Why do quasicrystal patterns not repeat at regular intervals?

Quasicrystal patterns do not repeat at regular intervals because they possess quasiperiodic rather than periodic translational order.1

In ordinary crystals, atoms are arranged in a repeating unit cell, which allows for translational periodicity—a structure that remains invariant when shifted by specific, regular distances. Quasicrystals, however, are composed of different structures assembled in a nonrepeating array. While they exhibit long-range order governed by deterministic rules, they lack a finite unit cell that repeats at equal intervals.123

The absence of periodicity is fundamentally linked to the presence of rotational symmetries that are forbidden in standard crystallography, such as fivefold symmetry. If a structure were to simultaneously possess both fivefold rotational symmetry and translational periodicity, it would result in a mathematical paradox where there is no lower limit to the spacing between atoms. Consequently, quasicrystals utilize quasiperiodic patterns—such as the Fibonacci sequence or Penrose patterns—to maintain order without the repetition required by periodic lattices.21