Tiling

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

Definition
There isn't a standardized definition of tilings, but the following covers regular, semiregular and uniform tilings: an n-dimensional tiling is an abstract (n + 1)-polytope along with a realization that maps vertices to distinct points in some n-dimensional space such that all elements have "valid" realizations by some definition (in Euclidean space planarity is standard). Tilings are always dyadic, which prevents open edges.

In Euclidean space, tilings that are studied are almost always countably infinite with an unbounded set of vertex points. However, spherical and projective polytopes are often defined as tilings and are not necessarily infinite. The important distinction between a tiling and an ordinary polytope is that an n-polytope's space is n-dimensional, whereas a tiling's space is (n - 1)-dimensional. A notable corollary is that (assuming standard constraints of polytope realizations) while the elements of a polytope are polytopes, the elements of a tiling are generally not tilings, but simply polytopes -- therefore tilings do not "recurse" with increasing rank the way polytopes do.

Properties
A tiling is non-self-intersecting if all facets (also called tiles) are non-self-intersecting and the interiors of all elements are disjoint. This depends on a definition of "interior" which is virtually unambiguous for non-self-intersecting finite polygons, but not as easy to define for cases like an apeirogon. The vast majority of tilings studied in mathematics research are non-self-intersecting.

A tiling is convex if it is non-self-intersecting and all elements are convex.

In the case of non-self-intersecting tilings, it's often implicit that every point in the space belongs to the interior of an element. The above definition of self-intersecting tilings allows for holes, overlaps, and elements with no clearly defined interior.

A tiling is called dense if there is a bounded set in the space that contains an infinite number of vertices.

Isotopic tilings are face-transitive tilings, as for any other polytope. A tiling is monohedral if its tiles are merely all congruent.

For non-self-intersecting 2D tilings, the condition of dyadicity requires "T-intersections" at edges to be interpreted with an additional vertex. For example, for the classic offset "brick wall" design to produce a valid abstract polytope, the horizontal edges must be split into two edges each, making the tiles abstract hexagons rather than quadrilaterals. If no tile has two adjacent edges in the same line, it is an edge-to-edge tiling. In non-edge-to-edge tilings, the vertices that join colinear adjacent edges are ignored for conditions such as congruency. For example, in the brick wall design, the bricks are abstract hexagons but the split edges are practically ignored, and the tiles are usually called rectangles.

Tilings have symmetries, which are defined identically to ordinary polytopes. Of particular interest are aperiodic tilings that have no nontrivial translational symmetries, and are not "close" to any periodic tiling by small alterations.

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.

The order-2 apeirogonal tiling is sometimes considered a valid regular tiling.

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. There are eight such tilings in Euclidean 2D space, one of which does not maintain the symmetries of its "parent" regular tiling.

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 solids), although this name can exclude the elongated triangular tiling because it lacks the full symmetry of its "parent" regular tiling. Together, the regular and semiregular tilings make up the uniform tilings.

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. Naturally, these become more numerous with higher values of k.

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.

Monohedral non-self-intersecting tilings
Restricting discussion to non-self-intersecting tilings, all triangles and quadrilaterals form isohedral edge-to-edge tilings of the Euclidean plane.

The classification of convex pentagonal monohedral tilings is an open problem. There are 15 known pentagons that produce periodic tilings, some of which have degrees of freedom and many of which are not edge-to-edge or isohedral, and an infinite family of pentagons that produce aperiodic tilings.

Aperiodic tilings
Aperiodic tilings (again restricting discussion to non-self-intersecting tilings) have been intensively studied. Non-edge-to-edge aperiodic tilings constructed from a finite set of tiles are easy to find, as e.g. each column in a square tiling can be displaced by a random amount. Robert Berger's 1964 discovery of a set of 20,426 s sparked interest in the study of aperiodic tilings, as they can form an edge-to-edge aperiodic tiling (Berger's tiles are squares with colored edges, but they can be "notched"). Raphael Robinson found an aperiodic edge-to-edge tiling with just six tiles.

The Penrose tilings, introduced in 1974, are perhaps the best known examples of aperiodic edge-to-edge tilings. The P1 tiling consists of a filled pentagram, a rhombus, and a nonconvex "boat" shape. The P2 and P3 tilings each consists of two quadrilaterals. These tile sets can also produce periodic tilings as well, but "matching rules" (or an equivalent set of notches) can be used to enforce aperiodicity.

A rep-tile is a shape where finitely many non-overlapping copies can be combined to produce a uniformly scaled version of the original shape. Repeating the process recursively may result in a monohedral aperiodic tiling. A particularly well-known example is the sphinx. However, the sphinx is also capable of tiling space periodically, since two copies can form a parallelogram.

The einstein problem asks if there exists a tile which can only tile 2D space aperiodically. The Socolar-Taylor tile found in 2010 almost solved the problem, but is disconnected. The GKMS aperiodic monotile, discovered in 2023, is generally considered the first legitimate solution.

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

A polyhedron that can fill space on its own, using only more copies of itself, is regarded as space-filling. The honeycomb it forms, being made of only one type of cell, is therefore cell-transitive. The cube is the only Platonic solid that can do this; other polyhedra with this property include the hexagonal prism, truncated octahedron, and the rhombic dodecahedron. These can all fill space with only translational symmetries, while the triangular prism requires rotational symmetries to do so.

Regular honeycombs
There is only one regular honeycomb in Euclidean 3D space: the cubic honeycomb.

Uniform honeycombs
Including the cubic honeycomb, there are 28 "uniform" honeycombs in Euclidean 3D space: that is to say, vertex-transitive honeycombs made of uniform components. They can be separated into direct modifications of the cubic honeycomb, modifications of the tetrahedral-octahedral honeycomb (itself a modification of the cubic honeycomb), prisms of uniform 2D tilings, and gyrations and elongations of existing honeycombs.

Since the uniform honeycombs are vertex-transitive, their duals are cell-transitive.

Hyperbolic honeycombs
In hyperbolic 3D space, the dihedral angles of cells around an edge can add up to more than 360°. Not only does this permit conventional polyhedra (usually Platonic solids) to be packed more tightly around an edge than they could be in Euclidean space, but also allows for the use of unconventional forms that could only exist in hyperbolic space, such as packing more than 6 triangles around a vertex.

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.