3127:
3085:
3118:
782:
2914:
814:
805:
796:
4301:
837:
828:
932:
923:
882:
873:
864:
855:
846:
896:
968:
950:
941:
914:
905:
977:
959:
3058:
3076:
3067:
2887:
2905:
2896:
3211:
90:
3222:
3240:
3202:
3231:
252:
645:
606:
684:
3044:
2873:
526:
448:
409:
487:
306:
567:
3193:
279:
146:
vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 3; 3 (both of different transitivity class), or (3), tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 3.6, 4 more contiguous equilateral triangles and a single regular hexagon.
2864:
3026:
2855:
3035:
2757:
2734:
2638:
2548:
2528:
2505:
2480:
2458:
2389:
2281:
1137:
677:
229:
3017:
2846:
2711:
2688:
2593:
2568:
2367:
2347:
2325:
2303:
1128:
638:
599:
560:
519:
480:
441:
402:
243:
236:
70:
51:
38:
2411:
2663:
2793:-uniform tilings have been enumerated up to 6. There are 673 6-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 10 6-uniform tilings with 6 distinct vertex types, as well as finding 92 of them with 5 vertex types, 187 of them with 4 vertex types, 284 of them with 3 vertex types, and 100 with 2 vertex types.
4308:
149:
However, this notation has two main problems related to ambiguous conformation and uniqueness First, when it comes to k-uniform tilings, the notation does not explain the relationships between the vertices. This makes it impossible to generate a covered plane given the notation alone. And second,
3149:
each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles. Two of the families are generated from shifted square, either progressive or zig-zagging positions. Grünbaum and
Shephard call these tilings
2816:
tiling has a square lattice, the 4(3-1)-uniform tiling has a snub square lattice, and the 5(3-1-1)-uniform tiling has an elongated triangular lattice. These higher-order uniform tilings use the same lattice but possess greater complexity. The fractalizing basis for theses tilings is as follows:
145:
Euclidean tilings are usually named after Cundy & Rollett’s notation. This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 3; 3; 3.6, tells us there are 3
153:
In order to solve those problems, GomJau-Hogg’s notation is a slightly modified version of the research and notation presented in 2012, about the generation and nomenclature of tessellations and double-layer grids. Antwerp v3.0, a free online application, allows for the infinite generation of
1040:
1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the number
768:(3.3.4.12, 3.4.3.12, 3.3.6.6, 3.4.4.6), while six can not be used to completely tile the plane by regular polygons with no gaps or overlaps - they only tessellate space entirely when irregular polygons are included (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, 5.5.10).
150:
some tessellations have the same nomenclature, they are very similar but it can be noticed that the relative positions of the hexagons are different. Therefore, the second problem is that this nomenclature is not unique for each tessellation.
3154:
although it contradicts
Coxeter's definition for uniformity which requires edge-to-edge regular polygons. Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions.
364:
If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as
734:
as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are
Archimedean tilings which are not uniform.
186:, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an
3142:
Convex regular polygons can also form plane tilings that are not edge-to-edge. Such tilings can be considered edge-to-edge as nonregular polygons with adjacent colinear edges.
2985:
2952:
154:
regular polygon tilings through a set of shape placement stages and iterative rotation and reflection operations, obtained directly from the GomJau-Hogg’s notation.
5241:
4506:
3660:
4439:
3775:
386:) forms of 3.6 (snub hexagonal) tiling, only one of which is shown in the following table. All other regular and semiregular tilings are achiral.
207:
5246:
4461:
4195:
3962:
5056:
4891:
3126:
5206:
5181:
5171:
5141:
5096:
5046:
5026:
4841:
4726:
3718:
3314:
5216:
5211:
5151:
5146:
5101:
5051:
5036:
4059:
5275:
5236:
5021:
4269:
3900:
5076:
5011:
4996:
4831:
4451:
5176:
5136:
5091:
5031:
5016:
5006:
4981:
4342:
3609:
5041:
4961:
4816:
4083:
349:
3084:
3117:
4971:
4956:
4916:
4846:
4796:
4711:
4531:
2393:
781:
492:
4941:
4906:
4896:
4756:
4300:
2913:
813:
804:
795:
5081:
4911:
4901:
4881:
4861:
4836:
4781:
4761:
4746:
4736:
4671:
4337:
836:
827:
650:
3910:
Ren, Ding; Reay, John R. (1987). "The boundary characteristic and Pick's theorem in the
Archimedean planar tilings".
2262:) Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.
931:
922:
881:
872:
863:
854:
845:
5231:
5226:
5221:
5126:
4886:
4851:
4811:
4791:
4766:
4751:
4741:
4701:
4188:
3284:
2329:
2307:
996:
967:
949:
940:
913:
904:
895:
730:
as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as
414:
175:
4332:
976:
958:
5166:
5161:
5071:
5066:
5061:
4856:
4826:
4821:
4801:
4786:
4776:
4771:
4691:
2572:
2285:
453:
5280:
5201:
5196:
5191:
5121:
5116:
5111:
5106:
4806:
4686:
4681:
4354:
3057:
761:, (3.12.12, 4.6.12, 4.8.8, (3.6), 3.4.6.4, 3.3.4.3.4, 3.3.3.4.4, 3.3.3.3.6). Four of them can exist in higher
2957:
This can similarly be done with the truncated trihexagonal tiling as a basis, with corresponding dilation of
4866:
4716:
4666:
3319:
3279:
3075:
572:
383:
3579:
3066:
4986:
4976:
4946:
4628:
4243:
3369:
3349:
2886:
5285:
5086:
4991:
4951:
4936:
4931:
4926:
4921:
4676:
4466:
4181:
2904:
689:
191:
4021:
3221:
2895:
3210:
89:
5131:
4871:
4584:
4572:
4456:
4385:
4361:
4286:
3705:
3566:
3554:
3467:
3414:
3309:
616:
195:
60:
497:
4876:
4696:
4542:
4501:
4496:
4376:
3359:
3294:
2960:
2927:
531:
187:
105:
4661:
4430:
4228:
3933:
3868:
3804:
3758:
3339:
3215:
2667:
2642:
2258:
1074:), as different types of vertices necessarily have different orbits, but not vice versa. Setting
694:
611:
536:
458:
358:
3880:
3201:
419:
163:
117:
have been widely used since antiquity. The first systematic mathematical treatment was that of
3239:
655:
577:
5156:
4706:
4633:
4476:
4259:
4156:
4137:
4118:
4055:
3958:
3896:
3539:
3456:"GomJau-Hogg's Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0"
3353:
743:
354:
257:
199:
183:
4159:
4140:
3725:
750:, one has 6 polygons, three have 5 polygons, seven have 4 polygons, and ten have 3 polygons.
316:
289:
262:
5186:
5001:
4966:
4643:
4607:
4552:
4518:
4471:
4445:
4434:
4349:
4321:
4264:
4238:
4233:
4033:
3991:
3946:
3919:
3860:
3829:
3796:
3485:
3475:
3434:
3426:
3289:
3269:
2484:
2462:
284:
124:
3843:
3430:
3230:
17:
4547:
4371:
4281:
3839:
3682:
3613:
3374:
3334:
3160:
3146:
2814:
2415:
2371:
2251:
1066:. In general, the uniformity is greater than or equal to the number of types of vertices (
1034:
118:
113:
3982:
Préa, P. (1997). "Distance sequences and percolation thresholds in
Archimedean Tilings".
3471:
251:
4484:
4397:
4366:
4255:
3364:
3343:
3324:
370:
171:
130:
3995:
3834:
3817:
3043:
2872:
683:
644:
605:
5269:
4638:
4602:
4402:
4390:
4248:
3950:
3923:
3884:
3872:
3274:
747:
311:
4103:
3762:
3603:
3192:
4537:
4274:
4204:
4047:
3749:
Chavey, Darrah (2014). "TILINGS BY REGULAR POLYGONS III: DODECAGON-DENSE TILINGS".
3329:
525:
447:
408:
379:
305:
109:
4121:
4087:
486:
1033:-uniform tilings with the same vertex figures can be further identified by their
4523:
3650:
566:
278:
3787:
Grünbaum, Branko; Shephard, Geoffrey C. (1977). "Tilings by regular polygons".
4592:
3889:
746:. Polygons in these meet at a point with no gap or overlap. Listing by their
4612:
4597:
4513:
4489:
4164:
4145:
4126:
4075:
3626:
3034:
2863:
3025:
2854:
2756:
2733:
2662:
2637:
2547:
2527:
2504:
2479:
2457:
2388:
2280:
1136:
1052:
Finally, if the number of types of vertices is the same as the uniformity (
742:
There are 17 combinations of regular convex polygons that form 21 types of
676:
228:
3016:
2845:
2710:
2687:
2592:
2567:
2410:
2366:
2346:
2324:
2302:
1127:
637:
598:
559:
518:
479:
440:
401:
242:
235:
182:
of the tiling, where a flag is a triple consisting of a mutually incident
69:
50:
37:
4381:
3851:
Debroey, I.; Landuyt, F. (1981). "Equitransitive edge-to-edge tilings".
3480:
3455:
3864:
3808:
3490:
203:
4307:
3439:
3125:
3116:
3083:
3074:
3065:
3056:
3042:
3033:
3024:
3015:
2912:
2903:
2894:
2885:
2871:
2862:
2853:
2844:
726:
Grünbaum and
Shephard distinguish the description of these tilings as
3415:"Generation and Nomenclature of Tessellations and Double-Layer Grids"
3800:
1141:
by 4-isohedral positions, 3 shaded colors of triangles (by orbits)
4038:
56:
1119:
3413:
Gomez-Jauregui, Valentin al.; Otero, Cesar; et al. (2012).
758:
4569:
4419:
4319:
4215:
4177:
4173:
3454:
Gomez-Jauregui, Valentin; Hogg, Harrison; et al. (2021).
4004:
3957:
Order in Space: A design source book, Keith
Critchlow, 1970
3934:"Tilings by Regular Polygons—II: A Catalog of Tilings"
3655:"Sequence A068599 (Number of n-uniform tilings.)"
3654:
3506:
1062:
85:
types of vertices, and two or more types of regular faces.
27:
Subdivision of the plane into polygons that are all regular
4005:"Symmetry-type graphs of Platonic and Archimedean solids"
2813:-uniform tilings. For example, notice that the 2-uniform
995:
Such periodic tilings may be classified by the number of
3818:"The ninety-one types of isogonal tilings in the plane"
1132:
by sides, yellow triangles, red squares (by polygons)
95:
3408:
3406:
2963:
2930:
1146:
3163:
tilings by non-edge-to-edge convex regular polygons
3157:
4725:
4652:
4621:
4583:
4022:"Minimal Covers of the Archimedean Tilings, Part 1"
166:and Shephard (section 1.3), a tiling is said to be
3932:
3888:
2979:
2946:
1045:of distinct vertex figures, which are also called
357:means that for every pair of vertices there is a
3683:"Enumeration of n-uniform k-Archimedean tilings"
3518:
3516:
4189:
3939:Computers & Mathematics with Applications
3541:The Elements of Plane Practical Geometry, Etc
8:
3527:. London: Thames and Hudson. pp. 60–61.
2924:The side lengths are dilated by a factor of
999:of vertices, edges and tiles. If there are
4020:Pellicer, Daniel; Williams, Gordon (2012).
3968:Sommerville, Duncan MacLaren Young (1958).
344:Archimedean, uniform or semiregular tilings
122:
4580:
4566:
4416:
4316:
4212:
4196:
4182:
4174:
3816:Grünbaum, Branko; Shephard, G. C. (1978).
3505:Hogg, Harrison; Gomez-Jauregui, Valentin.
3206:Rows of triangles with horizontal offsets
1117:Below is an example of a 3-unifom tiling:
29:
4507:Dividing a square into similar rectangles
4037:
3833:
3661:On-Line Encyclopedia of Integer Sequences
3489:
3479:
3438:
2970:
2962:
2937:
2929:
1124:
1003:orbits of vertices, a tiling is known as
556:
398:
3399:. Stradbroke (UK): Tarquin Publications.
3197:Rows of squares with horizontal offsets
3099:
2990:
2819:
2264:
770:
388:
361:mapping the first vertex to the second.
212:
98:can have different-sized regular faces.
4026:The Electronic Journal of Combinatorics
3708:, Grünbaum and Shephard 1986, pp. 65-67
3544:, John W. Parker & Son, p. 134
3387:
3235:Six triangles surround every hexagon.
3226:Three hexagons surround each triangle
2805:There are many ways of generating new
1521:
1458:
1397:
1338:
1281:
1060:below), then the tiling is said to be
762:
718:C&R: Cundy & Rollet's notation
335:C&R: Cundy & Rollet's notation
75:
2248:of the Euclidean plane. (also called
1159:
194:regular polygons. There must be six
7:
4071:Euclidean and general tiling links:
3607:-uniform tilings by regular polygons
3525:Order in Space: A Design Source Book
754:
43:
3970:An Introduction to the Geometry of
3687:zenorogue.github.io/tes-catalog/?c=
3395:Cundy, H.M.; Rollett, A.P. (1981).
3315:Uniform tilings in hyperbolic plane
1121:Colored 3-uniform tiling #57 of 61
757:(6, 4, 3), and eight more can make
3719:"In Search of Demiregular Tilings"
3431:10.1061/(ASCE)ST.1943-541X.0000532
759:semiregular or archimedean tilings
378:tilings. Note that there are two
63:, but two or more types of faces.
25:
3835:10.1090/S0002-9947-1978-0496813-3
3419:Journal of Structural Engineering
3138:Tilings that are not edge-to-edge
4306:
4299:
3978:Chapter X: The Regular Polytopes
3538:Dallas, Elmslie William (1855),
3238:
3229:
3220:
3209:
3200:
3191:
2755:
2732:
2709:
2686:
2661:
2636:
2591:
2566:
2546:
2526:
2503:
2478:
2456:
2409:
2387:
2365:
2345:
2323:
2301:
2279:
1135:
1126:
1086:, there are 11 such tilings for
975:
966:
957:
948:
939:
930:
921:
912:
903:
894:
880:
871:
862:
853:
844:
835:
826:
812:
803:
794:
780:
682:
675:
643:
636:
604:
597:
565:
558:
524:
517:
485:
478:
446:
439:
407:
400:
304:
277:
250:
241:
234:
227:
88:
68:
49:
36:
2673:4-3,3,3-4,3/r(c2)/r(h13)/r(h45)
3108:Truncated Trihexagonal Tiling
350:List of convex uniform tilings
46:has one type of regular face.
1:
4532:Regular Division of the Plane
4052:Introduction to Tessellations
3996:10.1016/S0895-7177(97)00216-1
3895:. W. H. Freeman and Company.
3584:American Mathematical Society
3011:
3010:
2980:{\displaystyle 3+{\sqrt {3}}}
2947:{\displaystyle 2+{\sqrt {3}}}
2840:
2839:
974:
965:
956:
947:
938:
929:
920:
911:
902:
893:
879:
870:
861:
852:
843:
834:
825:
811:
802:
793:
779:
722:GJ-H: Notation of GomJau-Hogg
339:GJ-H: Notation of GomJau-Hogg
141:Notation of Euclidean tilings
57:semiregular or uniform tiling
3951:10.1016/0898-1221(89)90156-9
3924:10.1016/0097-3165(87)90063-X
3751:Symmetry-Culture and Science
3145:There are seven families of
1155:-Archimedean tiling counts
1110:= 6; and 7 such tilings for
889:
821:
789:
775:
393:
248:
225:
4440:Architectonic and catoptric
4338:Aperiodic set of prototiles
3776:Tilings by regular polygons
3105:Truncated Hexagonal Tiling
208:three regular tessellations
18:Tilings of regular polygons
5302:
4160:"Demiregular tessellation"
4141:"Semiregular tessellation"
3651:Sloane, N. J. A.
3580:"Pentagon-Decagon Packing"
3285:Truncated hexagonal tiling
3189:
2809:-uniform tilings from old
2634:
2614:
2578:6-3,4-6-3,4-6,4/m90/r(c6)
2454:
2431:
2277:
2269:
1219:
1161:
347:
206:at a vertex, yielding the
66:
34:
4579:
4565:
4426:
4415:
4328:
4315:
4297:
4224:
4211:
3268:
3260:
3190:
2270:
2152:
2097:
2042:
1987:
1918:
1849:
1782:
1715:
1650:
1585:
1520:
1457:
1396:
1337:
1280:
1225:
1106:= 5; 10 such tilings for
1102:= 4; 15 such tilings for
1098:= 3; 33 such tilings for
1094:= 2; 39 such tilings for
1090:= 1; 20 such tilings for
1019:-isohedral; if there are
394:
218:
31:Example periodic tilings
5276:Euclidean plane geometry
3984:Mathl. Comput. Modelling
1011:-isogonal; if there are
135:The Harmony of the World
3320:List of uniform tilings
3280:Truncated square tiling
2765:4-3-3-3/m90/r(h7)/r(h5)
2418:12-0,3,3-0,4/m45/m(h1)
2266:2-uniform tilings (20)
753:Three of them can make
96:non-edge-to-edge tiling
4104:"Semi-Regular Tilings"
3523:Critchlow, K. (1969).
3370:Tiling with rectangles
3350:Semiregular polyhedron
3130:
3121:
3088:
3079:
3070:
3061:
3047:
3038:
3029:
3020:
2981:
2948:
2917:
2908:
2899:
2890:
2876:
2867:
2858:
2849:
2244:There are twenty (20)
1049:-Archimedean tilings.
123:
4003:Kovic, Jurij (2011).
3976:. Dover Publications.
3631:probabilitysports.com
3244:Three size triangles
3129:
3120:
3096:Fractalizing examples
3087:
3078:
3069:
3060:
3046:
3037:
3028:
3019:
2982:
2949:
2916:
2907:
2898:
2889:
2875:
2866:
2857:
2848:
2696:4-3/m(h4)/m(h3)/r(h2)
772:Plane-vertex tilings
348:Further information:
196:equilateral triangles
3891:Tilings and Patterns
3822:Trans. Am. Math. Soc
3706:Tilings and patterns
3567:Tilings and patterns
3557:, Figure 2.1.1, p.60
3555:Tilings and patterns
3397:Mathematical Models;
3310:Grid (spatial index)
2961:
2928:
2648:4-3,3-4,3/r90/m(h3)
2513:6-3-3,3-3/r60/r(h8)
2352:6-4-3,4-6/m30/r(c4)
1023:orbits of edges, as
1015:orbits of tiles, as
744:plane-vertex tilings
738:Plane-vertex tilings
390:Uniform tilings (8)
214:Regular tilings (3)
3931:Chavey, D. (1989).
3853:Geometriae Dedicata
3627:"n-Uniform Tilings"
3616:Nils Lenngren, 2009
3481:10.3390/sym13122376
3472:2021Symm...13.2376G
3360:Hyperbolic geometry
3295:Trihexagonal tiling
3216:A tiling by squares
3164:
2742:4-3,4-3,3/m90/r(h3)
2396:12-3,4-3/m30/r(c3)
2374:12-4,6-3/m30/r(c3)
2267:
2259:demiregular tilings
1156:
1122:
773:
391:
382:(enantiomorphic or
355:Vertex-transitivity
215:
188:edge-to-edge tiling
32:
4157:Weisstein, Eric W.
4138:Weisstein, Eric W.
4119:Weisstein, Eric W.
3865:10.1007/BF00183189
3612:2015-06-30 at the
3507:< "Antwerp 3.0"
3354:Archimedean solids
3340:Regular polyhedron
3158:
3131:
3122:
3089:
3080:
3071:
3062:
3048:
3039:
3030:
3021:
2977:
2944:
2918:
2909:
2900:
2891:
2877:
2868:
2859:
2850:
2490:6-3,3-3/m30/r(h1)
2332:6-4-3-3/m30/r(h5)
2265:
1147:
1120:
771:
389:
359:symmetry operation
213:
30:
5263:
5262:
5259:
5258:
5255:
5254:
4561:
4560:
4452:Computer graphics
4411:
4410:
4295:
4294:
4084:"Uniform Tilings"
4076:n-uniform tilings
4046:Dale Seymour and
3990:(8–10): 317–320.
3963:978-0-670-52830-1
3912:J. Comb. Theory A
3664:. OEIS Foundation
3300:
3299:
3135:
3134:
3093:
3092:
2975:
2942:
2922:
2921:
2779:
2778:
2719:4-4-3-3/m90/r(h3)
2310:6-4-3,3/m30/r(h1)
2246:2-uniform tilings
2240:2-uniform tilings
2237:
2236:
1145:
1144:
985:
984:
715:
714:
617:4-3-3,4/r90/r(h2)
332:
331:
202:or three regular
176:acts transitively
102:
101:
16:(Redirected from
5293:
4581:
4567:
4519:Conway criterion
4446:Circle Limit III
4417:
4350:Einstein problem
4317:
4310:
4303:
4239:Schwarz triangle
4213:
4198:
4191:
4184:
4175:
4170:
4169:
4151:
4150:
4132:
4131:
4113:
4111:
4110:
4098:
4096:
4095:
4086:. Archived from
4078:, Brian Galebach
4062:, pp. 50–57
4043:
4041:
4016:
3999:
3977:
3954:
3936:
3927:
3906:
3894:
3881:Grünbaum, Branko
3876:
3847:
3837:
3812:
3779:
3773:
3767:
3766:
3746:
3740:
3739:
3737:
3736:
3730:
3724:. Archived from
3723:
3715:
3709:
3703:
3697:
3696:
3694:
3693:
3679:
3673:
3672:
3670:
3669:
3647:
3641:
3640:
3638:
3637:
3623:
3617:
3601:
3595:
3594:
3592:
3591:
3576:
3570:
3564:
3558:
3552:
3546:
3545:
3535:
3529:
3528:
3520:
3511:
3510:
3502:
3496:
3495:
3493:
3483:
3451:
3445:
3444:
3442:
3410:
3401:
3400:
3392:
3290:Hexagonal tiling
3270:Hexagonal tiling
3242:
3233:
3224:
3213:
3204:
3195:
3165:
3100:
2991:
2986:
2984:
2983:
2978:
2976:
2971:
2953:
2951:
2950:
2945:
2943:
2938:
2820:
2801:-uniform tilings
2786:-uniform tilings
2759:
2736:
2713:
2690:
2665:
2640:
2601:6-3,4/m90/r(h4)
2595:
2570:
2553:6-3,6/m90/r(h3)
2550:
2530:
2507:
2482:
2460:
2413:
2391:
2369:
2349:
2327:
2305:
2288:3-4-3/m30/r(c3)
2283:
2268:
1157:
1139:
1130:
1123:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
991:-uniform tilings
979:
970:
961:
952:
943:
934:
925:
916:
907:
898:
884:
875:
866:
857:
848:
839:
830:
816:
807:
798:
784:
774:
766:-uniform tilings
686:
679:
647:
640:
608:
601:
569:
562:
528:
521:
498:12-6,4/m30/r(c2)
489:
482:
450:
443:
411:
404:
392:
308:
281:
254:
245:
238:
231:
216:
128:
125:Harmonices Mundi
114:regular polygons
92:
72:
53:
40:
33:
21:
5301:
5300:
5296:
5295:
5294:
5292:
5291:
5290:
5281:Regular tilings
5266:
5265:
5264:
5251:
4728:
4721:
4654:
4648:
4617:
4575:
4557:
4422:
4407:
4324:
4311:
4305:
4304:
4291:
4282:Wallpaper group
4220:
4207:
4202:
4155:
4154:
4136:
4135:
4117:
4116:
4108:
4106:
4101:
4093:
4091:
4081:
4069:
4019:
4002:
3981:
3967:
3930:
3909:
3903:
3885:Shephard, G. C.
3879:
3850:
3815:
3801:10.2307/2689529
3786:
3783:
3782:
3774:
3770:
3748:
3747:
3743:
3734:
3732:
3728:
3721:
3717:
3716:
3712:
3704:
3700:
3691:
3689:
3681:
3680:
3676:
3667:
3665:
3649:
3648:
3644:
3635:
3633:
3625:
3624:
3620:
3614:Wayback Machine
3602:
3598:
3589:
3587:
3578:
3577:
3573:
3565:
3561:
3553:
3549:
3537:
3536:
3532:
3522:
3521:
3514:
3504:
3503:
3499:
3453:
3452:
3448:
3412:
3411:
3404:
3394:
3393:
3389:
3384:
3379:
3375:Lattice (group)
3352:(including the
3344:Platonic solids
3335:Wallpaper group
3305:
3243:
3234:
3225:
3214:
3205:
3196:
3140:
3098:
3006:
2988:
2959:
2958:
2926:
2925:
2835:
2803:
2788:
2766:
2764:
2763:
2760:
2743:
2741:
2740:
2737:
2720:
2718:
2717:
2714:
2697:
2695:
2694:
2691:
2674:
2672:
2670:
2666:
2649:
2647:
2645:
2641:
2602:
2600:
2599:
2596:
2579:
2577:
2576:
2571:
2554:
2552:
2551:
2534:
2532:
2531:
2514:
2512:
2511:
2508:
2491:
2489:
2487:
2483:
2466:
2464:
2461:
2419:
2417:
2414:
2397:
2395:
2392:
2375:
2373:
2370:
2353:
2351:
2350:
2333:
2331:
2328:
2311:
2309:
2306:
2289:
2287:
2284:
2242:
1140:
1131:
1035:wallpaper group
1024:
1020:
1016:
1012:
1008:
1004:
1000:
993:
980:
971:
962:
953:
944:
935:
926:
917:
908:
899:
885:
876:
867:
858:
849:
840:
831:
817:
808:
799:
785:
755:regular tilings
740:
725:
724:
720:
707:
697:
695:6-3-3/r60/r(h5)
692:
687:
681:
680:
668:
658:
653:
648:
642:
641:
629:
619:
614:
609:
603:
602:
590:
580:
575:
570:
564:
563:
549:
539:
537:6-3-6/m30/r(v4)
534:
529:
523:
522:
510:
500:
495:
490:
484:
483:
471:
461:
459:6-4-3/m30/r(c2)
456:
451:
445:
444:
432:
422:
417:
412:
406:
405:
352:
346:
337:
319:
314:
309:
292:
287:
282:
265:
260:
255:
160:
158:Regular tilings
143:
93:
79:-uniform tiling
73:
54:
41:
28:
23:
22:
15:
12:
11:
5:
5299:
5297:
5289:
5288:
5283:
5278:
5268:
5267:
5261:
5260:
5257:
5256:
5253:
5252:
5250:
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:
5029:
5024:
5019:
5014:
5009:
5004:
4999:
4994:
4989:
4984:
4979:
4974:
4969:
4964:
4959:
4954:
4949:
4944:
4939:
4934:
4929:
4924:
4919:
4914:
4909:
4904:
4899:
4894:
4889:
4884:
4879:
4874:
4869:
4864:
4859:
4854:
4849:
4844:
4839:
4834:
4829:
4824:
4819:
4814:
4809:
4804:
4799:
4794:
4789:
4784:
4779:
4774:
4769:
4764:
4759:
4754:
4749:
4744:
4739:
4733:
4731:
4723:
4722:
4720:
4719:
4714:
4709:
4704:
4699:
4694:
4689:
4684:
4679:
4674:
4669:
4664:
4658:
4656:
4650:
4649:
4647:
4646:
4641:
4636:
4631:
4625:
4623:
4619:
4618:
4616:
4615:
4610:
4605:
4600:
4595:
4589:
4587:
4577:
4576:
4570:
4563:
4562:
4559:
4558:
4556:
4555:
4550:
4545:
4540:
4535:
4528:
4527:
4526:
4521:
4511:
4510:
4509:
4504:
4499:
4494:
4493:
4492:
4479:
4474:
4469:
4464:
4459:
4454:
4449:
4442:
4437:
4427:
4424:
4423:
4420:
4413:
4412:
4409:
4408:
4406:
4405:
4400:
4395:
4394:
4393:
4379:
4374:
4369:
4364:
4359:
4358:
4357:
4355:Socolar–Taylor
4347:
4346:
4345:
4335:
4333:Ammann–Beenker
4329:
4326:
4325:
4320:
4313:
4312:
4298:
4296:
4293:
4292:
4290:
4289:
4284:
4279:
4278:
4277:
4272:
4267:
4256:Uniform tiling
4253:
4252:
4251:
4241:
4236:
4231:
4225:
4222:
4221:
4216:
4209:
4208:
4203:
4201:
4200:
4193:
4186:
4178:
4172:
4171:
4152:
4133:
4122:"Tessellation"
4114:
4099:
4082:Dutch, Steve.
4079:
4068:
4067:External links
4065:
4064:
4063:
4060:978-0866514613
4044:
4017:
4000:
3979:
3965:
3955:
3928:
3918:(1): 110–119.
3907:
3901:
3877:
3848:
3813:
3795:(5): 227–247.
3781:
3780:
3768:
3757:(3): 193–210.
3741:
3710:
3698:
3674:
3642:
3618:
3596:
3571:
3559:
3547:
3530:
3512:
3497:
3446:
3425:(7): 843–852.
3402:
3386:
3385:
3383:
3380:
3378:
3377:
3372:
3367:
3365:Penrose tiling
3362:
3357:
3347:
3337:
3332:
3327:
3325:Wythoff symbol
3322:
3317:
3312:
3306:
3304:
3301:
3298:
3297:
3292:
3287:
3282:
3277:
3272:
3266:
3265:
3262:
3259:
3256:
3253:
3250:
3246:
3245:
3236:
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2626:
2623:
2620:
2617:
2613:
2612:
2597:
2589:
2574:
2564:
2544:
2533:6-3/m90/r(h1)
2524:
2509:
2501:
2485:
2476:
2465:3-6/m30/r(c2)
2453:
2452:
2449:
2446:
2443:
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2434:
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2407:
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2024:
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2013:
2010:
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2001:
1998:
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1989:
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1984:
1981:
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1953:
1948:
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1933:
1928:
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819:
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792:
788:
787:
778:
748:vertex figures
739:
736:
713:
712:
673:
634:
595:
555:
554:
515:
476:
437:
420:12-3/m30/r(h3)
397:
396:
345:
342:
330:
329:
302:
275:
247:
246:
239:
232:
224:
223:
220:
174:of the tiling
172:symmetry group
159:
156:
142:
139:
100:
99:
86:
65:
64:
61:type of vertex
47:
44:regular tiling
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5298:
5287:
5284:
5282:
5279:
5277:
5274:
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5271:
5248:
5245:
5243:
5240:
5238:
5235:
5233:
5230:
5228:
5225:
5223:
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5210:
5208:
5205:
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5128:
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5118:
5115:
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5110:
5108:
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5018:
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5010:
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4993:
4990:
4988:
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4923:
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4910:
4908:
4905:
4903:
4900:
4898:
4895:
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4890:
4888:
4885:
4883:
4880:
4878:
4875:
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4778:
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4768:
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4758:
4755:
4753:
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4708:
4705:
4703:
4700:
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4688:
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4680:
4678:
4675:
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4630:
4627:
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4620:
4614:
4611:
4609:
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4601:
4599:
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4508:
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4432:
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4404:
4401:
4399:
4396:
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4389:
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4380:
4378:
4375:
4373:
4370:
4368:
4365:
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4360:
4356:
4353:
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4351:
4348:
4344:
4341:
4340:
4339:
4336:
4334:
4331:
4330:
4327:
4323:
4318:
4314:
4309:
4302:
4288:
4285:
4283:
4280:
4276:
4273:
4271:
4268:
4266:
4263:
4262:
4261:
4257:
4254:
4250:
4247:
4246:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4226:
4223:
4219:
4214:
4210:
4206:
4199:
4194:
4192:
4187:
4185:
4180:
4179:
4176:
4167:
4166:
4161:
4158:
4153:
4148:
4147:
4142:
4139:
4134:
4129:
4128:
4123:
4120:
4115:
4105:
4102:Mitchell, K.
4100:
4090:on 2006-09-09
4089:
4085:
4080:
4077:
4074:
4073:
4072:
4066:
4061:
4057:
4053:
4049:
4045:
4040:
4039:10.37236/2512
4035:
4031:
4027:
4023:
4018:
4015:(2): 491–507.
4014:
4010:
4006:
4001:
3997:
3993:
3989:
3985:
3980:
3975:
3973:
3966:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3935:
3929:
3925:
3921:
3917:
3913:
3908:
3904:
3902:0-7167-1193-1
3898:
3893:
3892:
3886:
3882:
3878:
3874:
3870:
3866:
3862:
3858:
3854:
3849:
3845:
3841:
3836:
3831:
3827:
3823:
3819:
3814:
3810:
3806:
3802:
3798:
3794:
3790:
3785:
3784:
3777:
3772:
3769:
3764:
3760:
3756:
3752:
3745:
3742:
3731:on 2016-05-07
3727:
3720:
3714:
3711:
3707:
3702:
3699:
3688:
3684:
3678:
3675:
3663:
3662:
3656:
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3643:
3632:
3628:
3622:
3619:
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3611:
3608:
3606:
3600:
3597:
3585:
3581:
3575:
3572:
3568:
3563:
3560:
3556:
3551:
3548:
3543:
3542:
3534:
3531:
3526:
3519:
3517:
3513:
3508:
3501:
3498:
3492:
3487:
3482:
3477:
3473:
3469:
3465:
3461:
3457:
3450:
3447:
3441:
3436:
3432:
3428:
3424:
3420:
3416:
3409:
3407:
3403:
3398:
3391:
3388:
3381:
3376:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3355:
3351:
3348:
3345:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3307:
3302:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3275:Square tiling
3273:
3271:
3267:
3263:
3257:
3254:
3251:
3248:
3247:
3241:
3237:
3232:
3228:
3223:
3219:
3217:
3212:
3208:
3203:
3199:
3194:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3166:
3162:
3156:
3153:
3148:
3143:
3137:
3128:
3124:
3119:
3115:
3113:Fractalizing
3112:
3111:
3107:
3104:
3102:
3101:
3095:
3086:
3082:
3077:
3073:
3068:
3064:
3059:
3055:
3053:Fractalizing
3052:
3051:
3045:
3041:
3036:
3032:
3027:
3023:
3018:
3014:
3004:
3001:
2998:
2995:
2993:
2992:
2989:
2972:
2967:
2964:
2955:
2939:
2934:
2931:
2915:
2911:
2906:
2902:
2897:
2893:
2888:
2884:
2882:Fractalizing
2881:
2880:
2874:
2870:
2865:
2861:
2856:
2852:
2847:
2843:
2833:
2830:
2827:
2824:
2822:
2821:
2818:
2815:
2812:
2808:
2800:
2797:Fractalizing
2796:
2794:
2792:
2785:
2781:
2774:
2770:
2758:
2754:
2751:
2747:
2735:
2731:
2728:
2724:
2712:
2708:
2705:
2701:
2689:
2685:
2682:
2678:
2671:
2664:
2660:
2657:
2653:
2646:
2639:
2635:
2630:
2627:
2624:
2621:
2618:
2615:
2610:
2606:
2594:
2590:
2587:
2583:
2573:
2569:
2565:
2562:
2558:
2549:
2545:
2542:
2538:
2529:
2525:
2522:
2518:
2506:
2502:
2499:
2495:
2488:
2481:
2477:
2474:
2470:
2463:
2459:
2455:
2450:
2447:
2444:
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2438:
2435:
2432:
2427:
2423:
2416:
2412:
2408:
2405:
2401:
2394:
2390:
2386:
2383:
2379:
2372:
2368:
2364:
2361:
2357:
2348:
2344:
2341:
2337:
2330:
2326:
2322:
2319:
2315:
2308:
2304:
2300:
2297:
2293:
2286:
2282:
2278:
2273:
2263:
2261:
2260:
2255:
2253:
2247:
2239:
2233:
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2151:
2147:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2096:
2092:
2089:
2086:
2083:
2080:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2041:
2037:
2034:
2031:
2028:
2025:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1986:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1921:
1917:
1913:
1910:
1907:
1904:
1901:
1898:
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1893:
1890:
1888:
1885:
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1880:
1878:
1875:
1873:
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1865:
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1860:
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1848:
1844:
1841:
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1832:
1829:
1826:
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1813:
1811:
1808:
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1803:
1801:
1798:
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1777:
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1768:
1765:
1762:
1759:
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1749:
1746:
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1736:
1734:
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1704:
1701:
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1659:
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1559:
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1543:
1541:
1538:
1536:
1533:
1531:
1528:
1525:
1523:
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1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
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1480:
1478:
1475:
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1470:
1468:
1465:
1462:
1460:
1456:
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1446:
1443:
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1434:
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1428:
1425:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1401:
1399:
1395:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1353:
1350:
1348:
1345:
1342:
1340:
1336:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1291:
1288:
1285:
1283:
1279:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1231:
1228:
1222:
1218:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1168:
1165:-Archimedean
1164:
1158:
1154:
1150:
1138:
1134:
1129:
1125:
1118:
1115:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1064:
1059:
1055:
1050:
1048:
1044:
1038:
1036:
1032:
1028:
998:
990:
987:
978:
969:
960:
951:
942:
933:
924:
915:
906:
897:
890:
883:
874:
865:
856:
847:
838:
829:
822:
815:
806:
797:
790:
783:
776:
769:
767:
765:
760:
756:
751:
749:
745:
737:
735:
733:
729:
723:
719:
710:
705:
701:
696:
691:
685:
678:
674:
672:
666:
662:
657:
656:4-3/m90/r(h2)
652:
646:
639:
635:
632:
627:
623:
618:
613:
607:
600:
596:
593:
588:
584:
579:
578:8-4/m90/r(h4)
574:
568:
561:
557:
552:
547:
543:
538:
533:
527:
520:
516:
513:
508:
504:
499:
494:
488:
481:
477:
474:
469:
465:
460:
455:
449:
442:
438:
435:
430:
426:
421:
416:
410:
403:
399:
387:
385:
381:
377:
373:
372:
367:
362:
360:
356:
351:
343:
341:
340:
336:
327:
323:
318:
313:
307:
303:
300:
296:
291:
286:
280:
276:
273:
269:
264:
259:
253:
249:
244:
240:
237:
233:
230:
226:
221:
217:
211:
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
157:
155:
151:
147:
140:
138:
136:
132:
127:
126:
120:
116:
115:
111:
107:
97:
91:
87:
84:
80:
78:
71:
67:
62:
58:
52:
48:
45:
39:
35:
19:
5286:Tessellation
4543:Substitution
4538:Regular grid
4530:
4444:
4377:Quaquaversal
4275:Kisrhombille
4217:
4205:Tessellation
4163:
4144:
4125:
4107:. Retrieved
4092:. Retrieved
4088:the original
4070:
4051:
4048:Jill Britton
4029:
4025:
4012:
4009:Math. Commun
4008:
3987:
3983:
3971:
3969:
3942:
3938:
3915:
3911:
3890:
3859:(1): 47–60.
3856:
3852:
3825:
3821:
3792:
3788:
3771:
3754:
3750:
3744:
3733:. Retrieved
3726:the original
3713:
3701:
3690:. Retrieved
3686:
3677:
3666:. Retrieved
3658:
3645:
3634:. Retrieved
3630:
3621:
3604:
3599:
3588:. Retrieved
3583:
3574:
3562:
3550:
3540:
3533:
3524:
3500:
3466:(12): 2376.
3463:
3459:
3449:
3422:
3418:
3396:
3390:
3330:Tessellation
3151:
3144:
3141:
2956:
2923:
2810:
2806:
2804:
2798:
2790:
2789:
2783:
2772:
2768:
2749:
2745:
2726:
2722:
2703:
2699:
2680:
2676:
2655:
2651:
2608:
2604:
2585:
2581:
2560:
2556:
2540:
2536:
2520:
2516:
2497:
2493:
2472:
2468:
2425:
2421:
2403:
2399:
2381:
2377:
2359:
2355:
2339:
2335:
2317:
2313:
2295:
2291:
2257:
2249:
2245:
2243:
2231:
2226:
2221:
2216:
2211:
2206:
2201:
2196:
2191:
2186:
2181:
2176:
2171:
2166:
2161:
2156:
2143:
2076:
2021:
1960:
1955:
1950:
1945:
1940:
1935:
1930:
1925:
1891:
1886:
1881:
1876:
1871:
1866:
1861:
1856:
1819:
1814:
1809:
1804:
1799:
1794:
1789:
1752:
1747:
1742:
1737:
1732:
1727:
1722:
1682:
1677:
1672:
1667:
1662:
1657:
1617:
1612:
1607:
1602:
1597:
1592:
1549:
1544:
1539:
1534:
1529:
1481:
1476:
1471:
1466:
1415:
1410:
1405:
1351:
1346:
1289:
1229:
1220:
1162:
1152:
1148:
1116:
1111:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:Krotenheerdt
1061:
1057:
1053:
1051:
1046:
1042:
1039:
1030:
1029:
1007:-uniform or
994:
988:
763:
752:
741:
731:
727:
721:
717:
716:
708:
703:
699:
670:
664:
660:
630:
625:
621:
591:
586:
582:
550:
545:
541:
511:
506:
502:
472:
467:
463:
433:
428:
424:
380:mirror image
375:
369:
365:
363:
353:
338:
334:
333:
325:
321:
298:
294:
271:
267:
179:
167:
161:
152:
148:
144:
134:
108:
103:
82:
76:
4573:vertex type
4431:Anisohedral
4386:Self-tiling
4229:Pythagorean
3945:: 147–165.
3828:: 335–353.
3491:10902/23907
3258:p4m (*442)
3255:cmm (2*22)
3249:cmm (2*22)
2451:pmm, *2222
1027:-isotoxal.
728:Archimedean
376:semiregular
366:Archimedean
317:4/m45/r(h1)
290:6/m30/r(h1)
263:3/m30/r(h2)
5270:Categories
4477:Pentagonal
4109:2006-09-09
4094:2006-09-09
4032:(3): #P6.
3974:Dimensions
3735:2015-06-04
3692:2024-08-24
3668:2023-01-07
3636:2019-06-21
3590:2022-03-07
3440:10902/5869
3382:References
3252:p2 (2222)
3007:Dodecagon
2836:Dodecagon
2631:cmm, 2*22
2628:pmm, *2222
2445:pmm, *2222
2274:p4m, *442
1151:-uniform,
1037:symmetry.
818:3.3.3.3.6
809:3.3.3.4.4
800:3.3.4.3.4
395:p6m, *632
222:p4m, *442
219:p6m, *632
162:Following
112:by convex
104:Euclidean
4585:Spherical
4553:Voderberg
4514:Prototile
4481:Problems
4457:Honeycomb
4435:Isohedral
4322:Aperiodic
4260:honeycomb
4244:Rectangle
4234:Rhombille
4165:MathWorld
4146:MathWorld
4127:MathWorld
3873:122636363
3789:Math. Mag
3569:, p.58-69
3264:p3 (333)
3261:p6 (632)
3159:Periodic
3005:Dissected
2996:Triangle
2834:Dissected
2825:Triangle
2625:cmm, 2*22
2622:cmm, 2*22
2448:cmm, 2*22
2442:cmm, 2*22
2433:p6m, *632
2271:p6m, *632
1223:-uniform
841:3.4.3.12
832:3.3.4.12
688:C&R:
649:C&R:
610:C&R:
571:C&R:
530:C&R:
491:C&R:
452:C&R:
413:C&R:
310:C&R:
283:C&R:
256:C&R:
192:congruent
137:, 1619).
4667:V3.4.3.4
4502:Squaring
4497:Heesch's
4462:Isotoxal
4382:Rep-tile
4372:Pinwheel
4265:Coloring
4218:Periodic
4054:, 1989,
3887:(1987).
3763:33928615
3610:Archived
3460:Symmetry
3303:See also
3161:isogonal
3147:isogonal
3002:Hexagon
2831:Hexagon
2619:pgg, 22×
2616:p4g, 4*2
2252:isogonal
936:3.12.12
927:3.10.15
877:3.4.6.4
868:3.4.4.6
850:3.3.6.6
204:hexagons
164:Grünbaum
59:has one
5127:6.4.8.4
5082:5.4.6.4
5042:4.12.16
5032:4.10.12
5002:V4.8.10
4977:V4.6.16
4967:V4.6.14
4867:3.6.4.6
4862:3.4.∞.4
4857:3.4.8.4
4852:3.4.7.4
4847:3.4.6.4
4797:3.∞.3.∞
4792:3.4.3.4
4787:3.8.3.8
4782:3.7.3.7
4777:3.6.3.8
4772:3.6.3.6
4767:3.5.3.6
4762:3.5.3.5
4757:3.4.3.∞
4752:3.4.3.8
4747:3.4.3.7
4742:3.4.3.6
4737:3.4.3.5
4692:3.4.6.4
4662:3.4.3.4
4655:regular
4622:Regular
4548:Voronoi
4472:Packing
4403:Truchet
4398:Socolar
4367:Penrose
4362:Gilbert
4287:Wythoff
3844:0496813
3809:2689529
3653:(ed.).
3468:Bibcode
3152:uniform
2999:Square
2828:Square
2782:Higher
2439:p6, 632
2436:p6, 632
2254:tilings
2038:103082
972:5.5.10
954:4.6.12
945:4.5.20
918:3.9.18
909:3.8.24
900:3.7.42
732:uniform
612:3.4.3.4
454:3.4.6.4
371:uniform
200:squares
198:, four
178:on the
170:if the
168:regular
121:in his
110:tilings
5017:4.8.16
5012:4.8.14
5007:4.8.12
4997:4.8.10
4972:4.6.16
4962:4.6.14
4957:4.6.12
4727:Hyper-
4712:4.6.12
4485:Domino
4391:Sphinx
4270:Convex
4249:Domino
4058:
3961:
3899:
3871:
3842:
3807:
3761:
3012:Shape
2841:Shape
2153:Total
1983:49794
1914:24459
1845:11866
1215:Total
997:orbits
963:4.8.8
859:(3.6)
711:{3,6}
693:GJ-H:
669:{3,6}:
654:GJ-H:
633:{4,4}
615:GJ-H:
594:{4,4}
576:GJ-H:
553:{6,3}
535:GJ-H:
514:{3,6}
496:GJ-H:
493:4.6.12
475:{3,6}
457:GJ-H:
436:{6,3}
418:GJ-H:
384:chiral
315:GJ-H:
288:GJ-H:
261:GJ-H:
184:vertex
119:Kepler
5132:(6.8)
5087:(5.6)
5022:4.8.∞
4992:(4.8)
4987:(4.7)
4982:4.6.∞
4952:(4.6)
4947:(4.5)
4917:4.∞.4
4912:4.8.4
4907:4.7.4
4902:4.6.4
4897:4.5.4
4877:(3.8)
4872:(3.7)
4842:(3.4)
4837:(3.4)
4729:bolic
4697:(3.6)
4653:Semi-
4524:Girih
4421:Other
3869:S2CID
3805:JSTOR
3778:p.236
3759:S2CID
3729:(PDF)
3722:(PDF)
3586:. AMS
3342:(the
2775:= 5)
2771:= 4,
2752:= 4)
2748:= 3,
2729:= 5)
2725:= 3,
2706:= 4)
2702:= 2,
2683:= 6)
2679:= 3,
2658:= 5)
2654:= 4,
2611:= 4)
2607:= 4,
2588:= 4)
2584:= 3,
2563:= 3)
2559:= 2,
2543:= 4)
2539:= 2,
2523:= 7)
2519:= 5,
2500:= 3)
2496:= 3,
2475:= 3)
2471:= 2,
2428:= 3)
2424:= 3,
2406:= 4)
2402:= 4,
2384:= 4)
2380:= 4,
2362:= 5)
2358:= 5,
2342:= 4)
2338:= 4,
2320:= 4)
2316:= 4,
2298:= 3)
2294:= 3,
2098:≥ 15
1941:12309
1936:11006
1931:13762
1778:5960
1711:2850
1646:1472
1212:≥ 15
1114:= 7.
702:= 3,
663:= 2,
624:= 2,
585:= 2,
544:= 2,
532:(3.6)
505:= 3,
466:= 3,
427:= 2,
328:= 1)
324:= 1,
301:= 1)
297:= 1,
274:= 1)
270:= 1,
180:flags
131:Latin
106:plane
5217:8.16
5212:8.12
5182:7.14
5152:6.16
5147:6.12
5142:6.10
5102:5.12
5097:5.10
5052:4.16
5047:4.14
5037:4.12
5027:4.10
4887:3.16
4882:3.14
4702:3.12
4687:V3.6
4613:V4.n
4603:V3.n
4490:Wang
4467:List
4433:and
4384:and
4343:List
4258:and
4056:ISBN
3959:ISBN
3897:ISBN
3659:The
1951:1736
1946:9230
1926:1607
1877:3711
1872:5993
1867:5798
1862:7171
1857:1086
1810:1468
1805:2745
1800:2979
1795:3772
1738:1278
1733:1608
1728:1992
1663:1037
1581:673
1516:332
1453:151
706:= 3)
667:= 3)
628:= 2)
589:= 2)
548:= 1)
509:= 3)
470:= 2)
431:= 2)
415:3.12
81:has
5247:∞.8
5242:∞.6
5207:8.6
5177:7.8
5172:7.6
5137:6.8
5092:5.8
5057:4.∞
4892:3.∞
4817:3.4
4812:3.∞
4807:3.8
4802:3.7
4717:4.8
4707:4.∞
4682:3.6
4677:3.∞
4672:3.4
4608:4.n
4598:3.n
4571:By
4034:doi
3992:doi
3947:doi
3920:doi
3861:doi
3830:doi
3826:252
3797:doi
3486:hdl
3476:doi
3435:hdl
3427:doi
3423:138
2256:or
2043:14
1988:13
1956:129
1919:12
1882:647
1850:11
1815:212
1790:663
1783:10
1743:570
1723:424
1678:203
1673:537
1668:795
1658:298
1608:218
1603:426
1598:572
1593:175
1540:187
1535:284
1530:100
1472:149
1392:61
1333:20
1276:11
1209:14
1206:13
1203:12
1200:11
1197:10
690:3.6
651:3.4
573:4.8
374:or
190:by
5272::
4162:.
4143:.
4124:.
4050:,
4030:19
4028:.
4024:.
4013:16
4011:.
4007:.
3988:26
3986:.
3943:17
3941:.
3937:.
3916:44
3914:.
3883:;
3867:.
3857:11
3855:.
3840:MR
3838:.
3824:.
3820:.
3803:.
3793:50
3791:.
3755:25
3753:.
3685:.
3657:.
3629:.
3582:.
3515:^
3484:.
3474:.
3464:13
3462:.
3458:.
3433:.
3421:.
3417:.
3405:^
3186:7
3183:6
3180:5
3177:4
3174:3
3171:2
3168:1
2987:.
2954:.
2250:2-
2157:11
2148:?
2093:?
2090:0
2035:0
2032:0
1980:0
1977:0
1974:0
1971:0
1968:0
1961:15
1911:0
1908:0
1905:0
1902:0
1899:0
1896:0
1887:52
1842:0
1839:0
1836:0
1833:0
1830:0
1827:0
1824:0
1820:27
1775:0
1772:0
1769:0
1766:0
1763:0
1760:0
1757:0
1748:80
1716:9
1708:0
1705:0
1702:0
1699:0
1696:0
1693:0
1690:0
1687:0
1683:20
1651:8
1643:0
1640:0
1637:0
1634:0
1631:0
1628:0
1625:0
1622:0
1613:74
1586:7
1578:0
1575:0
1572:0
1569:0
1566:0
1563:0
1560:0
1557:0
1550:10
1545:92
1513:0
1510:0
1507:0
1504:0
1501:0
1498:0
1495:0
1492:0
1482:15
1477:94
1467:74
1450:0
1447:0
1444:0
1441:0
1438:0
1435:0
1432:0
1429:0
1416:33
1411:85
1406:33
1389:0
1386:0
1383:0
1380:0
1377:0
1374:0
1371:0
1368:0
1352:39
1347:22
1330:0
1327:0
1324:0
1321:0
1318:0
1315:0
1312:0
1309:0
1290:20
1273:0
1270:0
1267:0
1264:0
1261:0
1258:0
1255:0
1252:0
1230:11
1226:1
1194:9
1191:8
1082:=
1078:=
1070:≥
1056:=
981:6
891:3
886:4
823:4
791:5
786:3
777:6
709:sr
512:tr
473:rr
368:,
210:.
133::
94:A
74:A
55:A
42:A
5237:∞
5232:∞
5227:∞
5222:∞
5202:8
5197:8
5192:8
5187:8
5167:7
5162:7
5157:7
5122:6
5117:6
5112:6
5107:6
5077:5
5072:5
5067:5
5062:5
4942:4
4937:4
4932:4
4927:4
4922:4
4832:3
4827:3
4822:3
4644:6
4639:4
4634:3
4629:2
4593:2
4197:e
4190:t
4183:v
4168:.
4149:.
4130:.
4112:.
4097:.
4042:.
4036::
3998:.
3994::
3972:n
3953:.
3949::
3926:.
3922::
3905:.
3875:.
3863::
3846:.
3832::
3811:.
3799::
3765:.
3738:.
3695:.
3671:.
3639:.
3605:k
3593:.
3509:.
3494:.
3488::
3478::
3470::
3443:.
3437::
3429::
3356:)
3346:)
2973:3
2968:+
2965:3
2940:3
2935:+
2932:2
2811:k
2807:k
2799:k
2791:k
2784:k
2773:e
2769:t
2767:(
2762:2
2750:e
2746:t
2744:(
2739:1
2727:e
2723:t
2721:(
2716:2
2704:e
2700:t
2698:(
2693:1
2681:e
2677:t
2675:(
2669:2
2656:e
2652:t
2650:(
2644:1
2609:e
2605:t
2603:(
2598:1
2586:e
2582:t
2580:(
2575:2
2561:e
2557:t
2555:(
2541:e
2537:t
2535:(
2521:e
2517:t
2515:(
2510:2
2498:e
2494:t
2492:(
2486:1
2473:e
2469:t
2467:(
2426:e
2422:t
2420:(
2404:e
2400:t
2398:(
2382:e
2378:t
2376:(
2360:e
2356:t
2354:(
2340:e
2336:t
2334:(
2318:e
2314:t
2312:(
2296:e
2292:t
2290:(
2232:∞
2227:0
2222:∞
2217:∞
2212:∞
2207:∞
2202:∞
2197:∞
2192:∞
2187:∞
2182:∞
2177:∞
2172:∞
2167:∞
2162:∞
2144:0
2140:?
2137:?
2134:?
2131:?
2128:?
2125:?
2122:?
2119:?
2116:?
2113:?
2110:?
2107:?
2104:?
2101:0
2087:0
2084:0
2081:0
2077:0
2073:?
2070:?
2067:?
2064:?
2061:?
2058:?
2055:?
2052:?
2049:?
2046:0
2029:0
2026:0
2022:0
2018:?
2015:?
2012:?
2009:?
2006:?
2003:?
2000:?
1997:?
1994:?
1991:0
1965:0
1922:0
1892:1
1853:0
1786:0
1753:8
1719:0
1654:0
1618:7
1589:0
1554:0
1526:0
1522:6
1489:0
1486:0
1463:0
1459:5
1426:0
1423:0
1420:0
1402:0
1398:4
1365:0
1362:0
1359:0
1356:0
1343:0
1339:3
1306:0
1303:0
1300:0
1297:0
1294:0
1286:0
1282:2
1249:0
1246:0
1243:0
1240:0
1237:0
1234:0
1221:k
1188:7
1185:6
1182:5
1179:4
1176:3
1173:2
1170:1
1163:m
1153:m
1149:k
1112:n
1108:n
1104:n
1100:n
1096:n
1092:n
1088:n
1084:k
1080:n
1076:m
1072:k
1068:m
1058:k
1054:m
1047:m
1043:m
1031:k
1025:e
1021:e
1017:t
1013:t
1009:k
1005:k
1001:k
989:k
764:k
704:e
700:t
698:(
671:e
665:e
661:t
659:(
631:s
626:e
622:t
620:(
592:t
587:e
583:t
581:(
551:r
546:e
542:t
540:(
507:e
503:t
501:(
468:e
464:t
462:(
434:t
429:e
425:t
423:(
326:e
322:t
320:(
312:4
299:e
295:t
293:(
285:6
272:e
268:t
266:(
258:3
129:(
83:k
77:k
20:)
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