Ecole Normale Supérieure de Lyon

We study some decision problems concerning the tiling of the plane with Wang tiles We present a proof for the undecidability of the tiling problem for the whole plane and also for the periodic tiling In these poofs an aperiodic tile set is constructed We are interested in the second part in the link between the cardinality of possible tilings of the plane with a given tile set and the existence of a periodic tiling We are also interested in the property of quasiperiodicity and prove that all tile sets that tile the plane may tile the plane quasiperiodically