A quasicrystal is a structure having long-range order but no translational periodicity. Mathematically, quasicrystalline point sets are commonly modeled by Delone sets with finite local complexity and pure point diffraction, or by cut-and-project sets obtained by projecting selected points of a higher-dimensional lattice.
Unlike a periodic crystal, a quasicrystal may possess rotational symmetries incompatible with a periodic lattice, such as fivefold or icosahedral symmetry. One-dimensional substitution sequences and Penrose tilings provide standard mathematical examples of quasicrystalline order.
