Triangular Symmetry Group


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


and so the only solutions are (2,2,n), (2,3,3), (2,3,4), and (2,3,5) (Ball and Coxeter 1987). The group (2,3,6) gives rise to the semiregular planar tessellations of types 1, 2, 5, and 7. The group (2,3,7) gives hyperbolic tessellations.

See also

Geodesic Dome

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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.

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Triangular Symmetry Group

Cite this as:

Weisstein, Eric W. "Triangular Symmetry Group." From MathWorld--A Wolfram Web Resource.

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