Does the signal have fivefold diffraction symmetry — the hallmark of Penrose quasicrystalline order?
Computes the power spectrum and autocorrelation, then tests whether their peak structure is invariant under scaling by the golden ratio (1.618...). True Penrose/Fibonacci quasicrystals have spectral peaks at positions related by powers of the golden ratio, creating self-similar diffraction patterns with discrete Bragg peaks — the defining property that separates quasicrystals from both periodic crystals and amorphous matter.
Spectral self-similarity under golden-ratio scaling. Circle Map Quasiperiodic, Phyllotaxis, and Fibonacci QC all score 1.0: their dynamics are governed by the golden ratio, so their spectra are exactly self-similar at that ratio. Thue-Morse scores 0.0 — its substitution ratio is 2, not the golden ratio, so it has no fivefold spectral symmetry. This is the most specific quasicrystal metric in the framework.
Autocorrelation self-similarity at golden-ratio-scaled lags. 1.0 means the autocorrelation pattern repeats at lags related by powers of the golden ratio. Circle Map QP, Phyllotaxis, and Fibonacci QC all score 1.0. Logistic Chaos, Henon Map, and Tent Map score 0.0 — chaotic systems have no long-range autocorrelation structure, let alone ratio-scaled self-similarity.
How concentrated is spectral energy in golden-ratio chains? Uses peak prominences (height above the spectral envelope) to find chains of peaks spaced by factors of the golden ratio. Phyllotaxis (0.506) and Circle Map QP (0.505) have the sharpest chains. Fibonacci QC (0.358) is lower because its spectrum is sparser. This metric was specifically designed to resist false positives from harmonic series, which contain 3-chains matching golden-ratio spacing by coincidence.
Linearity of binary subword complexity growth. Binarizes the signal and counts the number of distinct subwords of length n for n=3..11. Quasicrystalline signals have linearly growing complexity (p(n)=n+1 for Sturmian sequences); periodic signals saturate (p(n) constant); random signals grow exponentially and saturate at 2^n. Scores 1.0 when complexity growth is perfectly linear; 0.0 when it saturates or is degenerate.
| Source | Domain | Value |
|---|---|---|
| Circle Map Quasiperiodic | chaos | 1.0000 |
| Phyllotaxis | bio | 1.0000 |
| Fibonacci Quasicrystal | number_theory | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Thue-Morse | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Projectile with Drag | motion | 1.0000 |
| Damped Pendulum | motion | 1.0000 |
| Lotka-Volterra | bio | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Devil's Staircase | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Circle Map Quasiperiodic | chaos | 1.0000 |
| Phyllotaxis | bio | 1.0000 |
| Fibonacci Quasicrystal | number_theory | 1.0000 |
| ··· | ||
| Logistic Chaos | chaos | 0.0000 |
| Henon Map | chaos | 0.0000 |
| Tent Map | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Phyllotaxis | bio | 0.5060 |
| Circle Map Quasiperiodic | chaos | 0.5052 |
| Fibonacci Quasicrystal | number_theory | 0.3584 |
| ··· | ||
| Divisor Count | number_theory | 0.0000 |
| fBm (Antipersistent) | noise | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
Penrose is the primary golden-ratio quasicrystal detector. The combination of fivefold_symmetry = 1.0 and long_range_order = 1.0 is essentially unique to signals governed by the golden ratio. In the atlas, exactly three source families achieve both: Fibonacci/golden-ratio quasicrystals, phyllotaxis (sunflower-like spiral packing at the golden angle), and the critical circle map at the golden-mean rotation number. All three are manifestations of the same mathematical structure — the golden ratio's unique property as the "most irrational" number.