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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.

Convex uniform tilings of the Euclidean plane

The R₃ [4,4] group family

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings
Square tiling	4.4.4.4 4 | 2 4 p4m or *442	self-dual
Truncated square tiling	4.8.8 2 | 4 4 4 4 2 | p4m or *442	Tetrakis square tiling
Snub square tiling	3.3.4.3.4 | 4 4 2 p4g or 4*2 and 442	Cairo pentagonal tiling

The V₃ [6,3] group family

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings
Hexagonal tiling	6.6.6 3 | 6 2 2 6 | 3 3 3 3 | p6m or *632	Triangular tiling
Trihexagonal tiling	3.6.3.6 2 | 6 3 3 3 | 3 p6m or *632 and *333	Quasiregular rhombic tiling
Truncated hexagonal tiling	3.12.12 2 3 | 6 3 3 | 3 p6m or *632	Triakis triangular tiling
Triangular tiling	3.3.3.3.3.3 6 | 3 2 3 | 3 3 | 3 3 3 p6m or *632 and *333	Hexagonal tiling
Small rhombitrihexagonal tiling	3.4.6.4 3 | 6 2 p6m or *632	Deltoidal trihexagonal tiling
Great rhombitrihexagonal tiling	4.6.12 or *632 2 6 3 | p6m	Bisected hexagonal tiling
Snub hexagonal tiling	3.3.3.3.6 | 6 3 2 p6 or 632	Floret pentagonal tiling

Non-Wythoffian uniform tiling

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings
Elongated triangular tiling	3.3.3.4.4 2 | 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.

External links

• Eric W. Weisstein, Uniform tessellation at MathWorld.
• Uniform Tessellations on the Euclid plane
• Tessellations of the Plane
• David Bailey's World of Tessellations
• k-uniform tilings
• n-uniform tilings

Wikipedia content modification information:

• This page was last modified on 20 September 2008, at 17:32.

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