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triangular tilings as flat surfaces (not actually cells, corresponding to the equatorial hexagons). The honeycomb with octahemioctahedra as vertex figures has the tetrahedra and the triangular tilings; the one with cubohemioctahedra as vertex figures has the octrahedra and the triangular tilings. They're #4 and #5 on
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If the definition is the first sentence only, a requirement that all the edges must congruent should be added for consistency within the article. I'm not aware if there's a competing definition of uniformity that permits faces with an even number of edges to have two different alternating lengths,
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The cubohemioctahedron and octahemioctahedra are indeed vertex figures of nonconvex
Euclidean honeycombs. In the tetrahedral-octahedral honeycomb you can (as the name suggests) find tetrahedra (corresponding to the triangles of the cuboctahedron), octahedra (corresponding to the squares), but also
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Uniform polytopes whose circumradius is equal to the edge length can be used as vertex figures for uniform tessellations. For example, the regular hexagon divides into 6 equilateral triangles and is the vertex figure for the regular triangular tiling. Also the cuboctahedron divides into 8 regular
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I have been looking at uniform tessellations of
Euclidean spaces, and found many had vertex figures as uniform polytopes of circumradius 1 (edge length). I scanned all the uniform polytopes of Klitzing's lists, took the subset with circumradius 1, and mapped them all onto single-ringed uniform
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Additionally, the vertex figure of a tiling must be the dual polytope of the cell of the dual tiling. So, the snub dodecahedron can be the vertex figure of a uniform hyperbolic tiling if and only if the pentagonal hexecontahedron can tile hyperbolic space. Is that possible? It certainly seems
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This might be excessively pedantic, but the article appears to consider the line segment to be a uniform polytope. A single vertex and the null polytope might be considered (uniform) polytopes as well. The revised definition only refers to dimensions 2 and up.
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I'm going to remove the latter sentence. Such figures are vertex-transitive, but not uniform (faces are not regular) if the arcs from the ringed node are not alike; and they're not prismatic if the parent CD is multiply connected.
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The "scaliform" term is really only used by Bowers – unless it is adapted by someone more notable, I don't believe it belongs on this article. I deleted the corresponding section with this justification. Hope it isn't an issue. –
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How useful is this paragraph? Any polytope that can be inscribed in a sphere is a candidate for a vertex figure, provided that its edge lengths are in appropriate ratios; but is it proven that all polytopes described above
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529:. Are all of these known to be v.f. of (nonconvex) honeycombs? Also, 31 uniform polyhedra have circumradii smaller than their edge-length; are they all known to correspond to triangle-faced nonconvex polychora? —
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can be constructed from the diagram by removing the ringed node, and ringing neighboring nodes. Such vertex figures are also uniform polytopes, being prismatic if the ringed node was in contact with more than one
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Quite obviously, the polytopes with H3 and H4 symmetry cannot be used as the vertex figures of a uniform tiling of
Euclidean space. Even the regular pentagon cannot be the vertex figure of Euclidean space.
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Rewrite if you like. By symmetry they are all listable, but by geometry (of uniform solutions) and topological result, they're identical, so the question is how far you want to expand the lists.
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2236:"A uniform polytope is a vertex-transitive polytope made from uniform facets of a lower dimension. The uniform polytopes in two dimensions are the regular polygons."
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True, but the notation shown is for the symmetry groups, not for the polytopes themselves; this is wrong — the symmetry group is bigger than the product ××. —
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but if there is it's not the most commonly accepted one. If the second sentence is part of the definition, this should be rephrased to make that more clear.
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I created this stub as I started some articles on higher dimensional uniform polytopes. It's crude, and I appreciate any help to improve it. Thanks!
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tetrahedra and 6 square pyramids (half octahedron), and it is the vertex figure for the alternated cubic honeycomb.
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The regular pentagon's circumradius is less than its edge, so it is the v.f. of a tiling of S2, not E2. —
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61:New to Knowledge? Welcome!
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1065:
1060:
1055:
1050:
1043:
1042:
1041:
1036:
1031:
1026:
1021:
1014:
1013:
1012:
1007:
1002:
997:
992:
987:
980:
976:
974:
970:
968:
964:
962:
958:
955:
949:
948:
947:
946:
941:
936:
931:
926:
921:
914:
913:
912:
907:
902:
900:Truncated_cube
897:
892:
885:
884:
883:
878:
871:
870:
869:
864:
859:
854:
849:
842:
838:
836:
832:
830:
826:
823:
817:
816:
813:
811:
808:
805:
803:
800:
797:
795:
792:
789:
787:
784:
781:
779:
776:
773:
771:
768:
765:
763:
760:
757:
755:
752:
749:
747:
744:
741:
737:
712:
709:
693:
692:
680:
677:
676:
675:
646:
645:
638:
635:
611:
610:
592:kissing number
586:
585:
584:
583:
569:
563:
562:
546:
545:
499:
498:
490:
487:
443:
440:
410:
407:
404:
403:
400:
399:
396:
395:
393:
366:
354:
353:
341:
329:
328:
325:
324:
313:
307:
306:
304:
287:the discussion
274:
273:
257:
245:
244:
236:
224:
223:
217:
206:
192:
191:
129:
128:
124:
123:
118:
113:
104:
103:
101:
100:
93:
88:
79:
73:
71:
70:
59:
50:
49:
46:
45:
39:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2406:
2395:
2392:
2390:
2387:
2386:
2384:
2377:
2376:
2372:
2368:
2357:
2349:
2345:
2341:
2315:
2311:
2307:
2303:
2302:
2301:
2300:
2299:
2298:
2293:
2289:
2285:
2280:
2279:
2278:
2277:
2274:
2270:
2266:
2262:
2258:
2254:
2253:
2252:
2251:
2247:
2243:
2237:
2231:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2204:
2203:
2199:
2196:
2194:
2191:
2189:
2186:
2185:
2184:
2163:
2159:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2135:
2134:
2130:
2127:
2125:
2122:
2120:
2117:
2116:
2115:
2094:
2090:
2085:
2082:
2080:
2077:
2076:
2075:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2052:
2051:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2028:
2027:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2004:
2003:
1999:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1972:
1971:
1967:
1963:Cantellated 2
1961:
1959:
1955:Cantellated 4
1953:
1951:
1948:
1946:
1943:
1941:
1938:
1936:
1933:
1932:
1931:
1927:
1921:
1919:
1913:
1911:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1884:
1883:
1879:
1872:
1865:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1838:
1837:
1833:
1830:
1828:
1825:
1823:
1820:
1819:
1818:
1791:
1782:
1779:
1777:
1774:
1772:
1769:
1768:
1767:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1744:
1743:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1720:
1719:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1696:
1695:
1691:
1687:Cantellated 2
1685:
1683:
1679:Cantellated 3
1677:
1675:
1672:
1670:
1667:
1665:
1662:
1660:
1657:
1656:
1655:
1651:
1645:
1643:
1637:
1635:
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1608:
1607:
1603:
1596:
1589:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1562:
1561:
1557:
1554:
1552:
1549:
1547:
1544:
1543:
1542:
1515:
1506:
1503:
1501:
1498:
1496:
1493:
1492:
1491:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1468:
1467:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1444:
1443:
1439:
1435:Cantellated 2
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1412:
1411:
1407:
1401:
1399:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1372:
1371:
1367:
1360:
1354:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1333:
1332:
1328:
1325:
1323:
1320:
1318:
1315:
1314:
1313:
1286:
1277:
1274:
1272:
1269:
1267:
1264:
1263:
1262:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1239:
1238:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1215:
1214:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1191:
1190:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1167:
1166:
1162:
1159:
1157:
1154:
1152:
1149:
1148:
1147:
1126:
1117:
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1093:
1092:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1069:
1068:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1045:
1044:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1016:
1015:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
982:
981:
954:
945:
942:
940:
937:
935:
932:
930:
927:
925:
922:
920:
917:
916:
915:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
887:
886:
882:
879:
877:
876:cuboctahedron
874:
873:
872:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
844:
843:
822:
809:
801:
793:
785:
777:
769:
761:
753:
745:
743:
738:
735:
734:
731:
730:
726:
722:
718:
711:Summary table
710:
708:
707:
703:
699:
691:
688:
683:
682:
678:
674:
670:
666:
662:
661:
660:
659:
655:
651:
644:
641:
640:
636:
634:
633:
629:
625:
621:
617:
609:
605:
601:
597:
593:
588:
587:
582:
578:
574:
570:
567:
566:
565:
564:
561:
557:
553:
550:implausible.
548:
547:
543:
542:
541:
540:
536:
532:
528:
524:
520:
516:
512:
507:
505:
497:
493:
492:
488:
486:
485:
481:
477:
470:
463:
459:
452:
441:
439:
438:
435:
431:
427:
422:
421:
418:
414:
408:
394:
377:
376:
371:
367:
364:
360:
359:
355:
350:
345:
342:
339:
335:
322:
318:
312:
309:
308:
305:
288:
284:
280:
279:
271:
265:
260:
258:
255:
251:
250:
246:
240:
237:
234:
230:
225:
221:
215:
207:
203:
198:
197:
188:
184:
181:
178:
174:
170:
166:
163:
160:
157:
154:
151:
148:
145:
142:
138:
135:
134:Find sources:
131:
130:
122:
121:Verifiability
119:
117:
114:
112:
109:
108:
107:
98:
94:
92:
89:
87:
83:
80:
78:
75:
74:
68:
64:
63:Learn to edit
60:
57:
52:
51:
48:
47:
43:
37:
33:
29:
28:
19:
2361:
2309:
2284:173.227.48.5
2256:
2242:173.227.48.5
2238:
2235:
2193:10-orthoplex
1995:Runcinated 4
1106:Snub_24-cell
862:dodecahedron
754:Quasiregular
714:
694:
684:
647:
642:
624:Double sharp
612:
508:
503:
500:
494:
445:
429:
423:
415:
412:
373:
317:Mid-priority
316:
276:
242:Mid‑priority
220:WikiProjects
182:
176:
168:
161:
155:
149:
143:
133:
105:
30:This is the
2366:OfficialURL
2358:"Scaliform"
2304:Actually a
2222:10-demicube
2124:9-orthoplex
1923:Truncated 1
1915:Truncated 2
1907:Truncated 4
1827:8-orthoplex
1647:Truncated 1
1639:Truncated 2
1631:Truncated 3
1551:7-orthoplex
1403:Truncated 1
1395:Truncated 2
1322:6-orthoplex
1156:5-orthoplex
867:icosahedron
847:tetrahedron
810:Heptellated
794:Pentellated
770:Cantellated
292:Mathematics
283:mathematics
239:Mathematics
159:free images
42:not a forum
2383:Categories
2314:truncation
2188:10-simplex
2153:9-demicube
1856:8-demicube
1580:7-demicube
1351:6-demicube
1185:5-demicube
852:octahedron
786:Stericated
778:Runcinated
719:articles.
446:I added a
2306:apeirogon
2119:9-simplex
1822:8-simplex
1546:7-simplex
1317:6-simplex
1151:5-simplex
995:tesseract
939:Snub cube
802:Hexicated
762:Truncated
451:confusing
442:Confusing
99:if needed
82:Be polite
32:talk page
2340:Tom Ruen
2265:Tom Ruen
1010:600-cell
1005:120-cell
721:Tom Ruen
665:Tom Ruen
600:Tom Ruen
552:Calcyman
434:Tom Ruen
417:Tom Ruen
409:Untitled
375:inactive
349:inactive
67:get help
40:This is
38:article.
2310:uniform
2198:10-cube
1000:24-cell
990:16-cell
746:Regular
740:Coxeter
698:Tamfang
650:Tamfang
573:Tamfang
531:Tamfang
458:WP:LEAD
319:on the
210:B-class
165:WP refs
153:scholar
2316:t{∞}=t
2129:9-cube
1832:8-cube
1556:7-cube
1327:6-cube
1161:5-cube
985:5-cell
462:WP:MSM
216:scale.
137:Google
742:group
690:node.
469:prose
180:JSTOR
141:books
95:Seek
2371:talk
2344:talk
2320:{∞}=
2288:talk
2269:talk
2246:talk
857:cube
725:talk
702:talk
669:talk
654:talk
628:talk
604:talk
577:talk
556:talk
535:talk
480:talk
173:FENS
147:news
84:and
2318:0,1
2259:as
814:0,7
806:0,6
798:0,5
790:0,4
782:0,3
774:0,2
766:0,1
504:are
311:Mid
187:TWL
2385::
2373:)
2346:)
2338:.
2290:)
2271:)
2248:)
2182:=
2180:10
2174:10
2172:BC
2168:10
2162:10
2113:=
2103:BC
1997:21
1965:41
1957:21
1925:42
1917:41
1909:21
1877:42
1873:,
1870:41
1866:,
1863:21
1816:=
1800:BC
1689:31
1681:21
1649:32
1641:31
1633:21
1601:32
1597:,
1594:31
1590:,
1587:21
1540:=
1524:BC
1437:21
1405:22
1397:21
1365:22
1361:,
1358:21
1311:=
1295:BC
1145:=
1135:BC
979:=
969:BC
841:=
831:BC
736:n
727:)
704:)
671:)
656:)
630:)
606:)
579:)
558:)
537:)
525:,
521:,
517:,
513:,
482:)
472:}}
466:{{
454:}}
448:{{
167:)
65:;
2369:(
2342:(
2286:(
2267:(
2244:(
2178:D
2176:=
2170:=
2166:A
2111:9
2109:D
2107:=
2105:9
2101:=
2099:9
2097:A
2093:9
1875:1
1868:2
1861:4
1814:8
1812:E
1810:=
1808:8
1806:D
1804:=
1802:8
1798:=
1796:8
1794:A
1790:8
1599:1
1592:2
1585:3
1538:7
1536:E
1534:=
1532:7
1530:D
1528:=
1526:7
1522:=
1520:7
1518:A
1514:7
1363:1
1356:2
1309:6
1307:E
1305:=
1303:6
1301:D
1299:=
1297:6
1293:=
1291:6
1289:A
1285:6
1143:5
1141:D
1139:=
1137:5
1133:=
1131:5
1129:A
1125:5
977:4
975:H
973:=
971:4
967:=
965:4
963:F
961:=
959:4
957:A
953:4
839:3
837:H
835:=
833:3
829:=
827:3
825:A
821:3
812:t
804:t
796:t
788:t
780:t
772:t
764:t
758:1
756:t
750:0
748:t
723:(
700:(
696:—
667:(
652:(
626:(
602:(
575:(
554:(
533:(
478:(
378:.
351:)
347:(
323:.
222::
183:·
177:·
169:·
162:·
156:·
150:·
144:·
139:(
69:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
