Uniform coloring

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.


111
112
123
The hexagonal tiling has 3 uniform colorings.
The square tiling has 9 uniform colorings:
1111, 1112(a), 1112(b),
1122, 1123(a), 1123(b),
1212, 1213, 1234.

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 requires one vertex figure coloring repeated in a periodic arrangement. 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-Archimedean coloring
111112
2-uniform coloring
112344 and 121434

References
  • Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1. Uniform and Archimedean colorings, pp. 102–107


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