Whether the arithmetic structure of the signal's dominant frequency ratios is rational, irrational, or rose-curve-symmetric — the epicyclic decomposition is just a Fourier series viewed as nested rotating phasors, and these metrics probe the Diophantine side of the spectrum rather than the power side.
A spirograph with gear ratio p/q closes after q revolutions of the outer gear; irrational ratios never close and fill the plane densely. Applied to a signal, we take its top-k Fourier modes, treat them as phasors, and ask whether their frequency ratios lie near simple fractions, whether their combined trajectory closes, and whether it carries a visible rotational symmetry.
Mean closeness of pairwise frequency ratios among the top-k modes to simple fractions p/q with q ≤ 12. Saturates at 1.0 for many sources because strong modes tend to sit at small-integer harmonics; low values indicate irrational-ratio structure. Very low variance (CV=0.13) — this metric is tight across the atlas, so large deviations are genuinely meaningful. Correlates with Spectral:spectral_entropy (r=-0.48): high rationality tends to co-occur with low spectral entropy (concentrated modes).
Peak height × integer-quality of the strongest rotational symmetry in the trajectory's angular distribution. Mixes "how sharp" with "how clean an integer order" — does NOT carry the integer n itself. High = rose-curve-like pattern with clear n-fold symmetry; near zero = asymmetric or aperiodic. 14.2% floor pile-up reflects how many atlas sources have no detectable rotational symmetry in their top-k mode geometry. Bearing signals (domain mean +1.21) carry strong symmetry — rolling elements generate discrete-harmonic structure — while climate and astro sources are mostly asymmetric.
The integer n in n-fold rotational symmetry, derived from the dominant rotational harmonic of the trajectory's angular density (DFT of f(θ) at 1° resolution). Complements petal_symmetry, which discards the integer order. Returns 0 for circles, uniform annuli, and broadband noise; 3 for noble-ratio 3-fold filigree (Fibonacci, Phyllotaxis, Champernowne); 5 for Rule 110; 7 for Middle-Square PRNG — the first scalar in the atlas that flags Middle-Square's known short-cycle defect (Middle-Square = 7, all other PRNGs = 0). 34% of sources are at the zero floor; DNA composition bias is a downstream finding (Spearman ρ = +0.77 with DNA GC-content).
Max fraction of the top-k modes within ±5% of any single mode's frequency. 0.125 = perfectly spread (8 isolated modes); 1.0 = all 8 in one ±5% cluster. Discriminates two visually-identical 3-fold trajectory mechanisms that angular_order alone cannot tell apart: genuine noble-ratio modes (Fibonacci ≈ 0.4 — spread across the spectrum) vs picket-fence leakage around a single peak (Logistic period-3, Kolakoski ≈ 1.0 — all 8 modes clustered). Bimodal "low-freq trend + clustered peak" sources (Champernowne, DNA Plasmodium) land at ~0.6–0.8. Contributes to PC4.
| Source | Origin | Value |
|---|---|---|
| OTOC Growth | random-matrix-quantum | 15.0000 |
| Spectral Form Factor | random-matrix-quantum | 15.0000 |
| Critical Transition (Fold) | iterated-maps | 12.2000 |
| ··· | ||
| Logistic Chaos | iterated-maps | 0.0000 |
| Henon Map | iterated-maps | 0.0000 |
| Tent Map | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Spike Train | stochastic-process | 1.0000 |
| Quantum Walk | random-matrix-quantum | 1.0000 |
| Bearing Inner | machine-vibration | 1.0000 |
| ··· | ||
| fBm (Persistent) | stochastic-process | 0.1250 |
| fBm (Antipersistent) | stochastic-process | 0.1250 |
| Anderson 1D Localized | random-matrix-quantum | 0.1250 |
| Source | Origin | Value |
|---|---|---|
| 2-Torus Quasiperiodic | coupled-oscillators | 0.6793 |
| Gaussian Collatz Orbit | number-theory | 0.6780 |
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 0.6667 |
| ··· | ||
| Solar Wind IMF | astrophysical | 0.0000 |
| Pink Noise | stochastic-process | 0.0000 |
| Blood Pressure Waveform | physiological | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| fBm (Antipersistent) | stochastic-process | 1.0000 |
| fBm (Persistent) | stochastic-process | 1.0000 |
| Quartic Map (Feigenbaum) | iterated-maps | 1.0000 |
| ··· | ||
| Mian-Chowla | number-theory | 0.4352 |
| Heisenberg Walk | symbolic-dynamics | 0.4454 |
| Copeland-Erdős | number-theory | 0.4593 |
Spirograph asks an arithmetic question the rest of the framework doesn't: not "is this signal periodic" but "are its dominant frequencies rationally related, and if so, with what n-fold geometry." Periodic signals with small-integer harmonic structure saturate gear_rationality. Quasiperiodic signals (irrational rotation number) trace open phasor curves with angular_order = 0. The (angular_order, frequency_packing) joint distinguishes the two "triangle phenotypes" the phasor_trajectory renderer reveals: φ-noble-ratio Fibonacci-class vs picket-fence Logistic-period-3-class. The geometry is small but sits in a corner of the atlas that purely power-spectral geometries miss: the number-theoretic structure of the frequency support. A sister metric closure_deficit (1 − |closure| of the phasor sum) is class-only — kept in compute_metrics for direct callers but dropped from the atlas because r = +0.90 with Spectral Analysis:peak_frequency makes it a near-duplicate of dominant-peak structure; it was the most-discriminating Spirograph metric (F = 6.76) but the cleanest example of "high F via duplication" in the geometry.