Uniform coloring

111	112	123
The hexagonal tiling has 3 uniform colorings.

File:Square tiling uniform colorings.png

The square tiling has 9 uniform colorings:
1111, 1112(a), 1112(b),
1122, 1123(a), 1123(b),
1212, 1213, 1234.

In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following different uniform color patterns.

A uniform coloring can be specified by listing the different colors with indices around a vertex figure.

n-uniform figures

In addition, an n-uniform coloring is a property of a uniform figure which has n types vertex figure, that are collectively vertex transitive.

Archimedean coloring

A related term is Archimedean color which doesn't require transitivity but just requires one vertex figure coloring repeated in a periodic arrangement. And a more general term are k-Archimedean colorings which count k distinctly colored vertex figures.

For example this Archimedean coloring (left) of a triangular tiling has two colors, but requires 4 unique colors by symmetry positions and become a 2-uniform coloring (right):

1-Archmedean coloring 111112

160px 2-uniform coloring 112344 and 121434

References

• Uniform and Archimedean colorings, pp. 102–107

External links

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