List of uniform tilings

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

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

*
* H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, "Uniform polyhedra", Phil. Trans. 1954, 246 A, 401-50.

External links

*
* [http://www2u.biglobe.ne.jp/~hsaka/mandara/ue2 Uniform Tessellations on the Euclid plane]
* [http://web.ukonline.co.uk/polyhedra/tessellations/tessel.htm Tessellations of the Plane]
* [http://www.tess-elation.co.uk/index.htm David Bailey's World of Tessellations]
* [http://www.uwgb.edu/dutchs/symmetry/uniftil.htm k-uniform tilings]
* [http://probabilitysports.com/tilings.html n-uniform tilings]

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