Quasiperiodic

Quasiperiodic describes a phenomenon, signal, or pattern that involves a finite set of periodic components whose frequencies are incommensurate, so the overall repetition never occurs exactly. A function f(t) is quasiperiodic if it can be written as a finite sum of harmonics with incommensurate frequencies: f(t) = Σ_{k=1}^N A_k cos(2π ω_k t + φ_k), where the ω_k are rationally independent. Equivalently, quasiperiodic motion can be viewed as the projection of a periodic motion on a higher-dimensional torus onto a line or trajectory in ordinary space.

If the frequencies are incommensurate, the signal never repeats exactly, though it may come arbitrarily close to past values. Quasiperiodic systems often arise in nearly integrable dynamical systems, where motion on invariant tori can be described by a set of angles θ_i = ω_i t + θ_i0 on a torus.

In mathematics and physics, quasiperiodicity also appears in tiling problems and crystallography. Quasiperiodic tilings, such as Penrose tilings, cover space without translational symmetry but with long-range order; their Fourier spectra are discrete and nonperiodic. Related concepts include almost periodic functions, of which quasiperiodic functions are a common subclass, and aperiodic tilings, which lack translational symmetry in more general ways.

Quasiperiodicity has applications in signal processing, astronomy, and materials science, among others, where complex but non-repeating patterns are analyzed or modeled.