3573:
3234:
2935:
2718:
2431:
2203:
1983:
1759:
1461:
1154:
853:
589:
232:
252:
3584:
3245:
2946:
2729:
2442:
2214:
1994:
1770:
1472:
1165:
864:
600:
243:
1335:
1029:
736:
462:
109:
144:
2311:
2091:
1867:
1647:
3450:
3111:
2826:
2605:
504:
497:
3491:
3482:
3153:
3146:
1377:
1370:
1071:
1064:
771:
1445:, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an
1138:, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an
838:, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an
574:, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an
3558:{â,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 apeirogonal tilings existing around each vertex in an
3219:, {6,7}, 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:{3,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in a
3640:
3807:
2920:
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-7 pentagonal tilings existing around each edge and with an
2344:
3803:
3627:
3434:
1553:
1319:
3731:
3711:
3689:
3617:
3424:
3278:
3090:
2584:
1543:
1309:
1008:
633:
441:
3401:
3067:
2561:
1520:
1286:
985:
418:
3622:
3602:
3429:
3409:
3371:
3333:
3283:
3263:
3095:
3075:
3037:
2999:
2780:
2569:
2531:
2493:
2265:
2045:
1821:
1601:
1548:
1528:
1490:
1314:
1294:
1256:
1218:
1013:
993:
955:
917:
690:
618:
426:
388:
350:
63:
3607:
3414:
3396:
3376:
3358:
3338:
2589:
2270:
1515:
1281:
1243:
638:
446:
3612:
3419:
3391:
3381:
3363:
3353:
3343:
3273:
3085:
3057:
3047:
3029:
3019:
3009:
2810:
2800:
2790:
2579:
2551:
2541:
2523:
2513:
2503:
2295:
2285:
2275:
2075:
2065:
2055:
1851:
1841:
1831:
1631:
1621:
1611:
1538:
1510:
1500:
1387:
1304:
1276:
1266:
1248:
1238:
1228:
1003:
975:
965:
947:
937:
927:
720:
710:
700:
628:
436:
408:
398:
380:
370:
360:
93:
83:
73:
2123:
1899:
1679:
781:
3386:
3348:
3268:
3080:
3062:
3052:
3042:
3024:
3014:
3004:
2805:
2795:
2785:
2574:
2556:
2546:
2536:
2518:
2508:
2498:
2290:
2280:
2070:
2060:
2050:
1846:
1836:
1826:
1626:
1616:
1606:
1533:
1505:
1495:
1299:
1271:
1261:
1233:
1223:
998:
980:
970:
960:
942:
932:
922:
715:
705:
695:
623:
431:
413:
403:
393:
375:
365:
355:
88:
78:
68:
3826:
3721:
1081:
256:
3793:
3664:
3559:
3485:
1365:
2351:
2130:
1906:
1686:
1557:, with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is = .
514:
3645:
3306:
2972:
2755:
2704:
2635:
2468:
2419:
The "ideal surface" projection below is a plane-at-infinity, in the
Poincaré half-space model of H3. It shows an
2240:
2020:
1796:
1576:
1442:
1330:
1191:
1135:
1024:
890:
835:
731:
665:
571:
457:
325:
282:
214:
104:
38:
3555:
3445:
2394:
2306:
3680:
3220:
3141:
3106:
2921:
2856:
2821:
2173:
1949:
1862:
1059:
766:
575:
492:
3216:
2086:
3715:
3577:
3507:
3238:
3168:
2939:
2871:
2722:
2650:
2435:
2207:
1987:
1763:
1465:
1394:
1158:
1088:
857:
788:
593:
521:
236:
159:
3821:
3572:
3233:
2934:
2717:
2700:
2600:
2430:
2398:
2202:
2177:
1982:
1953:
1758:
1733:
1642:
1460:
1153:
852:
588:
231:
251:
3547:
3208:
2910:
2689:
2390:
2169:
1945:
1725:
1434:
1127:
827:
560:
198:
3742:
3223:
578:
3595:
3551:
3313:
3256:
3212:
2979:
2914:
2762:
2693:
2475:
2405:
2247:
2184:
2027:
1960:
1803:
1740:
1583:
1483:
1438:
1198:
1131:
897:
831:
672:
611:
564:
332:
202:
45:
3789:
3583:
3244:
2945:
2728:
2441:
2213:
1993:
1769:
1471:
1164:
863:
599:
242:
3727:
3707:
3699:
3685:
2420:
2334:
2113:
1889:
1729:
1669:
296:
218:
139:
3675:
3531:
3192:
2894:
2673:
2374:
2153:
1929:
1709:
1418:
1111:
811:
643:
544:
182:
3287:, with alternating types or colors of cells. In Coxeter notation the half symmetry is = .
1334:
1028:
735:
461:
108:
3325:
2991:
2772:
2485:
2257:
2037:
1813:
1593:
1210:
909:
682:
342:
285:
55:
17:
3799:
3759:
299:
3815:
3562:
2924:
2707:
2413:
2329:
2192:
2108:
1972:
1884:
1748:
1664:
1450:
1143:
842:
221:
3543:
3204:
2906:
2685:
2386:
2165:
1941:
1721:
1430:
1123:
823:
556:
194:
143:
3804:
Kleinian, a tool for visualizing
Kleinian groups, Geometry and the Imagination
3772:
3490:
3481:
3449:
3152:
3145:
3110:
2825:
2604:
2310:
2090:
1866:
1646:
1376:
1369:
1070:
1063:
770:
503:
496:
3781:
3470:
3460:
2699:
All vertices are ultra-ideal (existing beyond the ideal boundary) with four
2321:
2191:
is {6,7,3}, with three order-5 hexagonal tilings meeting at each edge. The
1747:
is {4,7,3}, with three order-4 heptagonal tilings meeting at each edge. The
1355:
642:, with alternating types or colors of order-7 triangular tiling cells. In
3527:
3188:
2890:
2846:
2836:
2669:
2370:
2149:
1925:
1877:
1705:
1414:
1345:
1107:
1039:
807:
756:
746:
540:
472:
178:
129:
119:
3775:
3692:. (Tables I and II: Regular polytopes and honeycombs, pp. 294â296)
3131:
3121:
2101:
1049:
2625:
2615:
1657:
482:
3703:
3755:
3751:
3294:
2960:
2743:
2456:
1179:
878:
653:
313:
26:
2408:
of the apeirogonal tiling honeycomb is {â,7,3}, with three
2401:, each of which has a limiting circle on the ideal sphere.
2180:, each of which has a limiting circle on the ideal sphere.
1956:, each of which has a limiting circle on the ideal sphere.
1736:, each of which has a limiting circle on the ideal sphere.
270:
It a part of a sequence of self-dual regular honeycombs: {
3748:
Lorentzian
Coxeter groups and Boyd-Maxwell ball packings
2228:
2008:
1784:
1564:
281:
It is a part of a sequence of regular honeycombs with
3734:(Chapters 16â17: Geometries on Three-manifolds I, II)
3594:
It has a second construction as a uniform honeycomb,
3255:
It has a second construction as a uniform honeycomb,
1482:
It has a second construction as a uniform honeycomb,
610:
It has a second construction as a uniform honeycomb,
295:
It isa part of a sequence of regular honeycombs with
1455:
1148:
847:
583:
226:
3756:Visualizing Hyperbolic Honeycombs arXiv:1511.02851
3773:Hyperbolic Catacombs Carousel: {3,7,3} honeycomb
2416:of this honeycomb is a heptagonal tiling, {7,3}.
2195:of this honeycomb is a heptagonal tiling, {7,3}.
1975:of this honeycomb is a heptagonal tiling, {7,3}.
1751:of this honeycomb is a heptagonal tiling, {7,3}.
3739:Sphere Packings and Hyperbolic Reflection Groups
3641:Convex uniform honeycombs in hyperbolic space
3631:, with alternating types or colors of cells.
8:
2423:pattern of circles inside a largest circle.
3297:
2963:
2746:
2459:
2231:
2011:
1787:
1567:
1182:
881:
656:
316:
29:
3794:{7,3,3} Honeycomb Meets Plane at Infinity
3567:
3228:
2929:
2712:
2425:
2197:
1977:
1753:
3657:
3298:Order-7-infinite apeirogonal honeycomb
3716:Regular Honeycombs in Hyperbolic Space
3684:, 3rd. ed., Dover Publications, 1973.
3536:order-7-infinite apeirogonal honeycomb
3291:Order-7-infinite apeirogonal honeycomb
2703:existing around each edge and with an
1183:Order-7-infinite triangular honeycomb
3741:, JOURNAL OF ALGEBRA 79,78-97 (1982)
3696:The Beauty of Geometry: Twelve Essays
3667:Roice Nelson and Henry Segerman, 2014
2393:). Each infinite cell consists of an
2172:). Each infinite cell consists of an
1948:). Each infinite cell consists of an
1423:order-7-infinite triangular honeycomb
1176:Order-7-infinite triangular honeycomb
7:
1728:). Each infinite cell consists of a
25:
3625:
3620:
3615:
3610:
3605:
3600:
3582:
3571:
3560:infinite-order heptagonal tiling
3554:{â,7,â}. It has infinitely many
3489:
3480:
3448:
3432:
3427:
3422:
3417:
3412:
3407:
3399:
3394:
3389:
3384:
3379:
3374:
3369:
3361:
3356:
3351:
3346:
3341:
3336:
3331:
3281:
3276:
3271:
3266:
3261:
3243:
3232:
3151:
3144:
3109:
3093:
3088:
3083:
3078:
3073:
3065:
3060:
3055:
3050:
3045:
3040:
3035:
3027:
3022:
3017:
3012:
3007:
3002:
2997:
2944:
2933:
2824:
2808:
2803:
2798:
2793:
2788:
2783:
2778:
2727:
2716:
2603:
2587:
2582:
2577:
2572:
2567:
2559:
2554:
2549:
2544:
2539:
2534:
2529:
2521:
2516:
2511:
2506:
2501:
2496:
2491:
2440:
2429:
2309:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2232:Order-7-3 apeirogonal honeycomb
2212:
2201:
2089:
2073:
2068:
2063:
2058:
2053:
2048:
2043:
1992:
1981:
1865:
1849:
1844:
1839:
1834:
1829:
1824:
1819:
1768:
1757:
1645:
1629:
1624:
1619:
1614:
1609:
1604:
1599:
1551:
1546:
1541:
1536:
1531:
1526:
1518:
1513:
1508:
1503:
1498:
1493:
1488:
1470:
1459:
1447:infinite-order heptagonal tiling
1441:{3,7,â}. It has infinitely many
1375:
1368:
1333:
1317:
1312:
1307:
1302:
1297:
1292:
1284:
1279:
1274:
1269:
1264:
1259:
1254:
1246:
1241:
1236:
1231:
1226:
1221:
1216:
1163:
1152:
1134:{3,7,6}. It has infinitely many
1069:
1062:
1027:
1011:
1006:
1001:
996:
991:
983:
978:
973:
968:
963:
958:
953:
945:
940:
935:
930:
925:
920:
915:
862:
851:
769:
734:
718:
713:
708:
703:
698:
693:
688:
636:
631:
626:
621:
616:
598:
587:
502:
495:
460:
444:
439:
434:
429:
424:
416:
411:
406:
401:
396:
391:
386:
378:
373:
368:
363:
358:
353:
348:
266:Related polytopes and honeycombs
250:
241:
230:
142:
107:
91:
86:
81:
76:
71:
66:
61:
3746:Hao Chen, Jean-Philippe Labbé,
3725:The Shape of Space, 2nd edition
3516:
3506:
3498:
3476:
3466:
3456:
3441:
3324:
3312:
3302:
3177:
3167:
3159:
3137:
3127:
3117:
3102:
2990:
2978:
2968:
2879:
2870:
2862:
2852:
2842:
2832:
2817:
2771:
2761:
2751:
2747:Order-7-5 pentagonal honeycomb
2658:
2649:
2641:
2631:
2621:
2611:
2596:
2484:
2474:
2464:
2379:order-7-3 apeirogonal honeycomb
2359:
2350:
2340:
2328:
2317:
2302:
2256:
2246:
2236:
2225:Order-7-3 apeirogonal honeycomb
2138:
2129:
2119:
2107:
2097:
2082:
2036:
2026:
2016:
1914:
1905:
1895:
1883:
1873:
1858:
1812:
1802:
1792:
1788:Order-7-3 pentagonal honeycomb
1694:
1685:
1675:
1663:
1653:
1638:
1592:
1582:
1572:
1403:
1393:
1383:
1361:
1351:
1341:
1326:
1209:
1197:
1187:
1096:
1087:
1077:
1055:
1045:
1035:
1020:
908:
896:
886:
882:Order-7-6 triangular honeycomb
796:
787:
777:
762:
752:
742:
727:
681:
671:
661:
657:Order-7-5 triangular honeycomb
529:
520:
510:
488:
478:
468:
453:
341:
331:
321:
317:Order-7-4 triangular honeycomb
167:
158:
150:
135:
125:
115:
100:
54:
44:
34:
30:Order-7-3 triangular honeycomb
3598:{â,(7,â,7)}, Coxeter diagram,
3259:{6,(7,3,7)}, Coxeter diagram,
2964:Order-7-6 hexagonal honeycomb
2899:order-7-5 pentagonal honeycomb
2740:Order-7-5 pentagonal honeycomb
2012:Order-7-3 hexagonal honeycomb
1965:order-6-3 pentagonal honeycomb
1934:order-7-3 pentagonal honeycomb
1781:Order-7-3 pentagonal honeycomb
1486:{3,(7,â,7)}, Coxeter diagram,
1116:order-7-6 triangular honeycomb
875:Order-7-6 triangular honeycomb
816:order-7-3 triangular honeycomb
650:Order-7-5 triangular honeycomb
549:order-7-4 triangular honeycomb
310:Order-7-4 triangular honeycomb
187:order-7-3 triangular honeycomb
1:
3542:) is a regular space-filling
3203:) is a regular space-filling
3197:order-7-6 hexagonal honeycomb
2957:Order-7-6 hexagonal honeycomb
2189:order-7-3 hexagonal honeycomb
2158:order-7-3 hexagonal honeycomb
2005:Order-7-3 hexagonal honeycomb
1429:) is a regular space-filling
1122:) is a regular space-filling
822:) is a regular space-filling
555:) is a regular space-filling
193:) is a regular space-filling
3698:(1999), Dover Publications,
2460:Order-7-4 square honeycomb
2412:meeting at each edge. The
2410:order-7 apeirogonal tilings
1568:Order-7-3 square honeycomb
259:with selective cells shown
3843:
3556:order-7 apeirogonal tiling
2905:) a regular space-filling
2684:) a regular space-filling
2678:order-7-4 square honeycomb
2453:Order-7-4 square honeycomb
2395:order-7 apeirogonal tiling
2385:) a regular space-filling
2164:) a regular space-filling
1971:meeting at each edge. The
1969:order-7 pentagonal tilings
1940:) a regular space-filling
1745:order-7-3 square honeycomb
1720:) a regular space-filling
1714:order-7-3 square honeycomb
1561:Order-7-3 square honeycomb
646:the half symmetry is = .
572:order-7 triangular tilings
18:Order-7-3 square honeycomb
3646:List of regular polytopes
3221:order-6 heptagonal tiling
3217:order-7 hexagonal tilings
2922:order-5 heptagonal tiling
2705:order-4 heptagonal tiling
1950:order-7 pentagonal tiling
1443:order-7 triangular tiling
1140:order-6 heptagonal tiling
1136:order-7 triangular tiling
840:order-5 heptagonal tiling
836:order-7 triangular tiling
283:order-7 triangular tiling
215:order-7 triangular tiling
2397:whose vertices lie on a
2176:whose vertices lie on a
2174:order-6 hexagonal tiling
1952:whose vertices lie on a
1732:whose vertices lie on a
614:{3,7}, Coxeter diagram,
576:order-4 hexagonal tiling
1967:is {5,7,3}, with three
2701:order-5 square tilings
257:Upper half space model
834:{3,7,5}. It has five
3827:Regular 3-honeycombs
3665:Hyperbolic Catacombs
3215:{6,7,6}. It has six
3578:Poincaré disk model
3239:Poincaré disk model
2940:Poincaré disk model
2723:Poincaré disk model
2436:Poincaré disk model
2208:Poincaré disk model
1988:Poincaré disk model
1764:Poincaré disk model
1466:Poincaré disk model
1159:Poincaré disk model
858:Poincaré disk model
594:Poincaré disk model
237:Poincaré disk model
3532:hyperbolic 3-space
3224:vertex arrangement
3193:hyperbolic 3-space
2895:hyperbolic 3-space
2674:hyperbolic 3-space
2375:hyperbolic 3-space
2154:hyperbolic 3-space
1930:hyperbolic 3-space
1710:hyperbolic 3-space
1419:hyperbolic 3-space
1112:hyperbolic 3-space
812:hyperbolic 3-space
579:vertex arrangement
545:hyperbolic 3-space
183:hyperbolic 3-space
3790:{7,3,3} Honeycomb
3681:Regular Polytopes
3592:
3591:
3524:
3523:
3307:Regular honeycomb
3253:
3252:
3185:
3184:
2973:Regular honeycomb
2954:
2953:
2887:
2886:
2756:Regular honeycomb
2737:
2736:
2666:
2665:
2469:Regular honeycomb
2450:
2449:
2421:Apollonian gasket
2367:
2366:
2241:Regular honeycomb
2222:
2221:
2146:
2145:
2021:Regular honeycomb
2002:
2001:
1922:
1921:
1797:Regular honeycomb
1778:
1777:
1730:heptagonal tiling
1702:
1701:
1577:Regular honeycomb
1480:
1479:
1411:
1410:
1192:Regular honeycomb
1173:
1172:
1104:
1103:
891:Regular honeycomb
872:
871:
804:
803:
666:Regular honeycomb
608:
607:
537:
536:
326:Regular honeycomb
297:heptagonal tiling
263:
262:
219:heptagonal tiling
175:
174:
39:Regular honeycomb
16:(Redirected from
3834:
3737:George Maxwell,
3722:Jeffrey R. Weeks
3668:
3662:
3630:
3629:
3628:
3624:
3623:
3619:
3618:
3614:
3613:
3609:
3608:
3604:
3603:
3586:
3575:
3568:
3493:
3484:
3452:
3437:
3436:
3435:
3431:
3430:
3426:
3425:
3421:
3420:
3416:
3415:
3411:
3410:
3404:
3403:
3402:
3398:
3397:
3393:
3392:
3388:
3387:
3383:
3382:
3378:
3377:
3373:
3372:
3366:
3365:
3364:
3360:
3359:
3355:
3354:
3350:
3349:
3345:
3344:
3340:
3339:
3335:
3334:
3326:Coxeter diagrams
3314:SchlÀfli symbols
3295:
3286:
3285:
3284:
3280:
3279:
3275:
3274:
3270:
3269:
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3247:
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3030:
3026:
3025:
3021:
3020:
3016:
3015:
3011:
3010:
3006:
3005:
3001:
3000:
2992:Coxeter diagrams
2980:SchlÀfli symbols
2961:
2948:
2937:
2930:
2828:
2813:
2812:
2811:
2807:
2806:
2802:
2801:
2797:
2796:
2792:
2791:
2787:
2786:
2782:
2781:
2773:Coxeter diagrams
2744:
2731:
2720:
2713:
2607:
2592:
2591:
2590:
2586:
2585:
2581:
2580:
2576:
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2533:
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2526:
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2524:
2520:
2519:
2515:
2514:
2510:
2509:
2505:
2504:
2500:
2499:
2495:
2494:
2486:Coxeter diagrams
2457:
2444:
2433:
2426:
2313:
2298:
2297:
2296:
2292:
2291:
2287:
2286:
2282:
2281:
2277:
2276:
2272:
2271:
2267:
2266:
2229:
2216:
2205:
2198:
2093:
2078:
2077:
2076:
2072:
2071:
2067:
2066:
2062:
2061:
2057:
2056:
2052:
2051:
2047:
2046:
2009:
1996:
1985:
1978:
1869:
1854:
1853:
1852:
1848:
1847:
1843:
1842:
1838:
1837:
1833:
1832:
1828:
1827:
1823:
1822:
1785:
1772:
1761:
1754:
1649:
1634:
1633:
1632:
1628:
1627:
1623:
1622:
1618:
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1613:
1612:
1608:
1607:
1603:
1602:
1565:
1556:
1555:
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1549:
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1544:
1540:
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1535:
1534:
1530:
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1523:
1522:
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1517:
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1512:
1511:
1507:
1506:
1502:
1501:
1497:
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1474:
1463:
1456:
1379:
1372:
1337:
1322:
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1316:
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1311:
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1301:
1300:
1296:
1295:
1289:
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1287:
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1282:
1278:
1277:
1273:
1272:
1268:
1267:
1263:
1262:
1258:
1257:
1251:
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1249:
1245:
1244:
1240:
1239:
1235:
1234:
1230:
1229:
1225:
1224:
1220:
1219:
1211:Coxeter diagrams
1199:SchlÀfli symbols
1180:
1167:
1156:
1149:
1073:
1066:
1031:
1016:
1015:
1014:
1010:
1009:
1005:
1004:
1000:
999:
995:
994:
988:
987:
986:
982:
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977:
976:
972:
971:
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966:
962:
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956:
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949:
948:
944:
943:
939:
938:
934:
933:
929:
928:
924:
923:
919:
918:
910:Coxeter diagrams
898:SchlÀfli symbols
879:
866:
855:
848:
773:
738:
723:
722:
721:
717:
716:
712:
711:
707:
706:
702:
701:
697:
696:
692:
691:
683:Coxeter diagrams
673:SchlÀfli symbols
654:
644:Coxeter notation
641:
640:
639:
635:
634:
630:
629:
625:
624:
620:
619:
602:
591:
584:
506:
499:
464:
449:
448:
447:
443:
442:
438:
437:
433:
432:
428:
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421:
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383:
382:
381:
377:
376:
372:
371:
367:
366:
362:
361:
357:
356:
352:
351:
343:Coxeter diagrams
333:SchlÀfli symbols
314:
254:
245:
234:
227:
146:
111:
96:
95:
94:
90:
89:
85:
84:
80:
79:
75:
74:
70:
69:
65:
64:
56:Coxeter diagrams
46:SchlÀfli symbols
27:
21:
3842:
3841:
3837:
3836:
3835:
3833:
3832:
3831:
3812:
3811:
3786:Visual insights
3769:
3672:
3671:
3663:
3659:
3654:
3637:
3626:
3621:
3616:
3611:
3606:
3601:
3599:
3596:SchlÀfli symbol
3587:
3576:
3552:SchlÀfli symbol
3540:â,7,â honeycomb
3512:
3488:
3433:
3428:
3423:
3418:
3413:
3408:
3406:
3400:
3395:
3390:
3385:
3380:
3375:
3370:
3368:
3367:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3330:
3319:
3293:
3282:
3277:
3272:
3267:
3262:
3260:
3257:SchlÀfli symbol
3248:
3237:
3213:SchlÀfli symbol
3201:6,7,6 honeycomb
3173:
3149:
3094:
3089:
3084:
3079:
3074:
3072:
3066:
3061:
3056:
3051:
3046:
3041:
3036:
3034:
3033:
3028:
3023:
3018:
3013:
3008:
3003:
2998:
2996:
2985:
2959:
2949:
2938:
2915:SchlÀfli symbol
2903:5,7,5 honeycomb
2809:
2804:
2799:
2794:
2789:
2784:
2779:
2777:
2763:SchlÀfli symbol
2742:
2732:
2721:
2694:SchlÀfli symbol
2682:4,7,4 honeycomb
2588:
2583:
2578:
2573:
2568:
2566:
2560:
2555:
2550:
2545:
2540:
2535:
2530:
2528:
2527:
2522:
2517:
2512:
2507:
2502:
2497:
2492:
2490:
2476:SchlÀfli symbol
2455:
2445:
2434:
2406:SchlÀfli symbol
2383:â,7,3 honeycomb
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2262:
2258:Coxeter diagram
2248:SchlÀfli symbol
2227:
2217:
2206:
2185:SchlÀfli symbol
2162:6,7,3 honeycomb
2074:
2069:
2064:
2059:
2054:
2049:
2044:
2042:
2038:Coxeter diagram
2028:SchlÀfli symbol
2007:
1997:
1986:
1961:SchlÀfli symbol
1938:5,7,3 honeycomb
1850:
1845:
1840:
1835:
1830:
1825:
1820:
1818:
1814:Coxeter diagram
1804:SchlÀfli symbol
1783:
1773:
1762:
1741:SchlÀfli symbol
1718:4,7,3 honeycomb
1630:
1625:
1620:
1615:
1610:
1605:
1600:
1598:
1594:Coxeter diagram
1584:SchlÀfli symbol
1563:
1552:
1547:
1542:
1537:
1532:
1527:
1525:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1487:
1484:SchlÀfli symbol
1475:
1464:
1439:SchlÀfli symbol
1427:3,7,â honeycomb
1399:
1373:
1318:
1313:
1308:
1303:
1298:
1293:
1291:
1285:
1280:
1275:
1270:
1265:
1260:
1255:
1253:
1252:
1247:
1242:
1237:
1232:
1227:
1222:
1217:
1215:
1204:
1178:
1168:
1157:
1132:SchlÀfli symbol
1120:3,7,6 honeycomb
1067:
1012:
1007:
1002:
997:
992:
990:
984:
979:
974:
969:
964:
959:
954:
952:
951:
946:
941:
936:
931:
926:
921:
916:
914:
903:
877:
867:
856:
832:SchlÀfli symbol
820:3,7,5 honeycomb
719:
714:
709:
704:
699:
694:
689:
687:
652:
637:
632:
627:
622:
617:
615:
612:SchlÀfli symbol
603:
592:
565:SchlÀfli symbol
553:3,7,4 honeycomb
500:
445:
440:
435:
430:
425:
423:
417:
412:
407:
402:
397:
392:
387:
385:
384:
379:
374:
369:
364:
359:
354:
349:
347:
312:
268:
255:
246:
235:
211:
203:SchlÀfli symbol
191:3,7,3 honeycomb
92:
87:
82:
77:
72:
67:
62:
60:
23:
22:
15:
12:
11:
5:
3840:
3838:
3830:
3829:
3824:
3814:
3813:
3810:
3809:
3806:4 March 2014.
3800:Danny Calegari
3797:
3779:
3778:, Roice Nelson
3768:
3767:External links
3765:
3764:
3763:
3760:Henry Segerman
3758:Roice Nelson,
3753:
3744:
3735:
3719:
3693:
3670:
3669:
3656:
3655:
3653:
3650:
3649:
3648:
3643:
3636:
3633:
3590:
3589:
3588:Ideal surface
3580:
3522:
3521:
3518:
3514:
3513:
3510:
3504:
3503:
3500:
3496:
3495:
3478:
3474:
3473:
3468:
3464:
3463:
3458:
3454:
3453:
3443:
3439:
3438:
3328:
3322:
3321:
3316:
3310:
3309:
3304:
3300:
3299:
3292:
3289:
3251:
3250:
3249:Ideal surface
3241:
3183:
3182:
3179:
3175:
3174:
3171:
3165:
3164:
3161:
3157:
3156:
3139:
3135:
3134:
3129:
3125:
3124:
3119:
3115:
3114:
3104:
3100:
3099:
2994:
2988:
2987:
2982:
2976:
2975:
2970:
2966:
2965:
2958:
2955:
2952:
2951:
2950:Ideal surface
2942:
2885:
2884:
2881:
2877:
2876:
2874:
2868:
2867:
2864:
2860:
2859:
2854:
2850:
2849:
2844:
2840:
2839:
2834:
2830:
2829:
2819:
2815:
2814:
2775:
2769:
2768:
2765:
2759:
2758:
2753:
2749:
2748:
2741:
2738:
2735:
2734:
2733:Ideal surface
2725:
2664:
2663:
2660:
2656:
2655:
2653:
2647:
2646:
2643:
2639:
2638:
2633:
2629:
2628:
2623:
2619:
2618:
2613:
2609:
2608:
2598:
2594:
2593:
2488:
2482:
2481:
2478:
2472:
2471:
2466:
2462:
2461:
2454:
2451:
2448:
2447:
2446:Ideal surface
2438:
2365:
2364:
2361:
2357:
2356:
2354:
2348:
2347:
2342:
2338:
2337:
2332:
2326:
2325:
2319:
2315:
2314:
2304:
2300:
2299:
2260:
2254:
2253:
2250:
2244:
2243:
2238:
2234:
2233:
2226:
2223:
2220:
2219:
2218:Ideal surface
2210:
2144:
2143:
2140:
2136:
2135:
2133:
2127:
2126:
2121:
2117:
2116:
2111:
2105:
2104:
2099:
2095:
2094:
2084:
2080:
2079:
2040:
2034:
2033:
2030:
2024:
2023:
2018:
2014:
2013:
2006:
2003:
2000:
1999:
1998:Ideal surface
1990:
1920:
1919:
1916:
1912:
1911:
1909:
1903:
1902:
1897:
1893:
1892:
1887:
1881:
1880:
1875:
1871:
1870:
1860:
1856:
1855:
1816:
1810:
1809:
1806:
1800:
1799:
1794:
1790:
1789:
1782:
1779:
1776:
1775:
1774:Ideal surface
1766:
1700:
1699:
1696:
1692:
1691:
1689:
1683:
1682:
1677:
1673:
1672:
1667:
1661:
1660:
1655:
1651:
1650:
1640:
1636:
1635:
1596:
1590:
1589:
1586:
1580:
1579:
1574:
1570:
1569:
1562:
1559:
1478:
1477:
1476:Ideal surface
1468:
1409:
1408:
1405:
1401:
1400:
1397:
1391:
1390:
1385:
1381:
1380:
1363:
1359:
1358:
1353:
1349:
1348:
1343:
1339:
1338:
1328:
1324:
1323:
1213:
1207:
1206:
1201:
1195:
1194:
1189:
1185:
1184:
1177:
1174:
1171:
1170:
1169:Ideal surface
1161:
1102:
1101:
1098:
1094:
1093:
1091:
1085:
1084:
1079:
1075:
1074:
1057:
1053:
1052:
1047:
1043:
1042:
1037:
1033:
1032:
1022:
1018:
1017:
912:
906:
905:
900:
894:
893:
888:
884:
883:
876:
873:
870:
869:
868:Ideal surface
860:
802:
801:
798:
794:
793:
791:
785:
784:
779:
775:
774:
764:
760:
759:
754:
750:
749:
744:
740:
739:
729:
725:
724:
685:
679:
678:
675:
669:
668:
663:
659:
658:
651:
648:
606:
605:
604:Ideal surface
596:
535:
534:
531:
527:
526:
524:
518:
517:
512:
508:
507:
490:
486:
485:
480:
476:
475:
470:
466:
465:
455:
451:
450:
345:
339:
338:
335:
329:
328:
323:
319:
318:
311:
308:
300:vertex figures
267:
264:
261:
260:
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:
3839:
3828:
3825:
3823:
3820:
3819:
3817:
3808:
3805:
3801:
3798:
3795:
3792:(2014/08/01)
3791:
3787:
3783:
3780:
3777:
3774:
3771:
3770:
3766:
3761:
3757:
3754:
3752:
3749:
3745:
3743:
3740:
3736:
3733:
3732:0-8247-0709-5
3729:
3726:
3723:
3720:
3717:
3714:(Chapter 10,
3713:
3712:0-486-40919-8
3709:
3705:
3701:
3697:
3694:
3691:
3690:0-486-61480-8
3687:
3683:
3682:
3677:
3674:
3673:
3666:
3661:
3658:
3651:
3647:
3644:
3642:
3639:
3638:
3634:
3632:
3597:
3585:
3581:
3579:
3574:
3570:
3569:
3566:
3564:
3563:vertex figure
3561:
3557:
3553:
3549:
3545:
3541:
3537:
3533:
3529:
3519:
3515:
3511:
3509:
3508:Coxeter group
3505:
3501:
3497:
3492:
3487:
3483:
3479:
3477:Vertex figure
3475:
3472:
3469:
3465:
3462:
3459:
3455:
3451:
3447:
3444:
3440:
3329:
3327:
3323:
3317:
3315:
3311:
3308:
3305:
3301:
3296:
3290:
3288:
3258:
3246:
3242:
3240:
3235:
3231:
3230:
3227:
3225:
3222:
3218:
3214:
3210:
3206:
3202:
3198:
3194:
3190:
3180:
3176:
3172:
3170:
3169:Coxeter group
3166:
3162:
3158:
3154:
3147:
3143:
3140:
3138:Vertex figure
3136:
3133:
3130:
3126:
3123:
3120:
3116:
3112:
3108:
3105:
3101:
2995:
2993:
2989:
2983:
2981:
2977:
2974:
2971:
2967:
2962:
2956:
2947:
2943:
2941:
2936:
2932:
2931:
2928:
2926:
2925:vertex figure
2923:
2918:
2916:
2912:
2908:
2904:
2900:
2896:
2892:
2882:
2878:
2875:
2873:
2872:Coxeter group
2869:
2865:
2861:
2858:
2855:
2853:Vertex figure
2851:
2848:
2845:
2841:
2838:
2835:
2831:
2827:
2823:
2820:
2816:
2776:
2774:
2770:
2766:
2764:
2760:
2757:
2754:
2750:
2745:
2739:
2730:
2726:
2724:
2719:
2715:
2714:
2711:
2709:
2708:vertex figure
2706:
2702:
2697:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2661:
2657:
2654:
2652:
2651:Coxeter group
2648:
2644:
2640:
2637:
2634:
2632:Vertex figure
2630:
2627:
2624:
2620:
2617:
2614:
2610:
2606:
2602:
2599:
2595:
2489:
2487:
2483:
2479:
2477:
2473:
2470:
2467:
2463:
2458:
2452:
2443:
2439:
2437:
2432:
2428:
2427:
2424:
2422:
2417:
2415:
2414:vertex figure
2411:
2407:
2402:
2400:
2396:
2392:
2388:
2384:
2380:
2376:
2372:
2362:
2358:
2355:
2353:
2352:Coxeter group
2349:
2346:
2343:
2339:
2336:
2333:
2331:
2330:Vertex figure
2327:
2323:
2320:
2316:
2312:
2308:
2305:
2301:
2261:
2259:
2255:
2251:
2249:
2245:
2242:
2239:
2235:
2230:
2224:
2215:
2211:
2209:
2204:
2200:
2199:
2196:
2194:
2193:vertex figure
2190:
2186:
2181:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2151:
2141:
2137:
2134:
2132:
2131:Coxeter group
2128:
2125:
2122:
2118:
2115:
2112:
2110:
2109:Vertex figure
2106:
2103:
2100:
2096:
2092:
2088:
2085:
2081:
2041:
2039:
2035:
2031:
2029:
2025:
2022:
2019:
2015:
2010:
2004:
1995:
1991:
1989:
1984:
1980:
1979:
1976:
1974:
1973:vertex figure
1970:
1966:
1962:
1957:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1917:
1913:
1910:
1908:
1907:Coxeter group
1904:
1901:
1898:
1894:
1891:
1888:
1886:
1885:Vertex figure
1882:
1879:
1876:
1872:
1868:
1864:
1861:
1857:
1817:
1815:
1811:
1807:
1805:
1801:
1798:
1795:
1791:
1786:
1780:
1771:
1767:
1765:
1760:
1756:
1755:
1752:
1750:
1749:vertex figure
1746:
1742:
1737:
1735:
1731:
1727:
1723:
1719:
1715:
1711:
1707:
1697:
1693:
1690:
1688:
1687:Coxeter group
1684:
1681:
1678:
1674:
1671:
1668:
1666:
1665:Vertex figure
1662:
1659:
1656:
1652:
1648:
1644:
1641:
1637:
1597:
1595:
1591:
1587:
1585:
1581:
1578:
1575:
1571:
1566:
1560:
1558:
1485:
1473:
1469:
1467:
1462:
1458:
1457:
1454:
1452:
1451:vertex figure
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1416:
1406:
1402:
1398:
1396:
1395:Coxeter group
1392:
1389:
1386:
1382:
1378:
1371:
1367:
1364:
1362:Vertex figure
1360:
1357:
1354:
1350:
1347:
1344:
1340:
1336:
1332:
1329:
1325:
1214:
1212:
1208:
1202:
1200:
1196:
1193:
1190:
1186:
1181:
1175:
1166:
1162:
1160:
1155:
1151:
1150:
1147:
1145:
1144:vertex figure
1141:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1099:
1095:
1092:
1090:
1089:Coxeter group
1086:
1083:
1080:
1076:
1072:
1065:
1061:
1058:
1056:Vertex figure
1054:
1051:
1048:
1044:
1041:
1038:
1034:
1030:
1026:
1023:
1019:
913:
911:
907:
901:
899:
895:
892:
889:
885:
880:
874:
865:
861:
859:
854:
850:
849:
846:
844:
843:vertex figure
841:
837:
833:
829:
825:
821:
817:
813:
809:
799:
795:
792:
790:
789:Coxeter group
786:
783:
780:
776:
772:
768:
765:
763:Vertex figure
761:
758:
755:
751:
748:
745:
741:
737:
733:
730:
726:
686:
684:
680:
676:
674:
670:
667:
664:
660:
655:
649:
647:
645:
613:
601:
597:
595:
590:
586:
585:
582:
580:
577:
573:
568:
566:
562:
558:
554:
550:
546:
542:
532:
528:
525:
523:
522:Coxeter group
519:
516:
513:
509:
505:
498:
494:
491:
489:Vertex figure
487:
484:
481:
477:
474:
471:
467:
463:
459:
456:
452:
346:
344:
340:
336:
334:
330:
327:
324:
320:
315:
309:
307:
305:
301:
298:
293:
291:
287:
284:
279:
277:
273:
265:
258:
253:
249:
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:
3822:3-honeycombs
3796:(2014/08/14)
3785:
3747:
3738:
3724:
3695:
3679:
3660:
3593:
3544:tessellation
3539:
3535:
3525:
3320:{â,(7,â,7)}
3254:
3205:tessellation
3200:
3196:
3186:
2986:{6,(7,3,7)}
2919:
2907:tessellation
2902:
2898:
2888:
2698:
2686:tessellation
2681:
2677:
2667:
2418:
2409:
2403:
2399:2-hypercycle
2387:tessellation
2382:
2378:
2368:
2188:
2182:
2178:2-hypercycle
2166:tessellation
2161:
2157:
2147:
1968:
1964:
1958:
1954:2-hypercycle
1942:tessellation
1937:
1933:
1923:
1744:
1738:
1734:2-hypercycle
1722:tessellation
1717:
1713:
1703:
1481:
1446:
1431:tessellation
1426:
1422:
1412:
1205:{3,(7,â,7)}
1139:
1124:tessellation
1119:
1115:
1105:
904:{3,(7,3,7)}
839:
824:tessellation
819:
815:
805:
609:
570:It has four
569:
557:tessellation
552:
548:
538:
303:
294:
289:
280:
275:
271:
269:
212:
195:tessellation
190:
186:
176:
3718:) Table III
3467:Edge figure
3128:Edge figure
2843:Edge figure
2622:Edge figure
1352:Edge figure
1046:Edge figure
753:Edge figure
479:Edge figure
126:Edge figure
3816:Categories
3652:References
3517:Properties
3502:self-dual
3494:{(7,â,7)}
3178:Properties
3163:self-dual
3150:{(5,3,5)}
2880:Properties
2866:self-dual
2659:Properties
2645:self-dual
2360:Properties
2139:Properties
1915:Properties
1695:Properties
1404:Properties
1374:{(7,â,7)}
1097:Properties
1068:{(7,3,7)}
797:Properties
530:Properties
168:Properties
154:Self-dual
3782:John Baez
3548:honeycomb
3209:honeycomb
2917:{5,7,5}.
2911:honeycomb
2696:{4,7,4}.
2690:honeycomb
2391:honeycomb
2322:Apeirogon
2170:honeycomb
1946:honeycomb
1726:honeycomb
1449:, {7,â},
1435:honeycomb
1142:, {7,6},
1128:honeycomb
828:honeycomb
567:{3,7,4}.
561:honeycomb
205:{3,7,3}.
199:honeycomb
3750:, (2013)
3704:99-35678
3635:See also
3528:geometry
3520:Regular
3189:geometry
3181:Regular
2891:geometry
2883:Regular
2767:{5,7,5}
2670:geometry
2662:Regular
2480:{4,7,4}
2371:geometry
2363:Regular
2252:{â,7,3}
2150:geometry
2142:Regular
2032:{6,7,3}
1926:geometry
1918:Regular
1808:{5,7,3}
1706:geometry
1698:Regular
1588:{4,7,3}
1415:geometry
1407:Regular
1108:geometry
1100:Regular
808:geometry
800:Regular
677:{3,7,5}
541:geometry
533:Regular
337:{3,7,4}
209:Geometry
179:geometry
171:Regular
50:{3,7,3}
3776:YouTube
3676:Coxeter
3550:) with
3526:In the
3318:{â,7,â}
3211:) with
3187:In the
2984:{6,7,6}
2913:) with
2889:In the
2692:) with
2668:In the
2369:In the
2345:{3,7,â}
2187:of the
2148:In the
2124:{3,7,6}
1963:of the
1924:In the
1900:{3,7,5}
1743:of the
1704:In the
1680:{3,7,4}
1437:) with
1413:In the
1388:{â,7,3}
1203:{3,7,â}
1130:) with
1106:In the
1082:{6,7,3}
902:{3,7,6}
830:) with
806:In the
782:{5,7,3}
563:) with
539:In the
515:{4,7,3}
501:r{7,7}
306:,7,3}.
288:: {3,7,
201:) with
177:In the
3762:(2015)
3730:
3710:
3702:
3688:
3534:, the
3195:, the
2897:, the
2676:, the
2377:, the
2156:, the
1932:, the
1712:, the
1421:, the
1114:, the
814:, the
547:, the
185:, the
3486:{7,â}
3457:Faces
3446:{â,7}
3442:Cells
3142:{7,6}
3118:Faces
3107:{6,7}
3103:Cells
2857:{7,5}
2833:Faces
2822:{5,7}
2818:Cells
2636:{7,4}
2612:Faces
2601:{4,7}
2597:Cells
2335:{7,3}
2318:Faces
2307:{â,7}
2303:Cells
2114:{7,3}
2098:Faces
2087:{6,7}
2083:Cells
1890:{7,3}
1874:Faces
1863:{5,7}
1859:Cells
1670:{7,3}
1654:Faces
1643:{4,7}
1639:Cells
1366:{7,â}
1342:Faces
1331:{3,7}
1327:Cells
1060:{7,6}
1036:Faces
1025:{3,7}
1021:Cells
767:{7,5}
743:Faces
732:{3,7}
728:Cells
493:{7,4}
469:Faces
458:{3,7}
454:Cells
286:cells
140:{7,3}
116:Faces
105:{3,7}
101:Cells
3728:ISBN
3708:ISBN
3700:LCCN
3686:ISBN
3546:(or
3538:(or
3499:Dual
3303:Type
3207:(or
3199:(or
3160:Dual
2969:Type
2909:(or
2901:(or
2863:Dual
2752:Type
2688:(or
2680:(or
2642:Dual
2465:Type
2404:The
2389:(or
2381:(or
2341:Dual
2324:{â}
2237:Type
2183:The
2168:(or
2160:(or
2120:Dual
2017:Type
1959:The
1944:(or
1936:(or
1896:Dual
1793:Type
1739:The
1724:(or
1716:(or
1676:Dual
1573:Type
1433:(or
1425:(or
1384:Dual
1188:Type
1126:(or
1118:(or
1078:Dual
887:Type
826:(or
818:(or
778:Dual
662:Type
559:(or
551:(or
511:Dual
322:Type
197:(or
189:(or
151:Dual
35:Type
3530:of
3471:{â}
3461:{â}
3191:of
3132:{6}
3122:{6}
2893:of
2847:{5}
2837:{5}
2672:of
2626:{4}
2616:{4}
2373:of
2152:of
2102:{6}
1928:of
1878:{5}
1708:of
1658:{4}
1417:of
1356:{â}
1346:{3}
1110:of
1050:{6}
1040:{3}
810:of
757:{5}
747:{3}
543:of
483:{4}
473:{3}
302:: {
292:}.
278:}.
274:,7,
181:of
130:{3}
120:{3}
3818::
3802:,
3788::
3784:,
3706:,
3678:,
3565:.
3405:â
3226:.
3071:=
2927:.
2710:.
2565:=
1524:=
1453:.
1290:=
1146:.
989:=
845:.
581:.
422:=
224:.
304:p
290:p
276:p
272:p
20:)
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