The Dutch artist M. C. Escher is known for his repeating patterns of interlocking motifs. Most of Escher's patterns are Euclidean patterns, but he also designed some for the surface of the sphere and others for the hyperbolic plane, thus making use of all three classical geometries: Euclidean, spherical, and hyperbolic. In some cases it is evident that he applied a transformation to one of his patterns to obtain a new pattern, thus changing the symmetry of the original pattern, sometimes even forcing it onto a different geometry. In fact Escher transformed his Euclidean Pattern Number 45 of angels and devils both onto the sphere, Heaven and Hell on a carved maple sphere, and onto the hyperbolic plane, Circle Limit IV. A computer program has been written that converts one hyperbolic pattern to another by applying a transformation to its motif. We will describe the method used by this program.