Uniform tilings in hyperbolic plane

Each symmetry family contains 7 uniform tilings, defined by a Wythoff symbol or Coxeter-Dynkin diagram, 7 representing combinations of 3 active mirrors.

Selected families of uniform tilings are shown below (using the Poincaré disk model for the hyperbolic plane).

Three of them – (7 3 2), (5 4 2), and (4 3 3) – and no others, are minimal in the sense that if any of their defining numbers is replaced by a smaller integer the resulting pattern is either Euclidean or spherical rather than hyperbolic; conversely, any of the numbers can be increased (even to infinity) to generate other hyperbolic patterns.

Because all the elements are even, each uniform dual tiling one represents the fundamental domain of a reflective symmetry: *3333, *662, *3232, *443, *222222, *3222, and *642 respectively.

Because all the elements are even, each uniform dual tiling one represents the fundamental domain of a reflective symmetry: *4444, *882, *4242, *444, *22222222, *4222, and *842 respectively.

For instance in the (4,3,3) triangle family, the snub form has six polygons around a vertex and its dual has hexagons rather than pentagons.

Quadrilateral fundamental domains also exist in the hyperbolic plane, with the *3222 orbifold ([∞,3,∞] Coxeter notation) as the smallest family.

The vertex figure can be extracted from a fundamental domain as 3 cases (1) Corner (2) Mid-edge, and (3) Center.

Snub and alternated uniform tilings can also be generated (not shown) if a vertex figure contains only even-sided faces.

Coxeter diagrams of quadrilateral domains are treated as a degenerate tetrahedron graph with 2 of 6 edges labeled as infinity, or as dotted lines.

A logical requirement of at least one of two parallel mirrors being active limits the uniform cases to 9, and other ringed patterns are not valid.

Example Wythoff construction with right triangles ( r = 2) and the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.
A quadrilateral domain has 9 generator point positions that define uniform tilings. Vertex figures are listed for general orbifold symmetry * pqrs , with 2-gonal faces degenerating into edges.
Example uniform tilings of *3222 symmetry