How does fivefold rotational symmetry appear in a quasicrystal?

Fivefold rotational symmetry appears in a quasicrystal as a result of its unique atomic structure, which possesses long-range order but lacks the translational periodicity found in ordinary crystals.1

In a quasicrystal, this symmetry is manifested through the arrangement of atoms in patterns that do not repeat at regular intervals. While fivefold symmetry is forbidden in periodic crystal lattices—because it cannot coexist with translational symmetry without violating the requirement for a minimum interatomic spacing—quasicrystals resolve this through quasiperiodicity.12

Key aspects of this symmetry include:

• Geometric Patterns: High-resolution electron microscopy reveals that atoms are organized in rows that intersect at 72° angles, reflecting the fivefold rotational symmetry observed in the material's diffraction patterns.13
• Quasiperiodic Order: The spacing between these rows follows non-repeating sequences, such as the Fibonacci sequence, where the ratio of interval lengths is defined by the golden mean ($\tau$).12
• Structural Representation: This is often modeled by Penrose patterns, which demonstrate how fivefold symmetry can be maintained across a structure without the need for a repeating unit cell.2