geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a uniform coloring is a property of a uniform figure (

uniform tiling In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to t ...

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...

) that is colored to be

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

. Different

symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

can be expressed on the same geometric figure with the

faces The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...

following different uniform color patterns. A ''uniform coloring'' can be specified by listing the different colors with indices around a

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

n-uniform figures

In addition, an ''n''-uniform coloring is a property of a ''uniform figure'' which has ''n'' types

, that are collectively

vertex transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

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 In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

has two colors, but requires 4 unique colors by symmetry positions and become a 2-uniform coloring (right):

References

* Uniform and Archimedean colorings, pp. 102–107

External links

*
Uniform Tessellations on the Euclid plane

Uniform tilings Polyhedra {{geometry-stub