Combinatorial and Computational aspects of Tilings

The talk is based on joint word with Michael Baake and Dirk Frettlöh. Diffraction images with continuous rotation symmetry emerge from amorphous systems, but also from regular crystals when investigated via powder diffraction. Pinwheel patterns also display this symmetry, in spite of being perfectly ordered. We consider statistical properties needed to understand and to compare the diffraction images. A new substitution rule for the pinwheel tiling, with two different prototiles, permits the derivation of several results for this still somewhat enigmatic example. They are contrasted with properties of the square lattice and its powder diffraction.