Uniform tessellation

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Uniform tessellation

In mathematicsMathematics

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change....
, a uniform tessellation is a tessellationTessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no ga...
of a d-dimensional space, or a (hyper)surfaceSurface Overview

In mathematics, specifically in topology, a surface is a two-dimensional manifold....
, such that all its vertices are identicalVertex-transitive

Vertex-transitive may refer to:* Vertex-transitive graph...
, i.e., there is the same combination and arrangement of faces at each vertex.

They can be named by a vertex figureFacts About Vertex figure

In geometry, a vertex figure represents the arrangement of a connected set of points of all the neighboring vertices, in a p...
, listing the sequence of faces around every vertex. For example 4.4.4.4 represents a regular tessellation, a square tilingSquare tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane....
, with 4 squares around each vertex. They can also be named by a Wythoff symbolWythoff symbol

In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regu...
as well as Coxeter-Dynkin diagramCoxeter-Dynkin diagram

In geometry, a Coxeter-Dynkin diagram is a graph representing a relational set of mirror in space for a kaleidoscopic constr...
s.

When applied to Euclidean spaceEuclidean space

Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh...
, the tessellation is most often assumed to be by polyhedraPolyhedra

Polyhedra may mean:* Polyhedra DBMS, a relational database system...
. Examples of 3D regular tessellations are those of layers of right prismPrism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces ...
s according to the three regular tessellations in 2D; that with square cuboidCuboid

In anatomy, the cuboid bone is a bone in the foot....
s is in a way the most regular, especially with cubeCube Overview

A cube is a three-dimensional Platonic solid composed of six square faces, facets or sides, with three meeting at each ver...
s, because then it is congruent in three independent directions.

Examples

The spherical truncated icosidodecahedronTruncated icosidodecahedron The truncated icosidodecahedron is an Archimedean solid.... is a uniform tessellation on the sphereSphere A sphere is a perfectly symmetrical geometrical object.... .	The great rhombitrihexagonal tilingGreat rhombitrihexagonal tiling In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane.... is a uniform tessellation of the plane.	The great rhombitriheptagonal tilingGreat rhombitriheptagonal tiling In geometry, the great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane.... is a uniform tessellation on the hyperbolic planeFacts About Hyperbolic plane In mathematics, the term hyperbolic plane may refer to:... .	The cantellated cubic honeycombCantellated cubic honeycomb The cantellated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space.... 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-splineNonuniform rational B-spline

NURBS, short for non-uniform, rational B-spline, is a mathematical model commonly used in computer graphics for genera...
s (NURBS).

Uniform tessellation

Examples

See also