How does the golden mean relate to fivefold symmetry in quasicrystals?

In the study of quasicrystals, the golden mean ($\tau =\frac{\sqrt{5}+1}{2}$) is fundamental to the existence of fivefold symmetry, which is geometrically incompatible with the translational periodicity found in ordinary crystals.1

The relationship between the golden mean and fivefold symmetry can be summarized as follows:

• Geometric Constraint: If a structure were to possess both fivefold rotational symmetry and translational periodicity, the combination of symmetry operations would result in a displacement smaller than the minimum allowed interatomic spacing. The golden mean acts as the scaling factor by which these displacements shrink, effectively forbidding periodic lattices from exhibiting fivefold symmetry.1
• Quasiperiodic Order: Quasicrystals resolve this paradox through quasiperiodicity. In patterns such as the Penrose pattern, the arrangement of atoms follows a non-repeating, deterministic sequence.2
• Scaling and Sequences: In these structures, the ratio of large-to-small intervals between atomic rows is equal to the golden mean. Furthermore, the sequence of these intervals follows the Fibonacci sequence, which is intrinsically linked to the golden mean.2

Consequently, the golden mean defines the structural proportions that allow quasicrystals to maintain long-range order and fivefold symmetry without requiring the repeating unit cells found in standard crystalline materials.12