Aperiodic Order: Thue-Morse & Fibonacci Diffraction

Aperiodic sequences exhibit singular continuous or pure-point diffraction spectra — neither crystal nor glass, a third type of order.

N=256
1.00
Thue-Morse: t(n) = parity of popcount(n).
Substitution: 0→01, 1→10.
Diffraction: singular continuous spectrum — pure point on a Cantor-like set.

Fibonacci: 0→01, 1→0. Ratio → φ. Pure point diffraction (quasicrystal) — peaks at k = (m + nφ)·k₀.

Ruler: singular continuous, multifractal spectral measure.

The diffraction measure is |F̂(k)|² where F̂ is the Fourier transform of the point measure Σ f(n)δ(x−na).