Non-Euclidean Symmetry and Indra's Pearls

Caroline Series and David Wright
Bridges London: Mathematics, Music, Art, Architecture, Culture (2006)
Pages 25–32


Escher's well known picture of devils and angels is an example of a symmetrical tiling of two dimensional hyperbolic space. We discuss similar symmetries of three dimensional hyperbolic space, modelled as the inside of a solid ball. The `shadows' of the solid tiles on the boundary of the ball themselves form patterns governed by a new kind of symmetry, that of Mobius maps on the complex plane. All aspects of such pictures, together with instructions for making them, are explored in the authors' book Indra's Pearls. We give examples of beautiful fractal patterns created in this way.