Does the signal show Golden Mean scaling at the squared level — the signature of decagonal quasicrystals?
Tests spectral self-similarity at the golden-ratio-squared (phi^2 = 2.618...) rather than the golden ratio itself. Decagonal quasicrystals (like Al-Ni-Co alloys) have 10-fold rotational symmetry. They are periodic along one axis and aperiodic in the perpendicular plane, creating columnar structures. The phi^2 test separates genuine decagonal order from the simpler fivefold Penrose structure.
Spectral self-similarity under phi^2 scaling. Circle Map QP, Phyllotaxis, and Fibonacci QC all score 1.0 — exactly the same three sources that max out Penrose's fivefold_symmetry. This is mathematically inevitable: if the spectrum is self-similar at ratio phi, it's automatically self-similar at phi^2 (two applications of the same scaling). Henon Map, Lorenz, and Divisor Count score 0.0 — no golden-ratio structure at any power.
| Source | Domain | Value |
|---|---|---|
| Circle Map Quasiperiodic | chaos | 1.0000 |
| Phyllotaxis | bio | 1.0000 |
| Fibonacci Quasicrystal | number_theory | 1.0000 |
| ··· | ||
| Henon Map | chaos | 0.0000 |
| Lorenz Attractor | chaos | 0.0000 |
| Divisor Count | number_theory | 0.0000 |
Decagonal's perfect overlap with Penrose on the top-scoring sources reveals an important structural fact: no source in the atlas has phi^2 symmetry without also having phi symmetry. In real materials, decagonal quasicrystals differ from Penrose tilings because of the additional periodicity along the third axis — a distinction that cannot manifest in 1D data. Decagonal is most useful as a consistency check: if a signal scores high on Penrose fivefold_symmetry but zero on Decagonal phi_squared_ratio, that would indicate partial golden-ratio structure (perhaps phi-scaled spacing without phi^2 scaling), which would be a novel finding. No such case exists in the current atlas.