Uniform tessellation

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In mathematics, a uniform tessellation is a tessellation of a d-dimensional space, or a (hyper)surface, such that all its vertices are identical, i.e., there is the same combination and arrangement of faces at each vertex.

They can be named by a vertex figure, 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. They can also be named by a Wythoff symbol as well as Coxeter-Dynkin diagrams.

When applied to Euclidean space, the tessellation is most often assumed to be by polyhedra. Examples of 3D regular tessellations are those of layers of right prisms according to the three regular tessellations in 2D; that with square cuboids is in a way the most regular, especially with cubes, because then it is congruent in three independent directions.

[edit] Examples

The spherical truncated icosidodecahedron is a uniform tessellation on the sphere.

The great rhombitrihexagonal tiling is a uniform tessellation of the plane.

The great rhombitriheptagonal tiling is a uniform tessellation on the hyperbolic plane.

The cantellated cubic honeycomb is a uniform tessellation of 3-space.

The regular order-4 dodecahedral honeycomb is a uniform tessellation of hyperbolic 3-space.

When applied to surfaces, uniform tessellations are an important notion for Nonuniform rational B-splines (NURBS).

[edit] See also

• List of uniform tilings
• Honeycomb (geometry)
• Wythoff construction
• Convex uniform honeycomb
• List of regular polytopes