The uniform locally finite tilings of the plane

We characterize the uniform locally finite tilings of the plane by finite combinatorial objects called labeling schemes. This representation is based on the analysis of the group of symmetries of a tiling which is realized independently of its geometry. As a consequence, we are able to give an exhaustive enumeration of these tilings. This description encompasses the regular polyhedra and the Archimedean tilings, but most of these are tilings of the hyperbolic plane.