Whether 3-byte patterns prefer icosahedral directions.
Projects each triple of consecutive bytes onto the 30 roots of H3 — the vertices of an icosidodecahedron. These roots split into 6 axis-aligned vectors (permutations of (1,0,0)) and 24 "golden" vectors involving the golden ratio (even permutations of (1/2, phi/2, 1/2phi)). H3 is the symmetry group of the icosahedron, featuring 5-fold rotational symmetry that no crystallographic lattice can achieve. The geometry asks whether the data's 3-byte structure has any preference for these non-crystallographic directions.
Variation in root alignment quality. Symbolic Lorenz (0.856), fBm persistent (0.854), and Potomac river flow (0.849) cluster at the top — continuous signals with smooth trends produce 3-byte windows of varying alignment. Thue-Morse scores near 0 (its binary structure creates windows with almost identical alignment). The tight spread at the top (0.85 ± 0.01 for many natural signals) suggests that alignment variability is more about smoothness than about icosahedral affinity per se.
Fraction of the 30 roots actually used. BTC close, wave height, and GOES X-ray all score 0.333 (10 of 30 roots). Logistic period-3 uses only 1 root (0.033). The 1/3 ceiling for natural signals means they occupy roughly a 10-dimensional subspace of the full H3 root system — the golden roots are partially redundant for typical 3-byte data.
Uniformity of root usage. Windows PE (0.524) and surface wind (0.516) have the most uniform root distributions. Periodic orbits score near 0.0 (all weight on one root). Moderate entropy combined with low diversity means data is spread across a few roots but not uniformly.
| Source | Domain | Value |
|---|---|---|
| Symbolic Lorenz | exotic | 0.8563 |
| fBm (Persistent) | noise | 0.8540 |
| Potomac River Flow | geophysics | 0.8493 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Thue-Morse | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| BTC Close Price | financial | 0.3333 |
| Wave Height (Buoy) | geophysics | 0.3333 |
| GOES X-Ray Flux | astro | 0.3333 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.0333 |
| Source | Domain | Value |
|---|---|---|
| Windows PE x86-64 | binary | 0.5238 |
| Surface Wind (ORD 5-min) | climate | 0.5161 |
| Nikkei Returns | financial | 0.5118 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | chaos | -0.0000 |
| Logistic r=3.2 (Period-2) | chaos | -0.0000 |
| Logistic r=3.5 (Period-4) | chaos | -0.0000 |
H3 is the only geometry in the framework probing 5-fold symmetry — the symmetry of quasicrystals, viruses, and Penrose tilings. Its 3-byte window size puts it between G2 (pairs) and D4/H4 (quadruples), catching intermediate-scale correlations. In practice, H3's discrimination power is modest: the diversity_ratio ceiling at 0.333 and the narrow alignment_std range for natural signals limit its ability to separate sources. Its primary role is structural: it rounds out the Coxeter family (G2, H3, D4, H4, E8) across window sizes 2 through 8.