2463:
4596:
500:
1073:
36:
1760:
2056:
3126:
3072:
520:
1886:
1879:
1872:
1455:
2049:
960:
665:
1900:
3105:
3051:
967:
915:
686:
672:
3058:
953:
2084:
471:
3119:
3112:
3098:
455:
1039:
658:
3157:
3065:
2621:
2512:
1483:
1462:
3230:
3044:
651:
2810:
2900:
2491:
2477:
1753:
1119:
946:
679:
644:
2931:
2922:
2911:
2891:
2880:
2871:
2838:
2831:
2817:
2770:
2070:
2519:
2498:
2484:
2470:
1893:
1865:
1746:
1739:
1732:
1298:
1128:
974:
2824:
2803:
2796:
2763:
2756:
2749:
2742:
2735:
102:
2063:
2042:
2505:
1767:
1476:
1469:
1725:
1287:
1490:
2659:
2637:
2648:
3216:
129:
1034:
3223:
1907:
1781:
1511:
2690:
1176:
1167:
1158:
1146:
2681:
2672:
2077:
1788:
1532:
1525:
1518:
1064:
1055:
1046:
486:
1795:
1774:
1367:
1353:
1342:
1331:
1320:
1309:
1025:
1914:
1504:
1497:
1110:
4603:
2974:
2967:
2960:
2628:
2614:
2607:
538:
are common crystal structures. They are the densest sphere packings in three dimensions. Structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, with the face-centered cubic being the
2705:
Hexagonal tilings can be made with the identical {6,3} topology as the regular tiling (3 hexagons around every vertex). With isohedral faces, there are 13 variations. Symmetry given assumes all faces are the same color. Colors here represent the lattice positions. Single-color (1-tile) lattices are
402:
states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by
3207:
The first is made of 2-edges, three around every vertex, the second has hexagonal edges, three around every vertex. A third complex apeirogon, sharing the same vertices, is quasiregular, which alternates 2-edges and 6-edges.
2982:
The 2-uniform and 3-uniform tessellations have a rotational degree of freedom which distorts 2/3 of the hexagons, including a colinear case that can also be seen as a non-edge-to-edge tiling of hexagons and larger triangles.
906:
hexagonal tiling replaces edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a
2154:
Drawing the tiles colored red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 of which are topologically distinct. (The
1982:
symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces.
499:
3916:
3801:
3758:
3715:
3672:
3874:
3838:
1262:
1634:
2846:
Other isohedrally-tiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edge-to-edge, but interpreted as colinear adjacent edges:
3172:, sharing the vertices of the hexagonal tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons
1422:
4512:
5536:
1842:
454:
4801:
4734:
1255:
1080:
5541:
4756:
4490:
3499:
1627:
5351:
5186:
3602:
2126:
57:
5501:
5476:
5466:
5436:
5391:
5341:
5321:
5136:
5021:
1837:
1832:
5511:
5506:
5446:
5441:
5396:
5346:
5331:
1415:
1248:
5531:
5316:
4564:
3477:
3450:
3427:
1656:
852:
246:
79:
3280:
3261:
5371:
5306:
5291:
5126:
4746:
3999:
3982:
3290:
3241:
2453:
2443:
2433:
2424:
2414:
2404:
2395:
2375:
2366:
2337:
2327:
2298:
2269:
2259:
2230:
1620:
1206:
847:
842:
823:
813:
784:
241:
236:
228:
218:
180:
3271:
3251:
5471:
5431:
5386:
5326:
5311:
5301:
5276:
4637:
4420:
4058:
3143:, placing equal-diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (
2385:
2356:
2346:
2317:
2308:
2288:
2279:
2250:
2240:
1371:
1226:
1216:
833:
804:
794:
208:
200:
190:
5595:
5336:
5256:
5111:
3246:
2995:
2174:
1661:
620:
254:
3502:
3285:
3266:
2462:
2448:
2438:
2419:
2409:
2390:
2380:
2361:
2351:
2332:
2322:
2303:
2293:
2274:
2264:
2245:
2235:
1221:
1211:
828:
818:
799:
789:
223:
213:
195:
185:
5266:
5251:
5211:
5141:
5091:
5006:
4826:
3310:
2215:
1408:
1138:
It is also possible to subdivide the prototiles of certain hexagonal tilings by two, three, four or nine equal pentagons:
1099:
519:
5236:
5201:
5191:
5051:
4595:
4432:
416:
5376:
5206:
5196:
5176:
5156:
5131:
5076:
5056:
5041:
5031:
4966:
4632:
3315:
1180:
Pentagonal tiling type 3 with overlays of two sizes of regular hexagons (comprising 3 and 9 pentagons respectively).
172:
50:
44:
5585:
5575:
5570:
5526:
5521:
5516:
5421:
5181:
5146:
5106:
5086:
5061:
5046:
5036:
4996:
4483:
3465:
3330:
2190:
1574:
864:
470:
430:
resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised, known as
4627:
5580:
5461:
5456:
5366:
5361:
5356:
5151:
5121:
5116:
5096:
5081:
5071:
5066:
4986:
4214:
4159:
4110:
3325:
3169:
2210:
1952:
375:
61:
5590:
5496:
5491:
5486:
5416:
5411:
5406:
5401:
5101:
4981:
4976:
3977:
3595:
3418:
2220:
1357:
1335:
1324:
1313:
1094:, where each vertex of the rhombic tiling is stretched into a new edge. This is similar to the relation of the
4649:
1346:
1072:
5161:
5011:
4961:
4009:
3320:
2987:
2106:
535:
113:
5281:
5271:
5241:
4923:
4538:
776:
5381:
5286:
5246:
5231:
5226:
5221:
5216:
4971:
4761:
4476:
4458:
4451:
4444:
4266:
4204:
4149:
4100:
4038:
3879:
3764:
3721:
3678:
3635:
3487:
1947:
1822:
1812:
1759:
1595:
353:
101:
3843:
3807:
2055:
5426:
5166:
4879:
4867:
4751:
4680:
4656:
4581:
4408:
4401:
4396:
3384:
3370:
3356:
2526:
2116:
2101:
2001:
1817:
1802:
1095:
556:
399:
120:
3125:
3071:
5171:
4991:
4837:
4796:
4791:
4671:
4311:
4249:
4244:
4187:
4182:
4132:
4127:
4083:
4078:
4026:
3588:
3229:
2195:
2121:
1942:
1937:
1885:
531:
510:
412:
2048:
374:
of the hexagon is 120 degrees, so three hexagons at a point make a full 360 degrees. It is one of
4956:
4725:
4523:
2999:
2140:
2091:
1927:
903:
576:
360:
136:
3434:
1878:
1871:
1454:
1196:
736:
338:
146:
5451:
5001:
4928:
4771:
4554:
4256:
4194:
4139:
4090:
4068:
4048:
3930:
3616:
3612:
3553:
3534:
3515:
3495:
3473:
3446:
3423:
3305:
3148:
2205:
2148:
1996:
1564:
1171:
Pentagonal tiling type 4 with overlays of semiregular hexagons (each comprising 4 pentagons).
1150:
992:
966:
914:
379:
293:
283:
3147:). The gap inside each hexagon allows for one circle, creating the densest packing from the
5481:
5296:
5261:
4902:
4847:
4813:
4766:
4740:
4729:
4644:
4616:
4559:
4533:
4528:
4031:
3057:
2568:
2564:
2111:
2083:
1899:
1569:
1387:
1007:
988:
959:
952:
923:
664:
552:
435:
408:
3111:
3104:
3050:
1079:
685:
671:
485:
4842:
4666:
4576:
3989:
3215:
3064:
2620:
1482:
1461:
1200:
1162:
pentagonal tiling type 3 with overlays of regular hexagons (each comprising 3 pentagons).
506:
431:
330:
301:
297:
278:
3556:
3537:
3222:
3118:
3097:
2899:
1978:
This tiling is also part of a sequence of truncated rhombic polyhedra and tilings with
1038:
657:
4779:
4692:
4661:
4550:
4298:
4291:
4284:
4231:
4224:
4169:
3925:
3574:
3156:
3144:
3140:
2930:
2921:
2910:
2890:
2879:
2870:
2837:
2830:
2816:
2769:
2518:
2511:
2144:
1745:
1738:
1731:
1437:
908:
756:
461:
395:
371:
326:
158:
3043:
3002:
pattern has pmg (22*) symmetry, which is lowered to p1 (°) with 3 or 4 colored tiles.
2823:
2809:
2802:
2795:
2762:
2755:
2748:
2741:
2734:
2504:
2490:
2476:
2041:
1752:
1475:
1468:
1127:
650:
5564:
4933:
4897:
4697:
4685:
4543:
3957:
3947:
3937:
3628:
2991:
2915:
1979:
1380:
1191:
This tiling is topologically related as a part of a sequence of regular tilings with
891:
567:) represent the periodic repeat of one colored tile, counting hexagonal distances as
383:
2497:
2483:
2469:
2069:
1892:
1864:
1297:
1118:
945:
678:
643:
4832:
4569:
4499:
3518:
2707:
2062:
1766:
1554:
973:
476:
445:
270:
259:
2658:
2647:
2636:
1724:
1286:
987:
The hexagons can be dissected into sets of 6 triangles. This process leads to two
4818:
3397:
Order in Space: A design source book, Keith
Critchlow, pp. 74–75, pattern 2
1780:
1544:
1510:
1489:
404:
2689:
2680:
2671:
1787:
1531:
1524:
1517:
1175:
1166:
1157:
1145:
128:
4887:
3439:
1932:
1906:
1794:
1773:
1539:
1366:
1352:
1341:
1330:
1319:
1308:
1033:
4907:
4892:
4808:
4784:
3561:
3542:
3523:
2904:
2884:
2076:
1913:
1063:
1054:
1045:
2567:. Each has parametric variations within a fixed symmetry. Type 2 contains
1109:
1024:
4676:
1807:
1291:
491:
439:
427:
423:
314:
3470:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
2990:
4-colored tri-directional weaved pattern, distorting some hexagons into
1503:
1496:
1153:
type 1 with overlays of regular hexagons (each comprising 2 pentagons).
530:
The hexagonal tiling appears in many crystals. In three dimensions, the
17:
3414:
3335:
2563:
There are 3 types of monohedral convex hexagonal tilings. They are all
1192:
334:
4602:
540:
2973:
2966:
2959:
2627:
2613:
2606:
913:
555:
of a hexagonal tiling, all generated from reflective symmetry of
2096:
1549:
543:, amongst other materials, forms a face-centered cubic lattice.
4864:
4714:
4614:
4510:
4472:
4468:
2147:
that can be based on the regular hexagonal tiling (or the dual
1379:
This tiling is topologically related to regular polyhedra with
890:
The 3-color tiling is a tessellation generated by the order-3
448:
consists of a hexagonal lattice (often not regular) of wires.
29:
3151:, with each circle in contact with a maximum of 6 circles.
2135:
Wythoff constructions from hexagonal and triangular tilings
434:. They have many potential applications, due to their high
2994:. The weaved pattern with 2 colored faces has rotational
3406:
Coxeter, Regular
Complex Polytopes, pp. 111–112, p. 136.
3374:, from list of 107 isohedral tilings, pp. 473–481
3360:, Sec. 9.3 Other Monohedral tilings by convex polygons
3882:
3846:
3810:
3767:
3724:
3681:
3638:
2158:
is topologically identical to the hexagonal tiling.)
2571:, and is 2-isohedral keeping chiral pairs distinct.
1140:
599:
91:
5020:
4947:
4916:
4878:
3910:
3868:
3832:
3795:
3752:
3709:
3666:
3438:
1386:, as a part of a sequence that continues into the
1988:Symmetry mutations of dual quasiregular tilings:
1195:faces, starting with the hexagonal tiling, with
2575:3 types of monohedral convex hexagonal tilings
593:, and can be applied to hyperbolic tilings for
422:This structure exists naturally in the form of
415:lattice) is optimal. However, the less regular
4484:
3596:
1628:
1416:
1256:
8:
1240:62 symmetry mutation of regular tilings: {6,
464:is arranged like the hexagons in this tiling
3388:, uniform tilings that are not edge-to-edge
1612:32 symmetry mutation of truncated tilings:
4875:
4861:
4711:
4611:
4507:
4491:
4477:
4469:
3603:
3589:
3581:
2160:
1984:
1635:
1621:
1604:
1423:
1409:
1400:32 symmetry mutation of regular tilings: {
1392:
1263:
1249:
1232:
1090:The hexagonal tiling can be considered an
4802:Dividing a square into similar rectangles
3896:
3885:
3884:
3881:
3860:
3849:
3848:
3845:
3824:
3813:
3812:
3809:
3781:
3770:
3769:
3766:
3738:
3727:
3726:
3723:
3695:
3684:
3683:
3680:
3652:
3641:
3640:
3637:
3575:"2D Euclidean tilings o3o6x – hexat – O3"
3430:p. 296, Table II: Regular honeycombs
575:second. The same counting is used in the
80:Learn how and when to remove this message
27:Regular tiling of a two-dimensional space
3210:
3004:
2940:
2848:
2712:
2573:
1104:
997:
929:
43:This article includes a list of general
3472:. Dover Publications, Inc. p. 35.
3348:
3139:The hexagonal tiling can be used as a
1618:
1594:It is similarly related to the uniform
1406:
1246:
450:
394:Hexagonal tiling is the densest way to
3422:, (3rd edition, 1973), Dover edition,
2164:Uniform hexagonal/triangular tilings
509:can be seen as a hexagon tiling on a
7:
2559:Monohedral convex hexagonal tilings
3911:{\displaystyle {\tilde {E}}_{n-1}}
3796:{\displaystyle {\tilde {D}}_{n-1}}
3753:{\displaystyle {\tilde {B}}_{n-1}}
3710:{\displaystyle {\tilde {C}}_{n-1}}
3667:{\displaystyle {\tilde {A}}_{n-1}}
3164:Related regular complex apeirogons
376:three regular tilings of the plane
49:it lacks sufficient corresponding
25:
3200:vertices, and vertex figures are
2850:Isohedrally-tiled quadrilaterals
4601:
4594:
3869:{\displaystyle {\tilde {F}}_{4}}
3833:{\displaystyle {\tilde {G}}_{2}}
3288:
3283:
3278:
3269:
3264:
3259:
3249:
3244:
3239:
3228:
3221:
3214:
3155:
3124:
3117:
3110:
3103:
3096:
3070:
3063:
3056:
3049:
3042:
2986:It can also be distorted into a
2972:
2965:
2958:
2929:
2920:
2909:
2898:
2889:
2878:
2869:
2836:
2829:
2822:
2815:
2808:
2801:
2794:
2768:
2761:
2754:
2747:
2740:
2733:
2701:Topologically equivalent tilings
2688:
2679:
2670:
2657:
2646:
2635:
2626:
2619:
2612:
2605:
2517:
2510:
2503:
2496:
2489:
2482:
2475:
2468:
2461:
2451:
2446:
2441:
2436:
2431:
2422:
2417:
2412:
2407:
2402:
2393:
2388:
2383:
2378:
2373:
2364:
2359:
2354:
2349:
2344:
2335:
2330:
2325:
2320:
2315:
2306:
2301:
2296:
2291:
2286:
2277:
2272:
2267:
2262:
2257:
2248:
2243:
2238:
2233:
2228:
2082:
2075:
2068:
2061:
2054:
2047:
2040:
1912:
1905:
1898:
1891:
1884:
1877:
1870:
1863:
1793:
1786:
1779:
1772:
1765:
1758:
1751:
1744:
1737:
1730:
1723:
1530:
1523:
1516:
1509:
1502:
1495:
1488:
1481:
1474:
1467:
1460:
1453:
1365:
1351:
1340:
1329:
1318:
1307:
1296:
1285:
1224:
1219:
1214:
1209:
1204:
1174:
1165:
1156:
1144:
1126:
1117:
1108:
1078:
1071:
1062:
1053:
1044:
1037:
1032:
1023:
972:
965:
958:
951:
944:
850:
845:
840:
831:
826:
821:
816:
811:
802:
797:
792:
787:
782:
684:
677:
670:
663:
656:
649:
642:
518:
498:
484:
469:
453:
244:
239:
234:
226:
221:
216:
211:
206:
198:
193:
188:
183:
178:
127:
100:
34:
3486:John H. Conway, Heidi Burgiel,
1102:tessellations in 3 dimensions.
3890:
3854:
3818:
3775:
3732:
3689:
3646:
2714:13 isohedrally-tiled hexagons
539:more regular of the two. Pure
1:
4827:Regular Division of the Plane
3311:Hexagonal prismatic honeycomb
1598:polyhedra with vertex figure
1100:rhombo-hexagonal dodecahedron
525:Hexagonal Persian tile c.1955
2956:
2945:
2942:Isohedrally-tiled pentagons
2867:
2853:
2792:
2775:
2731:
2717:
2668:
2633:
1106:
1021:
862:
774:
754:
734:
717:
691:
618:
337:meet at each vertex. It has
4735:Architectonic and catoptric
4633:Aperiodic set of prototiles
3459:Regular and uniform tilings
3445:. New York: W. H. Freeman.
3316:Tilings of regular polygons
2156:truncated triangular tiling
1230:, progressing to infinity.
1132:Fencing uses this relation
438:and electrical properties.
5612:
3437:; Shephard, G. C. (1987).
3331:Hexagonal tiling honeycomb
3170:regular complex apeirogons
1987:
1607:
1395:
1142:
920:chamfered hexagonal tiling
898:Chamfered hexagonal tiling
4874:
4860:
4721:
4710:
4623:
4610:
4592:
4519:
4506:
3584:
3326:List of regular polytopes
2857:
2854:
2776:
2727:
2718:
2678:
2645:
2582:
2173:
2168:
2163:
2008:
2000:
1995:
1671:
1665:
1655:
1645:
1447:
1441:
1433:
1279:
1235:
1006:
935:
859:
857:
771:
769:
751:
749:
730:
727:
724:
633:
630:
624:
614:
611:
608:
551:There are three distinct
333:, in which exactly three
307:
109:
99:
94:
3978:Uniform convex honeycomb
3492:The Symmetries of Things
1092:elongated rhombic tiling
936:Chamfered hexagons (cH)
407:, who believed that the
378:. The other two are the
3321:List of uniform tilings
536:hexagonal close packing
417:Weaire–Phelan structure
398:in two dimensions. The
64:more precise citations.
3912:
3870:
3834:
3797:
3754:
3711:
3668:
3557:"Uniform tessellation"
3538:"Regular tessellation"
3184:are constrained by: 1/
1672:Noncompact hyperbolic
1448:Noncompact hyperbolic
927:
426:, where each sheet of
359:English mathematician
323:hexagonal tessellation
5596:Regular tessellations
4352:Uniform 10-honeycomb
3913:
3871:
3835:
3798:
3755:
3712:
3669:
3488:Chaim Goodman-Strauss
917:
557:Wythoff constructions
141:V3.3.3.3.3.3 (or V3)
3880:
3844:
3808:
3765:
3722:
3679:
3636:
3441:Tilings and Patterns
3385:Tilings and patterns
3371:Tilings and patterns
3357:Tilings and patterns
1096:rhombic dodecahedron
419:is slightly better.
400:honeycomb conjecture
356:triangular tiling).
121:Vertex configuration
4312:Uniform 9-honeycomb
4245:Uniform 8-honeycomb
4183:Uniform 7-honeycomb
4128:Uniform 6-honeycomb
4079:Uniform 5-honeycomb
4027:Uniform 4-honeycomb
3611:Fundamental convex
3573:Klitzing, Richard.
2943:
2851:
2715:
2663:a = f, b = c, d = e
2576:
1280:Hyperbolic tilings
1084:E to IH to FH to H
922:degenerates into a
597: > 6.
579:, with a notation {
532:face-centered cubic
413:body-centered cubic
3908:
3866:
3830:
3793:
3750:
3707:
3664:
3617:uniform honeycombs
3554:Weisstein, Eric W.
3535:Weisstein, Eric W.
3516:Weisstein, Eric W.
2941:
2849:
2713:
2574:
1187:Symmetry mutations
928:
577:Goldberg polyhedra
173:Coxeter diagram(s)
147:Schläfli symbol(s)
137:Face configuration
5586:Isohedral tilings
5576:Hexagonal tilings
5571:Euclidean tilings
5558:
5557:
5554:
5553:
5550:
5549:
4856:
4855:
4747:Computer graphics
4706:
4705:
4590:
4589:
4467:
4466:
4069:24-cell honeycomb
3893:
3857:
3821:
3778:
3735:
3692:
3649:
3619:in dimensions 2–9
3500:978-1-56881-220-5
3461:, pp. 58–65)
3419:Regular Polytopes
3338:board game design
3306:Hexagonal lattice
3297:
3296:
3149:triangular tiling
3132:
3131:
2996:632 (p6) symmetry
2980:
2979:
2939:
2938:
2844:
2843:
2698:
2697:
2665:B = D = F = 120°
2654:B + C + E = 360°
2643:B + C + D = 360°
2569:glide reflections
2556:
2555:
2149:triangular tiling
2141:uniform polyhedra
2132:
2131:
1976:
1975:
1592:
1591:
1377:
1376:
1184:
1183:
1151:Pentagonal tiling
1136:
1135:
1123:Hexagonal tiling
1088:
1087:
1008:2-uniform tilings
993:triangular tiling
989:2-uniform tilings
980:
979:
888:
887:
553:uniform colorings
547:Uniform colorings
380:triangular tiling
311:
310:
294:Vertex-transitive
284:Triangular tiling
267:Rotation symmetry
159:Wythoff symbol(s)
95:Hexagonal tiling
90:
89:
82:
16:(Redirected from
5603:
5581:Isogonal tilings
4876:
4862:
4814:Conway criterion
4741:Circle Limit III
4712:
4645:Einstein problem
4612:
4605:
4598:
4534:Schwarz triangle
4508:
4493:
4486:
4479:
4470:
3917:
3915:
3914:
3909:
3907:
3906:
3895:
3894:
3886:
3875:
3873:
3872:
3867:
3865:
3864:
3859:
3858:
3850:
3839:
3837:
3836:
3831:
3829:
3828:
3823:
3822:
3814:
3802:
3800:
3799:
3794:
3792:
3791:
3780:
3779:
3771:
3759:
3757:
3756:
3751:
3749:
3748:
3737:
3736:
3728:
3716:
3714:
3713:
3708:
3706:
3705:
3694:
3693:
3685:
3673:
3671:
3670:
3665:
3663:
3662:
3651:
3650:
3642:
3605:
3598:
3591:
3582:
3578:
3567:
3566:
3548:
3547:
3529:
3528:
3519:"Hexagonal Grid"
3483:
3466:Williams, Robert
3456:
3444:
3435:Grünbaum, Branko
3407:
3404:
3398:
3395:
3389:
3381:
3375:
3367:
3361:
3353:
3293:
3292:
3291:
3287:
3286:
3282:
3281:
3274:
3273:
3272:
3268:
3267:
3263:
3262:
3254:
3253:
3252:
3248:
3247:
3243:
3242:
3232:
3225:
3218:
3211:
3196:= 1. Edges have
3159:
3128:
3121:
3114:
3107:
3100:
3074:
3067:
3060:
3053:
3046:
3005:
2976:
2969:
2962:
2944:
2933:
2924:
2913:
2902:
2893:
2882:
2873:
2852:
2840:
2833:
2826:
2819:
2812:
2805:
2798:
2772:
2765:
2758:
2751:
2744:
2737:
2716:
2692:
2683:
2674:
2661:
2650:
2639:
2630:
2623:
2616:
2609:
2577:
2521:
2514:
2507:
2500:
2493:
2486:
2479:
2472:
2465:
2456:
2455:
2454:
2450:
2449:
2445:
2444:
2440:
2439:
2435:
2434:
2427:
2426:
2425:
2421:
2420:
2416:
2415:
2411:
2410:
2406:
2405:
2398:
2397:
2396:
2392:
2391:
2387:
2386:
2382:
2381:
2377:
2376:
2369:
2368:
2367:
2363:
2362:
2358:
2357:
2353:
2352:
2348:
2347:
2340:
2339:
2338:
2334:
2333:
2329:
2328:
2324:
2323:
2319:
2318:
2311:
2310:
2309:
2305:
2304:
2300:
2299:
2295:
2294:
2290:
2289:
2282:
2281:
2280:
2276:
2275:
2271:
2270:
2266:
2265:
2261:
2260:
2253:
2252:
2251:
2247:
2246:
2242:
2241:
2237:
2236:
2232:
2231:
2161:
2143:there are eight
2086:
2079:
2072:
2065:
2058:
2051:
2044:
1985:
1916:
1909:
1902:
1895:
1888:
1881:
1874:
1867:
1797:
1790:
1783:
1776:
1769:
1762:
1755:
1748:
1741:
1734:
1727:
1637:
1630:
1623:
1605:
1534:
1527:
1520:
1513:
1506:
1499:
1492:
1485:
1478:
1471:
1464:
1457:
1442:Compact hyperb.
1425:
1418:
1411:
1393:
1388:hyperbolic plane
1369:
1355:
1344:
1333:
1322:
1311:
1300:
1289:
1265:
1258:
1251:
1233:
1229:
1228:
1227:
1223:
1222:
1218:
1217:
1213:
1212:
1208:
1207:
1178:
1169:
1160:
1148:
1141:
1130:
1121:
1112:
1105:
1082:
1075:
1068:fully dissected
1066:
1057:
1048:
1041:
1036:
1027:
998:
976:
969:
962:
955:
948:
930:
924:rhombille tiling
855:
854:
853:
849:
848:
844:
843:
836:
835:
834:
830:
829:
825:
824:
820:
819:
815:
814:
807:
806:
805:
801:
800:
796:
795:
791:
790:
786:
785:
688:
681:
674:
667:
660:
653:
646:
600:
522:
502:
488:
473:
457:
436:tensile strength
432:carbon nanotubes
409:Kelvin structure
351:
344:
319:hexagonal tiling
249:
248:
247:
243:
242:
238:
237:
231:
230:
229:
225:
224:
220:
219:
215:
214:
210:
209:
203:
202:
201:
197:
196:
192:
191:
187:
186:
182:
181:
131:
104:
92:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
5611:
5610:
5606:
5605:
5604:
5602:
5601:
5600:
5591:Regular tilings
5561:
5560:
5559:
5546:
5023:
5016:
4949:
4943:
4912:
4870:
4852:
4717:
4702:
4619:
4606:
4600:
4599:
4586:
4577:Wallpaper group
4515:
4502:
4497:
4462:
4455:
4448:
4440:
4439:
4428:
4427:
4416:
4415:
4404:
4381:
4380:
4373:
4372:
4365:
4364:
4357:
4342:
4341:
4334:
4333:
4326:
4325:
4318:
4302:
4295:
4288:
4281:
4280:
4272:
4271:
4262:
4261:
4252:
4235:
4228:
4220:
4219:
4210:
4209:
4200:
4199:
4190:
4173:
4165:
4164:
4155:
4154:
4145:
4144:
4135:
4116:
4115:
4106:
4105:
4096:
4095:
4086:
4064:
4063:
4054:
4053:
4044:
4043:
4034:
4015:
4014:
4005:
4004:
3995:
3994:
3985:
3963:
3962:
3953:
3952:
3943:
3942:
3933:
3883:
3878:
3877:
3847:
3842:
3841:
3811:
3806:
3805:
3768:
3763:
3762:
3725:
3720:
3719:
3682:
3677:
3676:
3639:
3634:
3633:
3620:
3609:
3572:
3552:
3551:
3533:
3532:
3514:
3513:
3510:
3480:
3464:
3453:
3433:
3415:Coxeter, H.S.M.
3411:
3410:
3405:
3401:
3396:
3392:
3382:
3378:
3368:
3364:
3354:
3350:
3345:
3302:
3289:
3284:
3279:
3277:
3270:
3265:
3260:
3258:
3250:
3245:
3240:
3238:
3166:
3137:
2934:
2925:
2914:
2903:
2894:
2883:
2874:
2703:
2694:3-tile lattice
2693:
2685:4-tile lattice
2684:
2676:2-tile lattice
2675:
2664:
2662:
2653:
2651:
2642:
2640:
2561:
2452:
2447:
2442:
2437:
2432:
2430:
2423:
2418:
2413:
2408:
2403:
2401:
2394:
2389:
2384:
2379:
2374:
2372:
2365:
2360:
2355:
2350:
2345:
2343:
2336:
2331:
2326:
2321:
2316:
2314:
2307:
2302:
2297:
2292:
2287:
2285:
2278:
2273:
2268:
2263:
2258:
2256:
2249:
2244:
2239:
2234:
2229:
2227:
2170:
2145:uniform tilings
2137:
1859:
1719:
1707:
1702:
1698:
1694:
1690:
1686:
1682:
1678:
1653:
1647:
1641:
1429:
1370:
1356:
1345:
1334:
1323:
1312:
1301:
1290:
1269:
1225:
1220:
1215:
1210:
1205:
1203:
1201:Coxeter diagram
1197:Schläfli symbol
1189:
1179:
1170:
1161:
1149:
1131:
1122:
1114:Rhombic tiling
1113:
1083:
1067:
1058:
1049:
1028:
1012:Regular tiling
1001:Regular tiling
985:
983:Related tilings
900:
851:
846:
841:
839:
832:
827:
822:
817:
812:
810:
803:
798:
793:
788:
783:
781:
592:
549:
526:
523:
514:
507:carbon nanotube
503:
494:
489:
480:
474:
465:
458:
396:arrange circles
392:
346:
342:
339:Schläfli symbol
331:Euclidean plane
302:face-transitive
298:edge-transitive
245:
240:
235:
233:
232:
227:
222:
217:
212:
207:
205:
204:
199:
194:
189:
184:
179:
177:
166:
164:
152:
126:
105:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
5609:
5607:
5599:
5598:
5593:
5588:
5583:
5578:
5573:
5563:
5562:
5556:
5555:
5552:
5551:
5548:
5547:
5545:
5544:
5539:
5534:
5529:
5524:
5519:
5514:
5509:
5504:
5499:
5494:
5489:
5484:
5479:
5474:
5469:
5464:
5459:
5454:
5449:
5444:
5439:
5434:
5429:
5424:
5419:
5414:
5409:
5404:
5399:
5394:
5389:
5384:
5379:
5374:
5369:
5364:
5359:
5354:
5349:
5344:
5339:
5334:
5329:
5324:
5319:
5314:
5309:
5304:
5299:
5294:
5289:
5284:
5279:
5274:
5269:
5264:
5259:
5254:
5249:
5244:
5239:
5234:
5229:
5224:
5219:
5214:
5209:
5204:
5199:
5194:
5189:
5184:
5179:
5174:
5169:
5164:
5159:
5154:
5149:
5144:
5139:
5134:
5129:
5124:
5119:
5114:
5109:
5104:
5099:
5094:
5089:
5084:
5079:
5074:
5069:
5064:
5059:
5054:
5049:
5044:
5039:
5034:
5028:
5026:
5018:
5017:
5015:
5014:
5009:
5004:
4999:
4994:
4989:
4984:
4979:
4974:
4969:
4964:
4959:
4953:
4951:
4945:
4944:
4942:
4941:
4936:
4931:
4926:
4920:
4918:
4914:
4913:
4911:
4910:
4905:
4900:
4895:
4890:
4884:
4882:
4872:
4871:
4865:
4858:
4857:
4854:
4853:
4851:
4850:
4845:
4840:
4835:
4830:
4823:
4822:
4821:
4816:
4806:
4805:
4804:
4799:
4794:
4789:
4788:
4787:
4774:
4769:
4764:
4759:
4754:
4749:
4744:
4737:
4732:
4722:
4719:
4718:
4715:
4708:
4707:
4704:
4703:
4701:
4700:
4695:
4690:
4689:
4688:
4674:
4669:
4664:
4659:
4654:
4653:
4652:
4650:Socolar–Taylor
4642:
4641:
4640:
4630:
4628:Ammann–Beenker
4624:
4621:
4620:
4615:
4608:
4607:
4593:
4591:
4588:
4587:
4585:
4584:
4579:
4574:
4573:
4572:
4567:
4562:
4551:Uniform tiling
4548:
4547:
4546:
4536:
4531:
4526:
4520:
4517:
4516:
4511:
4504:
4503:
4498:
4496:
4495:
4488:
4481:
4473:
4465:
4464:
4460:
4453:
4446:
4442:
4435:
4433:
4430:
4423:
4421:
4418:
4411:
4409:
4406:
4403:
4399:
4389:
4385:
4384:
4382:
4378:
4376:
4374:
4370:
4368:
4366:
4362:
4360:
4358:
4356:
4353:
4350:
4346:
4345:
4343:
4339:
4337:
4335:
4331:
4329:
4327:
4323:
4321:
4319:
4317:
4314:
4309:
4305:
4304:
4300:
4293:
4286:
4282:
4278:
4276:
4274:
4269:
4267:
4264:
4259:
4257:
4254:
4251:
4247:
4242:
4238:
4237:
4233:
4226:
4222:
4217:
4215:
4212:
4207:
4205:
4202:
4197:
4195:
4192:
4189:
4185:
4180:
4176:
4175:
4171:
4167:
4162:
4160:
4157:
4152:
4150:
4147:
4142:
4140:
4137:
4134:
4130:
4125:
4121:
4120:
4118:
4113:
4111:
4108:
4103:
4101:
4098:
4093:
4091:
4088:
4085:
4081:
4076:
4072:
4071:
4066:
4061:
4059:
4056:
4051:
4049:
4046:
4041:
4039:
4036:
4033:
4029:
4024:
4020:
4019:
4017:
4012:
4010:
4007:
4002:
4000:
3997:
3992:
3990:
3987:
3984:
3980:
3975:
3971:
3970:
3965:
3960:
3958:
3955:
3950:
3948:
3945:
3940:
3938:
3935:
3932:
3928:
3926:Uniform tiling
3923:
3919:
3918:
3905:
3902:
3899:
3892:
3889:
3863:
3856:
3853:
3827:
3820:
3817:
3803:
3790:
3787:
3784:
3777:
3774:
3760:
3747:
3744:
3741:
3734:
3731:
3717:
3704:
3701:
3698:
3691:
3688:
3674:
3661:
3658:
3655:
3648:
3645:
3631:
3626:
3622:
3621:
3610:
3608:
3607:
3600:
3593:
3585:
3580:
3579:
3570:
3569:
3568:
3549:
3509:
3508:External links
3506:
3505:
3504:
3484:
3478:
3462:
3457:(Chapter 2.1:
3451:
3431:
3409:
3408:
3399:
3390:
3376:
3362:
3347:
3346:
3344:
3341:
3340:
3339:
3333:
3328:
3323:
3318:
3313:
3308:
3301:
3298:
3295:
3294:
3275:
3255:
3234:
3233:
3226:
3219:
3165:
3162:
3161:
3160:
3145:kissing number
3141:circle packing
3136:
3135:Circle packing
3133:
3130:
3129:
3122:
3115:
3108:
3101:
3093:
3092:
3089:
3086:
3083:
3080:
3076:
3075:
3068:
3061:
3054:
3047:
3039:
3038:
3035:
3032:
3029:
3026:
3022:
3021:
3018:
3015:
3012:
3009:
2992:parallelograms
2978:
2977:
2970:
2963:
2955:
2954:
2951:
2948:
2937:
2936:
2927:
2918:
2907:
2896:
2895:Parallelogram
2887:
2876:
2875:Parallelogram
2866:
2865:
2862:
2859:
2856:
2842:
2841:
2834:
2827:
2820:
2813:
2806:
2799:
2791:
2790:
2787:
2784:
2781:
2778:
2774:
2773:
2766:
2759:
2752:
2745:
2738:
2730:
2729:
2726:
2723:
2720:
2702:
2699:
2696:
2695:
2686:
2677:
2667:
2666:
2655:
2644:
2632:
2631:
2624:
2617:
2610:
2602:
2601:
2598:
2595:
2592:
2588:
2587:
2584:
2581:
2560:
2557:
2554:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2523:
2522:
2515:
2508:
2501:
2494:
2487:
2480:
2473:
2466:
2458:
2457:
2428:
2399:
2370:
2341:
2312:
2283:
2254:
2224:
2223:
2218:
2213:
2208:
2203:
2198:
2193:
2188:
2182:
2181:
2178:
2172:
2166:
2165:
2136:
2133:
2130:
2129:
2124:
2119:
2114:
2109:
2104:
2099:
2094:
2088:
2087:
2080:
2073:
2066:
2059:
2052:
2045:
2038:
2034:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2011:
2010:
2007:
2004:
1999:
1993:
1992:
1974:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1950:
1945:
1940:
1935:
1930:
1924:
1923:
1921:
1919:
1917:
1910:
1903:
1896:
1889:
1882:
1875:
1868:
1861:
1855:
1854:
1851:
1848:
1845:
1840:
1835:
1830:
1825:
1820:
1815:
1810:
1805:
1799:
1798:
1791:
1784:
1777:
1770:
1763:
1756:
1749:
1742:
1735:
1728:
1721:
1715:
1714:
1712:
1710:
1708:
1704:
1699:
1695:
1691:
1687:
1683:
1679:
1674:
1673:
1670:
1667:
1664:
1659:
1654:
1643:
1642:
1640:
1639:
1632:
1625:
1617:
1590:
1589:
1586:
1583:
1580:
1577:
1572:
1567:
1562:
1557:
1552:
1547:
1542:
1536:
1535:
1528:
1521:
1514:
1507:
1500:
1493:
1486:
1479:
1472:
1465:
1458:
1450:
1449:
1446:
1443:
1440:
1435:
1431:
1430:
1428:
1427:
1420:
1413:
1405:
1375:
1374:
1363:
1360:
1349:
1338:
1327:
1316:
1305:
1294:
1282:
1281:
1278:
1275:
1271:
1270:
1268:
1267:
1260:
1253:
1245:
1188:
1185:
1182:
1181:
1172:
1163:
1154:
1134:
1133:
1124:
1115:
1086:
1085:
1076:
1069:
1060:
1059:2/3 dissected
1051:
1050:1/3 dissected
1042:
1030:
1020:
1019:
1016:
1013:
1010:
1005:
1002:
984:
981:
978:
977:
970:
963:
956:
949:
941:
940:
937:
934:
909:rhombic tiling
899:
896:
892:permutohedrons
886:
885:
883:
880:
878:
875:
873:
870:
867:
861:
860:
858:
856:
837:
808:
779:
773:
772:
770:
768:
765:
762:
759:
753:
752:
750:
748:
745:
742:
739:
733:
732:
729:
726:
723:
720:
716:
715:
712:
709:
706:
703:
700:
697:
694:
690:
689:
682:
675:
668:
661:
654:
647:
640:
636:
635:
632:
629:
626:
623:
617:
616:
613:
610:
607:
584:
548:
545:
528:
527:
524:
517:
515:
504:
497:
495:
490:
483:
481:
475:
468:
466:
462:circle packing
459:
452:
391:
388:
372:internal angle
327:regular tiling
309:
308:
305:
304:
291:
287:
286:
281:
275:
274:
268:
264:
263:
257:
251:
250:
175:
169:
168:
161:
155:
154:
149:
143:
142:
139:
133:
132:
123:
117:
116:
114:Regular tiling
111:
107:
106:
97:
96:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5608:
5597:
5594:
5592:
5589:
5587:
5584:
5582:
5579:
5577:
5574:
5572:
5569:
5568:
5566:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5510:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5383:
5380:
5378:
5375:
5373:
5370:
5368:
5365:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5233:
5230:
5228:
5225:
5223:
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5030:
5029:
5027:
5025:
5019:
5013:
5010:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4978:
4975:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4954:
4952:
4946:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4921:
4919:
4915:
4909:
4906:
4904:
4901:
4899:
4896:
4894:
4891:
4889:
4886:
4885:
4883:
4881:
4877:
4873:
4869:
4863:
4859:
4849:
4846:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4828:
4824:
4820:
4817:
4815:
4812:
4811:
4810:
4807:
4803:
4800:
4798:
4795:
4793:
4790:
4786:
4783:
4782:
4781:
4778:
4777:
4775:
4773:
4770:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4742:
4738:
4736:
4733:
4731:
4727:
4724:
4723:
4720:
4713:
4709:
4699:
4696:
4694:
4691:
4687:
4684:
4683:
4682:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4658:
4655:
4651:
4648:
4647:
4646:
4643:
4639:
4636:
4635:
4634:
4631:
4629:
4626:
4625:
4622:
4618:
4613:
4609:
4604:
4597:
4583:
4580:
4578:
4575:
4571:
4568:
4566:
4563:
4561:
4558:
4557:
4556:
4552:
4549:
4545:
4542:
4541:
4540:
4537:
4535:
4532:
4530:
4527:
4525:
4522:
4521:
4518:
4514:
4509:
4505:
4501:
4494:
4489:
4487:
4482:
4480:
4475:
4474:
4471:
4463:
4456:
4449:
4443:
4441:
4438:
4431:
4429:
4426:
4419:
4417:
4414:
4407:
4405:
4400:
4398:
4394:
4390:
4387:
4386:
4383:
4375:
4367:
4359:
4354:
4351:
4348:
4347:
4344:
4336:
4328:
4320:
4315:
4313:
4310:
4307:
4306:
4303:
4296:
4289:
4283:
4275:
4273:
4265:
4263:
4255:
4253:
4248:
4246:
4243:
4240:
4239:
4236:
4229:
4223:
4221:
4213:
4211:
4203:
4201:
4193:
4191:
4186:
4184:
4181:
4178:
4177:
4174:
4168:
4166:
4158:
4156:
4148:
4146:
4138:
4136:
4131:
4129:
4126:
4123:
4122:
4119:
4117:
4109:
4107:
4099:
4097:
4089:
4087:
4082:
4080:
4077:
4074:
4073:
4070:
4067:
4065:
4057:
4055:
4047:
4045:
4037:
4035:
4030:
4028:
4025:
4022:
4021:
4018:
4016:
4008:
4006:
3998:
3996:
3988:
3986:
3981:
3979:
3976:
3973:
3972:
3969:
3966:
3964:
3956:
3954:
3946:
3944:
3936:
3934:
3929:
3927:
3924:
3921:
3920:
3903:
3900:
3897:
3887:
3861:
3851:
3825:
3815:
3804:
3788:
3785:
3782:
3772:
3761:
3745:
3742:
3739:
3729:
3718:
3702:
3699:
3696:
3686:
3675:
3659:
3656:
3653:
3643:
3632:
3630:
3627:
3624:
3623:
3618:
3614:
3606:
3601:
3599:
3594:
3592:
3587:
3586:
3583:
3576:
3571:
3564:
3563:
3558:
3555:
3550:
3545:
3544:
3539:
3536:
3531:
3530:
3526:
3525:
3520:
3517:
3512:
3511:
3507:
3503:
3501:
3497:
3493:
3489:
3485:
3481:
3479:0-486-23729-X
3475:
3471:
3467:
3463:
3460:
3454:
3452:0-7167-1193-1
3448:
3443:
3442:
3436:
3432:
3429:
3428:0-486-61480-8
3425:
3421:
3420:
3416:
3413:
3412:
3403:
3400:
3394:
3391:
3387:
3386:
3380:
3377:
3373:
3372:
3366:
3363:
3359:
3358:
3352:
3349:
3342:
3337:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3312:
3309:
3307:
3304:
3303:
3299:
3276:
3256:
3236:
3235:
3231:
3227:
3224:
3220:
3217:
3213:
3212:
3209:
3205:
3203:
3199:
3195:
3191:
3187:
3183:
3179:
3175:
3171:
3163:
3158:
3154:
3153:
3152:
3150:
3146:
3142:
3134:
3127:
3123:
3120:
3116:
3113:
3109:
3106:
3102:
3099:
3095:
3094:
3090:
3087:
3084:
3081:
3079:p3m1, (*333)
3078:
3077:
3073:
3069:
3066:
3062:
3059:
3055:
3052:
3048:
3045:
3041:
3040:
3036:
3033:
3030:
3027:
3024:
3023:
3019:
3016:
3013:
3010:
3007:
3006:
3003:
3001:
2997:
2993:
2989:
2984:
2975:
2971:
2968:
2964:
2961:
2957:
2952:
2949:
2946:
2932:
2928:
2923:
2919:
2917:
2916:Parallelogram
2912:
2908:
2906:
2901:
2897:
2892:
2888:
2886:
2881:
2877:
2872:
2868:
2863:
2860:
2847:
2839:
2835:
2832:
2828:
2825:
2821:
2818:
2814:
2811:
2807:
2804:
2800:
2797:
2793:
2788:
2785:
2782:
2779:
2771:
2767:
2764:
2760:
2757:
2753:
2750:
2746:
2743:
2739:
2736:
2732:
2724:
2721:
2711:
2709:
2700:
2691:
2687:
2682:
2673:
2669:
2660:
2656:
2649:
2638:
2634:
2629:
2625:
2622:
2618:
2615:
2611:
2608:
2604:
2603:
2599:
2596:
2593:
2590:
2589:
2585:
2579:
2578:
2572:
2570:
2566:
2558:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2528:
2525:
2524:
2520:
2516:
2513:
2509:
2506:
2502:
2499:
2495:
2492:
2488:
2485:
2481:
2478:
2474:
2471:
2467:
2464:
2460:
2459:
2429:
2400:
2371:
2342:
2313:
2284:
2255:
2226:
2225:
2222:
2219:
2217:
2214:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2183:
2179:
2176:
2167:
2162:
2159:
2157:
2152:
2150:
2146:
2142:
2134:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2089:
2085:
2081:
2078:
2074:
2071:
2067:
2064:
2060:
2057:
2053:
2050:
2046:
2043:
2039:
2036:
2035:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2012:
2005:
2003:
1998:
1994:
1991:
1986:
1983:
1981:
1980:Coxeter group
1971:
1968:
1965:
1963:V∞.6.6
1962:
1959:
1956:
1954:
1951:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1929:
1926:
1925:
1922:
1920:
1918:
1915:
1911:
1908:
1904:
1901:
1897:
1894:
1890:
1887:
1883:
1880:
1876:
1873:
1869:
1866:
1862:
1857:
1856:
1852:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1800:
1796:
1792:
1789:
1785:
1782:
1778:
1775:
1771:
1768:
1764:
1761:
1757:
1754:
1750:
1747:
1743:
1740:
1736:
1733:
1729:
1726:
1722:
1717:
1716:
1713:
1711:
1709:
1705:
1700:
1696:
1692:
1688:
1684:
1680:
1676:
1675:
1668:
1663:
1660:
1658:
1651:
1644:
1638:
1633:
1631:
1626:
1624:
1619:
1615:
1611:
1606:
1603:
1601:
1597:
1587:
1584:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1537:
1533:
1529:
1526:
1522:
1519:
1515:
1512:
1508:
1505:
1501:
1498:
1494:
1491:
1487:
1484:
1480:
1477:
1473:
1470:
1466:
1463:
1459:
1456:
1452:
1451:
1444:
1439:
1436:
1432:
1426:
1421:
1419:
1414:
1412:
1407:
1403:
1399:
1394:
1391:
1389:
1385:
1382:
1381:vertex figure
1373:
1368:
1364:
1361:
1359:
1354:
1350:
1348:
1343:
1339:
1337:
1332:
1328:
1326:
1321:
1317:
1315:
1310:
1306:
1304:
1299:
1295:
1293:
1288:
1284:
1283:
1276:
1273:
1272:
1266:
1261:
1259:
1254:
1252:
1247:
1243:
1239:
1234:
1231:
1202:
1198:
1194:
1186:
1177:
1173:
1168:
1164:
1159:
1155:
1152:
1147:
1143:
1139:
1129:
1125:
1120:
1116:
1111:
1107:
1103:
1101:
1097:
1093:
1081:
1077:
1074:
1070:
1065:
1061:
1056:
1052:
1047:
1043:
1040:
1035:
1031:
1026:
1022:
1018:Dual Tilings
1017:
1014:
1011:
1009:
1003:
1000:
999:
996:
994:
990:
982:
975:
971:
968:
964:
961:
957:
954:
950:
947:
943:
942:
939:Rhombi (daH)
938:
933:Hexagons (H)
932:
931:
925:
921:
916:
912:
910:
905:
897:
895:
893:
884:
881:
879:
876:
874:
871:
868:
866:
863:
838:
809:
780:
778:
775:
767:3 3 3 |
766:
764:2 6 | 3
763:
761:3 | 6 2
760:
758:
755:
746:
743:
740:
738:
735:
721:
718:
713:
710:
707:
704:
701:
698:
695:
692:
687:
683:
680:
676:
673:
669:
666:
662:
659:
655:
652:
648:
645:
641:
638:
637:
628:p3m1, (*333)
627:
622:
619:
605:
602:
601:
598:
596:
591:
587:
582:
578:
574:
570:
566:
562:
558:
554:
546:
544:
542:
537:
533:
521:
516:
512:
508:
501:
496:
493:
487:
482:
478:
472:
467:
463:
456:
451:
449:
447:
443:
441:
437:
433:
429:
425:
420:
418:
414:
410:
406:
401:
397:
389:
387:
385:
384:square tiling
381:
377:
373:
368:
366:
362:
357:
355:
349:
340:
336:
332:
328:
324:
320:
316:
306:
303:
299:
295:
292:
289:
288:
285:
282:
280:
277:
276:
272:
269:
266:
265:
261:
258:
256:
253:
252:
176:
174:
171:
170:
167:3 3 3 |
162:
160:
157:
156:
150:
148:
145:
144:
140:
138:
135:
134:
130:
124:
122:
119:
118:
115:
112:
108:
103:
98:
93:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
4938:
4838:Substitution
4833:Regular grid
4825:
4739:
4672:Quaquaversal
4570:Kisrhombille
4500:Tessellation
4436:
4424:
4412:
4392:
3967:
3560:
3541:
3522:
3491:
3469:
3458:
3440:
3417:
3402:
3393:
3383:
3379:
3369:
3365:
3355:
3351:
3206:
3201:
3197:
3193:
3189:
3185:
3181:
3177:
3173:
3168:There are 2
3167:
3138:
3025:p6m, (*632)
2985:
2981:
2845:
2708:parallelogon
2704:
2652:b = e, d = f
2562:
2200:
2185:
2155:
2153:
2138:
2127:V(3.∞)
1989:
1977:
1827:
1649:
1613:
1609:
1599:
1593:
1559:
1401:
1397:
1383:
1378:
1302:
1241:
1237:
1190:
1137:
1091:
1089:
986:
926:at the limit
919:
901:
889:
631:p6m, (*632)
625:p6m, (*632)
603:
594:
589:
585:
580:
572:
568:
564:
560:
550:
529:
477:Chicken wire
460:The densest
446:Chicken wire
444:
442:is similar.
421:
393:
390:Applications
369:
364:
363:called it a
358:
347:
322:
318:
312:
165:2 6 | 3
163:3 | 6 2
125:6.6.6 (or 6)
76:
67:
48:
4868:vertex type
4726:Anisohedral
4681:Self-tiling
4524:Pythagorean
3085:p6m (*632)
3031:p6m (*632)
2789:p6m (*632)
2710:hexagons.
2177:: , (*632)
2169:Fundamental
2032:*∞32
2009:Hyperbolic
1843:∞.6.6
1199:{6,n}, and
1004:Dissection
571:first, and
511:cylindrical
405:Lord Kelvin
361:John Conway
262:, , (*632)
62:introducing
5565:Categories
4772:Pentagonal
3343:References
3237:2{12}3 or
3088:p2 (2222)
3082:p3, (333)
3028:p6, (632)
2935:Rectangle
2926:Rectangle
2864:p2 (2222)
2861:cmm (2*22)
2786:cmm (2*22)
2780:p31m (3*3)
2728:pmg (22*)
2552:3.3.3.3.6
2006:Euclidean
1706:*∞32
1434:Spherical
1277:Euclidean
1274:Spherical
991:, and the
634:p6, (632)
615:3-uniform
612:2-uniform
609:1-uniform
290:Properties
273:, , (632)
70:March 2011
45:references
4880:Spherical
4848:Voderberg
4809:Prototile
4776:Problems
4752:Honeycomb
4730:Isohedral
4617:Aperiodic
4555:honeycomb
4539:Rectangle
4529:Rhombille
4397:honeycomb
4391:Uniform (
3968:Hexagonal
3901:−
3891:~
3855:~
3819:~
3786:−
3776:~
3743:−
3733:~
3700:−
3690:~
3657:−
3647:~
3562:MathWorld
3543:MathWorld
3524:MathWorld
3257:6{4}3 or
3034:p6 (632)
2953:p3 (333)
2950:pgg (22×)
2947:p2 (2222)
2905:Rectangle
2885:Trapezoid
2858:pgg (22×)
2855:pmg (22*)
2783:p2 (2222)
2777:pgg (22×)
2722:p2 (2222)
2565:isohedral
2139:Like the
2002:Spherical
1966:V12i.6.6
1718:Truncated
1657:Spherical
1596:truncated
1438:Euclidean
1193:hexagonal
1029:Original
904:chamfered
882:wH=t6dsH
877:cH=t6daH
606:-uniform
354:truncated
4962:V3.4.3.4
4797:Squaring
4792:Heesch's
4757:Isotoxal
4677:Rep-tile
4667:Pinwheel
4560:Coloring
4513:Periodic
3468:(1979).
3300:See also
3204:-gonal.
3020:Chevron
3014:Regular
3011:Gyrated
3008:Regular
2725:p3 (333)
2600:p3, 333
2597:p2, 2222
2594:pgg, 22×
2591:p2, 2222
2546:3.4.6.4
2534:3.12.12
2180:, (632)
2175:Symmetry
2171:domains
2029:*832...
1972:V6i.6.6
1969:V9i.6.6
1860:figures
1847:12i.6.6
1720:figures
1666:Compact
1579:{12i,3}
1445:Paraco.
1098:and the
737:Schläfli
639:Picture
621:Symmetry
492:Graphene
440:Silicene
428:graphene
424:graphite
382:and the
365:hextille
335:hexagons
315:geometry
255:Symmetry
18:Hextille
5422:6.4.8.4
5377:5.4.6.4
5337:4.12.16
5327:4.10.12
5297:V4.8.10
5272:V4.6.16
5262:V4.6.14
5162:3.6.4.6
5157:3.4.∞.4
5152:3.4.8.4
5147:3.4.7.4
5142:3.4.6.4
5092:3.∞.3.∞
5087:3.4.3.4
5082:3.8.3.8
5077:3.7.3.7
5072:3.6.3.8
5067:3.6.3.6
5062:3.5.3.6
5057:3.5.3.5
5052:3.4.3.∞
5047:3.4.3.8
5042:3.4.3.7
5037:3.4.3.6
5032:3.4.3.5
4987:3.4.6.4
4957:3.4.3.4
4950:regular
4917:Regular
4843:Voronoi
4767:Packing
4698:Truchet
4693:Socolar
4662:Penrose
4657:Gilbert
4582:Wythoff
4434:qδ
4422:hδ
4377:qδ
4369:hδ
4338:qδ
4330:hδ
4277:qδ
4268:hδ
4216:qδ
4206:hδ
4161:qδ
4151:hδ
4112:qδ
4102:hδ
4060:qδ
4050:hδ
4011:qδ
4001:hδ
3959:qδ
3949:hδ
3613:regular
3336:Hex map
3091:p1 (°)
3037:p1 (°)
3017:Weaved
3000:chevron
2719:pg (××)
2549:4.6.12
2527:Config.
2221:sr{6,3}
2216:tr{6,3}
2211:rr{6,3}
2037:Tiling
1960:V8.6.6
1957:V7.6.6
1928:Config.
1853:6i.6.6
1850:9i.6.6
1803:Config.
1669:Parac.
1662:Euclid.
1588:{3i,3}
1585:{6i,3}
1582:{9i,3}
777:Coxeter
757:Wythoff
744:t{3,6}
693:Colors
559:. The (
513:surface
479:fencing
329:of the
153:t{3,6}
58:improve
5312:4.8.16
5307:4.8.14
5302:4.8.12
5292:4.8.10
5267:4.6.16
5257:4.6.14
5252:4.6.12
5022:Hyper-
5007:4.6.12
4780:Domino
4686:Sphinx
4565:Convex
4544:Domino
4410:δ
4361:δ
4322:δ
4258:δ
4196:δ
4141:δ
4092:δ
4040:δ
3991:δ
3939:δ
3629:Family
3625:Space
3498:
3494:2008,
3476:
3449:
3426:
2988:chiral
2540:6.6.6
2537:(6.3)
2201:t{3,6}
2196:r{6,3}
2191:t{6,3}
2122:V(3.8)
2117:V(3.7)
2112:V(3.6)
2107:V(3.5)
2102:V(3.4)
2097:V(3.3)
1990:V(3.n)
1953:V6.6.6
1948:V5.6.6
1943:V4.6.6
1938:V3.6.6
1933:V2.6.6
1602:.6.6.
1015:Inset
865:Conway
741:{6,3}
731:(2,1)
728:(2,0)
725:(1,1)
722:(1,0)
719:(h,k)
541:copper
352:(as a
350:{3,6}
343:{6,3}
317:, the
47:, but
5427:(6.8)
5382:(5.6)
5317:4.8.∞
5287:(4.8)
5282:(4.7)
5277:4.6.∞
5247:(4.6)
5242:(4.5)
5212:4.∞.4
5207:4.8.4
5202:4.7.4
5197:4.6.4
5192:4.5.4
5172:(3.8)
5167:(3.7)
5137:(3.4)
5132:(3.4)
5024:bolic
4992:(3.6)
4948:Semi-
4819:Girih
4716:Other
2641:b = e
2206:{3,6}
2186:{6,3}
2092:Conf.
2026:*732
2023:*632
2020:*532
2017:*432
2014:*332
1858:n-kis
1838:8.6.6
1833:7.6.6
1828:6.6.6
1823:5.6.6
1818:4.6.6
1813:3.6.6
1808:2.6.6
1616:.6.6
1575:{∞,3}
1570:{8,3}
1565:{7,3}
1560:{6,3}
1555:{5,3}
1550:{4,3}
1545:{3,3}
1540:{2,3}
1372:{6,∞}
1358:{6,8}
1347:{6,7}
1336:{6,6}
1325:{6,5}
1314:{6,4}
1303:{6,3}
1292:{6,2}
747:t{3}
325:is a
151:{6,3}
5512:8.16
5507:8.12
5477:7.14
5447:6.16
5442:6.12
5437:6.10
5397:5.12
5392:5.10
5347:4.16
5342:4.14
5332:4.12
5322:4.10
5182:3.16
5177:3.14
4997:3.12
4982:V3.6
4908:V4.n
4898:V3.n
4785:Wang
4762:List
4728:and
4679:and
4638:List
4553:and
4395:-1)-
3615:and
3496:ISBN
3474:ISBN
3447:ISBN
3424:ISBN
3192:+ 1/
3188:+ 2/
2998:. A
1997:*n32
1703:...
1701:*832
1697:*732
1693:*632
1689:*532
1685:*432
1681:*332
1677:*232
1646:Sym.
1404:,3}
1362:...
918:The
583:+,3}
534:and
411:(or
370:The
279:Dual
110:Type
5542:∞.8
5537:∞.6
5502:8.6
5472:7.8
5467:7.6
5432:6.8
5387:5.8
5352:4.∞
5187:3.∞
5112:3.4
5107:3.∞
5102:3.8
5097:3.7
5012:4.8
5002:4.∞
4977:3.6
4972:3.∞
4967:3.4
4903:4.n
4893:3.n
4866:By
2151:).
872:tΔ
345:or
341:of
321:or
313:In
260:p6m
5567::
4461:21
4457:•
4454:k1
4450:•
4447:k2
4388:E
4379:11
4371:11
4363:11
4349:E
4340:10
4332:10
4324:10
4308:E
4301:21
4297:•
4294:51
4290:•
4287:52
4241:E
4234:31
4230:•
4227:33
4179:E
4172:22
4124:E
4075:E
4023:E
3974:E
3922:E
3876:/
3840:/
3559:.
3540:.
3521:.
3490:,
2586:3
2543:3
2531:6
1652:42
1390:.
1244:}
995::
911:.
902:A
894:.
869:H
714:7
711:2
708:4
705:2
702:3
699:2
696:1
505:A
386:.
367:.
300:,
296:,
271:p6
5532:∞
5527:∞
5522:∞
5517:∞
5497:8
5492:8
5487:8
5482:8
5462:7
5457:7
5452:7
5417:6
5412:6
5407:6
5402:6
5372:5
5367:5
5362:5
5357:5
5237:4
5232:4
5227:4
5222:4
5217:4
5127:3
5122:3
5117:3
4939:6
4934:4
4929:3
4924:2
4888:2
4492:e
4485:t
4478:v
4459:k
4452:2
4445:1
4437:n
4425:n
4413:n
4402:0
4393:n
4355:0
4316:0
4299:5
4292:2
4285:1
4279:9
4270:9
4260:9
4250:0
4232:3
4225:1
4218:8
4208:8
4198:8
4188:0
4170:2
4163:7
4153:7
4143:7
4133:0
4114:6
4104:6
4094:6
4084:0
4062:5
4052:5
4042:5
4032:0
4013:4
4003:4
3993:4
3983:0
3961:3
3951:3
3941:3
3931:0
3904:1
3898:n
3888:E
3862:4
3852:F
3826:2
3816:G
3789:1
3783:n
3773:D
3746:1
3740:n
3730:B
3703:1
3697:n
3687:C
3660:1
3654:n
3644:A
3604:e
3597:t
3590:v
3577:.
3565:.
3546:.
3527:.
3482:.
3455:.
3202:r
3198:p
3194:r
3190:q
3186:p
3182:r
3180:}
3178:q
3176:{
3174:p
2583:2
2580:1
1650:n
1648:*
1636:e
1629:t
1622:v
1614:n
1610:n
1608:*
1600:n
1424:e
1417:t
1410:v
1402:n
1398:n
1396:*
1384:n
1264:e
1257:t
1250:v
1242:n
1238:n
1236:*
604:k
595:p
590:k
588:,
586:h
581:p
573:k
569:h
565:k
563:,
561:h
348:t
83:)
77:(
72:)
68:(
54:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
