Regular Tessellations of the Hyperbolic Plane and the Art of M. C. Escher

A regular tessellation is a covering of the plane by non-overlapping tiles that are all congruent to the same regular polygon.  We will begin by observing that there are only three different regular tessellations of the Euclidean plane.  The search for more interesting regular tessellations will lead us to consider non-Euclidean geometries.  There are two 2-dimensional non-Euclidean geometries: the sphere and the hyperbolic plane.  We will construct an infinite number of different regular tessellations in those geometries.  Finally we will examine some of the drawings of M.C. Escher and discover how they are designed around the non-Euclidean regular tessellations.