Uniform tessellation

In geometry, a uniform tessellation is a vertex-transitive tessellation made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex.

An n-dimensional uniform tessellation can be constructed on the surface of n-spheres, in n-dimensional Euclidean space, and n-dimensional hyperbolic space.

Nearly all uniform tessellations can be generated by a Wythoff construction, and represented by a Coxeter-Dynkin diagram. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform polychoron, uniform polyteron, uniform polypeton, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson.

Wythoffian tessellations can be defined by a vertex figure. For 2-dimensional tilings, they can be given by a vertex configuration listing the sequence of faces around every vertex. For example 4.4.4.4 represents a regular tessellation, a square tiling, with 4 squares around each vertex. In general an n-dimensional uniform tessellation vertex figures are define by an (n-1)-polytope with edges labeled with integers, representing the number of sides of the polygonal face at each edge radiating from the vertex.

Examples

2-dimensional tessellations
Spherical Euclidean Hyperbolic

Coxeter-Dynkin
Picture
Truncated icosidodecahedron.

Truncated trihexagonal tiling.

Truncated triheptagonal tiling
(Poincaré disk model)

Truncated triapeirogonal tiling
Vertex figure
3-dimensional tessellations
3-spherical 3-Euclidean 3-hyperbolic
and paracompact uniform honeycomb
Coxeter diagram
Picture
(Stereographic projection)
16-cell

cubic honeycomb

order-4 dodecahedral honeycomb
(Beltrami-Klein model)

order-4 hexagonal tiling honeycomb
(Poincaré disk model)
Vertex figure
(Octahedron)

(Octahedron)

(Octahedron)
(Octahedron)