Why can a quasicrystal have long-range order without translational periodicity?

Quasicrystals possess long-range order without translational periodicity because their atomic structures are quasiperiodic rather than periodic.1

While periodic structures rely on a repeating unit cell, quasiperiodic patterns are governed by deterministic rules that dictate the placement of atoms across the entire material. These rules ensure that the pattern is globally ordered and predictable, yet it never repeats itself at regular intervals.1

This phenomenon can be understood through the following principles:

• Deterministic Rules: Similar to the Fibonacci sequence, quasicrystals follow specific mathematical growth rules that create long-range order without requiring a finite, repeating unit cell.1
• Non-repeating Arrays: Quasicrystals are often composed of different structural units assembled in a nonrepeating array, similar to a Penrose pattern, which maintains orientational order while lacking translational symmetry.12
• Diffraction Patterns: Despite the lack of periodicity, quasicrystals exhibit discrete Bragg peaks in their diffraction patterns, confirming that they possess a high degree of structural order rather than the random arrangement found in amorphous solids.1

In essence, the long-range order is a result of these strict, deterministic placement rules that extend throughout the material, even though the specific local arrangement of atoms does not recur periodically.1