In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 space, hyperbolic plane. Uniform tilings ar ...
or
uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also ...
) 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 fa ...
. Different
symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
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
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
.
n-uniform figures
In addition, an ''n''-uniform coloring is a property of a ''uniform figure'' which has ''n'' types
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
, 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
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
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