The piecewise-linear skeleton of the signal.
Tropical geometry — named for the Brazilian mathematician Imre Simon — replaces addition with max and multiplication with addition, turning smooth curves into piecewise-linear ones. This geometry detects how many distinct linear regimes the signal passes through, how diverse their slopes are, and how much area the signal sweeps above its running minimum envelope.
How many times does the signal change its linear regime? Stern-Brocot walk, logistic edge-of-chaos, and Thue-Morse all hit 16,382 (maximum for the sample size — essentially every point is a slope change). Rössler attractor and constants score 0 (smooth or flat). This metric separates "jagged" dynamics (symbolic, chaotic, quasicrystalline) from "smooth" dynamics (oscillators, attractors, drifts).
How many distinct slope values appear? Arnold cat map and beta noise reach 21 (maximum diversity — the signal visits every possible local gradient). fBm scores 1.0 (its smooth increments have one dominant slope). This measures the "vocabulary" of the signal's piecewise-linear representation.
Area between the signal and its running minimum. Forest fire (15,966) dominates: its bursty avalanche dynamics create tall, wide excursions above the baseline. Stern-Brocot walk (11,609) is next — its fractal staircase accumulates massive area. Rainfall (32) is among the lowest nonzero scores: precipitation is close to its own minimum most of the time (many dry hours).
| Source | Domain | Value |
|---|---|---|
| Forest Fire | exotic | 15966.0588 |
| Stern-Brocot Walk | number_theory | 11608.6434 |
| Symbolic Henon | exotic | 10846.8000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Rainfall (ORD Hourly) | climate | 31.8596 |
| Source | Domain | Value |
|---|---|---|
| Stern-Brocot Walk | number_theory | 16382.0000 |
| Logistic Edge-of-Chaos | chaos | 16382.0000 |
| Thue-Morse | exotic | 16382.0000 |
| ··· | ||
| Rossler Attractor | chaos | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Arnold Cat Map | chaos | 21.0000 |
| Beta Noise | noise | 21.0000 |
| ECG Normal | medical | 21.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| fBm (Persistent) | noise | 1.0000 |
Tropical geometry provides a "complexity fingerprint" that's independent of the signal's amplitude distribution. Two signals with identical histograms can have completely different tropical profiles if one is smooth and the other is jagged. In the atlas, slope_changes is tightly correlated with ordinal transition_entropy (both measure local variability), but envelope_area captures a global property (excursion magnitude) that no ordinal metric touches. Tropical is most useful in the symmetry view, where it separates bursty processes (forest fire, Stern-Brocot) from smooth oscillators (Lorenz, Rössler) along the envelope axis.