A solid structure in which there is

1. (1) long-range incommensurate translational order (see incommensurate lattice) and

2. (2) long-range orientational order with a point group, which is not allowed in crystallography.

Condition (1) is called quasiperiodicity. In two dimensions, the fivefold symmetry of a pentagon is an example of a point symmetry, which is not allowed in crystallography, but for which quasicrystals exist. In three dimensions the symmetry of an icosahedron is not allowed in crystallography, but quasicrystals with this symmetry exist (e.g. AlMn). Diffraction patterns for quasicrystals have Bragg peaks, with the density of Bragg peaks in each plane being higher than would be expected for a perfect periodic crystal.

The first known quasicrystals were made artificially in the laboratory, but examples were subsequently found to occur naturally. A notable example of a naturally occurring quasicrystal is icosahedrite, a mineral in which there are atoms that form an icosahedron.