2306:
2011:
1788:
1526:
230:
842:
835:
828:
821:
814:
915:
800:
2319:
1801:
1539:
243:
2185:
1898:
1675:
1405:
112:
1447:
807:
1187:
1125:
1086:
1029:
990:
147:
2227:
2220:
1933:
1710:
1440:
856:
2413:
2291:, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an
1996:, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an
1773:, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an
1511:, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an
210:{3,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an
2553:
2237:
717:
368:
2549:
2400:
2169:
1241:
982:
972:
787:
2486:
2466:
2444:
2390:
2159:
1572:
1384:
1231:
1169:
1073:
959:
777:
700:
586:
444:
897:
2367:
2136:
1361:
477:
457:
373:
2395:
2375:
2337:
2164:
2144:
2106:
2068:
1852:
1629:
1577:
1557:
1389:
1369:
1331:
1293:
1236:
1226:
1198:
1179:
1174:
1164:
1136:
1097:
1078:
1068:
1040:
1001:
954:
926:
907:
867:
782:
762:
724:
710:
705:
685:
647:
604:
591:
571:
533:
490:
472:
467:
462:
429:
391:
363:
323:
66:
2362:
2131:
2093:
1213:
964:
749:
449:
2385:
2357:
2347:
2154:
2126:
2116:
2098:
2088:
2078:
1882:
1872:
1862:
1659:
1649:
1639:
1567:
1379:
1351:
1341:
1323:
1313:
1303:
1218:
1208:
1156:
1146:
1117:
1107:
1060:
1050:
1021:
1011:
946:
936:
887:
877:
772:
754:
744:
734:
695:
677:
667:
657:
634:
624:
614:
581:
563:
553:
543:
520:
510:
500:
439:
421:
411:
401:
378:
353:
343:
333:
96:
86:
76:
1720:
157:
2380:
2352:
2342:
2149:
2121:
2111:
2083:
2073:
1877:
1867:
1857:
1654:
1644:
1634:
1562:
1374:
1356:
1346:
1336:
1318:
1308:
1298:
1203:
1151:
1141:
1112:
1102:
1055:
1045:
1016:
1006:
977:
941:
931:
902:
882:
872:
767:
739:
729:
690:
672:
662:
652:
629:
619:
609:
576:
558:
548:
538:
515:
505:
495:
434:
416:
406:
396:
348:
338:
328:
91:
81:
71:
2567:
2476:
1943:
1457:
2292:
2215:
1191:
640:
597:
526:
384:
2539:
2418:
2041:
1827:
1604:
1266:
41:
2435:
2305:
2010:
1787:
1525:
841:
834:
827:
820:
813:
229:
2470:
2310:
2244:
2015:
1950:
1792:
1727:
1530:
1464:
234:
164:
1997:
1928:
1774:
1705:
1512:
1435:
1129:
1090:
1033:
994:
211:
142:
2404:, with alternating types or colors of octahedral cells. In Coxeter notation the half symmetry is = .
2280:
1985:
1762:
1500:
199:
2497:
914:
17:
2295:
2000:
1777:
1515:
214:
2330:
2284:
2048:
1989:
1834:
1766:
1611:
1550:
1504:
1273:
799:
203:
48:
2535:
2318:
1800:
1538:
242:
2482:
2462:
2454:
2440:
259:
2430:
2268:
1973:
1750:
1582:
1488:
292:
187:
2184:
2060:
1897:
1844:
1674:
1621:
1404:
1285:
266:
111:
58:
2545:
2514:
1446:
2561:
919:
2276:
1981:
1758:
1496:
195:
806:
2550:
Kleinian, a tool for visualizing
Kleinian groups, Geometry and the Imagination
2226:
2219:
2180:
1932:
1893:
1709:
1670:
1439:
1400:
1186:
1124:
1085:
1028:
263:
107:
2527:
2288:
2205:
1993:
1770:
1508:
207:
989:
146:
2264:
2195:
1969:
1908:
1746:
1695:
1685:
1484:
1415:
287:
183:
132:
122:
2447:. (Tables I and II: Regular polytopes and honeycombs, pp. 294â296)
1918:
1425:
855:
316:
2458:
2510:
2506:
2029:
1815:
1592:
1254:
29:
860:
1581:, with alternating types or colors of octahedral cells. In
2503:
Lorentzian
Coxeter groups and Boyd-Maxwell ball packings
2489:(Chapters 16â17: Geometries on Three-manifolds I, II)
2329:
It has a second construction as a uniform honeycomb,
1549:
It has a second construction as a uniform honeycomb,
2300:
1520:
224:
2511:Visualizing Hyperbolic Honeycombs arXiv:1511.02851
2494:Sphere Packings and Hyperbolic Reflection Groups
2414:Convex uniform honeycombs in hyperbolic space
8:
2032:
1818:
1595:
1257:
32:
275:
2540:{7,3,3} Honeycomb Meets Plane at Infinity
2005:
1782:
2471:Regular Honeycombs in Hyperbolic Space
2439:, 3rd. ed., Dover Publications, 1973.
2496:, JOURNAL OF ALGEBRA 79,78-97 (1982)
2451:The Beauty of Geometry: Twelve Essays
7:
2033:Infinite-order octahedral honeycomb
2273:infinite-order octahedral honeycomb
2026:Infinite-order octahedral honeycomb
18:Infinite-order octahedral honeycomb
25:
2398:
2393:
2388:
2383:
2378:
2373:
2365:
2360:
2355:
2350:
2345:
2340:
2335:
2317:
2304:
2287:{3,4,â}. It has infinitely many
2225:
2218:
2183:
2167:
2162:
2157:
2152:
2147:
2142:
2134:
2129:
2124:
2119:
2114:
2109:
2104:
2096:
2091:
2086:
2081:
2076:
2071:
2066:
2009:
1931:
1896:
1880:
1875:
1870:
1865:
1860:
1855:
1850:
1799:
1786:
1708:
1673:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1575:
1570:
1565:
1560:
1555:
1537:
1524:
1445:
1438:
1403:
1387:
1382:
1377:
1372:
1367:
1359:
1354:
1349:
1344:
1339:
1334:
1329:
1321:
1316:
1311:
1306:
1301:
1296:
1291:
1239:
1234:
1229:
1224:
1216:
1211:
1206:
1201:
1196:
1185:
1177:
1172:
1167:
1162:
1154:
1149:
1144:
1139:
1134:
1123:
1115:
1110:
1105:
1100:
1095:
1084:
1076:
1071:
1066:
1058:
1053:
1048:
1043:
1038:
1027:
1019:
1014:
1009:
1004:
999:
988:
980:
975:
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962:
957:
952:
944:
939:
934:
929:
924:
913:
905:
900:
895:
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880:
875:
870:
865:
854:
840:
833:
826:
819:
812:
805:
798:
785:
780:
775:
770:
765:
760:
752:
747:
742:
737:
732:
727:
722:
708:
703:
698:
693:
688:
683:
675:
670:
665:
660:
655:
650:
645:
632:
627:
622:
617:
612:
607:
602:
589:
584:
579:
574:
569:
561:
556:
551:
546:
541:
536:
531:
518:
513:
508:
503:
498:
493:
488:
475:
470:
465:
460:
455:
447:
442:
437:
432:
427:
419:
414:
409:
404:
399:
394:
389:
376:
371:
366:
361:
351:
346:
341:
336:
331:
326:
321:
254:Related polytopes and honeycombs
241:
228:
145:
110:
94:
89:
84:
79:
74:
69:
64:
2501:Hao Chen, Jean-Philippe Labbé,
2480:The Shape of Space, 2nd edition
2253:
2243:
2233:
2211:
2201:
2191:
2176:
2059:
2047:
2037:
1958:
1949:
1939:
1924:
1914:
1904:
1889:
1843:
1833:
1823:
1735:
1726:
1716:
1701:
1691:
1681:
1666:
1620:
1610:
1600:
1473:
1463:
1453:
1431:
1421:
1411:
1396:
1284:
1272:
1262:
172:
163:
153:
138:
128:
118:
103:
57:
47:
37:
2333:{3,(4,â,4)}, Coxeter diagram,
1553:{3,(4,3,4)}, Coxeter diagram,
1:
1819:Order-8 octahedral honeycomb
1596:Order-7 octahedral honeycomb
1258:Order-6 octahedral honeycomb
33:Order-5 octahedral honeycomb
2453:(1999), Dover Publications,
2293:infinite-order square tiling
1978:order-8 octahedral honeycomb
1812:Order-8 octahedral honeycomb
1755:order-7 octahedral honeycomb
1589:Order-7 octahedral honeycomb
1493:order-6 octahedral honeycomb
1251:Order-6 octahedral honeycomb
192:order-5 octahedral honeycomb
27:Tesselation in regular space
2275:is a regular space-filling
1980:is a regular space-filling
1757:is a regular space-filling
1495:is a regular space-filling
258:It a part of a sequence of
194:is a regular space-filling
2584:
1585:the half symmetry is = .
278:
2419:List of regular polytopes
307:
291:
1992:{3,4,8}. It has eight
1769:{3,4,7}. It has seven
1998:order-8 square tiling
1775:order-7 square tiling
1513:order-6 square tiling
212:order-5 square tiling
206:{3,4,5}. It has five
2568:Regular 3-honeycombs
1507:{3,4,6}. It has six
262:and honeycombs with
2311:Poincaré disk model
2016:Poincaré disk model
1793:Poincaré disk model
1531:Poincaré disk model
235:Poincaré disk model
2296:vertex arrangement
2269:hyperbolic 3-space
2001:vertex arrangement
1974:hyperbolic 3-space
1778:vertex arrangement
1751:hyperbolic 3-space
1516:vertex arrangement
1489:hyperbolic 3-space
279:{3,4,p} polytopes
215:vertex arrangement
188:hyperbolic 3-space
2536:{7,3,3} Honeycomb
2436:Regular Polytopes
2327:
2326:
2261:
2260:
2042:Regular honeycomb
2023:
2022:
1966:
1965:
1828:Regular honeycomb
1809:
1808:
1743:
1742:
1605:Regular honeycomb
1547:
1546:
1481:
1480:
1267:Regular honeycomb
1248:
1247:
260:regular polychora
251:
250:
180:
179:
42:Regular honeycomb
16:(Redirected from
2575:
2492:George Maxwell,
2477:Jeffrey R. Weeks
2403:
2402:
2401:
2397:
2396:
2392:
2391:
2387:
2386:
2382:
2381:
2377:
2376:
2370:
2369:
2368:
2364:
2363:
2359:
2358:
2354:
2353:
2349:
2348:
2344:
2343:
2339:
2338:
2321:
2314:(cell centered)
2308:
2301:
2229:
2222:
2187:
2172:
2171:
2170:
2166:
2165:
2161:
2160:
2156:
2155:
2151:
2150:
2146:
2145:
2139:
2138:
2137:
2133:
2132:
2128:
2127:
2123:
2122:
2118:
2117:
2113:
2112:
2108:
2107:
2101:
2100:
2099:
2095:
2094:
2090:
2089:
2085:
2084:
2080:
2079:
2075:
2074:
2070:
2069:
2061:Coxeter diagrams
2049:SchlÀfli symbols
2030:
2019:(cell centered)
2013:
2006:
1935:
1900:
1885:
1884:
1883:
1879:
1878:
1874:
1873:
1869:
1868:
1864:
1863:
1859:
1858:
1854:
1853:
1845:Coxeter diagrams
1835:SchlÀfli symbols
1816:
1803:
1796:(cell centered)
1790:
1783:
1712:
1677:
1662:
1661:
1660:
1656:
1655:
1651:
1650:
1646:
1645:
1641:
1640:
1636:
1635:
1631:
1630:
1622:Coxeter diagrams
1612:SchlÀfli symbols
1593:
1583:Coxeter notation
1580:
1579:
1578:
1574:
1573:
1569:
1568:
1564:
1563:
1559:
1558:
1541:
1534:(cell centered)
1528:
1521:
1449:
1442:
1407:
1392:
1391:
1390:
1386:
1385:
1381:
1380:
1376:
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1320:
1319:
1315:
1314:
1310:
1309:
1305:
1304:
1300:
1299:
1295:
1294:
1286:Coxeter diagrams
1274:SchlÀfli symbols
1255:
1244:
1243:
1242:
1238:
1237:
1233:
1232:
1228:
1227:
1221:
1220:
1219:
1215:
1214:
1210:
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1199:
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1166:
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1157:
1153:
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1148:
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1120:
1119:
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1109:
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1103:
1099:
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1080:
1079:
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1070:
1069:
1063:
1062:
1061:
1057:
1056:
1052:
1051:
1047:
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1042:
1041:
1031:
1024:
1023:
1022:
1018:
1017:
1013:
1012:
1008:
1007:
1003:
1002:
992:
985:
984:
983:
979:
978:
974:
973:
967:
966:
965:
961:
960:
956:
955:
949:
948:
947:
943:
942:
938:
937:
933:
932:
928:
927:
917:
910:
909:
908:
904:
903:
899:
898:
890:
889:
888:
884:
883:
879:
878:
874:
873:
869:
868:
858:
844:
837:
830:
823:
816:
809:
802:
790:
789:
788:
784:
783:
779:
778:
774:
773:
769:
768:
764:
763:
757:
756:
755:
751:
750:
746:
745:
741:
740:
736:
735:
731:
730:
726:
725:
713:
712:
711:
707:
706:
702:
701:
697:
696:
692:
691:
687:
686:
680:
679:
678:
674:
673:
669:
668:
664:
663:
659:
658:
654:
653:
649:
648:
637:
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635:
631:
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625:
621:
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616:
615:
611:
610:
606:
605:
594:
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583:
582:
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572:
566:
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559:
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534:
523:
522:
521:
517:
516:
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480:
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469:
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381:
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350:
349:
345:
344:
340:
339:
335:
334:
330:
329:
325:
324:
276:
245:
238:(cell centered)
232:
225:
149:
114:
99:
98:
97:
93:
92:
88:
87:
83:
82:
78:
77:
73:
72:
68:
67:
59:Coxeter diagrams
49:SchlÀfli symbols
30:
21:
2583:
2582:
2578:
2577:
2576:
2574:
2573:
2572:
2558:
2557:
2532:Visual insights
2524:
2427:
2410:
2399:
2394:
2389:
2384:
2379:
2374:
2372:
2366:
2361:
2356:
2351:
2346:
2341:
2336:
2334:
2331:SchlÀfli symbol
2322:
2313:
2309:
2285:SchlÀfli symbol
2249:
2223:
2168:
2163:
2158:
2153:
2148:
2143:
2141:
2135:
2130:
2125:
2120:
2115:
2110:
2105:
2103:
2102:
2097:
2092:
2087:
2082:
2077:
2072:
2067:
2065:
2054:
2028:
2018:
2014:
1990:SchlÀfli symbol
1881:
1876:
1871:
1866:
1861:
1856:
1851:
1849:
1814:
1804:
1795:
1791:
1767:SchlÀfli symbol
1658:
1653:
1648:
1643:
1638:
1633:
1628:
1626:
1591:
1576:
1571:
1566:
1561:
1556:
1554:
1551:SchlÀfli symbol
1542:
1533:
1529:
1505:SchlÀfli symbol
1469:
1443:
1388:
1383:
1378:
1373:
1368:
1366:
1360:
1355:
1350:
1345:
1340:
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1111:
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1072:
1067:
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1054:
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1044:
1039:
1037:
1036:
1032:
1020:
1015:
1010:
1005:
1000:
998:
997:
993:
981:
976:
971:
969:
968:
963:
958:
953:
951:
950:
945:
940:
935:
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925:
923:
922:
918:
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896:
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893:
891:
886:
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786:
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766:
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723:
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320:
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256:
246:
237:
233:
223:
204:SchlÀfli symbol
95:
90:
85:
80:
75:
70:
65:
63:
28:
23:
22:
15:
12:
11:
5:
2581:
2579:
2571:
2570:
2560:
2559:
2556:
2555:
2552:4 March 2014.
2546:Danny Calegari
2543:
2523:
2522:External links
2520:
2519:
2518:
2515:Henry Segerman
2513:Roice Nelson,
2508:
2499:
2490:
2474:
2448:
2426:
2423:
2422:
2421:
2416:
2409:
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2324:
2323:Ideal surface
2315:
2259:
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2255:
2251:
2250:
2247:
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2240:
2235:
2231:
2230:
2213:
2209:
2208:
2203:
2199:
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2178:
2174:
2173:
2063:
2057:
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2027:
2024:
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1964:
1963:
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1956:
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1953:
1947:
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1941:
1937:
1936:
1926:
1922:
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1916:
1912:
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1906:
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1891:
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1847:
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1840:
1837:
1831:
1830:
1825:
1821:
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1810:
1807:
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1805:Ideal surface
1797:
1741:
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1737:
1733:
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1730:
1724:
1723:
1718:
1714:
1713:
1703:
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1624:
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1614:
1608:
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1598:
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1545:
1544:
1543:Ideal surface
1535:
1479:
1478:
1475:
1471:
1470:
1467:
1461:
1460:
1455:
1451:
1450:
1433:
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1398:
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1183:
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638:
595:
524:
481:
382:
314:
310:
309:
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296:
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285:
281:
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255:
252:
249:
248:
247:Ideal surface
239:
222:
219:
178:
177:
174:
170:
169:
167:
161:
160:
155:
151:
150:
140:
136:
135:
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126:
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116:
115:
105:
101:
100:
61:
55:
54:
51:
45:
44:
39:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2580:
2569:
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2563:
2554:
2551:
2547:
2544:
2541:
2538:(2014/08/01)
2537:
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2525:
2521:
2516:
2512:
2509:
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2504:
2500:
2498:
2495:
2491:
2488:
2487:0-8247-0709-5
2484:
2481:
2478:
2475:
2472:
2469:(Chapter 10,
2468:
2467:0-486-40919-8
2464:
2460:
2456:
2452:
2449:
2446:
2445:0-486-61480-8
2442:
2438:
2437:
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2278:
2274:
2270:
2266:
2256:
2252:
2248:
2246:
2245:Coxeter group
2242:
2239:
2236:
2232:
2228:
2221:
2217:
2214:
2212:Vertex figure
2210:
2207:
2204:
2200:
2197:
2194:
2190:
2186:
2182:
2179:
2175:
2064:
2062:
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2046:
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2040:
2036:
2031:
2025:
2017:
2012:
2008:
2007:
2004:
2002:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1961:
1957:
1954:
1952:
1951:Coxeter group
1948:
1945:
1942:
1938:
1934:
1930:
1927:
1925:Vertex figure
1923:
1920:
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1772:
1768:
1764:
1760:
1756:
1752:
1748:
1738:
1734:
1731:
1729:
1728:Coxeter group
1725:
1722:
1719:
1715:
1711:
1707:
1704:
1702:Vertex figure
1700:
1697:
1694:
1690:
1687:
1684:
1680:
1676:
1672:
1669:
1665:
1625:
1623:
1619:
1615:
1613:
1609:
1606:
1603:
1599:
1594:
1588:
1586:
1584:
1552:
1540:
1536:
1532:
1527:
1523:
1522:
1519:
1517:
1514:
1510:
1506:
1502:
1498:
1494:
1490:
1486:
1476:
1472:
1468:
1466:
1465:Coxeter group
1462:
1459:
1456:
1452:
1448:
1441:
1437:
1434:
1432:Vertex figure
1430:
1427:
1424:
1420:
1417:
1414:
1410:
1406:
1402:
1399:
1395:
1289:
1287:
1283:
1277:
1275:
1271:
1268:
1265:
1261:
1256:
1250:
1193:
1188:
1184:
1131:
1126:
1122:
1092:
1087:
1083:
1035:
1030:
1026:
996:
991:
987:
921:
916:
912:
862:
857:
853:
848:
847:
843:
839:
836:
832:
829:
825:
822:
818:
815:
811:
808:
804:
801:
797:
794:
793:
719:
718:{3,4,∞}
715:
642:
639:
599:
596:
528:
525:
485:
482:
386:
383:
318:
315:
312:
311:
304:
301:
298:
297:
294:
289:
286:
283:
282:
277:
274:
272:
268:
265:
261:
253:
244:
240:
236:
231:
227:
226:
220:
218:
216:
213:
209:
205:
201:
197:
193:
189:
185:
175:
171:
168:
166:
165:Coxeter group
162:
159:
156:
152:
148:
144:
141:
139:Vertex figure
137:
134:
131:
127:
124:
121:
117:
113:
109:
106:
102:
62:
60:
56:
52:
50:
46:
43:
40:
36:
31:
19:
2542:(2014/08/14)
2531:
2502:
2493:
2479:
2450:
2434:
2328:
2277:tessellation
2272:
2262:
2055:{3,(4,â,4)}
1982:tessellation
1977:
1967:
1759:tessellation
1754:
1744:
1548:
1497:tessellation
1492:
1482:
1280:{3,(3,4,3)}
483:
305:Paracompact
270:
257:
196:tessellation
191:
181:
2473:) Table III
2202:Edge figure
1915:Edge figure
1692:Edge figure
1422:Edge figure
1192:{4,∞}
308:Noncompact
129:Edge figure
2425:References
2254:Properties
2224:{(4,â,4)}
1959:Properties
1736:Properties
1474:Properties
1444:{(4,3,4)}
264:octahedral
173:Properties
2528:John Baez
2289:octahedra
2281:honeycomb
1994:octahedra
1986:honeycomb
1771:octahedra
1763:honeycomb
1509:octahedra
1501:honeycomb
208:octahedra
200:honeycomb
2562:Category
2505:, (2013)
2459:99-35678
2408:See also
2265:geometry
2257:Regular
1970:geometry
1962:Regular
1839:{3,4,8}
1747:geometry
1739:Regular
1616:{3,4,7}
1485:geometry
1477:Regular
184:geometry
176:Regular
53:{3,4,5}
2431:Coxeter
2283:) with
2263:In the
2238:{â,4,3}
2053:{3,4,â}
1988:) with
1968:In the
1944:{8,4,3}
1765:) with
1745:In the
1721:{7,4,3}
1503:) with
1483:In the
1458:{6,4,3}
1278:{3,4,6}
851:figure
641:{3,4,8}
598:{3,4,7}
527:{3,4,6}
484:{3,4,5}
385:{3,4,4}
317:{3,4,3}
302:Finite
269:: {3,4,
202:) with
182:In the
158:{5,4,3}
2517:(2015)
2485:
2465:
2457:
2443:
2271:, the
1976:, the
1753:, the
1491:, the
892:
849:Vertex
795:Image
358:
284:Space
221:Images
190:, the
2216:{4,â}
2192:Faces
2181:{3,4}
2177:Cells
1929:{4,8}
1905:Faces
1894:{3,4}
1890:Cells
1706:{4,7}
1682:Faces
1671:{3,4}
1667:Cells
1436:{4,6}
1412:Faces
1401:{3,4}
1397:Cells
1130:{4,8}
1091:{4,7}
1034:{4,6}
995:{4,5}
920:{4,4}
861:{4,3}
313:Name
299:Form
267:cells
143:{4,5}
119:Faces
108:{3,4}
104:Cells
2483:ISBN
2463:ISBN
2455:LCCN
2441:ISBN
2279:(or
2234:Dual
2038:Type
1984:(or
1940:Dual
1824:Type
1761:(or
1717:Dual
1601:Type
1499:(or
1454:Dual
1263:Type
716:...
198:(or
154:Dual
38:Type
2267:of
2206:{â}
2196:{3}
1972:of
1919:{8}
1909:{3}
1749:of
1696:{7}
1686:{3}
1487:of
1426:{6}
1416:{3}
186:of
133:{5}
123:{3}
2564::
2548:,
2534::
2530:,
2461:,
2433:,
2371:=
2298:.
2140:=
2003:.
1780:.
1518:.
1365:=
273:}
217:.
293:H
288:S
271:p
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
