# Technical Report No. 2013-3 : From Complex Analysis and Group Theory to Geometry and Art (AU-CAS-MathStats)

Every college dorm displays at least one print by M.C. Escher. Many people admire Escher's work, but do not know its mathematical roots. Escher's interest in divisions of planes goes back to his early work, but the mathematical in uence in his work did not fully appear until he journeyed through the Mediterranean around 1936. Particularly, when visiting La Alhambra in Spain, he became fascinated with the order and symmetry of the tiling. He then studied mathematical papers on topics such as symmetry groups, non-Euclidean geometries, and impossible shapes, later incorporating them into his artwork. When one looks at an Escher print, e.g. Angels and Demons, one immediately sees the complicated tiling within the circle. However, what is not necessarily realized is that the work is a representation of a hyperbolic geometric space. Though inspired by the at tiling of the Alhambra, Escher strayed away from Euclidean geometry in many of his works, creating tiling in spherical and hyperbolic geometries. These three types of geometries make up the world we live in. On a local scale, we live on a at surface, i.e. in Euclidean geometry, but calculating distances on Earth's surface requires spherical geometry. On a larger scale, the universe acts under the laws of hyperbolic geometry, the same as in Escher's Angels and Demons. In a similar manner, much of Escher's work is the visualization of key mathematical concepts from Complex Analysis and Group Theory.