Materials with an order between crystalline and amorphous are called quasicrystals (Nelson 1986). In contrast to a crystal, which is formed by a periodic repetition of one unit cell, these quasicrystals can be assembled by an aperiodic repetition of two different unit cells. One or both of these unit cells may have fivefold symmetry as first shown by Shechtman et al. (1984). This symmetry is forbidden for crystals since it cannot fill space without overlap or voids.

Quasicrystals have no long-range order (Sokoloff 1985) even though there is no variation of bond length or bond angle: all atoms fit into one of the two unit cells (Kramer and Neri 1984; Levine and Steinhardt 1984; Cahn et al. 1986; Henley 1987; Steinhardt 1987). Similar quasicrystalline behavior can be modeled with specific (Fibonacci) sequences of superlattices (Todd et al. 1986) (see the following section).

In two dimensions, the arrangement of tiles into a completely filled pattern pro­vides an instructive example. While only three types of regular polygons (namely triangles, squares and hexagons) can fill a given flat surface by themselves, the use of two types allows many other tile shapes, including one with fivefold symmetry, to fill the same surface completely (Penrose tiling: Penrose 1974; Nelson 1986).

Updated: July 31, 2015 — 2:03 am