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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

 1/p+1/q+1/r>1,

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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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 Group

Cite this as:

Weisstein, Eric W. "Triangular Symmetry Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TriangularSymmetryGroup.html

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