4237:
3686:
3387:
3135:
2846:
2409:
2189:
1965:
1667:
1145:
844:
580:
232:
4248:
3944:
3697:
3398:
3146:
2857:
2629:
2420:
2200:
1976:
1678:
1371:
1156:
855:
591:
243:
2726:
2517:
2297:
2073:
1853:
4114:
3829:
3563:
3278:
3020:
494:
487:
4155:
4146:
3864:
3605:
3598:
1583:
1576:
1541:
1288:
1253:
1062:
1055:
1020:
762:
727:
452:
109:
144:
1651:, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an
1355:, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an
1129:, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an
829:, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an
565:, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an
4222:{∞,∞} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order apeirogonal tilings existing around each vertex in an
3929:, {7,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an
3671:, {6,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an
217:{3,∞} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an
4304:
2759:
3372:
All vertices are ultra-ideal (existing beyond the ideal boundary) with five infinite-order pentagonal tilings existing around each edge and with an
4449:
1593:
2549:
2329:
2105:
1885:
1298:
772:
1072:
4445:
4291:
4098:
1759:
1525:
4377:
4357:
4335:
4281:
4223:
4219:
4149:
4088:
3730:
3542:
3179:
2999:
2809:
2749:
2721:
1749:
1571:
1515:
999:
624:
504:
431:
4065:
3519:
2976:
1726:
1492:
976:
408:
4286:
4266:
4093:
4073:
4035:
3997:
3783:
3735:
3715:
3547:
3527:
3489:
3451:
3273:
3232:
3164:
2984:
2946:
2908:
2680:
2471:
2251:
2155:
2068:
2027:
1807:
1754:
1734:
1696:
1648:
1536:
1520:
1500:
1462:
1424:
1352:
1248:
1207:
1126:
1015:
1004:
984:
946:
908:
826:
722:
681:
609:
562:
447:
416:
378:
340:
259:
214:
104:
63:
4271:
4078:
4060:
4050:
4040:
4022:
4012:
4002:
3798:
3504:
3466:
3247:
3184:
3004:
2961:
2923:
2695:
2685:
2486:
2266:
2042:
1822:
1721:
1711:
1487:
1477:
1449:
1439:
1222:
961:
923:
696:
629:
436:
393:
355:
78:
4276:
4083:
4055:
4045:
4027:
4017:
4007:
3813:
3803:
3793:
3725:
3668:
3558:
3537:
3509:
3499:
3481:
3471:
3461:
3262:
3252:
3242:
3174:
2994:
2966:
2956:
2938:
2928:
2918:
2710:
2700:
2690:
2501:
2491:
2481:
2292:
2281:
2271:
2261:
2057:
2047:
2037:
1837:
1827:
1817:
1744:
1716:
1706:
1510:
1482:
1472:
1454:
1444:
1434:
1237:
1227:
1217:
994:
966:
956:
938:
928:
918:
711:
701:
691:
619:
426:
398:
388:
370:
360:
350:
93:
83:
73:
3808:
3788:
3720:
3532:
3514:
3494:
3476:
3456:
3257:
3237:
3169:
2989:
2971:
2951:
2933:
2913:
2705:
2496:
2476:
2276:
2256:
2052:
2032:
1832:
1812:
1763:, with alternating types or colors of infinite-order triangular tiling cells. In Coxeter notation the half symmetry is = .
1739:
1701:
1505:
1467:
1429:
1232:
1212:
989:
971:
951:
933:
913:
706:
686:
614:
421:
403:
383:
365:
345:
88:
68:
4463:
4367:
3117:
3015:
1848:
4435:
3926:
2599:
2512:
2766:
2556:
2336:
2112:
1892:
3672:
3373:
3308:
3121:
3050:
2539:
2379:
2319:
1875:
1050:
757:
566:
482:
273:
218:
139:
4309:
3970:
3758:
3424:
3207:
2883:
2834:
The "ideal surface" projection below is a plane-at-infinity, in the
Poincaré half-space model of H3. It shows an
2655:
2446:
2226:
2002:
1782:
1397:
1182:
881:
656:
313:
38:
3930:
3859:
1283:
4326:
3824:
3593:
4361:
4241:
4171:
3879:
3690:
3620:
3391:
3323:
3139:
3067:
2850:
2413:
2193:
1969:
1671:
1600:
1305:
1149:
1079:
848:
779:
584:
511:
236:
159:
4236:
3685:
3386:
3134:
2845:
2813:
2603:
2408:
2383:
2188:
2159:
1964:
1939:
1666:
1144:
843:
579:
231:
4211:
3918:
3660:
3362:
3106:
2805:
2595:
2375:
2151:
1931:
1640:
1344:
1118:
818:
551:
198:
4388:
4109:
17:
4259:
4215:
3977:
3922:
3765:
3708:
3664:
3431:
3366:
3214:
3157:
3110:
2890:
2820:
2662:
2610:
2453:
2390:
2233:
2166:
2009:
1946:
1789:
1689:
1644:
1404:
1348:
1189:
1122:
888:
822:
663:
602:
555:
320:
202:
45:
4431:
4247:
3943:
3696:
3397:
3145:
2856:
2628:
2419:
2199:
1975:
1677:
1370:
1155:
854:
590:
242:
4373:
4353:
4345:
4331:
2835:
2095:
1935:
4321:
4195:
3902:
3644:
3346:
3090:
2789:
2579:
2359:
2135:
1915:
1624:
1328:
1102:
802:
634:
535:
182:
3739:, with alternating types or colors of cells. In Coxeter notation the half symmetry is = .
3188:, with alternating types or colors of cells. In Coxeter notation the half symmetry is = .
3989:
3775:
3443:
3224:
2900:
2672:
2463:
2243:
2019:
1799:
1416:
1199:
900:
673:
332:
262:
55:
4441:
276:
4457:
4226:
3933:
3675:
3376:
3124:
2828:
2744:
2618:
2534:
2398:
2314:
2178:
2090:
1954:
1870:
1656:
1360:
1134:
833:
569:
221:
2617:
is {7,∞,3}, with three infinite-order heptagonal tilings meeting at each edge. The
4207:
3914:
3656:
3358:
3102:
2801:
2591:
2397:
is {6,∞,3}, with three infinite-order hexagonal tilings meeting at each edge. The
2371:
2147:
1927:
1636:
1340:
1114:
814:
547:
194:
633:, with alternating types or colors of infinite-order triangular tiling cells. In
1953:
is {4,∞,3}, with three infinite-order square tilings meeting at each edge. The
4446:
Kleinian, a tool for visualizing
Kleinian groups, Geometry and the Imagination
4414:
4154:
4145:
4113:
3863:
3828:
3604:
3597:
3562:
3277:
3019:
2725:
2516:
2296:
2072:
1852:
1582:
1575:
1540:
1287:
1252:
1061:
1054:
1019:
761:
726:
493:
486:
451:
108:
4423:
4134:
4124:
3116:
All vertices are ultra-ideal (existing beyond the ideal boundary) with four
2736:
1561:
4191:
3898:
3849:
3839:
3640:
3342:
3298:
3288:
3086:
2785:
2575:
2527:
2355:
2131:
2083:
1911:
1620:
1551:
1324:
1273:
1263:
1098:
1030:
798:
747:
737:
531:
462:
178:
129:
119:
143:
4417:
4338:. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
3583:
3573:
2307:
1040:
3040:
3030:
1863:
472:
4349:
4401:
4397:
3958:
3746:
3412:
3195:
2871:
2831:
of this honeycomb is an infinite-order apeirogonal tiling, {∞,3}.
1385:
1170:
869:
644:
301:
26:
286:
It is a part of a sequence of self-dual regular honeycombs: {
2823:
of the apeirogonal tiling honeycomb is {∞,∞,3}, with three
2816:, each of which has a limiting circle on the ideal sphere.
2621:
of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.
2606:, each of which has a limiting circle on the ideal sphere.
2401:
of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.
2386:, each of which has a limiting circle on the ideal sphere.
2162:, each of which has a limiting circle on the ideal sphere.
1957:
of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.
1942:, each of which has a limiting circle on the ideal sphere.
4394:
Lorentzian
Coxeter groups and Boyd-Maxwell ball packings
2643:
2434:
2214:
1990:
1770:
272:
It is a part of a sequence of regular honeycombs with
258:
It is a part of a sequence of regular honeycombs with
4380:(Chapters 16–17: Geometries on Three-manifolds I, II)
4258:
It has a second construction as a uniform honeycomb,
3707:
It has a second construction as a uniform honeycomb,
3156:
It has a second construction as a uniform honeycomb,
1688:
It has a second construction as a uniform honeycomb,
601:
It has a second construction as a uniform honeycomb,
1661:
1139:
838:
574:
226:
4402:Visualizing Hyperbolic Honeycombs arXiv:1511.02851
4415:Hyperbolic Catacombs Carousel: {3,∞,3} honeycomb
2181:of this honeycomb is a heptagonal tiling, {∞,3}.
4385:Sphere Packings and Hyperbolic Reflection Groups
3962:Order-infinite-infinite apeirogonal honeycomb
4305:Convex uniform honeycombs in hyperbolic space
4295:, with alternating types or colors of cells.
4200:order-infinite-infinite apeirogonal honeycomb
3955:Order-infinite-infinite apeirogonal honeycomb
1389:Order-infinite-infinite triangular honeycomb
8:
2838:pattern of circles inside a largest circle.
1629:order-infinite-infinite triangular honeycomb
1382:Order-infinite-infinite triangular honeycomb
3961:
3749:
3415:
3198:
2874:
2646:
2437:
2217:
1993:
1773:
1388:
1173:
872:
647:
304:
29:
4436:{7,3,3} Honeycomb Meets Plane at Infinity
4231:
3938:
3680:
3381:
3129:
2840:
2623:
2403:
2183:
1959:
1365:
2647:Order-infinite-3 apeirogonal honeycomb
4362:Regular Honeycombs in Hyperbolic Space
4330:, 3rd. ed., Dover Publications, 1973.
3750:Order-infinite-7 heptagonal honeycomb
3199:Order-infinite-5 pentagonal honeycomb
3120:existing around each edge and with an
2794:order-infinite-3 apeirogonal honeycomb
2640:Order-infinite-3 apeirogonal honeycomb
2438:Order-infinite-3 heptagonal honeycomb
1994:Order-infinite-3 pentagonal honeycomb
1174:Order-infinite-7 triangular honeycomb
873:Order-infinite-6 triangular honeycomb
648:Order-infinite-5 triangular honeycomb
305:Order-infinite-4 triangular honeycomb
30:Order-infinite-3 triangular honeycomb
4387:, JOURNAL OF ALGEBRA 79,78-97 (1982)
4342:The Beauty of Geometry: Twelve Essays
3907:order-infinite-7 heptagonal honeycomb
3743:Order-infinite-7 heptagonal honeycomb
3416:Order-infinite-6 hexagonal honeycomb
3351:order-infinite-5 pentagonal honeycomb
3192:Order-infinite-5 pentagonal honeycomb
2808:). Each infinite cell consists of an
2615:order-infinite-3 heptagonal honeycomb
2598:). Each infinite cell consists of an
2584:order-infinite-3 heptagonal honeycomb
2431:Order-infinite-3 heptagonal honeycomb
2378:). Each infinite cell consists of an
2218:Order-infinite-3 hexagonal honeycomb
2154:). Each infinite cell consists of an
2140:order-infinite-3 pentagonal honeycomb
1987:Order-infinite-3 pentagonal honeycomb
1333:order-infinite-7 triangular honeycomb
1167:Order-infinite-7 triangular honeycomb
1107:order-infinite-6 triangular honeycomb
866:Order-infinite-6 triangular honeycomb
807:order-infinite-3 triangular honeycomb
641:Order-infinite-5 triangular honeycomb
540:order-infinite-4 triangular honeycomb
298:Order-infinite-4 triangular honeycomb
187:order-infinite-3 triangular honeycomb
7:
3649:order-infinite-6 hexagonal honeycomb
3409:Order-infinite-6 hexagonal honeycomb
2395:order-infinite-3 hexagonal honeycomb
2364:order-infinite-3 hexagonal honeycomb
2211:Order-infinite-3 hexagonal honeycomb
1934:). Each infinite cell consists of a
4404:Roice Nelson, Henry Segerman (2015)
2875:Order-infinite-4 square honeycomb
2825:infinite-order apeirogonal tilings
1774:Order-infinite-3 square honeycomb
25:
4224:infinite-order apeirogonal tiling
4220:infinite-order apeirogonal tiling
3927:infinite-order heptagonal tilings
3095:order-infinite-4 square honeycomb
2868:Order-infinite-4 square honeycomb
2810:infinite-order apeirogonal tiling
2175:infinite-order pentagonal tilings
1951:order-infinite-3 square honeycomb
1920:order-infinite-3 square honeycomb
1767:Order-infinite-3 square honeycomb
1653:infinite-order apeirogonal tiling
563:infinite-order triangular tilings
18:Order-infinite-3 square honeycomb
4289:
4284:
4279:
4274:
4269:
4264:
4246:
4235:
4218:{∞,∞,∞}. It has infinitely many
4153:
4144:
4112:
4096:
4091:
4086:
4081:
4076:
4071:
4063:
4058:
4053:
4048:
4043:
4038:
4033:
4025:
4020:
4015:
4010:
4005:
4000:
3995:
3942:
3862:
3827:
3811:
3806:
3801:
3796:
3791:
3786:
3781:
3733:
3728:
3723:
3718:
3713:
3695:
3684:
3669:infinite-order hexagonal tilings
3603:
3596:
3561:
3545:
3540:
3535:
3530:
3525:
3517:
3512:
3507:
3502:
3497:
3492:
3487:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3396:
3385:
3276:
3260:
3255:
3250:
3245:
3240:
3235:
3230:
3182:
3177:
3172:
3167:
3162:
3144:
3133:
3018:
3002:
2997:
2992:
2987:
2982:
2974:
2969:
2964:
2959:
2954:
2949:
2944:
2936:
2931:
2926:
2921:
2916:
2911:
2906:
2855:
2844:
2724:
2708:
2703:
2698:
2693:
2688:
2683:
2678:
2627:
2600:infinite-order heptagonal tiling
2515:
2499:
2494:
2489:
2484:
2479:
2474:
2469:
2418:
2407:
2295:
2279:
2274:
2269:
2264:
2259:
2254:
2249:
2198:
2187:
2156:infinite-order pentagonal tiling
2071:
2055:
2050:
2045:
2040:
2035:
2030:
2025:
1974:
1963:
1851:
1835:
1830:
1825:
1820:
1815:
1810:
1805:
1757:
1752:
1747:
1742:
1737:
1732:
1724:
1719:
1714:
1709:
1704:
1699:
1694:
1676:
1665:
1649:infinite-order triangular tiling
1647:{3,∞,∞}. It has infinitely many
1581:
1574:
1539:
1523:
1518:
1513:
1508:
1503:
1498:
1490:
1485:
1480:
1475:
1470:
1465:
1460:
1452:
1447:
1442:
1437:
1432:
1427:
1422:
1369:
1353:infinite-order triangular tiling
1351:{3,∞,7}. It has infinitely many
1286:
1251:
1235:
1230:
1225:
1220:
1215:
1210:
1205:
1154:
1143:
1127:infinite-order triangular tiling
1125:{3,∞,6}. It has infinitely many
1060:
1053:
1018:
1002:
997:
992:
987:
982:
974:
969:
964:
959:
954:
949:
944:
936:
931:
926:
921:
916:
911:
906:
853:
842:
827:infinite-order triangular tiling
760:
725:
709:
704:
699:
694:
689:
684:
679:
627:
622:
617:
612:
607:
589:
578:
492:
485:
450:
434:
429:
424:
419:
414:
406:
401:
396:
391:
386:
381:
376:
368:
363:
358:
353:
348:
343:
338:
260:Infinite-order triangular tiling
254:Related polytopes and honeycombs
241:
230:
215:Infinite-order triangular tiling
142:
107:
91:
86:
81:
76:
71:
66:
61:
4392:Hao Chen, Jean-Philippe Labbé,
4371:The Shape of Space, 2nd edition
4180:
4170:
4162:
4140:
4130:
4120:
4105:
3988:
3976:
3966:
3887:
3878:
3870:
3855:
3845:
3835:
3820:
3774:
3764:
3754:
3629:
3619:
3611:
3589:
3579:
3569:
3554:
3442:
3430:
3420:
3331:
3322:
3314:
3304:
3294:
3284:
3269:
3223:
3213:
3203:
3075:
3066:
3058:
3046:
3036:
3026:
3011:
2899:
2889:
2879:
2774:
2765:
2755:
2743:
2732:
2717:
2671:
2661:
2651:
2564:
2555:
2545:
2533:
2523:
2508:
2462:
2452:
2442:
2344:
2335:
2325:
2313:
2303:
2288:
2242:
2232:
2222:
2120:
2111:
2101:
2089:
2079:
2064:
2018:
2008:
1998:
1900:
1891:
1881:
1869:
1859:
1844:
1798:
1788:
1778:
1609:
1599:
1589:
1567:
1557:
1547:
1532:
1415:
1403:
1393:
1313:
1304:
1294:
1279:
1269:
1259:
1244:
1198:
1188:
1178:
1087:
1078:
1068:
1046:
1036:
1026:
1011:
899:
887:
877:
787:
778:
768:
753:
743:
733:
718:
672:
662:
652:
520:
510:
500:
478:
468:
458:
443:
331:
319:
309:
167:
158:
150:
135:
125:
115:
100:
54:
44:
34:
4262:{∞,(∞,∞,∞)}, Coxeter diagram,
3711:{6,(∞,3,∞)}, Coxeter diagram,
2171:order-6-3 pentagonal honeycomb
1692:{3,(∞,∞,∞)}, Coxeter diagram,
1:
4206:) is a regular space-filling
3913:) is a regular space-filling
3655:) is a regular space-filling
3118:infinite-order square tilings
1635:) is a regular space-filling
1339:) is a regular space-filling
1113:) is a regular space-filling
813:) is a regular space-filling
546:) is a regular space-filling
193:) is a regular space-filling
4344:(1999), Dover Publications,
2827:meeting at each edge. The
4480:
3931:order-7 apeirogonal tiling
3673:order-6 apeirogonal tiling
3374:order-5 apeirogonal tiling
3357:) a regular space-filling
3122:order-4 apeirogonal tiling
3101:) a regular space-filling
2800:) a regular space-filling
2590:) a regular space-filling
2380:order-3 apeirogonal tiling
2370:) a regular space-filling
2177:meeting at each edge. The
2146:) a regular space-filling
1926:) a regular space-filling
1357:order-7 apeirogonal tiling
1131:order-6 apeirogonal tiling
831:order-5 apeirogonal tiling
637:the half symmetry is = .
567:order-4 apeirogonal tiling
274:order-3 apeirogonal tiling
219:order-3 apeirogonal tiling
4310:List of regular polytopes
3160:{4,∞}, Coxeter diagram,
2812:whose vertices lie on a
2602:whose vertices lie on a
2382:whose vertices lie on a
2158:whose vertices lie on a
1938:whose vertices lie on a
605:{3,∞}, Coxeter diagram,
2173:is {5,∞,3}, with three
3925:{7,∞,7}. It has seven
825:{3,∞,5}. It has five
4464:Regular 3-honeycombs
3667:{6,∞,6}. It has six
4242:Poincaré disk model
3691:Poincaré disk model
3392:Poincaré disk model
3140:Poincaré disk model
2851:Poincaré disk model
2414:Poincaré disk model
2194:Poincaré disk model
1970:Poincaré disk model
1672:Poincaré disk model
1150:Poincaré disk model
849:Poincaré disk model
585:Poincaré disk model
237:Poincaré disk model
4196:hyperbolic 3-space
3903:hyperbolic 3-space
3645:hyperbolic 3-space
3347:hyperbolic 3-space
3091:hyperbolic 3-space
2790:hyperbolic 3-space
2580:hyperbolic 3-space
2360:hyperbolic 3-space
2136:hyperbolic 3-space
1916:hyperbolic 3-space
1625:hyperbolic 3-space
1329:hyperbolic 3-space
1103:hyperbolic 3-space
803:hyperbolic 3-space
536:hyperbolic 3-space
183:hyperbolic 3-space
4432:{7,3,3} Honeycomb
4327:Regular Polytopes
4256:
4255:
4188:
4187:
3971:Regular honeycomb
3952:
3951:
3895:
3894:
3759:Regular honeycomb
3705:
3704:
3637:
3636:
3425:Regular honeycomb
3406:
3405:
3339:
3338:
3208:Regular honeycomb
3154:
3153:
3083:
3082:
2884:Regular honeycomb
2865:
2864:
2836:Apollonian gasket
2782:
2781:
2656:Regular honeycomb
2637:
2636:
2572:
2571:
2447:Regular honeycomb
2428:
2427:
2352:
2351:
2227:Regular honeycomb
2208:
2207:
2128:
2127:
2003:Regular honeycomb
1984:
1983:
1936:heptagonal tiling
1908:
1907:
1783:Regular honeycomb
1686:
1685:
1617:
1616:
1398:Regular honeycomb
1379:
1378:
1321:
1320:
1183:Regular honeycomb
1164:
1163:
1095:
1094:
882:Regular honeycomb
863:
862:
795:
794:
657:Regular honeycomb
599:
598:
528:
527:
314:Regular honeycomb
251:
250:
175:
174:
39:Regular honeycomb
16:(Redirected from
4471:
4383:George Maxwell,
4368:Jeffrey R. Weeks
4294:
4293:
4292:
4288:
4287:
4283:
4282:
4278:
4277:
4273:
4272:
4268:
4267:
4250:
4239:
4232:
4157:
4148:
4116:
4101:
4100:
4099:
4095:
4094:
4090:
4089:
4085:
4084:
4080:
4079:
4075:
4074:
4068:
4067:
4066:
4062:
4061:
4057:
4056:
4052:
4051:
4047:
4046:
4042:
4041:
4037:
4036:
4030:
4029:
4028:
4024:
4023:
4019:
4018:
4014:
4013:
4009:
4008:
4004:
4003:
3999:
3998:
3990:Coxeter diagrams
3978:Schläfli symbols
3959:
3946:
3939:
3866:
3831:
3816:
3815:
3814:
3810:
3809:
3805:
3804:
3800:
3799:
3795:
3794:
3790:
3789:
3785:
3784:
3776:Coxeter diagrams
3766:Schläfli symbols
3747:
3738:
3737:
3736:
3732:
3731:
3727:
3726:
3722:
3721:
3717:
3716:
3699:
3688:
3681:
3607:
3600:
3565:
3550:
3549:
3548:
3544:
3543:
3539:
3538:
3534:
3533:
3529:
3528:
3522:
3521:
3520:
3516:
3515:
3511:
3510:
3506:
3505:
3501:
3500:
3496:
3495:
3491:
3490:
3484:
3483:
3482:
3478:
3477:
3473:
3472:
3468:
3467:
3463:
3462:
3458:
3457:
3453:
3452:
3444:Coxeter diagrams
3432:Schläfli symbols
3413:
3400:
3389:
3382:
3280:
3265:
3264:
3263:
3259:
3258:
3254:
3253:
3249:
3248:
3244:
3243:
3239:
3238:
3234:
3233:
3225:Coxeter diagrams
3196:
3187:
3186:
3185:
3181:
3180:
3176:
3175:
3171:
3170:
3166:
3165:
3148:
3137:
3130:
3022:
3007:
3006:
3005:
3001:
3000:
2996:
2995:
2991:
2990:
2986:
2985:
2979:
2978:
2977:
2973:
2972:
2968:
2967:
2963:
2962:
2958:
2957:
2953:
2952:
2948:
2947:
2941:
2940:
2939:
2935:
2934:
2930:
2929:
2925:
2924:
2920:
2919:
2915:
2914:
2910:
2909:
2901:Coxeter diagrams
2872:
2859:
2848:
2841:
2728:
2713:
2712:
2711:
2707:
2706:
2702:
2701:
2697:
2696:
2692:
2691:
2687:
2686:
2682:
2681:
2644:
2631:
2624:
2519:
2504:
2503:
2502:
2498:
2497:
2493:
2492:
2488:
2487:
2483:
2482:
2478:
2477:
2473:
2472:
2435:
2422:
2411:
2404:
2299:
2284:
2283:
2282:
2278:
2277:
2273:
2272:
2268:
2267:
2263:
2262:
2258:
2257:
2253:
2252:
2215:
2202:
2191:
2184:
2075:
2060:
2059:
2058:
2054:
2053:
2049:
2048:
2044:
2043:
2039:
2038:
2034:
2033:
2029:
2028:
1991:
1978:
1967:
1960:
1855:
1840:
1839:
1838:
1834:
1833:
1829:
1828:
1824:
1823:
1819:
1818:
1814:
1813:
1809:
1808:
1771:
1762:
1761:
1760:
1756:
1755:
1751:
1750:
1746:
1745:
1741:
1740:
1736:
1735:
1729:
1728:
1727:
1723:
1722:
1718:
1717:
1713:
1712:
1708:
1707:
1703:
1702:
1698:
1697:
1680:
1669:
1662:
1585:
1578:
1543:
1528:
1527:
1526:
1522:
1521:
1517:
1516:
1512:
1511:
1507:
1506:
1502:
1501:
1495:
1494:
1493:
1489:
1488:
1484:
1483:
1479:
1478:
1474:
1473:
1469:
1468:
1464:
1463:
1457:
1456:
1455:
1451:
1450:
1446:
1445:
1441:
1440:
1436:
1435:
1431:
1430:
1426:
1425:
1417:Coxeter diagrams
1405:Schläfli symbols
1386:
1373:
1366:
1290:
1255:
1240:
1239:
1238:
1234:
1233:
1229:
1228:
1224:
1223:
1219:
1218:
1214:
1213:
1209:
1208:
1200:Coxeter diagrams
1190:Schläfli symbols
1171:
1158:
1147:
1140:
1064:
1057:
1022:
1007:
1006:
1005:
1001:
1000:
996:
995:
991:
990:
986:
985:
979:
978:
977:
973:
972:
968:
967:
963:
962:
958:
957:
953:
952:
948:
947:
941:
940:
939:
935:
934:
930:
929:
925:
924:
920:
919:
915:
914:
910:
909:
901:Coxeter diagrams
889:Schläfli symbols
870:
857:
846:
839:
764:
729:
714:
713:
712:
708:
707:
703:
702:
698:
697:
693:
692:
688:
687:
683:
682:
674:Coxeter diagrams
664:Schläfli symbols
645:
635:Coxeter notation
632:
631:
630:
626:
625:
621:
620:
616:
615:
611:
610:
593:
582:
575:
496:
489:
454:
439:
438:
437:
433:
432:
428:
427:
423:
422:
418:
417:
411:
410:
409:
405:
404:
400:
399:
395:
394:
390:
389:
385:
384:
380:
379:
373:
372:
371:
367:
366:
362:
361:
357:
356:
352:
351:
347:
346:
342:
341:
333:Coxeter diagrams
321:Schläfli symbols
302:
245:
234:
227:
146:
111:
96:
95:
94:
90:
89:
85:
84:
80:
79:
75:
74:
70:
69:
65:
64:
56:Coxeter diagrams
46:Schläfli symbols
27:
21:
4479:
4478:
4474:
4473:
4472:
4470:
4469:
4468:
4454:
4453:
4428:Visual insights
4411:
4318:
4301:
4290:
4285:
4280:
4275:
4270:
4265:
4263:
4260:Schläfli symbol
4251:
4240:
4216:Schläfli symbol
4204:∞,∞,∞ honeycomb
4176:
4152:
4097:
4092:
4087:
4082:
4077:
4072:
4070:
4064:
4059:
4054:
4049:
4044:
4039:
4034:
4032:
4031:
4026:
4021:
4016:
4011:
4006:
4001:
3996:
3994:
3983:
3957:
3947:
3923:Schläfli symbol
3911:7,∞,7 honeycomb
3812:
3807:
3802:
3797:
3792:
3787:
3782:
3780:
3745:
3734:
3729:
3724:
3719:
3714:
3712:
3709:Schläfli symbol
3700:
3689:
3665:Schläfli symbol
3653:6,∞,6 honeycomb
3625:
3601:
3546:
3541:
3536:
3531:
3526:
3524:
3518:
3513:
3508:
3503:
3498:
3493:
3488:
3486:
3485:
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3448:
3437:
3411:
3401:
3390:
3367:Schläfli symbol
3355:5,∞,5 honeycomb
3261:
3256:
3251:
3246:
3241:
3236:
3231:
3229:
3215:Schläfli symbol
3194:
3183:
3178:
3173:
3168:
3163:
3161:
3158:Schläfli symbol
3149:
3138:
3111:Schläfli symbol
3099:4,∞,4 honeycomb
3053:
3003:
2998:
2993:
2988:
2983:
2981:
2975:
2970:
2965:
2960:
2955:
2950:
2945:
2943:
2942:
2937:
2932:
2927:
2922:
2917:
2912:
2907:
2905:
2891:Schläfli symbol
2870:
2860:
2849:
2821:Schläfli symbol
2798:∞,∞,3 honeycomb
2709:
2704:
2699:
2694:
2689:
2684:
2679:
2677:
2673:Coxeter diagram
2663:Schläfli symbol
2642:
2632:
2611:Schläfli symbol
2588:7,∞,3 honeycomb
2500:
2495:
2490:
2485:
2480:
2475:
2470:
2468:
2464:Coxeter diagram
2454:Schläfli symbol
2433:
2423:
2412:
2391:Schläfli symbol
2368:6,∞,3 honeycomb
2280:
2275:
2270:
2265:
2260:
2255:
2250:
2248:
2244:Coxeter diagram
2234:Schläfli symbol
2213:
2203:
2192:
2167:Schläfli symbol
2144:5,∞,3 honeycomb
2056:
2051:
2046:
2041:
2036:
2031:
2026:
2024:
2020:Coxeter diagram
2010:Schläfli symbol
1989:
1979:
1968:
1947:Schläfli symbol
1924:4,∞,3 honeycomb
1836:
1831:
1826:
1821:
1816:
1811:
1806:
1804:
1800:Coxeter diagram
1790:Schläfli symbol
1769:
1758:
1753:
1748:
1743:
1738:
1733:
1731:
1725:
1720:
1715:
1710:
1705:
1700:
1695:
1693:
1690:Schläfli symbol
1681:
1670:
1645:Schläfli symbol
1633:3,∞,∞ honeycomb
1605:
1579:
1524:
1519:
1514:
1509:
1504:
1499:
1497:
1491:
1486:
1481:
1476:
1471:
1466:
1461:
1459:
1458:
1453:
1448:
1443:
1438:
1433:
1428:
1423:
1421:
1410:
1384:
1374:
1349:Schläfli symbol
1337:3,∞,6 honeycomb
1236:
1231:
1226:
1221:
1216:
1211:
1206:
1204:
1169:
1159:
1148:
1123:Schläfli symbol
1111:3,∞,6 honeycomb
1058:
1003:
998:
993:
988:
983:
981:
975:
970:
965:
960:
955:
950:
945:
943:
942:
937:
932:
927:
922:
917:
912:
907:
905:
894:
868:
858:
847:
823:Schläfli symbol
811:3,∞,5 honeycomb
710:
705:
700:
695:
690:
685:
680:
678:
643:
628:
623:
618:
613:
608:
606:
603:Schläfli symbol
594:
583:
556:Schläfli symbol
544:3,∞,4 honeycomb
516:
490:
435:
430:
425:
420:
415:
413:
407:
402:
397:
392:
387:
382:
377:
375:
374:
369:
364:
359:
354:
349:
344:
339:
337:
326:
300:
256:
246:
235:
211:
203:Schläfli symbol
191:3,∞,3 honeycomb
92:
87:
82:
77:
72:
67:
62:
60:
23:
22:
15:
12:
11:
5:
4477:
4475:
4467:
4466:
4456:
4455:
4452:
4451:
4448:4 March 2014.
4442:Danny Calegari
4439:
4421:
4420:, Roice Nelson
4410:
4409:External links
4407:
4406:
4405:
4399:
4390:
4381:
4365:
4339:
4317:
4314:
4313:
4312:
4307:
4300:
4297:
4254:
4253:
4252:Ideal surface
4244:
4186:
4185:
4182:
4178:
4177:
4174:
4168:
4167:
4164:
4160:
4159:
4142:
4138:
4137:
4132:
4128:
4127:
4122:
4118:
4117:
4107:
4103:
4102:
3992:
3986:
3985:
3980:
3974:
3973:
3968:
3964:
3963:
3956:
3953:
3950:
3949:
3948:Ideal surface
3893:
3892:
3889:
3885:
3884:
3882:
3876:
3875:
3872:
3868:
3867:
3857:
3853:
3852:
3847:
3843:
3842:
3837:
3833:
3832:
3822:
3818:
3817:
3778:
3772:
3771:
3768:
3762:
3761:
3756:
3752:
3751:
3744:
3741:
3703:
3702:
3701:Ideal surface
3693:
3635:
3634:
3631:
3627:
3626:
3623:
3617:
3616:
3613:
3609:
3608:
3591:
3587:
3586:
3581:
3577:
3576:
3571:
3567:
3566:
3556:
3552:
3551:
3446:
3440:
3439:
3434:
3428:
3427:
3422:
3418:
3417:
3410:
3407:
3404:
3403:
3402:Ideal surface
3394:
3337:
3336:
3333:
3329:
3328:
3326:
3320:
3319:
3316:
3312:
3311:
3306:
3302:
3301:
3296:
3292:
3291:
3286:
3282:
3281:
3271:
3267:
3266:
3227:
3221:
3220:
3217:
3211:
3210:
3205:
3201:
3200:
3193:
3190:
3152:
3151:
3150:Ideal surface
3142:
3081:
3080:
3077:
3073:
3072:
3070:
3064:
3063:
3060:
3056:
3055:
3048:
3044:
3043:
3038:
3034:
3033:
3028:
3024:
3023:
3013:
3009:
3008:
2903:
2897:
2896:
2893:
2887:
2886:
2881:
2877:
2876:
2869:
2866:
2863:
2862:
2861:Ideal surface
2853:
2780:
2779:
2776:
2772:
2771:
2769:
2763:
2762:
2757:
2753:
2752:
2747:
2741:
2740:
2734:
2730:
2729:
2719:
2715:
2714:
2675:
2669:
2668:
2665:
2659:
2658:
2653:
2649:
2648:
2641:
2638:
2635:
2634:
2633:Ideal surface
2570:
2569:
2566:
2562:
2561:
2559:
2553:
2552:
2547:
2543:
2542:
2537:
2531:
2530:
2525:
2521:
2520:
2510:
2506:
2505:
2466:
2460:
2459:
2456:
2450:
2449:
2444:
2440:
2439:
2432:
2429:
2426:
2425:
2424:Ideal surface
2416:
2350:
2349:
2346:
2342:
2341:
2339:
2333:
2332:
2327:
2323:
2322:
2317:
2311:
2310:
2305:
2301:
2300:
2290:
2286:
2285:
2246:
2240:
2239:
2236:
2230:
2229:
2224:
2220:
2219:
2212:
2209:
2206:
2205:
2204:Ideal surface
2196:
2126:
2125:
2122:
2118:
2117:
2115:
2109:
2108:
2103:
2099:
2098:
2093:
2087:
2086:
2081:
2077:
2076:
2066:
2062:
2061:
2022:
2016:
2015:
2012:
2006:
2005:
2000:
1996:
1995:
1988:
1985:
1982:
1981:
1980:Ideal surface
1972:
1906:
1905:
1902:
1898:
1897:
1895:
1889:
1888:
1883:
1879:
1878:
1873:
1867:
1866:
1861:
1857:
1856:
1846:
1842:
1841:
1802:
1796:
1795:
1792:
1786:
1785:
1780:
1776:
1775:
1768:
1765:
1684:
1683:
1682:Ideal surface
1674:
1615:
1614:
1611:
1607:
1606:
1603:
1597:
1596:
1591:
1587:
1586:
1569:
1565:
1564:
1559:
1555:
1554:
1549:
1545:
1544:
1534:
1530:
1529:
1419:
1413:
1412:
1407:
1401:
1400:
1395:
1391:
1390:
1383:
1380:
1377:
1376:
1375:Ideal surface
1319:
1318:
1315:
1311:
1310:
1308:
1302:
1301:
1296:
1292:
1291:
1281:
1277:
1276:
1271:
1267:
1266:
1261:
1257:
1256:
1246:
1242:
1241:
1202:
1196:
1195:
1192:
1186:
1185:
1180:
1176:
1175:
1168:
1165:
1162:
1161:
1160:Ideal surface
1152:
1093:
1092:
1089:
1085:
1084:
1082:
1076:
1075:
1070:
1066:
1065:
1048:
1044:
1043:
1038:
1034:
1033:
1028:
1024:
1023:
1013:
1009:
1008:
903:
897:
896:
891:
885:
884:
879:
875:
874:
867:
864:
861:
860:
859:Ideal surface
851:
793:
792:
789:
785:
784:
782:
776:
775:
770:
766:
765:
755:
751:
750:
745:
741:
740:
735:
731:
730:
720:
716:
715:
676:
670:
669:
666:
660:
659:
654:
650:
649:
642:
639:
597:
596:
595:Ideal surface
587:
526:
525:
522:
518:
517:
514:
508:
507:
502:
498:
497:
480:
476:
475:
470:
466:
465:
460:
456:
455:
445:
441:
440:
335:
329:
328:
323:
317:
316:
311:
307:
306:
299:
296:
277:vertex figures
255:
252:
249:
248:
247:Ideal surface
239:
210:
207:
173:
172:
169:
165:
164:
162:
156:
155:
152:
148:
147:
137:
133:
132:
127:
123:
122:
117:
113:
112:
102:
98:
97:
58:
52:
51:
48:
42:
41:
36:
32:
31:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4476:
4465:
4462:
4461:
4459:
4450:
4447:
4443:
4440:
4437:
4434:(2014/08/01)
4433:
4429:
4425:
4422:
4419:
4416:
4413:
4412:
4408:
4403:
4400:
4398:
4395:
4391:
4389:
4386:
4382:
4379:
4378:0-8247-0709-5
4375:
4372:
4369:
4366:
4363:
4360:(Chapter 10,
4359:
4358:0-486-40919-8
4355:
4351:
4347:
4343:
4340:
4337:
4336:0-486-61480-8
4333:
4329:
4328:
4323:
4320:
4319:
4315:
4311:
4308:
4306:
4303:
4302:
4298:
4296:
4261:
4249:
4245:
4243:
4238:
4234:
4233:
4230:
4228:
4227:vertex figure
4225:
4221:
4217:
4213:
4209:
4205:
4201:
4197:
4193:
4183:
4179:
4175:
4173:
4172:Coxeter group
4169:
4165:
4161:
4156:
4151:
4147:
4143:
4141:Vertex figure
4139:
4136:
4133:
4129:
4126:
4123:
4119:
4115:
4111:
4108:
4104:
3993:
3991:
3987:
3981:
3979:
3975:
3972:
3969:
3965:
3960:
3954:
3945:
3941:
3940:
3937:
3935:
3934:vertex figure
3932:
3928:
3924:
3920:
3916:
3912:
3908:
3904:
3900:
3890:
3886:
3883:
3881:
3880:Coxeter group
3877:
3873:
3869:
3865:
3861:
3858:
3856:Vertex figure
3854:
3851:
3848:
3844:
3841:
3838:
3834:
3830:
3826:
3823:
3819:
3779:
3777:
3773:
3769:
3767:
3763:
3760:
3757:
3753:
3748:
3742:
3740:
3710:
3698:
3694:
3692:
3687:
3683:
3682:
3679:
3677:
3676:vertex figure
3674:
3670:
3666:
3662:
3658:
3654:
3650:
3646:
3642:
3632:
3628:
3624:
3622:
3621:Coxeter group
3618:
3614:
3610:
3606:
3599:
3595:
3592:
3590:Vertex figure
3588:
3585:
3582:
3578:
3575:
3572:
3568:
3564:
3560:
3557:
3553:
3447:
3445:
3441:
3435:
3433:
3429:
3426:
3423:
3419:
3414:
3408:
3399:
3395:
3393:
3388:
3384:
3383:
3380:
3378:
3377:vertex figure
3375:
3370:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3334:
3330:
3327:
3325:
3324:Coxeter group
3321:
3317:
3313:
3310:
3307:
3305:Vertex figure
3303:
3300:
3297:
3293:
3290:
3287:
3283:
3279:
3275:
3272:
3268:
3228:
3226:
3222:
3218:
3216:
3212:
3209:
3206:
3202:
3197:
3191:
3189:
3159:
3147:
3143:
3141:
3136:
3132:
3131:
3128:
3126:
3125:vertex figure
3123:
3119:
3114:
3112:
3108:
3104:
3100:
3096:
3092:
3088:
3078:
3074:
3071:
3069:
3068:Coxeter group
3065:
3061:
3057:
3052:
3049:
3047:Vertex figure
3045:
3042:
3039:
3035:
3032:
3029:
3025:
3021:
3017:
3014:
3010:
2904:
2902:
2898:
2894:
2892:
2888:
2885:
2882:
2878:
2873:
2867:
2858:
2854:
2852:
2847:
2843:
2842:
2839:
2837:
2832:
2830:
2829:vertex figure
2826:
2822:
2817:
2815:
2811:
2807:
2803:
2799:
2795:
2791:
2787:
2777:
2773:
2770:
2768:
2767:Coxeter group
2764:
2761:
2758:
2754:
2751:
2748:
2746:
2745:Vertex figure
2742:
2738:
2735:
2731:
2727:
2723:
2720:
2716:
2676:
2674:
2670:
2666:
2664:
2660:
2657:
2654:
2650:
2645:
2639:
2630:
2626:
2625:
2622:
2620:
2619:vertex figure
2616:
2612:
2607:
2605:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2567:
2563:
2560:
2558:
2557:Coxeter group
2554:
2551:
2548:
2544:
2541:
2538:
2536:
2535:Vertex figure
2532:
2529:
2526:
2522:
2518:
2514:
2511:
2507:
2467:
2465:
2461:
2457:
2455:
2451:
2448:
2445:
2441:
2436:
2430:
2421:
2417:
2415:
2410:
2406:
2405:
2402:
2400:
2399:vertex figure
2396:
2392:
2387:
2385:
2381:
2377:
2373:
2369:
2365:
2361:
2357:
2347:
2343:
2340:
2338:
2337:Coxeter group
2334:
2331:
2328:
2324:
2321:
2318:
2316:
2315:Vertex figure
2312:
2309:
2306:
2302:
2298:
2294:
2291:
2287:
2247:
2245:
2241:
2237:
2235:
2231:
2228:
2225:
2221:
2216:
2210:
2201:
2197:
2195:
2190:
2186:
2185:
2182:
2180:
2179:vertex figure
2176:
2172:
2168:
2163:
2161:
2157:
2153:
2149:
2145:
2141:
2137:
2133:
2123:
2119:
2116:
2114:
2113:Coxeter group
2110:
2107:
2104:
2100:
2097:
2094:
2092:
2091:Vertex figure
2088:
2085:
2082:
2078:
2074:
2070:
2067:
2063:
2023:
2021:
2017:
2013:
2011:
2007:
2004:
2001:
1997:
1992:
1986:
1977:
1973:
1971:
1966:
1962:
1961:
1958:
1956:
1955:vertex figure
1952:
1948:
1943:
1941:
1937:
1933:
1929:
1925:
1921:
1917:
1913:
1903:
1899:
1896:
1894:
1893:Coxeter group
1890:
1887:
1884:
1880:
1877:
1874:
1872:
1871:Vertex figure
1868:
1865:
1862:
1858:
1854:
1850:
1847:
1843:
1803:
1801:
1797:
1793:
1791:
1787:
1784:
1781:
1777:
1772:
1766:
1764:
1691:
1679:
1675:
1673:
1668:
1664:
1663:
1660:
1658:
1657:vertex figure
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1626:
1622:
1612:
1608:
1604:
1602:
1601:Coxeter group
1598:
1595:
1592:
1588:
1584:
1577:
1573:
1570:
1568:Vertex figure
1566:
1563:
1560:
1556:
1553:
1550:
1546:
1542:
1538:
1535:
1531:
1420:
1418:
1414:
1408:
1406:
1402:
1399:
1396:
1392:
1387:
1381:
1372:
1368:
1367:
1364:
1362:
1361:vertex figure
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1316:
1312:
1309:
1307:
1306:Coxeter group
1303:
1300:
1297:
1293:
1289:
1285:
1282:
1280:Vertex figure
1278:
1275:
1272:
1268:
1265:
1262:
1258:
1254:
1250:
1247:
1243:
1203:
1201:
1197:
1193:
1191:
1187:
1184:
1181:
1177:
1172:
1166:
1157:
1153:
1151:
1146:
1142:
1141:
1138:
1136:
1135:vertex figure
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1090:
1086:
1083:
1081:
1080:Coxeter group
1077:
1074:
1071:
1067:
1063:
1056:
1052:
1049:
1047:Vertex figure
1045:
1042:
1039:
1035:
1032:
1029:
1025:
1021:
1017:
1014:
1010:
904:
902:
898:
892:
890:
886:
883:
880:
876:
871:
865:
856:
852:
850:
845:
841:
840:
837:
835:
834:vertex figure
832:
828:
824:
820:
816:
812:
808:
804:
800:
790:
786:
783:
781:
780:Coxeter group
777:
774:
771:
767:
763:
759:
756:
754:Vertex figure
752:
749:
746:
742:
739:
736:
732:
728:
724:
721:
717:
677:
675:
671:
667:
665:
661:
658:
655:
651:
646:
640:
638:
636:
604:
592:
588:
586:
581:
577:
576:
573:
571:
570:vertex figure
568:
564:
559:
557:
553:
549:
545:
541:
537:
533:
523:
519:
515:
513:
512:Coxeter group
509:
506:
503:
499:
495:
488:
484:
481:
479:Vertex figure
477:
474:
471:
467:
464:
461:
457:
453:
449:
446:
442:
336:
334:
330:
324:
322:
318:
315:
312:
308:
303:
297:
295:
293:
289:
284:
282:
278:
275:
270:
268:
264:
261:
253:
244:
240:
238:
233:
229:
228:
225:
223:
222:vertex figure
220:
216:
213:It has three
208:
206:
204:
200:
196:
192:
188:
184:
180:
170:
166:
163:
161:
160:Coxeter group
157:
153:
149:
145:
141:
138:
136:Vertex figure
134:
131:
128:
124:
121:
118:
114:
110:
106:
103:
99:
59:
57:
53:
49:
47:
43:
40:
37:
33:
28:
19:
4438:(2014/08/14)
4427:
4393:
4384:
4370:
4341:
4325:
4257:
4208:tessellation
4203:
4199:
4189:
3984:{∞,(∞,∞,∞)}
3915:tessellation
3910:
3906:
3896:
3706:
3657:tessellation
3652:
3648:
3638:
3438:{6,(∞,3,∞)}
3371:
3359:tessellation
3354:
3350:
3340:
3155:
3115:
3103:tessellation
3098:
3094:
3084:
2833:
2824:
2818:
2814:2-hypercycle
2802:tessellation
2797:
2793:
2783:
2614:
2608:
2604:2-hypercycle
2592:tessellation
2587:
2583:
2573:
2394:
2388:
2384:2-hypercycle
2372:tessellation
2367:
2363:
2353:
2174:
2170:
2164:
2160:2-hypercycle
2148:tessellation
2143:
2139:
2129:
1950:
1944:
1940:2-hypercycle
1928:tessellation
1923:
1919:
1909:
1687:
1652:
1637:tessellation
1632:
1628:
1618:
1411:{3,(∞,∞,∞)}
1356:
1341:tessellation
1336:
1332:
1322:
1130:
1115:tessellation
1110:
1106:
1096:
895:{3,(∞,3,∞)}
830:
815:tessellation
810:
806:
796:
600:
561:It has four
560:
548:tessellation
543:
539:
529:
291:
287:
285:
280:
271:
266:
257:
212:
195:tessellation
190:
186:
176:
4364:) Table III
4131:Edge figure
3846:Edge figure
3580:Edge figure
3295:Edge figure
3037:Edge figure
1558:Edge figure
1270:Edge figure
1037:Edge figure
744:Edge figure
469:Edge figure
126:Edge figure
4316:References
4181:Properties
4166:self-dual
4158:{(∞,∞,∞)}
3888:Properties
3874:self-dual
3630:Properties
3615:self-dual
3602:{(5,3,5)}
3332:Properties
3318:self-dual
3076:Properties
3062:self-dual
2775:Properties
2565:Properties
2345:Properties
2121:Properties
1901:Properties
1610:Properties
1580:{(∞,∞,∞)}
1314:Properties
1088:Properties
1059:{(∞,3,∞)}
788:Properties
521:Properties
168:Properties
154:Self-dual
4424:John Baez
4212:honeycomb
3919:honeycomb
3661:honeycomb
3369:{5,∞,5}.
3363:honeycomb
3113:{4,∞,4}.
3107:honeycomb
2806:honeycomb
2737:Apeirogon
2596:honeycomb
2376:honeycomb
2152:honeycomb
1932:honeycomb
1655:, {∞,∞},
1641:honeycomb
1359:, {∞,7},
1345:honeycomb
1133:, {∞,6},
1119:honeycomb
819:honeycomb
558:{3,∞,4}.
552:honeycomb
205:{3,∞,3}.
199:honeycomb
4458:Category
4396:, (2013)
4350:99-35678
4299:See also
4192:geometry
4184:Regular
3899:geometry
3891:Regular
3770:{7,∞,7}
3641:geometry
3633:Regular
3343:geometry
3335:Regular
3219:{5,∞,5}
3087:geometry
3079:Regular
2895:{4,∞,4}
2786:geometry
2778:Regular
2667:{∞,∞,3}
2576:geometry
2568:Regular
2458:{7,∞,3}
2356:geometry
2348:Regular
2238:{6,∞,3}
2132:geometry
2124:Regular
2014:{5,∞,3}
1912:geometry
1904:Regular
1794:{4,∞,3}
1621:geometry
1613:Regular
1325:geometry
1317:Regular
1194:{3,∞,7}
1099:geometry
1091:Regular
799:geometry
791:Regular
668:{3,∞,5}
532:geometry
524:Regular
209:Geometry
179:geometry
171:Regular
50:{3,∞,3}
4418:YouTube
4322:Coxeter
4214:) with
4190:In the
3982:{∞,∞,∞}
3921:) with
3897:In the
3663:) with
3639:In the
3436:{6,∞,6}
3365:) with
3341:In the
3109:) with
3085:In the
2784:In the
2760:{3,∞,∞}
2613:of the
2574:In the
2550:{3,∞,7}
2393:of the
2354:In the
2330:{3,∞,6}
2169:of the
2130:In the
2106:{3,∞,5}
1949:of the
1910:In the
1886:{3,∞,4}
1643:) with
1619:In the
1594:{∞,∞,3}
1409:{3,∞,∞}
1347:) with
1323:In the
1299:{7,∞,3}
1121:) with
1097:In the
1073:{6,∞,3}
893:{3,∞,6}
821:) with
797:In the
773:{5,∞,3}
554:) with
530:In the
505:{4,∞,3}
491:r{∞,∞}
325:{3,∞,4}
283:,∞,3}.
265:: {3,∞,
201:) with
177:In the
4376:
4356:
4348:
4334:
4198:, the
3905:, the
3647:, the
3349:, the
3093:, the
3054:{∞,∞}
2792:, the
2582:, the
2362:, the
2138:, the
1918:, the
1627:, the
1331:, the
1105:, the
805:, the
538:, the
327:{3,∞}
185:, the
4150:{∞,∞}
4121:Faces
4110:{∞,∞}
4106:Cells
3860:{∞,7}
3836:Faces
3825:{7,∞}
3821:Cells
3594:{∞,6}
3570:Faces
3559:{6,∞}
3555:Cells
3309:{∞,5}
3285:Faces
3274:{5,∞}
3270:Cells
3051:{∞,4}
3027:Faces
3016:{4,∞}
3012:Cells
2750:{∞,3}
2733:Faces
2722:{∞,∞}
2718:Cells
2540:{∞,3}
2524:Faces
2513:{7,∞}
2509:Cells
2320:{∞,3}
2304:Faces
2293:{6,∞}
2289:Cells
2096:{∞,3}
2080:Faces
2069:{5,∞}
2065:Cells
1876:{∞,3}
1860:Faces
1849:{4,∞}
1845:Cells
1572:{∞,∞}
1548:Faces
1537:{3,∞}
1533:Cells
1284:{∞,7}
1260:Faces
1249:{3,∞}
1245:Cells
1051:{∞,6}
1027:Faces
1016:{3,∞}
1012:Cells
758:{∞,5}
734:Faces
723:{3,∞}
719:Cells
483:{∞,4}
459:Faces
448:{3,∞}
444:Cells
263:cells
140:{∞,3}
116:Faces
105:{3,∞}
101:Cells
4374:ISBN
4354:ISBN
4346:LCCN
4332:ISBN
4210:(or
4202:(or
4163:Dual
3967:Type
3917:(or
3909:(or
3871:Dual
3755:Type
3659:(or
3651:(or
3612:Dual
3421:Type
3361:(or
3353:(or
3315:Dual
3204:Type
3105:(or
3097:(or
3059:Dual
2880:Type
2819:The
2804:(or
2796:(or
2756:Dual
2739:{∞}
2652:Type
2609:The
2594:(or
2586:(or
2546:Dual
2443:Type
2389:The
2374:(or
2366:(or
2326:Dual
2223:Type
2165:The
2150:(or
2142:(or
2102:Dual
1999:Type
1945:The
1930:(or
1922:(or
1882:Dual
1779:Type
1639:(or
1631:(or
1590:Dual
1394:Type
1343:(or
1335:(or
1295:Dual
1179:Type
1117:(or
1109:(or
1069:Dual
878:Type
817:(or
809:(or
769:Dual
653:Type
550:(or
542:(or
501:Dual
310:Type
197:(or
189:(or
151:Dual
35:Type
4194:of
4135:{∞}
4125:{∞}
3901:of
3850:{7}
3840:{7}
3643:of
3584:{6}
3574:{6}
3345:of
3299:{5}
3289:{5}
3089:of
3041:{4}
3031:{4}
2788:of
2578:of
2528:{7}
2358:of
2308:{6}
2134:of
2084:{5}
1914:of
1864:{4}
1623:of
1562:{∞}
1552:{3}
1327:of
1274:{7}
1264:{3}
1101:of
1041:{6}
1031:{3}
801:of
748:{5}
738:{3}
534:of
473:{4}
463:{3}
294:}.
290:,∞,
279:: {
269:}.
181:of
130:{3}
120:{3}
4460::
4444:,
4430::
4426:,
4352:,
4324:,
4229:.
4069:↔
3936:.
3678:.
3523:=
3379:.
3127:.
2980:=
1730:=
1659:.
1496:=
1363:.
1137:.
980:=
836:.
572:.
412:=
224:.
292:p
288:p
281:p
267:p
20:)
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