Résumé
Colorings of the discrete plane (i.e., tilings) are a geometrical model which is intimately linked with computability theory. We show in this manuscript how many recent results in tiling theory can be unified through the concept of basis and antibasis: A property P is a basis if any tiling space contains a point with property P. We then discuss the various ways to encode computation in tilings. We introduce a new encoding that gave a sparse grid, and explain how to characterize Turing degrees of tilings using this grid. Finally we discuss tilings for the point of view of model theory. We characterize various important classes of tilings by logical fragments of monadic second order theory