The countability of a tiling family and the periodicity of a tiling

If a given finite protoset, together with a given finite matching rule, gives rise to at most countably many different tilings ofd-dimensional space, then at least one of them is periodic. This research was supported in part by the SFB 343 “Diskrete Strukturen in der Mathematik” at Bielefeld University and the Russian Foundation for Fundamental Research.