How does the Fibonacci sequence model quasiperiodic order?

The Fibonacci sequence models quasiperiodic order by providing a deterministic rule for arranging elements without repeating a finite unit cell. In this model, two distinct intervals—labeled large (L) and small (S)—are arranged such that the sequence never repeats periodically, yet maintains long-range order.1

The sequence is generated by iterative substitution rules: every L is replaced by LS, and every S is replaced by L. As the generations progress, the ratio of the number of L to S symbols approaches the golden mean ($\tau$). Because there is no fundamental unit that repeats at equal intervals, the pattern is quasiperiodic rather than periodic.1

In quasicrystals, this mathematical structure manifests in the physical spacing of atomic rows. For example, in Penrose patterns, the widths of large and small rows follow the Fibonacci sequence, with their ratio equal to $\tau$. This quasiperiodic arrangement allows for symmetries, such as fivefold rotational symmetry, that are impossible in standard crystallography, where periodic translational order requires a repeating unit cell.1