Triangle Group - Overlapping Tilings

Overlapping Tilings

Triangle groups preserve a tiling by triangles, namely a fundamental domain for the action (the triangle defined by the lines of reflection), called a Möbius triangle, and are given by a triple of integers, (l,m,n), – integers correspond to (2l,2m,2n) triangles coming together at a vertex. There are also tilings by overlapping triangles, which correspond to Schwarz triangles with rational numbers (l/a,m/b,n/c), where the denominators are coprime to the numerators. This corresponds to edges meeting at angles of aπ/l (resp.), which corresponds to a rotation of 2aπ/l (resp.), which has order l and is thus identical as an abstract group element, but distinct when represented by a reflection.

For example, the Schwartz triangle (2 3 3) yields a density 1 tiling of the sphere, while the triangle (2 3/2 3) yields a density 3 tiling of the sphere, but with the same abstract group. These symmetries of overlapping tilings are not considered triangle groups.