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.