Diffraction and dynamical spectrum
How infinite-volume correlations become Bragg and continuous diffraction, and what finite FFTs can actually show.
From a tiling to a diffraction measure
Diffraction belongs to a specified decoration, not to a tile name. Choose control points xj and weights wj, then form the weighted Dirac comb ω=Σwjδxj. Its autocorrelation γ is the volume-averaged limit of ω restricted to larger regions convolved with its reflected conjugate. The mathematical diffraction measure is the Fourier transform γ̂.[6][35]
Atomic or resonator motifs modify amplitudes through form factors. Moving a repeated decoration within every Spectre changes nearest-neighbor distances, extinctions, and Fourier intensity even when the underlying tile adjacency is unchanged.[30] A control-point theorem therefore cannot be copied unchanged to holes, antennas, struts, or a multiphase specimen.
Bragg and continuous components
The pure-point part of γ̂ consists of delta peaks, Bragg diffraction. Singular-continuous and absolutely-continuous parts are spread rather than concentrated, but they encode different kinds of order and disorder. Pure-point diffraction is not synonymous with “aperiodic,” “substitution,” or “quasicrystalline-looking.” CAP and CASPr model-set constructions provide the extra hypotheses needed for exact Hat- and Spectre-family Fourier-Bohr amplitudes.[31][32][35]
Dynamical spectrum concerns translation acting on the entire tiling hull; diffraction concerns a chosen weighted point set. They are related under standard ergodic hypotheses but are not interchangeable labels. Substitution matrices first provide growth and frequencies; pair correlations and geometric displacement data are additional inputs to spectral classification.[36]
What finite Fourier transforms miss
A finite FFT multiplies the infinite structure by a window. Peak width, sidelobes, pixel smoothing, clipping, and boundary shape can create or conceal weak components. A sharp image is evidence of organized correlations in that approximant, not by itself a proof of pure-point diffraction. Report point weights, physical motif, aperture, normalization, boundary, patch generation, and convergence across increasing windows.
Exact renormalization is stronger. Inflation maps displacement classes across scale, yielding matrix recursions for pair-correlation measures and Fourier cocycles. These equations let large-patch FFTs be checked against hierarchy-implied frequencies rather than interpreted only by eye.[36]
Hat and Spectre evidence
The Hat has several distinct diffraction statements. Socolar gives a six-dimensional Golden Key construction; CAP supplies a regular-model-set description; one vertex decoration also has diffraction on an underlying periodic framework.[6][31][33] Spectre/CASPr has Rauzy-fractal windows and pure-point order, while crystallographic work finds non-periodic chiral sixfold diffraction for a specified decoration.[32][34][35]
Moritake and colleagues fabricated 372,100 circular holes of radius 100 nm at Hat centroids in a roughly 500 × 500 μm, 350-nm silicon-nitride specimen, with pseudo-period swept from 600 to 750 nm. The measured pinwheel twist was 15.52° relative to radial directions. Exact model-set calculations supply complementary intensities: a CAP equal-weight central peak of order 1/(75φ4)≈0.001945 and a CASPr brightest equal-weight peak of order (31−8√15)/972≈1.66×10−5. Position-independent peaks, mirror-reversed pinwheels, and circular- polarization contrast establish planar-chiral optical diffraction for that specimen. They do not prove a band gap or universal Spectre response.[19][35]
See also
Waves, acoustics, and photonics, Cut-and-project schemes and model sets, Hat tile, Spectre tile
Categories: Mathematics · Research frontiers