Since the discovery of quasicrystals began to percolate through the scientific community, thirty years ago, Delaunay sets have been the tool of choice for describing their structures geometrically. These descriptions have gradually evolved from tiling vertex models to random cluster models, as the structures of real and simulated quasicrystals have been clarified experimentally. In this paper I outline these developments and explain why this productive dialogue between mathematicians and materials scientists will continue. © 2015, Pleiades Publishing, Ltd.
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