Triangular Symmetry Group
TriangleSymmetryGroups
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.
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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.Referenced on Wolfram|Alpha
Triangular Symmetry GroupCite this as:
Weisstein, Eric W. "Triangular Symmetry Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangularSymmetryGroup.html