Given any natural numbers l, m, n > 1 exactly one of the classical two-dimensional geometries (Euclidean, spherical, or hyperbolic) admits a triangle with the angles (π/l, π/m, π/n), and the space is tiled by reflections of the triangle.

Spherical triangle groups can be identified with the symmetry groups of regular polyhedra in the three-dimensional Euclidean space: Δ(2,3,3) corresponds to the tetrahedron, Δ(2,3,4) to both the cube and the octahedron (which have the same symmetry group), Δ(2,3,5) to both the dodecahedron and the icosahedron.

The groups Δ(2,2,n), n > 1 of dihedral symmetry can be interpreted as the symmetry groups of the family of dihedra, which are degenerate solids formed by two identical regular n-gons joined together, or dually hosohedra, which are formed by joining n digons together at two vertices.

The spherical tiling corresponding to a regular polyhedron is obtained by forming the barycentric subdivision of the polyhedron and projecting the resulting points and lines onto the circumscribed sphere.

[1] The triangle group is the infinite symmetry group of a tiling of the hyperbolic plane by hyperbolic triangles whose angles add up to a number less than π.

All triples not already listed represent tilings of the hyperbolic plane.

[3] The group D(l,m,n) is defined by the following presentation: In terms of the generators above, these are x = ab, y = ca, yx = cb.

Geometrically, the three elements x, y, xy correspond to rotations by 2π/l, 2π/m and 2π/n about the three vertices of the triangle.

), which has order l and is thus identical as an abstract group element, but distinct when represented by a reflection.

For example, the Schwarz 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.

In Grothendieck's theory of dessins d'enfants, a Belyi function gives rise to a tessellation of a Riemann surface by reflection domains of a triangle group.