Given a triangle with angles (, , ), the resulting symmetry group
is called a
triangle group (also known as a spherical tessellation). In three dimensions, such
groups must satisfy

and so the only solutions are , , , and (Ball and Coxeter 1987). The group gives rise to the semiregular planar tessellations
of types 1, 2, 5, and 7. The group gives hyperbolic tessellations.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, pp. 155-161, 1987.Coxeter,
H. S. M. "The Partition of a Sphere According to the Icosahedral Group."
Scripta Math4, 156-157, 1936.Coxeter, H. S. M.
Regular
Polytopes, 3rd ed. New York: Dover, 1973.Kraitchik, M. "A
Mosaic on the Sphere." §7.3 in Mathematical
Recreations. New York: W. W. Norton, pp. 208-209, 1942.