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.
How far the top-k phasor trajectory is from closing on itself. Saturates at 1.0 for nearly-periodic signals (Constant 0x00, Constant 0xFF, Logistic Period-2, Noisy Period-2, Logistic Period-4 all at ≥0.96) — their phasor sum traces a closed or near-closed loop. Also high for Fibonacci Word (0.76), Critical Circle Map (0.72), Circle Map Quasiperiodic (0.68). Low values mean the dominant modes have incommensurate frequencies, so the trajectory never returns. The most discriminating Spirograph metric (F=6.76); redundant with Spectral:peak_frequency (r=+0.88) for many sources but retained because it captures closure specifically, not just peak location.
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).
Discrete rotational symmetry order of the phasor trajectory's angular distribution. 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.
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| Constant 0x00 | noise | 1.0000 |
| Noisy Period-2 | chaos | 0.9743 |
| ··· | ||
| fBm (Persistent) | noise | 0.0003 |
| Perlin Noise | noise | 0.0003 |
| fBm (Antipersistent) | noise | 0.0003 |
| Source | Domain | Value |
|---|---|---|
| Gray Code Counter | exotic | 1.0000 |
| Partition Function | number_theory | 1.0000 |
| Primes | number_theory | 1.0000 |
| ··· | ||
| Earthquake Depths | geophysics | 0.4563 |
| Earthquake Magnitudes | geophysics | 0.4564 |
| Collatz Stopping Times | number_theory | 0.5139 |
| Source | Domain | Value |
|---|---|---|
| 2-Torus Quasiperiodic | chaos | 0.6806 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.6667 |
| Kolakoski Sequence | exotic | 0.6660 |
| ··· | ||
| Divisor Count | number_theory | 0.0000 |
| Rossler Attractor | chaos | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
Spirograph asks an arithmetic question the rest of the framework doesn't: not "is this signal periodic" but "are its dominant frequencies rationally related." Periodic signals with small-integer harmonic structure saturate gear_rationality and closure_deficit together. Quasiperiodic signals (irrational rotation number) show low closure_deficit even when individual modes are strong — their phasor sum never returns. The geometry is small (three metrics) but sits in a corner of the atlas that purely power-spectral geometries miss: the number-theoretic structure of the frequency support.
# Topological Lens