Uniform tiling

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere.

Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by the group name represented by the order of the mirrors in sequential vertices.

A fundamental domain triangle is (p q r), and a right triangle (p q 2), where p, q, r are whole numbers greater than 1. The triangle may exist as a spherical triangle, a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of p, q and r.

Finally tilings can be described by their vertex configuration, the sequence of polygons around each vertex.

All uniform tilings can be constructed from various operations applied to regular tilings. These operations as named by Norman Johnson are called truncation (cutting vertices), rectification (cutting vertices until edges disappear), and cantellation (cutting edges). Omnitruncation is an operation that combines truncation and cantellation. Snubbing is an operation of alternate truncation of the omnitruncated form. (See Uniform polyhedron#Wythoff construction operators for more details.)

Coxeter groups for the plane define the Wythoff construction and can be represented by Coxeter-Dynkin diagrams:

There are symmetry groups on the Euclidean plane constructed from fundamental triangles: (4 4 2), (6 3 2), and (3 3 3). Each is represented by a set of lines of reflection that divide the plane into fundamental triangles.

These symmetry groups create 3 regular tilings, and 7 semiregular ones. A number of the semiregular tilings are repeated from different symmetry constructors.

A prismatic symmetry group represented by (2 2 2 2) represents by two sets of parallel mirrors, which in general can have a rectangular fundamental domain. It generates no new tilings.

A further prismatic symmetry group represented by (∞ 2 2) which has an infinite fundamental domain. It constructs two uniform tilings, the apeirogonal prism and apeirogonal antiprism.

The stacking of the finite faces of these two prismatic tilings constructs one non-Wythoffian uniform tiling of the plane. It is called the elongated triangular tiling, composed of alternating layers of squares and triangles.

The only possible fundamental domain in Euclidean 2-space that is not a simplex is the rectangle (∞ 2 ∞ 2), with Coxeter diagram: CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png. All forms generated from it become a square tiling.

There are infinitely many uniform tilings of convex regular polygons on the hyperbolic plane, each based on a different reflective symmetry group (p q r).

The Coxeter-Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.

Further symmetry groups exist in the hyperbolic plane with quadrilateral fundamental domains starting with (2 2 2 3), etc., that can generate new forms. As well there's fundamental domains that place vertices at infinity, such as (∞ 2 3), etc.

Branko Grünbaum, in the 1987 book Tilings and patterns, in section 12.3 enumerates a list of 25 uniform tilings, including the 11 convex forms, and adds 14 more he calls hollow tilings which included the first two expansions above, star polygon faces and vertex figures.

H.S.M. Coxeter et al., in the 1954 paper 'Uniform polyhedra', in Table 8: Uniform Tessellations, uses the first three expansions and enumerates a total of 38 uniform tilings. If a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings.

Besides the 11 convex solutions, the 28 uniform star tilings listed by Coxeter et al., grouped by shared edge graphs, are shown below. For clarity, apeirogons are not coloured in the first seven tilings, and thereafter only the polygons around one vertex are coloured.

All of these tilings are topologically related to the ordinary uniform tilings with convex regular polygons, with 2-valence vertices ignored, and square faces as digons, reduced to a single edge.