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; these 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 its group notation: the sequence of the reflection orders of the fundamental domain vertices.

A fundamental domain triangle is denoted (p q r), where p, q, r are whole numbers > 1, i.e. ≥ 2; a fundamental domain right triangle is denoted (p q 2). 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.

There are several symbolic schemes for denoting these figures:

The modified Schläfli symbol for a right triangle domain: (p q 2) → {p, q}.
The Coxeter-Dynkin diagram is a triangular graph with p, q, r labeled on the edges. If r = 2, then the graph is linear, since diagram nodes with connectivity 2 are not connected to each other by a diagram branch (since domain mirrors meeting at 90 degrees generate no new mirrors).
The Wythoff symbol takes the three integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite to a domain vertex, then the reflection order of this domain vertex is given before the bar.
Finally, a uniform tiling can be described by its vertex configuration: the (identical) sequence of polygons around each (equivalent) 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 and vertices). 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.)