7435:
7430:
7426:
7425:
7422:
7419:
7416:
7405:
7394:
7383:
7382:{5,3}=tr{5,3}
7379:
7348:
7341:
7334:
7327:
7320:
7313:
7304:
7300:
7299:
7296:
7293:
7290:
7279:
7268:
7257:
7256:{5,3}=rr{5,3}
7253:
7222:
7215:
7208:
7201:
7194:
7187:
7178:
7174:
7173:
7170:
7167:
7164:
7153:
7151:
7140:
7136:
7105:
7098:
7091:
7084:
7077:
7070:
7065:
7061:
7060:
7057:
7054:
7051:
7040:
7038:
7027:
7023:
6992:
6985:
6978:
6971:
6964:
6957:
6952:
6948:
6947:
6944:
6941:
6938:
6927:
6925:
6914:
6910:
6879:
6872:
6865:
6858:
6851:
6844:
6835:
6831:
6830:
6827:
6824:
6821:
6810:
6808:
6806:
6775:
6768:
6761:
6754:
6747:
6740:
6735:
6732:
6731:
6728:
6725:
6722:
6720:
6718:
6707:
6676:
6669:
6662:
6655:
6648:
6641:
6636:
6632:
6631:
6628:
6625:
6622:
6593:
6569:
6539:
6538:
6535:
6532:
6520:
6513:
6510:
6507:
6502:
6497:
6492:
6487:
6484:
6419:
6404:
6400:
6397:
6394:
6393:
6390:
6387:
6384:
6373:
6356:
6345:
6314:
6307:
6300:
6293:
6286:
6279:
6274:
6270:
6269:
6266:
6263:
6260:
6258:
6256:
6254:
6253:{3,4}=rr{3,4}
6250:
6219:
6212:
6205:
6198:
6191:
6184:
6175:
6172:
6171:
6168:
6165:
6162:
6160:
6158:
6156:
6106:
6099:
6092:
6085:
6078:
6071:
6064:
6061:
6060:
6057:
6054:
6051:
6049:
6047:
6045:
5995:
5988:
5981:
5974:
5967:
5960:
5953:
5950:
5949:
5946:
5943:
5940:
5934:
5925:
5923:
5917:
5908:
5904:
5854:
5847:
5840:
5833:
5826:
5819:
5810:
5807:
5806:
5803:
5800:
5797:
5795:
5793:
5787:
5778:
5728:
5721:
5714:
5707:
5700:
5693:
5684:
5681:
5680:
5677:
5674:
5671:
5660:
5649:
5647:
5597:
5590:
5583:
5576:
5569:
5562:
5553:
5550:
5549:
5546:
5543:
5540:
5529:
5518:
5507:
5506:{4,3}=tr{4,3}
5503:
5472:
5465:
5458:
5451:
5444:
5437:
5428:
5424:
5423:
5420:
5417:
5414:
5403:
5392:
5381:
5380:{4,3}=rr{4,3}
5377:
5346:
5339:
5332:
5325:
5318:
5311:
5302:
5298:
5297:
5294:
5291:
5288:
5277:
5275:
5264:
5260:
5229:
5222:
5215:
5208:
5201:
5194:
5189:
5186:
5185:
5182:
5179:
5176:
5165:
5163:
5152:
5148:
5117:
5110:
5103:
5096:
5089:
5082:
5080:Truncated cube
5077:
5073:
5072:
5069:
5066:
5063:
5052:
5050:
5039:
5008:
5001:
4994:
4987:
4980:
4973:
4964:Rectified cube
4962:
4959:
4958:
4955:
4952:
4949:
4938:
4936:
4934:
4903:
4896:
4889:
4882:
4875:
4868:
4863:
4860:
4859:
4856:
4853:
4850:
4848:
4846:
4835:
4804:
4797:
4790:
4783:
4776:
4769:
4764:
4760:
4759:
4756:
4753:
4750:
4721:
4692:
4662:
4661:
4658:
4655:
4643:
4636:
4633:
4630:
4627:
4622:
4619:
4614:
4611:
4605:
4604:
4543:
4528:
4524:
4521:
4518:
4517:
4514:
4511:
4508:
4497:
4480:
4469:
4438:
4431:
4424:
4417:
4410:
4403:
4394:
4390:
4389:
4386:
4383:
4380:
4369:
4358:
4347:
4346:{3,3}=tr{3,3}
4343:
4312:
4305:
4298:
4291:
4284:
4277:
4266:
4262:
4261:
4258:
4255:
4252:
4241:
4230:
4219:
4218:{3,3}=rr{3,3}
4215:
4184:
4177:
4170:
4163:
4156:
4149:
4138:
4134:
4133:
4130:
4127:
4124:
4113:
4111:
4100:
4096:
4065:
4058:
4051:
4044:
4037:
4030:
4021:
4018:
4017:
4014:
4011:
4008:
3997:
3995:
3984:
3980:
3949:
3942:
3935:
3928:
3921:
3914:
3909:
3905:
3904:
3901:
3898:
3895:
3884:
3882:
3871:
3867:
3836:
3829:
3822:
3815:
3808:
3801:
3790:
3786:
3785:
3782:
3779:
3776:
3765:
3763:
3761:
3757:
3726:
3719:
3712:
3705:
3698:
3691:
3682:
3679:
3678:
3675:
3672:
3669:
3667:
3665:
3654:
3623:
3616:
3609:
3602:
3595:
3588:
3583:
3579:
3578:
3575:
3572:
3569:
3540:
3516:
3491:
3490:
3487:
3484:
3472:
3465:
3462:
3459:
3456:
3451:
3448:
3443:
3440:
3434:
3433:
3372:
3356:
3352:
3349:
3346:
3345:
3336:
3327:
3318:
3307:
3298:
3289:
3280:
3269:
3261:
3260:
3251:
3242:
3233:
3222:
3213:
3204:
3201:
3190:
3182:
3181:
3172:
3163:
3154:
3143:
3134:
3125:
3122:
3111:
3103:
3102:
3093:
3084:
3075:
3064:
3055:
3046:
3037:
3026:
3018:
3017:
3008:
2999:
2990:
2979:
2970:
2961:
2952:
2941:
2933:
2932:
2929:
2926:
2925:p. 4.2.4
2923:
2920:
2917:
2914:
2911:
2908:
2902:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2875:Wythoff symbol
2871:
2870:
2866:
2863:
2859:
2856:
2852:
2849:
2845:
2842:
2838:
2835:
2831:
2828:
2824:
2821:
2817:
2813:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2787:
2786:
2773:
2767:
2764:
2763:
2760:
2757:
2756:
2754:
2749:
2739:
2726:
2720:
2717:
2716:
2713:
2710:
2709:
2707:
2702:
2692:
2679:
2673:
2670:
2669:
2666:
2663:
2662:
2660:
2655:
2645:
2632:
2626:
2623:
2620:
2617:
2616:
2614:
2602:
2589:
2583:
2580:
2577:
2574:
2573:
2571:
2566:
2556:
2543:
2537:
2534:
2533:
2530:
2527:
2526:
2524:
2512:
2499:
2493:
2490:
2487:
2484:
2483:
2481:
2476:
2466:
2453:
2447:
2444:
2441:
2438:
2437:
2435:
2423:
2415:
2414:
2385:
2356:
2327:
2298:
2269:
2240:
2211:
2182:
2176:
2175:
2172:
2169:cantitruncated
2163:
2160:
2155:
2150:
2147:
2144:
2141:
2125:
2124:
2113:
2102:
2091:
2080:
2069:
2058:
2047:
2036:
2028:
2027:
2016:
2005:
1994:
1983:
1972:
1961:
1950:
1939:
1931:
1930:
1919:
1908:
1897:
1886:
1875:
1864:
1853:
1842:
1834:
1833:
1830:
1827:
1826:p. 4.q.4
1824:
1821:
1820:p. 2q.2q
1818:
1815:
1812:
1809:
1803:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1774:Wythoff symbol
1770:
1769:
1765:
1762:
1758:
1755:
1751:
1748:
1744:
1741:
1737:
1734:
1730:
1727:
1723:
1720:
1716:
1712:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1686:
1685:
1672:
1666:
1663:
1662:
1659:
1656:
1655:
1653:
1648:
1638:
1625:
1619:
1616:
1615:
1612:
1609:
1608:
1606:
1601:
1591:
1578:
1572:
1569:
1568:
1565:
1562:
1561:
1559:
1554:
1544:
1531:
1525:
1522:
1519:
1516:
1515:
1513:
1501:
1488:
1482:
1479:
1476:
1473:
1472:
1470:
1465:
1455:
1442:
1436:
1433:
1432:
1429:
1426:
1425:
1423:
1411:
1398:
1392:
1389:
1386:
1383:
1382:
1380:
1375:
1365:
1352:
1346:
1343:
1340:
1337:
1336:
1334:
1322:
1314:
1313:
1266:
1219:
1172:
1143:
1114:
1067:
1038:
1009:
1003:
1002:
999:
996:cantitruncated
990:
987:
982:
977:
974:
971:
968:
958:
957:Summary tables
955:
954:
953:
949:
941:
937:
929:
925:
917:
913:
864:
863:
846:
840:
834:
717:
714:
690:Main article:
687:
684:
683:
682:
679:Wythoff symbol
675:
668:
657:
646:
636:
633:Norman Johnson
619:
613:
607:
592:H.S.M. Coxeter
588:
579:
576:
575:
574:
563:
549:
529:
526:
525:
524:
510:
486:
474:
460:
447:
444:
443:
442:
409:
406:
404:
401:
400:
399:
396:
393:
389:
386:
334:
332:
329:
314:vertex figures
306:Dual polyhedra
257:star polyhedra
167:Platonic solid
152:
151:
142:
141:
133:
132:
124:
123:
122:
113:
112:
104:
103:
95:
94:
93:
92:
91:
88:
87:
42:
40:
33:
26:
9:
6:
4:
3:
2:
14196:
14185:
14182:
14181:
14179:
14168:
14164:
14160:
14155:
14152:
14148:
14145:
14143:
14136:
14129:
14123:
14121:
14117:
14114:
14112:
14108:
14104:
14100:
14097:
14095:
14091:
14088:
14086:
14082:
14078:
14077:
14074:
14072:
14070:
14067:
14065:
14061:
14058:
14056:
14053:
14051:
14048:
14047:
14044:
14042:
14040:
14037:
14035:
14031:
14028:
14026:
14023:
14021:
14018:
14017:
14014:
14012:
14005:
13998:
13992:
13990:
13987:
13985:
13981:
13978:
13976:
13973:
13971:
13968:
13967:
13964:
13962:
13955:
13948:
13942:
13940:
13937:
13935:
13931:
13928:
13926:
13923:
13921:
13918:
13917:
13914:
13912:
13905:
13899:
13897:
13894:
13892:
13888:
13885:
13883:
13880:
13878:
13875:
13874:
13871:
13869:
13867:
13864:
13862:
13858:
13855:
13853:
13850:
13848:
13845:
13844:
13841:
13837:
13834:
13832:
13829:
13827:
13826:Demitesseract
13824:
13822:
13818:
13815:
13813:
13810:
13808:
13805:
13804:
13801:
13797:
13794:
13792:
13790:
13787:
13785:
13781:
13778:
13776:
13773:
13771:
13768:
13767:
13764:
13761:
13759:
13756:
13754:
13751:
13749:
13746:
13744:
13741:
13739:
13736:
13735:
13732:
13726:
13723:
13719:
13712:
13708:
13701:
13697:
13692:
13688:
13683:
13679:
13674:
13672:
13670:
13666:
13656:
13652:
13650:
13648:
13644:
13640:
13638:
13636:
13632:
13628:
13626:
13623:
13622:
13617:
13613:
13605:
13600:
13598:
13593:
13591:
13586:
13585:
13582:
13577:
13574:
13571:
13569:
13566:
13563:
13560:
13558:
13555:
13553:
13550:
13545:
13544:
13539:
13536:
13531:
13530:
13520:
13514:
13510:
13506:
13502:
13498:
13494:
13490:
13485:
13481:
13477:
13473:
13469:
13465:
13461:
13457:
13453:
13449:
13445:
13441:
13437:
13433:
13429:
13424:
13420:
13416:
13412:
13409:
13403:
13399:
13395:
13391:
13387:
13383:
13380:
13374:
13370:
13366:
13362:
13358:
13357:
13351:
13347:
13343:
13339:
13335:
13331:
13327:
13323:
13319:
13315:
13311:
13307:
13303:
13299:
13295:
13294:
13286:
13282:
13278:
13274:
13270:
13268:
13265:
13262:
13259:
13258:
13244:
13239:
13235:
13231:
13230:
13225:
13218:
13212:
13209:
13206:
13201:
13195:
13191:
13187:
13183:
13177:
13172:
13168:
13164:
13163:
13155:
13148:
13142:
13137:
13131:
13126:
13120:
13115:
13106:
13099:
13094:
13087:
13083:
13082:Diudea (2018)
13078:
13074:
13064:
13061:
13059:
13056:
13054:
13051:
13049:
13046:
13044:
13041:
13039:
13036:
13034:
13031:
13027:
13024:
13022:
13019:
13017:
13014:
13013:
13012:
13009:
13005:
13002:
13000:
12997:
12995:
12992:
12991:
12990:
12987:
12986:
12913:For example,
12883:
12844:
12841:
12838:
12837:
12771:For example,
12636:
12633:
12630:
12629:
12603:
12569:
12562:
12555:
12554:
12495:
12458:
12455:
12452:
12451:
12421:For example,
12381:
12342:
12339:
12336:
12335:
12332:
12298:
12291:
12284:
12283:
12279:
12245:
12242:
12239:
12238:
12234:
12227:
12198:
12195:
12192:
12191:
12187:
12185:
12180:
12173:
12161:
12132:
12123:
12120:
12119:omnitruncated
12114:
12113:
12109:
12102:
12098:
12069:
12066:
12063:
12057:
12054:
12053:
12049:
12020:
12011:
12006:
12005:
12001:
11994:
11965:
11956:
11953:
11950:
11949:
11941:
11937:
11908:
11899:
11896:
11890:
11889:
11884:
11855:
11846:
11843:
11840:
11839:
11835:
11806:
11793:
11789:
11787:
11782:
11779:
11776:
11775:
11762:
11759:
11756:
11754:
11752:
11750:
11716:
11712:
11709:
11705:
11702:
11698:
11695:
11689:
11676:
11671:
11668:
11665:
11663:
11661:
11659:
11657:{6,2}=t{2,6}
11621:
11617:
11614:
11610:
11607:
11603:
11600:
11588:
11583:
11580:
11577:
11575:
11573:
11571:
11569:{6,2}=t{2,3}
11505:
11501:
11498:
11494:
11491:
11487:
11484:
11472:
11467:
11464:
11461:
11458:
11456:
11450:
11445:
11441:
11439:
11434:
11430:
11396:
11392:
11389:
11385:
11382:
11378:
11375:
11363:
11358:
11355:
11352:
11350:
11345:
11341:
11339:
11334:
11330:
11328:
11323:
11319:
11281:
11277:
11274:
11270:
11267:
11263:
11260:
11248:
11243:
11240:
11237:
11235:
11233:
11228:
11224:
11222:
11217:
11213:
11179:
11175:
11172:
11168:
11165:
11161:
11158:
11146:
11141:
11138:
11135:
11133:
11131:
11129:
11124:
11120:
11087:
11084:
11080:
11078:
11075:
11063:
11058:
11055:
11052:
11050:
11045:
11041:
11039:
11037:
11004:
11001:
10997:
10995:
10993:
10984:
10979:
10976:
10973:
10971:
10969:
10967:
10962:
10958:
10925:
10922:
10918:
10916:
10914:
10905:
10900:
10897:
10894:
10865:
10841:
10812:
10811:
10800:
10795:
10790:
10772:
10769:
10767:
10748:
10745:
10742:
10740:
10738:
10736:
10734:{5,2}=t{2,5}
10698:
10694:
10691:
10687:
10684:
10680:
10677:
10665:
10660:
10657:
10654:
10651:
10649:
10643:
10638:
10634:
10632:
10627:
10623:
10589:
10585:
10582:
10578:
10575:
10571:
10568:
10556:
10551:
10548:
10545:
10543:
10538:
10534:
10532:
10527:
10523:
10521:
10516:
10512:
10474:
10470:
10467:
10463:
10460:
10456:
10453:
10441:
10436:
10433:
10430:
10428:
10426:
10421:
10417:
10415:
10410:
10406:
10372:
10368:
10365:
10361:
10358:
10354:
10351:
10339:
10334:
10331:
10328:
10326:
10324:
10322:
10317:
10313:
10280:
10278:
10276:
10273:
10261:
10256:
10253:
10250:
10248:
10243:
10239:
10237:
10235:
10202:
10199:
10195:
10193:
10191:
10182:
10177:
10174:
10171:
10169:
10167:
10165:
10160:
10156:
10123:
10120:
10116:
10114:
10112:
10103:
10098:
10095:
10092:
10063:
10039:
10010:
10009:
9998:
9993:
9988:
9970:
9964:
9960:
9959:
9958:
9956:
9937:
9934:
9931:
9929:
9927:
9925:
9891:
9887:
9884:
9880:
9877:
9873:
9870:
9864:
9851:
9846:
9843:
9840:
9838:
9836:
9834:
9832:{4,2}=t{2,4}
9796:
9792:
9789:
9785:
9782:
9778:
9775:
9762:
9757:
9754:
9751:
9748:
9746:
9740:
9735:
9731:
9729:
9724:
9720:
9686:
9682:
9679:
9675:
9672:
9668:
9665:
9653:
9648:
9645:
9642:
9640:
9635:
9631:
9629:
9624:
9620:
9618:
9613:
9609:
9571:
9567:
9564:
9560:
9557:
9553:
9550:
9538:
9533:
9530:
9527:
9525:
9523:
9518:
9514:
9512:
9507:
9503:
9469:
9465:
9462:
9458:
9455:
9451:
9448:
9435:
9430:
9427:
9424:
9422:
9420:
9418:
9413:
9409:
9376:
9374:
9372:
9369:
9357:
9352:
9349:
9346:
9344:
9339:
9335:
9333:
9331:
9298:
9295:
9291:
9289:
9287:
9278:
9273:
9270:
9267:
9265:
9263:
9261:
9256:
9252:
9219:
9216:
9212:
9210:
9208:
9199:
9194:
9191:
9188:
9159:
9135:
9106:
9105:
9094:
9089:
9084:
9066:
9060:
9056:
9055:
9054:
9052:
9033:
9030:
9027:
9025:
9023:
9021:
9019:{2,3}=t{2,3}
8983:
8979:
8976:
8972:
8969:
8965:
8962:
8950:
8945:
8942:
8939:
8936:
8934:
8928:
8923:
8919:
8917:
8912:
8908:
8874:
8870:
8867:
8863:
8860:
8856:
8853:
8847:
8834:
8829:
8826:
8823:
8821:
8816:
8812:
8810:
8805:
8801:
8799:
8794:
8790:
8752:
8748:
8745:
8741:
8738:
8734:
8731:
8719:
8714:
8711:
8708:
8706:
8704:
8699:
8695:
8693:
8688:
8684:
8650:
8646:
8643:
8639:
8636:
8632:
8629:
8617:
8612:
8609:
8606:
8604:
8602:
8600:
8595:
8591:
8558:
8555:
8551:
8549:
8546:
8534:
8529:
8526:
8523:
8521:
8516:
8512:
8510:
8508:
8475:
8472:
8468:
8466:
8464:
8455:
8450:
8447:
8444:
8442:
8440:
8438:
8433:
8429:
8396:
8393:
8389:
8387:
8385:
8376:
8371:
8368:
8365:
8336:
8312:
8283:
8282:
8271:
8266:
8261:
8243:
8237:
8232:
8228:
8227:
8226:
8224:
8205:
8202:
8199:
8196:
8194:
8188:
8183:
8179:
8177:
8143:
8139:
8136:
8132:
8129:
8125:
8122:
8109:
8104:
8101:
8098:
8096:
8091:
8087:
8085:
8080:
8076:
8074:
8069:
8065:
8027:
8023:
8020:
8016:
8013:
8009:
8006:
7993:
7988:
7985:
7982:
7980:
7978:
7976:
7971:
7967:
7965:t{2,2}={4,2}
7934:
7931:
7927:
7925:
7922:
7910:
7905:
7902:
7899:
7897:
7895:
7893:
7888:
7884:
7851:
7848:
7844:
7842:
7840:
7835:
7821:
7816:
7813:
7810:
7781:
7757:
7728:
7727:
7716:
7711:
7706:
7688:
7682:
7677:
7673:
7672:
7671:
7669:
7659:
7657:
7653:
7636:
7606:
7598:
7597:Coxeter group
7593:
7590:
7584:
7574:
7563:
7559:(p) family (D
7547:
7544:
7541:
7539:
7534:
7530:
7528:
7522:
7517:
7513:
7511:
7506:
7502:
7468:
7464:
7461:
7457:
7454:
7450:
7447:
7443:
7440:
7436:
7434:
7428:
7423:
7420:
7417:
7415:
7410:
7406:
7404:
7399:
7395:
7393:
7388:
7384:
7346:
7342:
7339:
7335:
7332:
7328:
7325:
7321:
7318:
7314:
7312:
7302:
7297:
7294:
7291:
7289:
7284:
7280:
7278:
7273:
7269:
7267:
7262:
7258:
7220:
7216:
7213:
7209:
7206:
7202:
7199:
7195:
7192:
7188:
7186:
7176:
7171:
7168:
7165:
7163:
7158:
7154:
7152:
7150:
7145:
7141:
7139:{3,5}=t{3,5}
7103:
7099:
7096:
7092:
7089:
7085:
7082:
7078:
7075:
7071:
7069:
7063:
7058:
7055:
7052:
7050:
7045:
7041:
7039:
7037:
7032:
7028:
7026:{5,3}=t{5,3}
6990:
6986:
6983:
6979:
6976:
6972:
6969:
6965:
6962:
6958:
6956:
6950:
6945:
6942:
6939:
6937:
6932:
6928:
6926:
6924:
6919:
6915:
6913:{5,3}=r{5,3}
6877:
6873:
6870:
6866:
6863:
6859:
6856:
6852:
6849:
6845:
6843:
6833:
6828:
6825:
6822:
6820:
6815:
6811:
6809:
6807:
6773:
6769:
6766:
6762:
6759:
6755:
6752:
6748:
6745:
6741:
6739:
6734:
6729:
6726:
6723:
6721:
6719:
6717:
6712:
6708:
6674:
6670:
6667:
6663:
6660:
6656:
6653:
6649:
6646:
6642:
6640:
6634:
6629:
6626:
6623:
6594:
6570:
6541:
6540:
6529:
6524:
6519:
6481:
6478:
6476:
6471:
6466:
6460:
6455:
6425:
6417:
6416:Coxeter group
6412:
6410:
6391:
6388:
6385:
6383:
6378:
6374:
6372:
6366:
6361:
6357:
6355:
6350:
6346:
6312:
6308:
6305:
6301:
6298:
6294:
6291:
6287:
6284:
6280:
6278:
6272:
6267:
6264:
6261:
6259:
6257:
6255:
6217:
6213:
6210:
6206:
6203:
6199:
6196:
6192:
6189:
6185:
6182:
6176:
6174:
6169:
6166:
6163:
6161:
6159:
6157:
6104:
6100:
6097:
6093:
6090:
6086:
6083:
6079:
6076:
6072:
6069:
6065:
6063:
6058:
6055:
6052:
6050:
6048:
6046:
5993:
5989:
5986:
5982:
5979:
5975:
5972:
5968:
5965:
5961:
5958:
5957:Cuboctahedron
5954:
5952:
5947:
5944:
5941:
5939:
5930:
5926:
5924:
5922:
5913:
5909:
5907:{4,3}=t{3,3}
5852:
5848:
5845:
5841:
5838:
5834:
5831:
5827:
5824:
5820:
5817:
5809:
5804:
5801:
5798:
5796:
5794:
5792:
5783:
5779:
5777:h{4,3}={3,3}
5726:
5722:
5719:
5715:
5712:
5708:
5705:
5701:
5698:
5694:
5691:
5683:
5678:
5675:
5672:
5670:
5665:
5661:
5659:
5654:
5650:
5648:
5595:
5591:
5588:
5584:
5581:
5577:
5574:
5570:
5567:
5563:
5560:
5552:
5547:
5544:
5541:
5539:
5534:
5530:
5528:
5523:
5519:
5517:
5512:
5508:
5470:
5466:
5463:
5459:
5456:
5452:
5449:
5445:
5442:
5438:
5436:
5426:
5421:
5418:
5415:
5413:
5408:
5404:
5402:
5397:
5393:
5391:
5386:
5382:
5344:
5340:
5337:
5333:
5330:
5326:
5323:
5319:
5316:
5312:
5310:
5300:
5295:
5292:
5289:
5287:
5282:
5278:
5276:
5274:
5269:
5265:
5263:{3,4}=t{3,4}
5227:
5223:
5220:
5216:
5213:
5209:
5206:
5202:
5199:
5195:
5193:
5188:
5183:
5180:
5177:
5175:
5170:
5166:
5164:
5162:
5157:
5153:
5151:{4,3}=t{4,3}
5115:
5111:
5108:
5104:
5101:
5097:
5094:
5090:
5087:
5083:
5081:
5075:
5070:
5067:
5064:
5062:
5057:
5053:
5051:
5049:
5044:
5040:
5006:
5002:
4999:
4995:
4992:
4988:
4985:
4981:
4978:
4974:
4971:
4970:Cuboctahedron
4961:
4956:
4953:
4950:
4948:
4943:
4939:
4937:
4935:
4901:
4897:
4894:
4890:
4887:
4883:
4880:
4876:
4873:
4869:
4867:
4862:
4857:
4854:
4851:
4849:
4847:
4845:
4840:
4836:
4802:
4798:
4795:
4791:
4788:
4784:
4781:
4777:
4774:
4770:
4768:
4762:
4757:
4754:
4751:
4722:
4693:
4664:
4663:
4652:
4647:
4642:
4608:
4602:
4597:
4592:
4588:
4587:
4586:
4584:
4579:
4549:
4541:
4540:Coxeter group
4536:
4534:
4515:
4512:
4509:
4507:
4502:
4498:
4496:
4490:
4485:
4481:
4479:
4474:
4470:
4436:
4432:
4429:
4425:
4422:
4418:
4415:
4411:
4408:
4404:
4401:
4392:
4387:
4384:
4381:
4379:
4374:
4370:
4368:
4363:
4359:
4357:
4352:
4348:
4310:
4306:
4303:
4299:
4296:
4292:
4289:
4285:
4282:
4278:
4275:
4264:
4259:
4256:
4253:
4251:
4246:
4242:
4240:
4235:
4231:
4229:
4224:
4220:
4182:
4178:
4175:
4171:
4168:
4164:
4161:
4157:
4154:
4150:
4147:
4146:cuboctahedron
4136:
4131:
4128:
4125:
4123:
4118:
4114:
4112:
4110:
4105:
4101:
4099:{3,3}=t{3,3}
4063:
4059:
4056:
4052:
4049:
4045:
4042:
4038:
4035:
4031:
4028:
4020:
4015:
4012:
4009:
4007:
4002:
3998:
3996:
3994:
3989:
3985:
3983:{3,3}=t{3,3}
3947:
3943:
3940:
3936:
3933:
3929:
3926:
3922:
3919:
3915:
3913:
3907:
3902:
3899:
3896:
3894:
3889:
3885:
3883:
3881:
3876:
3872:
3870:{3,3}=r{3,3}
3834:
3830:
3827:
3823:
3820:
3816:
3813:
3809:
3806:
3802:
3799:
3788:
3783:
3780:
3777:
3775:
3770:
3766:
3764:
3762:
3724:
3720:
3717:
3713:
3710:
3706:
3703:
3699:
3696:
3692:
3689:
3681:
3676:
3673:
3670:
3668:
3666:
3664:
3659:
3655:
3621:
3617:
3614:
3610:
3607:
3603:
3600:
3596:
3593:
3589:
3587:
3581:
3576:
3573:
3570:
3541:
3517:
3493:
3492:
3481:
3476:
3471:
3437:
3431:
3426:
3421:
3417:
3416:
3415:
3413:
3408:
3378:
3370:
3369:Coxeter group
3364:
3362:
3341:
3337:
3332:
3328:
3323:
3319:
3317:
3312:
3308:
3303:
3299:
3294:
3290:
3285:
3281:
3279:
3274:
3270:
3266:
3263:
3256:
3252:
3247:
3243:
3238:
3234:
3232:
3227:
3223:
3218:
3214:
3209:
3205:
3202:
3200:
3195:
3191:
3187:
3184:
3177:
3173:
3168:
3164:
3159:
3155:
3153:
3148:
3144:
3139:
3135:
3130:
3126:
3123:
3121:
3116:
3112:
3108:
3105:
3098:
3094:
3089:
3085:
3080:
3076:
3074:
3069:
3065:
3060:
3056:
3051:
3047:
3042:
3038:
3036:
3031:
3027:
3023:
3020:
3013:
3009:
3004:
3000:
2995:
2991:
2989:
2984:
2980:
2975:
2971:
2966:
2962:
2957:
2953:
2951:
2946:
2942:
2938:
2935:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2907:
2906:Vertex figure
2904:
2903:
2900:| p 2 2
2899:
2897:p 2 2 |
2896:
2894:p 2 | 2
2893:
2891:p | 2 2
2890:
2888:2 p | 2
2887:
2885:2 | p 2
2884:
2882:2 2 | p
2881:
2879:2 | p 2
2878:
2876:
2873:
2872:
2864:
2857:
2850:
2843:
2836:
2829:
2822:
2815:
2814:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2788:
2771:
2765:
2758:
2752:
2747:
2740:
2724:
2718:
2711:
2705:
2700:
2693:
2677:
2671:
2664:
2658:
2653:
2646:
2630:
2624:
2621:
2618:
2612:
2603:
2587:
2581:
2578:
2575:
2569:
2564:
2557:
2541:
2535:
2528:
2522:
2513:
2497:
2491:
2488:
2485:
2479:
2474:
2467:
2451:
2445:
2442:
2439:
2433:
2424:
2422:
2416:
2386:
2357:
2328:
2299:
2270:
2241:
2212:
2183:
2181:
2178:
2177:
2173:
2165:Omnitruncated
2164:
2161:
2156:
2151:
2148:
2145:
2142:
2139:
2138:
2135:
2132:
2131:
2123:
2118:
2114:
2112:
2107:
2103:
2101:
2096:
2092:
2090:
2085:
2081:
2079:
2074:
2070:
2068:
2063:
2059:
2057:
2052:
2048:
2046:
2041:
2037:
2033:
2030:
2026:
2021:
2017:
2015:
2010:
2006:
2004:
1999:
1995:
1993:
1988:
1984:
1982:
1977:
1973:
1971:
1966:
1962:
1960:
1955:
1951:
1949:
1944:
1940:
1936:
1933:
1929:
1924:
1920:
1918:
1913:
1909:
1907:
1902:
1898:
1896:
1891:
1887:
1885:
1880:
1876:
1874:
1869:
1865:
1863:
1858:
1854:
1852:
1847:
1843:
1839:
1836:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1808:
1807:Vertex figure
1805:
1804:
1801:| p q 2
1800:
1798:p q 2 |
1797:
1795:p q | 2
1794:
1792:p | q 2
1791:
1789:2 p | q
1788:
1786:2 | p q
1785:
1783:2 q | p
1782:
1780:q | p 2
1779:
1775:
1772:
1771:
1763:
1756:
1749:
1742:
1735:
1728:
1721:
1714:
1713:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1687:
1670:
1664:
1657:
1651:
1646:
1639:
1623:
1617:
1610:
1604:
1599:
1592:
1576:
1570:
1563:
1557:
1552:
1545:
1529:
1523:
1520:
1517:
1511:
1502:
1486:
1480:
1477:
1474:
1468:
1463:
1456:
1440:
1434:
1427:
1421:
1412:
1396:
1390:
1387:
1384:
1378:
1373:
1366:
1350:
1344:
1341:
1338:
1332:
1323:
1321:
1315:
1267:
1220:
1173:
1144:
1115:
1068:
1039:
1010:
1008:
1005:
1000:
992:Omnitruncated
991:
988:
983:
978:
975:
972:
969:
966:
963:
952:(snub prisms)
948:
945:
942:
936:
933:
930:
924:
921:
918:
912:
909:
906:
905:
904:
901:
898:
896:
892:
891:
886:
885:
880:
876:
871:
869:
862:
858:
854:
850:
847:
844:
841:
838:
835:
832:
829:
828:
827:
823:
819:
815:
809:
805:
801:
797:
793:
789:
785:
780:
778:
774:
770:
765:
763:
759:
755:
750:
747:
745:
738:
734:
729:
722:
713:
711:
707:
698:
693:
680:
676:
673:
669:
666:
662:
658:
655:
651:
647:
644:
640:
637:
634:
630:
629:
624:
620:
617:
614:
611:
608:
605:
601:
597:
593:
590:The geometer
589:
586:
582:
581:
572:
571:Arthur Cayley
568:
564:
561:
557:
553:
552:Louis Poinsot
550:
547:
543:
539:
535:
532:
531:
522:
518:
514:
511:
508:
504:
500:
496:
495:
490:
487:
484:
483:
478:
475:
472:
468:
464:
461:
458:
455:was known by
454:
453:cuboctahedron
450:
449:
440:
436:
432:
428:
424:
420:
416:
412:
411:
397:
394:
390:
387:
384:
383:
382:
379:
375:
371:
365:
362:
358:
353:
350:
343:
328:
326:
321:
319:
318:Catalan solid
315:
311:
307:
303:
301:
297:
293:
289:
285:
281:
277:
273:
269:
265:
260:
258:
254:
250:
246:
245:quasi-regular
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
202:
192:
188:
184:
180:
176:
172:
171:cuboctahedron
168:
164:
160:
156:
146:
137:
128:
117:
108:
99:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
14146:
14115:
14106:
14098:
14089:
14080:
14060:10-orthoplex
13796:Dodecahedron
13769:
13717:
13706:
13695:
13686:
13677:
13668:
13664:
13654:
13646:
13642:
13634:
13630:
13575:
13541:
13508:
13488:
13431:
13427:
13418:
13389:
13386:GrĂĽnbaum, B.
13355:
13297:
13291:
13263:
13261:BrĂĽckner, M.
13233:
13227:
13217:
13207:
13205:Mäder, R. E.
13200:
13166:
13160:
13147:
13136:
13125:
13114:
13105:
13093:
13077:
12839:Quarter (q)
12188:Description
12100:
11790:Description
10763:
9952:
9048:
8220:
7665:
7651:
7637:
7594:
7586:
7561:
6639:Dodecahedron
6456:
6413:
6406:
4580:
4537:
4530:
3760:{3,3}={3,3}
3409:
3365:
3358:
2919:p. 4.4
2162:Cantellated
2133:
2129:
2128:
989:Cantellated
946:
934:
922:
910:
902:
899:
894:
888:
882:
878:
872:
865:
860:
856:
852:
821:
817:
813:
807:
803:
799:
795:
791:
787:
781:
766:
761:
751:
748:
741:
703:
664:
660:
653:
650:Edmond Bonan
627:
616:Sopov (1970)
502:
492:
489:Luca Pacioli
480:
419:Pythagoreans
366:
355:
345:
336:
322:
304:
296:quasiregular
280:quasiregular
261:
249:semi-regular
204:
198:
76:
70:October 2011
67:
48:
14069:10-demicube
14030:9-orthoplex
13980:8-orthoplex
13930:7-orthoplex
13887:6-orthoplex
13857:5-orthoplex
13812:Pentachoron
13800:Icosahedron
13775:Tetrahedron
13194:dual images
12494:Alternation
12340:hrr{2p,2q}
12056:Cantellated
9869:Tetrahedron
8121:tetrahedron
6738:Icosahedron
5812:Cantic cube
5690:Tetrahedron
5559:Icosahedron
4400:icosahedron
3688:tetrahedron
3586:Tetrahedron
3316:2.2.2.2.2.2
2157:Birectified
2154:(tr. dual)
2152:Bitruncated
2032:Icosahedral
1838:Tetrahedral
984:Birectified
981:(tr. dual)
979:Bitruncated
868:alternation
773:hosohedrons
672:Mathematica
585:Edmund Hess
284:semiregular
62:introducing
14055:10-simplex
14039:9-demicube
13989:8-demicube
13939:7-demicube
13896:6-demicube
13866:5-demicube
13780:Octahedron
13255:References
13182:Zvi Har’El
13169:: 57–110.
13084:, p.
12989:Polyhedron
12714:, same as
12634:hr{2p,2q}
12529:, same as
12175:Operation
11777:Operation
11694:octahedron
8852:octahedron
7650:has order
4866:Octahedron
3798:octahedron
2149:Rectified
2146:Truncated
1935:Octahedral
976:Rectified
973:Truncated
944:Antiprisms
855:2 2), for
760:is also a
758:octahedron
754:octahedron
737:octahedron
463:Archimedes
427:Theaetetus
392:polyhedra.
331:Definition
294:—14
268:antiprisms
227:. Uniform
45:references
14103:orthoplex
14025:9-simplex
13975:8-simplex
13925:7-simplex
13882:6-simplex
13852:5-simplex
13821:Tesseract
13543:MathWorld
13480:122634260
13456:0080-4614
13346:202575183
13322:0080-4614
12842:q{2p,2q}
12556:Cantic (h
12453:Half (h)
12240:Snub (s)
11952:Truncated
11842:Rectified
11692:(same as
11686:(same as
11155:(same as
11072:(same as
10901:Vertices
10756:(6 2 2) D
10348:(same as
10270:(same as
10099:Vertices
9945:(5 2 2) D
9366:(same as
9195:Vertices
9041:(4 2 2) D
8850:(same as
8844:(same as
8543:(same as
8372:Vertices
8213:(3 2 2) D
8119:(same as
8003:(same as
7919:(same as
7817:Vertices
7656:bipyramid
6630:Vertices
6399:(5 3 2) I
6179:(same as
6066:(same as
5955:(same as
5814:(same as
5688:(same as
5686:Half cube
5557:(same as
4758:Vertices
4523:(4 3 2) O
4398:(same as
4272:(same as
4144:(same as
4025:(same as
3796:(same as
3686:(same as
3577:Vertices
3351:(3 3 2) T
3231:2.2.2.2.2
2122:3.3.3.3.5
2089:3.3.3.3.3
2025:3.3.3.3.4
1928:3.3.3.3.3
908:Hosohedra
895:hosohedra
890:hosohedra
798:), where
777:dihedrons
648:In 1987,
621:In 1974,
439:Etruscans
278:—2
235:(if also
229:polyhedra
225:congruent
14178:Category
14157:Topics:
14120:demicube
14085:polytope
14079:Uniform
13840:600-cell
13836:120-cell
13789:Demicube
13763:Pentagon
13743:Triangle
13507:(1974).
13361:Springer
13283:(1954).
12983:See also
12604:Same as
12456:h{2p,q}
12243:s{p,2q}
12196:sr{p,q}
12104:rr{4,3}.
12067:rr{p,q}
12062:expanded
11428:sr{2,6}
10802:symbols
10799:Schläfli
10780:Picture
10621:sr{2,5}
10000:symbols
9997:Schläfli
9978:Picture
9718:sr{2,4}
9096:symbols
9093:Schläfli
9074:Picture
8906:sr{2,3}
8273:symbols
8270:Schläfli
8251:Picture
8175:sr{2,2}
7718:symbols
7715:Schläfli
7696:Picture
7500:sr{5,3}
6531:symbols
6528:Schläfli
6509:Picture
6344:sr{4,3}
6155:tr{3,3}
6044:rr{3,3}
4654:symbols
4651:Schläfli
4632:Picture
4468:sr{3,3}
3483:symbols
3480:Schläfli
3461:Picture
3344:3.3.3.6
3326:2.4.6.4
3297:2.6.2.6
3288:2.12.12
3268:(6 2 2)
3265:Dihedral
3259:3.3.3.5
3241:2.4.5.4
3212:2.5.2.5
3203:2.10.10
3189:(5 2 2)
3186:Dihedral
3180:3.3.3.4
3162:2.4.4.4
3133:2.4.2.4
3110:(4 2 2)
3107:Dihedral
3101:3.3.3.3
3083:2.4.3.4
3054:2.3.2.3
3025:(3 2 2)
3022:Dihedral
3016:3.3.3.2
2998:2.4.2.4
2969:2.2.2.2
2940:(2 2 2)
2937:Dihedral
2931:3.3.3.p
2913:2.2p.2p
2811:sr{p,2}
2808:tr{p,2}
2805:rr{p,2}
2802:2r{p,2}
2799:2t{p,2}
2418:Extended
2140:(p 2 2)
2035:(5 3 2)
1938:(4 3 2)
1841:(3 3 2)
1829:4.2p.2q
1814:q.2p.2p
1777:(p q 2)
1710:sr{p,q}
1707:tr{p,q}
1704:rr{p,q}
1701:2r{p,q}
1698:2t{p,q}
1317:Extended
887:and the
573:in 1859.
221:isometry
201:geometry
14094:simplex
14064:10-cube
13831:24-cell
13817:16-cell
13758:Hexagon
13612:regular
13497:0326550
13472:0365333
13436:Bibcode
13338:0062446
13302:Bibcode
12567:{2p,q}
12296:{p,2q}
12184:diagram
12182:Coxeter
12178:Symbol
12124:tr{p,q}
12101:beveled
12012:2t{p,q}
11900:2r{p,q}
11795:Parent
11786:diagram
11784:Coxeter
11780:Symbol
11459:
11211:t{2,6}
11118:t{6,2}
10794:Coxeter
10783:Tiling
10652:
10404:t{2,5}
10311:t{5,2}
9992:Coxeter
9981:Tiling
9749:
9501:t{2,4}
9407:t{4,2}
9088:Coxeter
9077:Tiling
8937:
8682:t{2,3}
8589:t{3,2}
8265:Coxeter
8254:Tiling
8197:
7710:Coxeter
7699:Tiling
6523:Coxeter
6512:Tiling
4646:Coxeter
4635:Tiling
3475:Coxeter
3464:Tiling
3335:4.4.12
3250:4.4.10
3152:2.2.2.2
2928:4.2p.4
2796:r{p,2}
2793:t{p,2}
2159:(dual)
2143:Parent
2100:3.4.5.4
2067:3.5.3.5
2056:3.10.10
2003:3.4.4.4
1992:3.3.3.3
1970:3.4.3.4
1906:3.4.3.4
1873:3.3.3.3
1695:r{p,q}
1692:t{p,q}
986:(dual)
970:Parent
965:Johnson
920:Dihedra
884:dihedra
879:regular
661:Kaleido
403:History
290:and 53
282:and 11
274:and 13
233:regular
231:may be
215:and is
58:improve
14034:9-cube
13984:8-cube
13934:7-cube
13891:6-cube
13861:5-cube
13748:Square
13625:Family
13515:
13495:
13478:
13470:
13462:
13454:
13404:
13375:
13344:
13336:
13328:
13320:
13190:Images
12130:{p,q}
12117:(Also
12060:(Also
12018:{p,q}
11963:{p,q}
11957:t{p,q}
11906:{p,q}
11893:(also
11853:{p,q}
11847:r{p,q}
11804:{p,q}
11035:{2,6}
10956:{6,2}
10898:Edges
10895:Faces
10866:Pos. 0
10842:Pos. 1
10813:Pos. 2
10789:figure
10787:Vertex
10233:{2,5}
10154:{5,2}
10096:Edges
10093:Faces
10064:Pos. 0
10040:Pos. 1
10011:Pos. 2
9987:figure
9985:Vertex
9329:{2,4}
9250:{4,2}
9192:Edges
9189:Faces
9160:Pos. 0
9136:Pos. 1
9107:Pos. 2
9083:figure
9081:Vertex
8506:{2,3}
8427:{3,2}
8369:Edges
8366:Faces
8337:Pos. 0
8313:Pos. 1
8284:Pos. 2
8260:figure
8258:Vertex
7882:{2,2}
7814:Edges
7811:Faces
7782:Pos. 0
7758:Pos. 1
7729:Pos. 2
7705:figure
7703:Vertex
6805:{3,5}
6706:{5,3}
6627:Edges
6624:Faces
6595:Pos. 0
6571:Pos. 1
6542:Pos. 2
6518:figure
6516:Vertex
5038:{4,3}
4933:{3,4}
4834:{4,3}
4755:Edges
4752:Faces
4723:Pos. 0
4694:Pos. 1
4665:Pos. 2
4641:figure
4639:Vertex
3653:{3,3}
3574:Edges
3571:Faces
3542:Pos. 0
3518:Pos. 1
3494:Pos. 2
3470:figure
3468:Vertex
3306:4.4.6
3221:4.4.5
3171:4.4.8
3142:4.4.4
3124:2.8.8
3092:4.4.6
3063:4.4.3
3045:2.6.6
3007:4.4.4
2978:4.4.2
2960:2.4.4
2869:{p,2}
2862:{p,2}
2855:{p,2}
2848:{p,2}
2841:{p,2}
2834:{p,2}
2827:{p,2}
2820:{p,2}
2790:{p,2}
2111:4.6.10
1817:(p.q)
1768:{p,q}
1761:{p,q}
1754:{p,q}
1747:{p,q}
1740:{p,q}
1733:{p,q}
1726:{p,q}
1719:{p,q}
1689:{p,q}
932:Prisms
824:< 1
794:
790:
654:Polyca
540:, the
534:Kepler
435:Euclid
340:
264:prisms
253:convex
185:, and
47:, but
13753:p-gon
13476:S2CID
13464:74475
13460:JSTOR
13342:S2CID
13330:91532
13326:JSTOR
13288:(PDF)
13157:(PDF)
13069:Notes
12128:0,1,2
11996:dual.
11798:{p,q}
11315:0,1,2
10777:Name
10508:0,1,2
9975:Name
9605:0,1,2
9071:Name
8786:0,1,2
8248:Name
8061:0,1,2
7693:Name
7642:... D
7380:0,1,2
6621:(20)
6592:(30)
6568:(12)
6499:Graph
6489:Graph
6486:Name
5504:0,1,2
4720:(12)
4624:Graph
4616:Graph
4613:Name
4344:0,1,2
3453:Graph
3445:Graph
3442:Name
3073:2.2.2
2950:{2,2}
2867:0,1,2
2860:0,1,2
2174:Snub
2078:5.6.6
2045:5.5.5
2014:4.6.8
1981:4.6.6
1959:3.8.8
1948:4.4.4
1917:4.6.6
1895:3.3.3
1884:3.6.6
1862:3.6.6
1851:3.3.3
1766:0,1,2
1759:0,1,2
1001:Snub
967:name
457:Plato
423:Plato
237:face-
213:faces
14111:cube
13784:Cube
13614:and
13513:ISBN
13452:ISSN
13402:ISBN
13373:ISBN
13318:ISSN
13088:40].
12058:(rr)
11954:(t)
11895:dual
11844:(r)
11327:{12}
11128:{12}
10892:(6)
10863:(6)
10839:(2)
10797:and
10520:{10}
10321:{10}
10090:(5)
10061:(5)
10037:(2)
9995:and
9774:Cube
9447:Cube
9186:(4)
9157:(4)
9133:(2)
9091:and
8363:(3)
8334:(3)
8310:(2)
8268:and
8005:cube
7808:(2)
7779:(2)
7755:(2)
7713:and
7587:The
7572:edit
7545:150
7424:120
7421:180
7392:{10}
7295:120
7036:{10}
6526:and
6407:The
4767:Cube
4749:(8)
4691:(6)
4649:and
4531:The
3568:(4)
3539:(6)
3515:(4)
3478:and
3359:The
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