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.

A tiling in n dimensions can be thought of as an infinite n+1-dimensional polytope.

2D
Two-dimensional fillings of the plane by polygons are generally referred to as tilings or tessellations.

Regular tilings
A regular tiling in 2D space is vertex-transitive, edge-transitive, 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 (because they can be derived from the regular tilings in the same way that the Archimedean solids are derived from the Platonic solid s) 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 "swapping" the facets and the vertices. Since the semiregular tilings are vertex-transitive, their duals are 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 hyperbolic (non-Euclidean) space, the interior angles of faces around a vertex can add up to more than 360°, permitting tilings that would have been impossible otherwise.

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

Regular and cell-transitive honeycombs
There is only 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.

Uniform honeycombs
Like in 2D, the regular (cubic) honeycomb can be modified like a polytope, yielding vertex-transitive honeycombs. Uniform 2D tilings can also be made into prism s that qualify as uniform honeycombs. There are 28 such honeycombs.

Hyperbolic honeycombs
In hyperbolic 3D space, the dihedral angles of cells around an edge can add up to more than 360°.

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.