Whether the signal's values cluster around a preferred direction.
Maps each pair of consecutive values to a point on the 2-sphere via spherical coordinates: the first value becomes the polar angle (north-south), the second becomes the azimuthal angle (around the equator). The resulting point cloud on the sphere reveals directional bias that a histogram would miss — two signals with identical value distributions can have completely different spherical profiles if their consecutive-pair correlations differ.
Standard deviation of the angles between each point and the cloud's mean direction. Thue-Morse, L-System Dragon, and Rule 30 all score 1.57 (maximum spread — points uniformly scattered, no preferred direction). Constants score 0.0 (all points at one pole). This is the spherical analogue of standard deviation.
Resultant length of the mean direction vector: how tightly do the points cluster? 1.0 means all points at the same location (Logistic period-2, Collatz gap lengths, Rainfall). Near 0.0 means perfectly diffuse (Thue-Morse at 0.0001). A signal can have high entropy in the Torus geometry but high concentration on the sphere if its consecutive pairs always point in the same angular direction.
How evenly are points split between the northern and southern hemispheres? 1.0 means perfect 50/50 (Thue-Morse, L-System Dragon, Clipped Sine). 0.0 means all points on one side (constants, logistic period-2). This detects asymmetry in the polar angle distribution — signals that spend more time in one "half" of their range.
Average z-coordinate on the sphere (cosine of polar angle). Positive means points cluster toward the north pole (low first-of-pair values, since small values map to theta near 0 where cos(theta) = 1). Collatz gap lengths (1.0) and Rainfall (1.0) are maximally northern: their values are dominated by small numbers, which map to the north pole. Forest fire (-0.96) is maximally southern: its large avalanche values map to the south. This metric encodes distributional skewness through geometry.
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 1.5708 |
| L-System (Dragon Curve) | exotic | 1.5708 |
| Rule 30 | exotic | 1.5708 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| Collatz Gap Lengths | number_theory | 0.9999 |
| Rainfall (ORD Hourly) | climate | 0.9987 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Thue-Morse | exotic | 0.0001 |
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 0.9999 |
| L-System (Dragon Curve) | exotic | 0.9999 |
| Clipped Sine | waveform | 0.9979 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Collatz Gap Lengths | number_theory | 0.9996 |
| Rainfall (ORD Hourly) | climate | 0.9987 |
| Geomagnetic ap Index | geophysics | 0.9798 |
| ··· | ||
| Forest Fire | exotic | -0.9638 |
| Stern-Brocot Walk | number_theory | -0.7557 |
| Logistic Edge-of-Chaos | chaos | -0.4606 |
Spherical geometry separates sources by their directional structure in a way that scalar statistics cannot. Two signals with the same mean and variance can have opposite mean_z values if one is right-skewed and the other is left-skewed. In the atlas, the concentration axis separates noise-like sources (diffuse, concentration near 0) from spike-dominated sources (concentrated, Collatz gaps and rainfall near 1.0). The hemisphere_balance axis is orthogonal to this, catching symmetry vs. asymmetry regardless of concentration.