Escher-Like Tessellations on Spherical Models

Abstract

Everyone has played with a kaleidoscope in which a few colored stones reflect through the mirrored edges of an equilateral triangle to create attractive ornaments in the plane. The surface of a sphere also has triangles with the same property — reflections through mirrored edges of these triangles cover the whole sphere. Those special spherical triangles are referred to as Möbius triangles. In his lifetime, M.C. Escher produced many interesting drawings of planar ornaments (tessellations) and also carved the surface of some wooden balls with interlocked designs that repeat by reflection or rotation according to the symmetry of an octahedron, tetrahedron, or dodecahedron.