Tiling

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

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

2D
Two-dimensional fillings of space are generally referred to as tilings or tessellations.

Regular tilings
A regular tiling in 2D space is vertex-, edge-, and face-transitive. There are only three such tilings in Euclidean 2D space.

Semiregular tilings
Like a regular tiling, a semiregular tiling has only one type of vertex, and edges that are all of equal length. However, it can have different kinds of faces. Semiregular tilings can also be called Archimedean tilings or 1-uniform tilings. There are eight such tilings in Euclidean 2D space.

Duals of tilings can be constructed in the same way they are for polytopes: by "exchanging" the facets and the vertices. Since the semiregular tilings are vertex-transitive, their duals will be face-transitive.

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

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

3D
Three-dimensional fillings of space are usually referred to as honeycombs.

Regular and cell-transitive honeycombs
There is just one regular honeycomb in Euclidean 3D space: the cubic honeycomb.

A polyhedron that can fill space on its own, using only more copies of itself, is regarded as space-filling. Other than the cube, such polyhedra include the hexagonal prism, truncated octahedron, and the rhombic dodecahedron and its elongation. These can all fill space with only translational symmetries. The triangular prism is also space-filling, but requires rotational symmetries to do so.

Semiregular honeycombs (based on cubic honeycomb)
Like in 2D, regular honeycombs can be modified in much the same way as polytopes, yielding semiregular honeycombs that are vertex-transitive.

Higher dimensions
A honeycomb analog in n dimensions can be referred to as an n-honeycomb, or by one of many competing unofficial names. An n-dimensional hypercube can, almost by definition, fill the space it inhabits.