http://en.wikipedia.org/wiki/List_of_uniform_tilings
Lista de Mosaicos uniformesFrom Wikipedia, the free encyclopedia
This table shows the 11 convex uniform tilings of the Euclidean plane, and their dual tilings.
There are three regular, and eight semiregular, tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face.
Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex.
These 11 uniform tilings have 32 different uniform colorings. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are not color uniform)
In addition to the 11 convex uniform tilings, there are also 14 nonconvex forms, using star polygons, and reverse orientation vertex configurations.
Dual tilings are listed by their face configuration, the number of faces at each vertex of a face. For example V4.8.8 means isosceles triangle tiles with one corner with 4 triangles, and two corners containing 8 triangles.
In the 1987 book, Tilings and Patterns, Branko Grünbaum calls the vertex uniform tilings Archimedean in parallel to the Archimedean solids, and the dual tilings Laves tilings in honor of crystalographer Fritz Laves.
Contents
1 Convex uniform tilings of the Euclidean plane o 1.1 The R 3 [4,4] group familyo 1.2 The V 3 [6,3] group familyo 1.3 Non-Wythoffian uniform tiling
2 See also 3 References
4 External links
Convex uniform tilings of the Euclidean plane
The R3 [4,4] group family
Platonic and Archimedean tilingsVertex figure
Wythoff symbol(s)Symmetry group
Dual Laves tilings
Square tiling
4.4.4.44 | 2 4p4mor *442
self-dual
Truncated square tiling
4.8.82 | 4 44 4 2 |p4mor *442
Tetrakis square tiling
Snub square tiling
3.3.4.3.4| 4 4 2p4gor 4*2 and 442
Cairo pentagonal tiling
[edit] The V3 [6,3] group family
Platonic and Archimedean tilingsVertex figure
Wythoff symbol(s)Symmetry group
Dual Laves tilings
Hexagonal tiling
6.6.63 | 6 22 6 | 33 3 3 |p6mor *632
Triangular tiling
Trihexagonal tiling
3.6.3.62 | 6 33 3 | 3p6mor *632 and *333
Quasiregular rhombic tiling
Truncated hexagonal tiling
3.12.122 3 | 63 3 | 3p6mor *632
Triakis triangular tiling
Triangular tiling
3.3.3.3.3.36 | 3 23 | 3 3| 3 3 3p6mor *632 and *333
Hexagonal tiling
Small rhombitrihexagonal tiling
3.4.6.43 | 6 2p6mor *632
Deltoidal trihexagonal tiling
Great rhombitrihexagonal tiling
4.6.12or *6322 6 3 |p6m
Bisected hexagonal tiling
Snub hexagonal tiling
3.3.3.3.6| 6 3 2p6or 632
Floret pentagonal tiling
Non-Wythoffian uniform tiling
Platonic and Archimedean tilingsVertex figure
Wythoff symbol(s)Symmetry group
Dual Laves tilings
Elongated triangular tiling
3.3.3.4.42 | 2 (2 2)cmm
Prismatic pentagonal tiling
See also
Convex uniform honeycomb - The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings.
Uniform tilings in hyperbolic plane
References
Grünbaum, Branko ; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
H.S.M. Coxeter , M.S. Longuet-Higgins, J.C.P. Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50.