Does the signal have twelvefold diffraction symmetry — the signature of square-triangle Stampfli tilings?
Tests for spectral self-similarity at the dodecagonal ratio (2 + sqrt(3) = 3.732...) and also checks sqrt(3) scaling, which relates to the triangle height in square-triangle tilings. Dodecagonal quasicrystals (found in Ta-Te and V-Ni-Si alloys) have the highest rotational symmetry known in real quasicrystals.
Autocorrelation self-similarity at dodecagonal-ratio lags. Essentially zero across the atlas: Duffing (0.002) and Solar Wind (0.0) are the highest. No 1D source in the atlas has 12-fold quasicrystalline order, which is expected — dodecagonal symmetry is a 2D spatial property that doesn't naturally manifest in 1D time series.
Spectral self-similarity at sqrt(3) scaling, corresponding to the height/base ratio of equilateral triangles in the Stampfli tiling. EEG Eyes Closed (0.466) leads, followed by EEG Eyes Open (0.358) and EEG Healthy (0.321). Brain signals have weak spectral correlations at sqrt(3)-spaced frequencies, possibly related to harmonic relationships in neural oscillation bands. Circle Map QP scores 0.0 — its golden-ratio structure has no sqrt(3) component.
| Source | Domain | Value |
|---|---|---|
| Duffing Oscillator | chaos | 0.0016 |
| Solar Wind Speed | astro | 0.0000 |
| Solar Wind IMF | astro | 0.0000 |
| ··· | ||
| Logistic Chaos | chaos | 0.0000 |
| Henon Map | chaos | 0.0000 |
| Tent Map | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| EEG Eyes Closed | medical | 0.4662 |
| EEG Eyes Open | medical | 0.3583 |
| EEG Healthy | medical | 0.3215 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Circle Map Quasiperiodic | chaos | 0.0000 |
Dodecagonal's most interesting finding is the EEG cluster at the top of triangle_height_ratio. While not strong enough to claim genuine dodecagonal quasicrystalline order, the sqrt(3) spectral correlation in brain signals is consistent across three independent EEG datasets (eyes closed, eyes open, healthy controls). This could reflect the known harmonic relationships between theta (4-8 Hz), alpha (8-13 Hz), and beta (13-30 Hz) bands, where frequency ratios of approximately 1.7 (close to sqrt(3) = 1.732) appear between adjacent bands. The long_range_order metric, like Ammann-Beenker, serves as a specificity control — its near-zero scores validate that high scores on Penrose are ratio-specific.