Structural transformations in quasicrystals induced by higher-dimensional lattice transitions

Quasicrystals constitute a state of matter that exhibits long-range order, but lacks the periodicity in the atomic positions characteristic of crystalline materials. The discovery of these aperiodic structures in 1984 astounded the scientific community, and last year's Nobel prize for Chemistry to Shechtman is testimony to the continuing interest in both the physical and mathematical properties of quasicrystals. In this paper we investigate the rules by which aperiodic tessellations of space (tilings) can deform into each other in ways that preserve as much as possible the properties (symmetries) of the original configuration. We illustrate our results with the famous Penrose and Ammann tilings.