Kaleidoscopes repeat a small part of an input image to make periodic images. In a flat Euclidean plane and using mirror symmetry at straight lines we can only have two-, three-, four- and sixfold rotational symmetries. With circle arcs instead of straight lines and inversion in circles instead of mirror symmetry we can freely choose rotational symmetries. This gives us Poincaré disc representations of periodic images in hyperbolic space or stereographic projections of repeated patterns on spheres. I present an efficient iterative computational method for creating such images.