2314:
1891:
2195:
459:
450:
1905:
1870:
1898:
649:
2811:
441:
432:
40:
1654:
2173:
1757:
2184:
665:
65:
2206:
1778:
2162:
2151:
640:
2226:
1570:
1535:
822:
730:
1563:
1241:
1989:
777:
624:
1220:
1206:
2307:
1996:
1982:
1884:
1863:
1556:
1549:
1542:
1248:
1227:
1213:
1199:
597:
1877:
1255:
1234:
1577:
771:
1372:
1365:
1344:
612:
2217:
1358:
1351:
1337:
1330:
1323:
1316:
724:
2300:
1584:
2818:
784:
2028:
1830:
715:
1954:
variations, that can distort the kites into bilateral trapezoids or more general quadrilaterals. Ignoring the face colors below, the fully symmetry is p6m, and the lower symmetry is p31m with three mirrors meeting at a point, and threefold rotation points.
762:
737:
123:
844:
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are eight forms, seven topologically distinct. (The
874:
1813:
face of this tiling has angles 120°, 90°, 60° and 90°. It is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.
2727:
3751:
3016:
2949:
3756:
2971:
2705:
2602:
793:
3566:
3401:
867:
3716:
3691:
3681:
3651:
3606:
3556:
3536:
3351:
3236:
3726:
3721:
3661:
3656:
3611:
3561:
3546:
2677:
746:
3746:
3531:
2779:
2583:
2556:
2089:
1719:
1477:
39:
2313:
2194:
1890:
458:
449:
1711:
1691:
812:, placing equal diameter circles at the center of every point. Every circle is in contact with four other circles in the packing (
3586:
3521:
3506:
3341:
2961:
1729:
1189:
1179:
1160:
1150:
1140:
1131:
1121:
1111:
1102:
1082:
1073:
1044:
1034:
1005:
976:
966:
937:
568:
558:
549:
539:
529:
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500:
382:
372:
348:
338:
328:
167:
147:
2377:
3686:
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3601:
3541:
3526:
3516:
3491:
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1904:
1701:
1169:
1092:
1063:
1053:
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995:
986:
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947:
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629:
578:
510:
392:
157:
1869:
3551:
3471:
3326:
2420:
2275:
2094:
1964:
1850:
1482:
886:
277:
175:
2605:
1897:
1706:
1696:
1184:
1174:
1155:
1145:
1126:
1116:
1097:
1087:
1068:
1058:
1039:
1029:
1010:
1000:
981:
971:
952:
942:
573:
563:
544:
534:
515:
505:
387:
377:
343:
333:
162:
152:
3481:
3466:
3426:
3356:
3306:
3221:
3041:
1932:
1404:
1292:
693:
648:
212:
82:
3795:
3451:
3416:
3406:
3266:
2810:
3591:
3421:
3411:
3391:
3371:
3346:
3291:
3271:
3256:
3246:
3181:
2847:
2341:
1631:
890:
139:
440:
3790:
3785:
3741:
3736:
3731:
3636:
3396:
3361:
3321:
3301:
3276:
3261:
3251:
3211:
2698:
2571:
2210:
2177:
1621:
1267:
2842:
431:
3676:
3671:
3581:
3576:
3571:
3366:
3336:
3331:
3311:
3296:
3286:
3281:
2188:
1917:
1626:
1384:
295:
3711:
3706:
3701:
3631:
3626:
3621:
3616:
3316:
3196:
3191:
1942:
1414:
1297:
669:
2864:
3376:
3226:
3176:
2346:
2022:
by dividing the triangles and hexagons into central triangles and merging neighboring triangles into kites.
1664:
1433:
603:
299:
51:
3496:
3486:
3456:
3138:
2753:
2172:
1611:
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3501:
3461:
3446:
3441:
3436:
3431:
3186:
2976:
2691:
2183:
2038:
is a part of a set of uniform dual tilings, corresponding to the dual of the rhombitrihexagonal tiling.
1653:
1756:
664:
3641:
3381:
3094:
3082:
2966:
2895:
2871:
2796:
2524:
2166:
2155:
1591:
281:
58:
3386:
3206:
3052:
3011:
3006:
2886:
2019:
1937:
1606:
1409:
1272:
361:
64:
2205:
2161:
3171:
2940:
2738:
2463:
2390:
2321:
2260:
2144:
2047:
1794:
1749:
288:
265:
2540:
1777:
468:
353:
314:
in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.)
259:
74:
2225:
2150:
639:
3666:
3216:
3143:
2986:
2769:
2673:
2631:
2612:
2598:
2579:
2552:
2500:
2264:
2059:
1912:
1802:
1445:
1441:
1394:
1379:
1302:
1282:
838:
617:
365:
321:. The hexagons can be considered as truncated triangles, t{3} with two types of edges. It has
318:
255:
247:
222:
2634:
17:
3696:
3511:
3476:
3153:
3117:
3062:
3028:
2981:
2955:
2944:
2859:
2831:
2774:
2748:
2743:
2473:
2402:
2051:
1927:
1922:
1841:
It is one of seven dual uniform tilings in hexagonal symmetry, including the regular duals.
1806:
1782:
1596:
1569:
1534:
1437:
1419:
1399:
1389:
1277:
1262:
685:
311:
284:
2485:
1562:
729:
3057:
2881:
2791:
2481:
2256:
2055:
1951:
1821:
1810:
1769:
1739:
1683:
1676:
1671:
490:
322:
239:
207:
2615:
1820:
is a dual of the semiregular tiling rhombitrihexagonal tiling. Its faces are deltoids or
696:
by replacing some of the hexagons and surrounding squares and triangles with dodecagons:
1988:
623:
2994:
2907:
2876:
2765:
2650:
2306:
1995:
1981:
1555:
1548:
1541:
1247:
1240:
834:
821:
813:
809:
633:
129:
2503:
2454:
Kirby, Matthew; Umble, Ronald (2011), "Edge tessellations and stamp folding puzzles",
2244:
Point symmetry allows the plane to be filled by growing kites, with the topology as a
1254:
1233:
1219:
1205:
776:
596:
3779:
3148:
3112:
2912:
2900:
2758:
2406:
2249:
2245:
1601:
1883:
1862:
1801:. The edges of this tiling can be formed by the intersection overlay of the regular
1226:
1212:
1198:
3047:
2784:
2714:
2665:
1876:
273:
191:
180:
2424:
1576:
2259:
tiling with kite faces, also a topological variation of a square tiling and with
816:). The translational lattice domain (red rhombus) contains six distinct circles.
3033:
1371:
1364:
1343:
770:
2216:
1785:, is composed by a collection of 8 kites from the deltoidal trihexagonal tiling
1357:
1350:
1336:
1329:
1322:
1315:
611:
3102:
2545:
2477:
1583:
723:
668:
The tiling can be replaced by circular edges, centered on the hexagons as an
3122:
3107:
3023:
2999:
2639:
2620:
2508:
1793:
is a dual of the semiregular tiling known as the rhombitrihexagonal tiling.
2299:
2046:
This tiling is topologically related as a part of sequence of tilings with
792:
745:
761:
736:
2891:
2444:
Order in Space: A design source book, Keith
Critchlow, p.74-75, pattern B
673:
243:
231:
2576:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
1436:
polyhedra with vertex figure (3.4.n.4), and continues as tilings of the
783:
714:
251:
2817:
2027:
1829:
2607:(Chapter 21, Naming Archimedean and Catalan polyhedra and tilings.
2468:
2267:, with every vertex containing all orientations of the kite face.
1776:
663:
837:
that can be based from the regular hexagonal tiling (or the dual
3079:
2929:
2829:
2725:
2687:
2683:
2661:, 1970, p. 69-61, Pattern N, Dual p. 77-76, pattern 2
1432:
This tiling is topologically related as a part of sequence of
364:. In the limit, where the rectangles degenerate into edges, a
2391:"Tilings by Regular Polygons—II: A Catalog of Tilings"
1755:
2364:
2362:
2252:. Below is an example with dihedral hexagonal symmetry.
317:
With edge-colorings there is a half symmetry form (3*3)
2248:, V4.4.4.4, and can be created by crossing string of a
2593:
John H. Conway, Heidi
Burgiel, Chaim Goodman-Strauss,
688:, having hexagons dissected into six triangles. The
94:
2515:(See comparative overlay of this tiling and its dual)
848:
is topologically identical to the hexagonal tiling.)
85:
2070:
32 symmetry mutation of dual expanded tilings: V3.4.
1957:
591:
29:
3235:
3162:
3131:
3093:
1765:
1748:
1738:
1728:
1718:
1682:
1670:
1660:
1646:
118:{\displaystyle r{\begin{Bmatrix}6\\3\end{Bmatrix}}}
2544:
1781:A 2023 discovered aperiodic monotile, solving the
368:results, constructed as a snub triangular tiling,
360:{3,6}. The bicolored square can be distorted into
117:
808:The rhombitrihexagonal tiling can be used as a
1458:32 symmetry mutation of expanded tilings: 3.4.
2699:
2395:Computers & Mathematics with Applications
868:
8:
3090:
3076:
2926:
2826:
2722:
2706:
2692:
2684:
2651:"2D Euclidean tilings x3o6x - rothat - O8"
2050:V3.4.n.4, and continues as tilings of the
1845:Dual uniform hexagonal/triangular tilings
1652:
1450:
875:
861:
850:
3017:Dividing a square into similar rectangles
2467:
89:
84:
27:Semiregular tiling of the Euclidean plane
2269:
2064:
1843:
698:
398:
2358:
858:
2241:Other deltoidal tilings are possible.
1643:
653:Roman floor mosaic in Castel di Guido
644:The Temple of Diana in Nîmes, France
7:
2659:Order in Space: A design source book
855:Uniform hexagonal/triangular tilings
2378:Ring Cycles a Jacks Chain variation
2680:, pp. 50–56, dual p. 116
25:
2058:figures have (*n32) reflectional
1444:figures have (*n32) reflectional
2816:
2809:
2312:
2305:
2298:
2224:
2215:
2204:
2193:
2182:
2171:
2160:
2149:
2026:
1994:
1987:
1980:
1903:
1896:
1889:
1882:
1875:
1868:
1861:
1828:
1709:
1704:
1699:
1694:
1689:
1582:
1575:
1568:
1561:
1554:
1547:
1540:
1533:
1370:
1363:
1356:
1349:
1342:
1335:
1328:
1321:
1314:
1253:
1246:
1239:
1232:
1225:
1218:
1211:
1204:
1197:
1187:
1182:
1177:
1172:
1167:
1158:
1153:
1148:
1143:
1138:
1129:
1124:
1119:
1114:
1109:
1100:
1095:
1090:
1085:
1080:
1071:
1066:
1061:
1056:
1051:
1042:
1037:
1032:
1027:
1022:
1013:
1008:
1003:
998:
993:
984:
979:
974:
969:
964:
955:
950:
945:
940:
935:
820:
791:
782:
775:
769:
760:
744:
735:
728:
722:
713:
647:
638:
622:
610:
595:
576:
571:
566:
561:
556:
547:
542:
537:
532:
527:
518:
513:
508:
503:
498:
457:
448:
439:
430:
390:
385:
380:
375:
370:
346:
341:
336:
331:
326:
165:
160:
155:
150:
145:
63:
38:
630:Archeological Museum of Seville
238:is a semiregular tiling of the
2018:This tiling is related to the
1:
3042:Regular Division of the Plane
2670:Introduction to Tessellations
2237:Other deltoidal (kite) tiling
2036:deltoidal trihexagonal tiling
1837:Related polyhedra and tilings
1818:deltoidal trihexagonal tiling
1791:deltoidal trihexagonal tiling
1647:Deltoidal trihexagonal tiling
1640:Deltoidal trihexagonal tiling
694:truncated trihexagonal tiling
213:Deltoidal trihexagonal tiling
18:Deltoidal trihexagonal tiling
2407:10.1016/0898-1221(89)90156-9
2001:
758:
711:
488:
466:
425:
2950:Architectonic and catoptric
2848:Aperiodic set of prototiles
2578:. Dover Publications, Inc.
2565:Regular and uniform tilings
2551:. New York: W. H. Freeman.
2342:Tilings of regular polygons
846:truncated triangular tiling
3812:
2635:"Semiregular tessellation"
2543:; Shephard, G. C. (1987).
1308:
853:
593:
33:Rhombitrihexagonal tiling
3089:
3075:
2936:
2925:
2838:
2825:
2807:
2734:
2721:
2478:10.4169/math.mag.84.4.283
2325:
2098:
2088:
2078:
1971:
1849:
1744:Rhombitrihexagonal tiling
1651:
1486:
1476:
1466:
1453:
885:
754:
707:
690:rhombitrihexagonal tiling
525:
477:
418:
407:
291:'s operational language.
272:. It can be considered a
236:rhombitrihexagonal tiling
37:
32:
2595:The Symmetries of Things
2368:Conway, 2008, p288 table
670:overlapping circles grid
2347:List of uniform tilings
1665:Dual semiregular tiling
692:is also related to the
604:The Grammar of Ornament
419:Cantic snub triangular
2616:"Uniform tessellation"
1786:
1760:
708:2-uniform dissections
681:
445:Uniform edge coloring
436:Uniform face coloring
119:
2263:V4.4.4.4. It is also
2009:Half regular hexagon
1959:Isohedral variations
1780:
1759:
684:There is one related
667:
120:
2547:Tilings and Patterns
2525:Tilings and patterns
2456:Mathematics Magazine
829:Wythoff construction
454:Nonuniform geometry
362:isosceles trapezoids
83:
59:Vertex configuration
3796:Semiregular tilings
2649:Klitzing, Richard.
2504:"Dual tessellation"
2389:Chavey, D. (1989).
2271:
2075:
2048:face configurations
2020:trihexagonal tiling
1960:
1846:
416:Rhombitrihexagonal
300:semiregular tilings
2632:Weisstein, Eric W.
2613:Weisstein, Eric W.
2501:Weisstein, Eric W.
2270:
2261:face configuration
2065:
2042:Symmetry mutations
1958:
1844:
1787:
1761:
1750:Face configuration
1428:Symmetry mutations
741:3.3.4.3.4 & 3
682:
310:There is only one
289:Alicia Boole Stott
280:terminology or an
270:rhombihexadeltille
115:
109:
52:Semiregular tiling
3791:Isohedral tilings
3786:Euclidean tilings
3773:
3772:
3769:
3768:
3765:
3764:
3071:
3070:
2962:Computer graphics
2921:
2920:
2805:
2804:
2664:Dale Seymour and
2657:Keith Critchlow,
2603:978-1-56881-220-5
2421:"Uniform Tilings"
2333:
2332:
2265:vertex transitive
2234:
2233:
2016:
2015:
1948:
1947:
1803:triangular tiling
1775:
1774:
1637:
1636:
1442:vertex-transitive
1425:
1424:
839:triangular tiling
801:
800:
657:
656:
585:
584:
366:triangular tiling
319:orbifold notation
306:Uniform colorings
228:
227:
223:Vertex-transitive
188:Rotation symmetry
16:(Redirected from
3803:
3091:
3077:
3029:Conway criterion
2956:Circle Limit III
2927:
2860:Einstein problem
2827:
2820:
2813:
2749:Schwarz triangle
2723:
2708:
2701:
2694:
2685:
2654:
2645:
2644:
2626:
2625:
2589:
2572:Williams, Robert
2567:, p. 58-65)
2562:
2550:
2541:Grünbaum, Branko
2527:
2522:
2516:
2514:
2513:
2496:
2490:
2488:
2471:
2451:
2445:
2442:
2436:
2435:
2433:
2432:
2423:. Archived from
2417:
2411:
2410:
2386:
2380:
2375:
2369:
2366:
2316:
2309:
2302:
2272:
2228:
2219:
2208:
2197:
2186:
2175:
2164:
2153:
2099:Compact hyperb.
2076:
2052:hyperbolic plane
2030:
1998:
1991:
1984:
1961:
1950:This tiling has
1907:
1900:
1893:
1886:
1879:
1872:
1865:
1847:
1832:
1807:hexagonal tiling
1783:Einstein problem
1714:
1713:
1712:
1708:
1707:
1703:
1702:
1698:
1697:
1693:
1692:
1656:
1644:
1586:
1579:
1572:
1565:
1558:
1551:
1544:
1537:
1487:Compact hyperb.
1451:
1438:hyperbolic plane
1374:
1367:
1360:
1353:
1346:
1339:
1332:
1325:
1318:
1257:
1250:
1243:
1236:
1229:
1222:
1215:
1208:
1201:
1192:
1191:
1190:
1186:
1185:
1181:
1180:
1176:
1175:
1171:
1170:
1163:
1162:
1161:
1157:
1156:
1152:
1151:
1147:
1146:
1142:
1141:
1134:
1133:
1132:
1128:
1127:
1123:
1122:
1118:
1117:
1113:
1112:
1105:
1104:
1103:
1099:
1098:
1094:
1093:
1089:
1088:
1084:
1083:
1076:
1075:
1074:
1070:
1069:
1065:
1064:
1060:
1059:
1055:
1054:
1047:
1046:
1045:
1041:
1040:
1036:
1035:
1031:
1030:
1026:
1025:
1018:
1017:
1016:
1012:
1011:
1007:
1006:
1002:
1001:
997:
996:
989:
988:
987:
983:
982:
978:
977:
973:
972:
968:
967:
960:
959:
958:
954:
953:
949:
948:
944:
943:
939:
938:
877:
870:
863:
851:
833:There are eight
824:
795:
786:
779:
773:
764:
748:
739:
732:
726:
717:
699:
686:2-uniform tiling
651:
642:
626:
614:
599:
592:
581:
580:
579:
575:
574:
570:
569:
565:
564:
560:
559:
552:
551:
550:
546:
545:
541:
540:
536:
535:
531:
530:
523:
522:
521:
517:
516:
512:
511:
507:
506:
502:
501:
461:
452:
443:
434:
422:Snub triangular
399:
395:
394:
393:
389:
388:
384:
383:
379:
378:
374:
373:
351:
350:
349:
345:
344:
340:
339:
335:
334:
330:
329:
312:uniform coloring
294:There are three
285:hexagonal tiling
278:Norman Johnson's
242:. There are one
170:
169:
168:
164:
163:
159:
158:
154:
153:
149:
148:
124:
122:
121:
116:
114:
113:
67:
42:
30:
21:
3811:
3810:
3806:
3805:
3804:
3802:
3801:
3800:
3776:
3775:
3774:
3761:
3238:
3231:
3164:
3158:
3127:
3085:
3067:
2932:
2917:
2834:
2821:
2815:
2814:
2801:
2792:Wallpaper group
2730:
2717:
2712:
2648:
2630:
2629:
2611:
2610:
2586:
2570:
2559:
2539:
2536:
2531:
2530:
2523:
2519:
2499:
2498:
2497:
2493:
2453:
2452:
2448:
2443:
2439:
2430:
2428:
2419:
2418:
2414:
2388:
2387:
2383:
2376:
2372:
2367:
2360:
2355:
2338:
2283:
2257:face transitive
2239:
2230:V3.4.∞.4
2229:
2220:
2209:
2198:
2187:
2176:
2165:
2154:
2143:
2137:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2086:
2080:
2056:face-transitive
2044:
2012:Quadrilaterals
1952:face transitive
1839:
1770:face-transitive
1740:Dual polyhedron
1710:
1705:
1700:
1695:
1690:
1688:
1684:Coxeter diagram
1642:
1525:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1474:
1468:
1430:
1188:
1183:
1178:
1173:
1168:
1166:
1159:
1154:
1149:
1144:
1139:
1137:
1130:
1125:
1120:
1115:
1110:
1108:
1101:
1096:
1091:
1086:
1081:
1079:
1072:
1067:
1062:
1057:
1052:
1050:
1043:
1038:
1033:
1028:
1023:
1021:
1014:
1009:
1004:
999:
994:
992:
985:
980:
975:
970:
965:
963:
956:
951:
946:
941:
936:
934:
899:
895:
881:
835:uniform tilings
831:
806:
796:
787:
774:
765:
749:
740:
727:
718:
662:
660:Related tilings
652:
643:
627:
615:
600:
590:
577:
572:
567:
562:
557:
555:
548:
543:
538:
533:
528:
526:
519:
514:
509:
504:
499:
497:
492:
481:
470:
462:
453:
444:
435:
391:
386:
381:
376:
371:
369:
359:
354:Schläfli symbol
347:
342:
337:
332:
327:
325:
323:Coxeter diagram
308:
260:Schläfli symbol
240:Euclidean plane
166:
161:
156:
151:
146:
144:
140:Coxeter diagram
108:
107:
101:
100:
90:
81:
80:
75:Schläfli symbol
68:
43:
28:
23:
22:
15:
12:
11:
5:
3809:
3807:
3799:
3798:
3793:
3788:
3778:
3777:
3771:
3770:
3767:
3766:
3763:
3762:
3760:
3759:
3754:
3749:
3744:
3739:
3734:
3729:
3724:
3719:
3714:
3709:
3704:
3699:
3694:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3634:
3629:
3624:
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3544:
3539:
3534:
3529:
3524:
3519:
3514:
3509:
3504:
3499:
3494:
3489:
3484:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3444:
3439:
3434:
3429:
3424:
3419:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3334:
3329:
3324:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3274:
3269:
3264:
3259:
3254:
3249:
3243:
3241:
3233:
3232:
3230:
3229:
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3168:
3166:
3160:
3159:
3157:
3156:
3151:
3146:
3141:
3135:
3133:
3129:
3128:
3126:
3125:
3120:
3115:
3110:
3105:
3099:
3097:
3087:
3086:
3080:
3073:
3072:
3069:
3068:
3066:
3065:
3060:
3055:
3050:
3045:
3038:
3037:
3036:
3031:
3021:
3020:
3019:
3014:
3009:
3004:
3003:
3002:
2989:
2984:
2979:
2974:
2969:
2964:
2959:
2952:
2947:
2937:
2934:
2933:
2930:
2923:
2922:
2919:
2918:
2916:
2915:
2910:
2905:
2904:
2903:
2889:
2884:
2879:
2874:
2869:
2868:
2867:
2865:Socolar–Taylor
2857:
2856:
2855:
2845:
2843:Ammann–Beenker
2839:
2836:
2835:
2830:
2823:
2822:
2808:
2806:
2803:
2802:
2800:
2799:
2794:
2789:
2788:
2787:
2782:
2777:
2766:Uniform tiling
2763:
2762:
2761:
2751:
2746:
2741:
2735:
2732:
2731:
2726:
2719:
2718:
2713:
2711:
2710:
2703:
2696:
2688:
2682:
2681:
2678:978-0866514613
2662:
2655:
2646:
2627:
2608:
2591:
2584:
2568:
2563:(Chapter 2.1:
2557:
2535:
2532:
2529:
2528:
2517:
2491:
2462:(4): 283–289,
2446:
2437:
2412:
2381:
2370:
2357:
2356:
2354:
2351:
2350:
2349:
2344:
2337:
2334:
2331:
2330:
2327:
2324:
2318:
2317:
2310:
2303:
2296:
2292:
2291:
2290:p6m, , (*632)
2288:
2285:
2281:
2278:
2238:
2235:
2232:
2231:
2222:
2213:
2202:
2191:
2180:
2169:
2158:
2147:
2139:
2138:
2134:
2129:
2125:
2121:
2117:
2113:
2109:
2104:
2103:
2100:
2097:
2092:
2087:
2043:
2040:
2032:
2031:
2014:
2013:
2010:
2007:
2004:
2000:
1999:
1992:
1985:
1978:
1974:
1973:
1972:p31m, , (3*3)
1970:
1969:p6m, , (*632)
1967:
1946:
1945:
1940:
1935:
1930:
1925:
1920:
1915:
1909:
1908:
1901:
1894:
1887:
1880:
1873:
1866:
1858:
1857:
1854:
1838:
1835:
1834:
1833:
1773:
1772:
1767:
1763:
1762:
1752:
1746:
1745:
1742:
1736:
1735:
1732:
1730:Rotation group
1726:
1725:
1722:
1720:Symmetry group
1716:
1715:
1686:
1680:
1679:
1674:
1668:
1667:
1662:
1658:
1657:
1649:
1648:
1641:
1638:
1635:
1634:
1629:
1624:
1619:
1614:
1609:
1604:
1599:
1594:
1588:
1587:
1580:
1573:
1566:
1559:
1552:
1545:
1538:
1531:
1527:
1526:
1522:
1517:
1513:
1509:
1505:
1501:
1497:
1492:
1491:
1488:
1485:
1480:
1475:
1464:
1463:
1429:
1426:
1423:
1422:
1417:
1412:
1407:
1402:
1397:
1392:
1387:
1382:
1376:
1375:
1368:
1361:
1354:
1347:
1340:
1333:
1326:
1319:
1311:
1310:
1309:Uniform duals
1306:
1305:
1300:
1295:
1290:
1285:
1280:
1275:
1270:
1265:
1259:
1258:
1251:
1244:
1237:
1230:
1223:
1216:
1209:
1202:
1194:
1193:
1164:
1135:
1106:
1077:
1048:
1019:
990:
961:
931:
930:
927:
924:
921:
918:
915:
912:
909:
906:
902:
901:
897:
893:
883:
882:
880:
879:
872:
865:
857:
830:
827:
826:
825:
814:kissing number
810:circle packing
805:
804:Circle packing
802:
799:
798:
789:
780:
767:
757:
756:
752:
751:
742:
733:
720:
710:
709:
706:
703:
661:
658:
655:
654:
645:
636:
634:Sevilla, Spain
628:Floor tiling,
620:
608:
589:
586:
583:
582:
553:
524:
495:
487:
486:
483:
479:
476:
473:
465:
464:
455:
446:
437:
428:
424:
423:
420:
417:
414:
410:
409:
406:
403:
357:
307:
304:
302:in the plane.
226:
225:
220:
216:
215:
210:
204:
203:
200:
199:Bowers acronym
196:
195:
189:
185:
184:
178:
172:
171:
142:
136:
135:
132:
130:Wythoff symbol
126:
125:
112:
106:
103:
102:
99:
96:
95:
93:
88:
77:
71:
70:
61:
55:
54:
49:
45:
44:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3808:
3797:
3794:
3792:
3789:
3787:
3784:
3783:
3781:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3244:
3242:
3240:
3234:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3169:
3167:
3161:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3136:
3134:
3130:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3100:
3098:
3096:
3092:
3088:
3084:
3078:
3074:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3043:
3039:
3035:
3032:
3030:
3027:
3026:
3025:
3022:
3018:
3015:
3013:
3010:
3008:
3005:
3001:
2998:
2997:
2996:
2993:
2992:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2957:
2953:
2951:
2948:
2946:
2942:
2939:
2938:
2935:
2928:
2924:
2914:
2911:
2909:
2906:
2902:
2899:
2898:
2897:
2893:
2890:
2888:
2885:
2883:
2880:
2878:
2875:
2873:
2870:
2866:
2863:
2862:
2861:
2858:
2854:
2851:
2850:
2849:
2846:
2844:
2841:
2840:
2837:
2833:
2828:
2824:
2819:
2812:
2798:
2795:
2793:
2790:
2786:
2783:
2781:
2778:
2776:
2773:
2772:
2771:
2767:
2764:
2760:
2757:
2756:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2736:
2733:
2729:
2724:
2720:
2716:
2709:
2704:
2702:
2697:
2695:
2690:
2689:
2686:
2679:
2675:
2671:
2667:
2663:
2660:
2656:
2652:
2647:
2642:
2641:
2636:
2633:
2628:
2623:
2622:
2617:
2614:
2609:
2606:
2604:
2600:
2596:
2592:
2587:
2585:0-486-23729-X
2581:
2577:
2573:
2569:
2566:
2560:
2558:0-7167-1193-1
2554:
2549:
2548:
2542:
2538:
2537:
2533:
2526:
2521:
2518:
2511:
2510:
2505:
2502:
2495:
2492:
2487:
2483:
2479:
2475:
2470:
2465:
2461:
2457:
2450:
2447:
2441:
2438:
2427:on 2006-09-09
2426:
2422:
2416:
2413:
2408:
2404:
2400:
2396:
2392:
2385:
2382:
2379:
2374:
2371:
2365:
2363:
2359:
2352:
2348:
2345:
2343:
2340:
2339:
2335:
2328:
2323:
2322:Configuration
2320:
2319:
2315:
2311:
2308:
2304:
2301:
2297:
2294:
2293:
2289:
2287:pmg, , (22*)
2286:
2279:
2277:
2274:
2273:
2268:
2266:
2262:
2258:
2253:
2251:
2250:dream catcher
2247:
2246:square tiling
2242:
2236:
2227:
2223:
2218:
2214:
2212:
2207:
2203:
2201:
2196:
2192:
2190:
2185:
2181:
2179:
2174:
2170:
2168:
2163:
2159:
2157:
2152:
2148:
2146:
2141:
2140:
2135:
2130:
2126:
2122:
2118:
2114:
2110:
2106:
2105:
2101:
2096:
2093:
2091:
2084:
2077:
2073:
2069:
2063:
2061:
2057:
2053:
2049:
2041:
2039:
2037:
2029:
2025:
2024:
2023:
2021:
2011:
2008:
2005:
2002:
1997:
1993:
1990:
1986:
1983:
1979:
1976:
1975:
1968:
1966:
1963:
1962:
1956:
1953:
1944:
1941:
1939:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1910:
1906:
1902:
1899:
1895:
1892:
1888:
1885:
1881:
1878:
1874:
1871:
1867:
1864:
1860:
1859:
1855:
1852:
1848:
1842:
1836:
1831:
1827:
1826:
1825:
1823:
1819:
1814:
1812:
1808:
1804:
1800:
1796:
1792:
1784:
1779:
1771:
1768:
1764:
1758:
1753:
1751:
1747:
1743:
1741:
1737:
1733:
1731:
1727:
1724:p6m, , (*632)
1723:
1721:
1717:
1687:
1685:
1681:
1678:
1675:
1673:
1669:
1666:
1663:
1659:
1655:
1650:
1645:
1639:
1633:
1632:3.4.∞.4
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1608:
1605:
1603:
1600:
1598:
1595:
1593:
1590:
1589:
1585:
1581:
1578:
1574:
1571:
1567:
1564:
1560:
1557:
1553:
1550:
1546:
1543:
1539:
1536:
1532:
1529:
1528:
1523:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1493:
1489:
1484:
1481:
1479:
1472:
1465:
1461:
1457:
1452:
1449:
1447:
1443:
1439:
1435:
1427:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
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790:
785:
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778:
772:
768:
763:
759:
755:Dual Tilings
753:
747:
743:
738:
734:
731:
725:
721:
716:
712:
704:
701:
700:
697:
695:
691:
687:
679:
676:it is called
675:
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143:
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134:3 | 6 2
133:
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97:
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72:
66:
62:
60:
57:
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53:
50:
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41:
36:
31:
19:
3201:
3053:Substitution
3048:Regular grid
3040:
2954:
2887:Quaquaversal
2785:Kisrhombille
2715:Tessellation
2669:
2666:Jill Britton
2658:
2638:
2619:
2594:
2575:
2564:
2546:
2520:
2507:
2494:
2459:
2455:
2449:
2440:
2429:. Retrieved
2425:the original
2415:
2398:
2394:
2384:
2373:
2254:
2243:
2240:
2199:
2082:
2071:
2067:
2045:
2035:
2033:
2017:
1949:
1840:
1817:
1815:
1798:
1790:
1788:
1616:
1470:
1459:
1455:
1431:
1287:
845:
843:
832:
807:
689:
683:
677:
602:
316:
309:
293:
269:
264:
262:of rr{3,6}.
235:
229:
3083:vertex type
2941:Anisohedral
2896:Self-tiling
2739:Pythagorean
2401:: 147–165.
1853:: , (*632)
1797:calls it a
1734:p6, , (632)
1434:cantellated
1303:3.3.3.3.3.3
705:Dissection
678:Jacks chain
274:cantellated
268:calls it a
266:John Conway
183:, , (*632)
79:rr{6,3} or
3780:Categories
2987:Pentagonal
2534:References
2431:2006-09-09
2284:, , (*66)
2136:*∞32
1766:Properties
1524:*∞32
1490:Paracomp.
702:1-uniform
618:Kensington
298:and eight
250:, and one
219:Properties
194:, , (632)
3095:Spherical
3063:Voderberg
3024:Prototile
2991:Problems
2967:Honeycomb
2945:Isohedral
2832:Aperiodic
2770:honeycomb
2754:Rectangle
2744:Rhombille
2640:MathWorld
2621:MathWorld
2509:MathWorld
2469:0908.3257
2329:V6.4.3.4
2326:V4.4.4.4
2221:V3.4.8.4
2090:Spherical
1478:Spherical
1298:3.3.3.3.6
616:The game
405:, (*632)
402:Symmetry
258:. It has
3177:V3.4.3.4
3012:Squaring
3007:Heesch's
2972:Isotoxal
2892:Rep-tile
2882:Pinwheel
2775:Coloring
2728:Periodic
2672:, 1989,
2574:(1979).
2336:See also
2276:Symmetry
2255:Another
2211:V3.4.7.4
2200:V3.4.6.4
2189:V3.4.5.4
2178:V3.4.4.4
2167:V3.4.3.4
2156:V3.4.2.4
2102:Paraco.
2079:Symmetry
2060:symmetry
2054:. These
1965:Symmetry
1938:V.4.6.12
1933:V3.4.6.4
1856:, (632)
1851:Symmetry
1799:tetrille
1754:V3.4.6.4
1467:Symmetry
1446:symmetry
1440:. These
1410:V.4.6.12
1405:V3.4.6.4
926:sr{6,3}
923:tr{6,3}
920:rr{6,3}
891:, (*632)
887:Symmetry
766:3.4.6.4
719:3.4.6.4
674:quilting
588:Examples
475:rr{3,6}
469:Schläfli
408:, (3*3)
282:expanded
254:on each
244:triangle
232:geometry
176:Symmetry
69:3.4.6.4
3637:6.4.8.4
3592:5.4.6.4
3552:4.12.16
3542:4.10.12
3512:V4.8.10
3487:V4.6.16
3477:V4.6.14
3377:3.6.4.6
3372:3.4.∞.4
3367:3.4.8.4
3362:3.4.7.4
3357:3.4.6.4
3307:3.∞.3.∞
3302:3.4.3.4
3297:3.8.3.8
3292:3.7.3.7
3287:3.6.3.8
3282:3.6.3.6
3277:3.5.3.6
3272:3.5.3.5
3267:3.4.3.∞
3262:3.4.3.8
3257:3.4.3.7
3252:3.4.3.6
3247:3.4.3.5
3202:3.4.6.4
3172:3.4.3.4
3165:regular
3132:Regular
3058:Voronoi
2982:Packing
2913:Truchet
2908:Socolar
2877:Penrose
2872:Gilbert
2797:Wythoff
2486:2843659
2295:Tiling
2145:Config.
2095:Euclid.
1809:. Each
1627:3.4.8.4
1622:3.4.7.4
1617:3.4.6.4
1612:3.4.5.4
1607:3.4.4.4
1602:3.4.3.4
1597:3.4.2.4
1592:Config.
1530:Figure
1483:Euclid.
1288:3.4.6.4
929:s{3,6}
914:t{3,6}
911:r{6,3}
908:t{6,3}
788:4.6.12
607:(1856)
493:diagram
491:Coxeter
485:s{3,6}
296:regular
252:hexagon
248:squares
202:Rothat
3527:4.8.16
3522:4.8.14
3517:4.8.12
3507:4.8.10
3482:4.6.16
3472:4.6.14
3467:4.6.12
3237:Hyper-
3222:4.6.12
2995:Domino
2901:Sphinx
2780:Convex
2759:Domino
2676:
2601:
2597:2008,
2582:
2555:
2484:
2142:Figure
2003:Faces
1923:V(3.6)
1805:and a
1795:Conway
1390:V(3.6)
1293:4.6.12
917:{3,6}
905:{6,3}
900:(3*3)
896:(632)
750:to CH
482:{3,6}
471:symbol
463:Limit
427:Image
256:vertex
246:, two
234:, the
3642:(6.8)
3597:(5.6)
3532:4.8.∞
3502:(4.8)
3497:(4.7)
3492:4.6.∞
3462:(4.6)
3457:(4.5)
3427:4.∞.4
3422:4.8.4
3417:4.7.4
3412:4.6.4
3407:4.5.4
3387:(3.8)
3382:(3.7)
3352:(3.4)
3347:(3.4)
3239:bolic
3207:(3.6)
3163:Semi-
3034:Girih
2931:Other
2464:arXiv
2353:Notes
2006:Kite
1977:Form
1918:V3.12
1822:kites
1672:Faces
1385:V3.12
1278:6.6.6
1273:(3.6)
797:to 3
672:. In
601:From
413:Name
3727:8.16
3722:8.12
3692:7.14
3662:6.16
3657:6.12
3652:6.10
3612:5.12
3607:5.10
3562:4.16
3557:4.14
3547:4.12
3537:4.10
3397:3.16
3392:3.14
3212:3.12
3197:V3.6
3123:V4.n
3113:V3.n
3000:Wang
2977:List
2943:and
2894:and
2853:List
2768:and
2674:ISBN
2599:ISBN
2580:ISBN
2553:ISBN
2133:...
2131:*832
2127:*732
2123:*632
2119:*532
2115:*432
2111:*332
2107:*232
2034:The
1943:V3.6
1816:The
1811:kite
1789:The
1677:kite
1661:Type
1521:...
1519:*832
1515:*732
1511:*632
1507:*532
1503:*432
1499:*332
1495:*232
1415:V3.6
1268:3.12
208:Dual
48:Type
3757:∞.8
3752:∞.6
3717:8.6
3687:7.8
3682:7.6
3647:6.8
3602:5.8
3567:4.∞
3402:3.∞
3327:3.4
3322:3.∞
3317:3.8
3312:3.7
3227:4.8
3217:4.∞
3192:3.6
3187:3.∞
3182:3.4
3118:4.n
3108:3.n
3081:By
2590:p40
2474:doi
2403:doi
2074:.4
1462:.4
841:).
287:by
276:by
230:In
181:p6m
3782::
2668:,
2637:.
2618:.
2506:.
2482:MR
2480:,
2472:,
2460:84
2458:,
2399:17
2397:.
2393:.
2361:^
2085:32
2062:.
1928:V3
1913:V6
1824:.
1473:32
1448:.
1420:V3
1400:V3
1395:V6
1380:V6
889::
632:,
396:.
352:,
192:p6
3747:∞
3742:∞
3737:∞
3732:∞
3712:8
3707:8
3702:8
3697:8
3677:7
3672:7
3667:7
3632:6
3627:6
3622:6
3617:6
3587:5
3582:5
3577:5
3572:5
3452:4
3447:4
3442:4
3437:4
3432:4
3342:3
3337:3
3332:3
3154:6
3149:4
3144:3
3139:2
3103:2
2707:e
2700:t
2693:v
2653:.
2643:.
2624:.
2588:.
2561:.
2512:.
2489:.
2476::
2466::
2434:.
2409:.
2405::
2282:6
2280:D
2083:n
2081:*
2072:n
2068:n
2066:*
1471:n
1469:*
1460:n
1456:n
1454:*
1283:3
1263:6
876:e
869:t
862:v
680:.
480:2
478:s
358:2
356:s
111:}
105:3
98:6
92:{
87:r
20:)
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