Lists of uniform tilings on the sphere, plane, and hyperbolic plane

Content sourced from Wikipedia, licensed under CC BY-SA 3.0.

Uniform tilings on the sphere, plane, and hyperbolic plane can all be built by a Wythoff construction inside a fundamental triangle (p, q, r) with angles π/p, π/q, and π/r. Right-triangle cases have p q 2.

A single generator point inside the triangle (with seven possible positions: the three corners, the three points on the edges, and the interior) creates the tiling. All tiling vertices are at the generator or its reflected copies, and edges connect a generator to its mirror across a triangle edge. Up to three different face types can be centered at the triangle’s corners. If the triangle is a right triangle, there can be as few as one face type, giving a regular tiling; more general triangles have at least two face types, leading to quasiregular tilings.

There are different notations for these tilings: Wythoff symbol, Coxeter diagram, and Coxeter’s t-notation. Simple tiles come from Möbius triangles with integer p, q, r; Schwarz triangles allow rational numbers p, q, r and can include star polygon faces and overlapping elements.

For each (p, q, r) (and a few special forms), there are seven generator positions. There are three symmetry classes of reflection on the sphere and three in the Euclidean plane; the hyperbolic plane also contains many more patterns.

Symmetry groups include Euclidean (affine) groups and hyperbolic groups. The basic discussion above covers the integer solutions on the sphere. If Schwarz triangles with rational numbers are allowed, you get the full set of uniform tilings, including nonconvex ones.

In these tilings, each triangle is a fundamental domain, often colored to show even and odd reflections. Selected tilings created by the Wythoff construction are shown as polyhedra; some forms are degenerate, with overlapping edges or vertices, and are shown with brackets in the vertex figures.

Spherical tilings with dihedral symmetry exist for p = 2, 3, 4, …, many with digon faces that degenerate into polyhedra. Two forms (Rectified and Cantellated) are repetitions and are not listed in the table. Some representative hyperbolic tilings are shown in the Poincaré disk projection.

The Coxeter–Dynkin diagram is given in a linear form here, but it is really a triangle with a trailing segment r that connects back to the first node.