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Schwarz Triangle

The Schwarz triangles are spherical triangles which, by repeated reflection in their indices, lead to a set of congruent spherical triangles covering the sphere a finite number of times.

Schwarz triangles are specified by triples of numbers (p,q,r). There are four "families" of Schwarz triangles, and the largest triangles from each of these families are

(2 2 n^'),(3/2 3/2 3/2),(3/2 4/3 4/3),(5/4 5/4 5/4).
(1)

The others can be derived from

(p q r)=(p x r_1)+(x q r_2),
(2)

where

1/(r_1)+1/(r_2)=1/r
(3)

and

cos(pi/x)=-cos(pi/(x^'))
(4)
=(cos(pi/q)sin(pi/(r_1))-cos(pi/p)sin(pi/(r_2)))/(sin(pi/r)).
(5)

See also

Colunar Triangle, Spherical Triangle, Wythoff Symbol

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References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 112-113 and 296, 1973.Schwarz, H. A. "Zur Theorie der hypergeometrischen Reihe." J. reine angew. Math. 75, 292-335, 1873.

Referenced on Wolfram|Alpha

Schwarz Triangle

Cite this as:

Weisstein, Eric W. "Schwarz Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchwarzTriangle.html

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