The hallmark of a conventional crystal such as table salt is translational symmetry; a single unit cell repeats at regular intervals. That defining symmetry imposes long-range correlations, but it also greatly restricts the rotational symmetry that crystals can exhibit. Celebrated theorems nearly 200 years old permit only two-, three-, four-, and sixfold symmetry axes.
Quasicrystals do not have the translational symmetry of crystals, but they do have a subtle long-range order. Along any symmetry axis of a quasicrystal, the placement of atoms can be expressed as a sum of two or more periodic functions. But the wavelength ratio for any two of those functions is irrational, and as a result, no single periodic function can represent the atomic positions. The atoms’ quasiperiodicity—indeed, “quasicrystal” is short for quasiperiodic crystal—implies long-range correlations and, as with conventional crystals, allows quasicrystals to display sharp, structure-revealing diffraction patterns. The freedom from translational invariance enables quasicrystals...
