Non-Euclidean Kaleidoscopes

Abstract

Tilings of the plane, for example by squares or equilateral triangles, can be produced by reflecting in lines. Symmetrical ‘beach ball’ patterns can be produced by reflecting in great circles on the surface of a sphere. Under stereographic projection, reflection in great circles transfers to a motion of the plane called inversion. Inversion distorts distances but preserves angles. It can be viewed as a generalisation of reflection in which the mirrors are circular. Certain special inversions are the congruences of two-dimensional non-Euclidean geometry. An example of a non-Euclidean tiling is the famous ‘devils and angels’ picture of Escher. Inversions in spheres produce the congruences of three-dimensional non-Euclidean geometry. Very beautiful fractals reminiscent of Paisley patterns can be produced in this way. Problems about such fractals and three-dimensional non-Euclidean geometry are an important topic of current research.