Tiling

A variety of tilings on display.

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 a countably infinite abstract (n + 1)-polytope along with a realization that maps vertices to distinct points in n-dimensional space such that the set of vertex coordinates is unbounded and all elements have "valid" realizations by some definition (in Euclidean space planarity is standard). Tilings are always dyadic, which prevents open edges. Elements may themselves be infinite.

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 defined as for any other polyhedron. 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.

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.

Regular tilings in Euclidean 2D space

Square tiling	Triangular tiling	Hexagonal tiling

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

Semiregular tilings

Main article: List of semiregular tilings of 2D Euclidean space

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.

Examples of k-uniform tilings

Tiling	k	Dual of tiling
	2-uniform
	4-uniform
	7-uniform

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.

Examples of hyperbolic tilings of 2D space

Order-7 triangular tiling	Small rhombitriheptagonal tiling	Deltoidal triheptagonal tiling
regular	semiregular	semiregular dual

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.

3D

A filling of space by bilunabirotundae, dodecahedra and cubes.

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

Main article: List of uniform honeycombs of 3D Euclidean space

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.

Examples of 3D hyperbolic honeycombs

Order-4 dodecahedral honeycomb	Hexagonal tiling honeycomb	Order-5 octahedral honeycomb
{5,3,4}	{6,3,3}	{3,4,5}
4 dodecahedra meet around an edge	3 hexagons around a vertex tile a plane. Here, these tilings also meet 3 to an edge.	5 octahedra meet around an edge

Higher dimensions

A honeycomb analog in n dimensions can be referred to as an n-honeycomb, or by one of many competing unofficial names.

One naming scheme

1D	Sequence
2D	Tiling
3D	Honeycomb
4D	Tetracomb
5D	Pentacomb
6D	Hexacomb
7D	Heptacomb
8D	Octacomb
9D	Enneacomb

An n-dimensional hypercube can, almost by definition, fill the space it inhabits.

External links

• Klitzing, Richard. "Euclidean Tesselations".
• Wikipedia Contributors, "Convex uniform honeycomb".
• Wikipedia Contributors. "Euclidean tilings by convex regular polygons".