Given a Triangle with angles (, , ), the resulting symmetry Group is called a
triangle group (also known as a Spherical Tessellation). In 3-D, 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.

**References**

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 Math* **4**, 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.

© 1996-9

1999-05-26