Aperiodic

Aperiodic describes systems or structures that do not admit a fixed repeating pattern. In mathematics, a sequence is periodic if there exists a positive integer p such that a_{n+p} = a_n for all large n; if no such p exists, the sequence is aperiodic. Many sequences defined by substitution rules, such as the Thue–Morse sequence, are aperiodic. Aperiodicity is a central notion in the study of order and complexity and contrasts with both purely random and strictly periodic patterns.

In tiling theory, an aperiodic tiling uses a finite set of tile shapes that can tile the plane only in non-periodic arrangements. The Penrose tilings are among the best known examples. The existence of aperiodic tile sets shows that global non-repeating order can be enforced by local matching rules, leading to rich mathematical structure and models of quasicrystalline order.

In crystallography and physics, aperiodicity occurs in quasicrystals, materials that display long-range order without translational symmetry. Discovered in 1982 by Dan Shechtman, quasicrystals produce diffraction patterns with sharp Bragg peaks and symmetries forbidden to periodic crystals, such as fivefold symmetry. This discovery expanded the understanding of solid-state order and earned the Nobel Prize in Chemistry in 2011.

In computer science, aperiodicity appears in formal language and automata theory: a language is aperiodic if its syntactic monoid is aperiodic, a condition related to star-free expressions and certain regular languages.