Tiling

A tiling, also known as a tessellation or honeycomb in the context of certain dimensions, is a complete filling of space by many copies of a polytope or set of polytopes.

Most tilings involve a high degree of repetition, and can be thought of as infinite polytopes.

Uniform tilings
A uniform tiling has only one type of vertex, and all edges are of equal length.

More complicated tilings called k-uniform tilings, defined as having exactly k types of vertex, are possible as well.

Regular tilings
Regular tilings are a subset of uniform tilings, with the added restriction that all their tiles are the same. Only three exist in Euclidean space: the square, triangular, and hexagonal tilings.

Non-Euclidean tilings
In a non-Euclidean plane, the angles around a vertex can add up to more than 360°, permitting tilings that would have been impossible otherwise.

In three dimensions
Three-dimensional fillings of space are usually referred to as honeycombs.

Polyhedra that can fill space on their own include the cube, triangular prism, hexagonal prism, truncated octahedron, and rhombic dodecahedron.

In higher dimensions
A hypercube of any dimension can fill the space it inhabits