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

$$(2\ 2\ n'), ({\textstyle{3\over 2}}\ {\textstyle{3\over 2}}\... ...e{5\over 4}}\ {\textstyle{5\over 4}}\ {\textstyle{5\over 4}}).$$

The others can be derived from

$$(p\ q\ r)=(p\ x\ r_1)+(x\ q\ r_2),$$

where

$${1\over r_1}+{1\over r_2}={1\over r}$$

and

$$\cos\left({\pi\over x}\right) = -\cos\left({\pi\over x'}\right)$$

$$= {\cos\left({\pi\over q}\right)\sin\left({\pi\over r_1}\right)-\co... ...er p}\right)\sin\left({\pi\over r_2}\right)\over\sin\left({\pi\over r}\right)}.$$

See also Colunar Triangle, Spherical Triangle

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.