This Demonstration shows Möbius triangles on a sphere. Möbius triangles are congruent spherical triangles that completely cover a sphere without overlap. The angles of these triangles are , , and , where , , and are in the set . Such a Möbius triangle is denoted by . There is an infinite family of dihedral Möbius triangles, and three triangles , , and that correspond to the tetrahedral, octahedral, and icosahedral symmetry groups, respectively.
The numbers must satisfy the inequality [1, p. 157]. The definition of Möbius triangle was taken from [2, p.189].
 W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover, 1987 pp. 133, 160.
 R. E. Maeder, The Mathematica Programmer II, San Diego: Academic Press, 1996.