Möbius Triangles

dihedral

tetrahedral

octahedral

icosahedral

n for dihedral

3

line

polygon

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

π/p

,

π/q

, and

π/r

, where

p

,

q

, and

r

are in the set

{2,3,4,5}

. Such a Möbius triangle is denoted by

(pqr)

. There is an infinite family

(22n)

of dihedral Möbius triangles, and three triangles

(233)

,

(234)

, and

(235)

that correspond to the tetrahedral, octahedral, and icosahedral symmetry groups, respectively.