Arnold's Cat: Cat Map Chaos
Mechanics, Chaos & Fluids arc, page 90 of 193. Back to The Pixelated Physics Universe
Who they are
Arnold's Cat gets sheared into stripes, scrambled into noise and then, on a 124×124 pixel grid, snaps back whole after exactly 15 steps. The map (x, y) → (x + y, x + 2y) mod 1 stretches by a factor of about 2.618 each step and squeezes by 0.382, so area is kept while the picture turns chaotic. On a finite grid no pixel is ever lost, only moved, so the shuffle must repeat. Arnold's Cat on a pixel-art unisex tee; equations, symbols and sources in the panel below.
Arnold's Cat is a character of the Mechanics, Chaos & Fluids arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
Each step stretches the picture in one direction and squeezes it in another, keeping its area, so the cat quickly turns to noise. On a finite grid of pixels the map only shuffles them, and a shuffle repeated enough times must bring every pixel home: on a 124×124 grid, after 15 steps.
$$ \begin{gathered} \Gamma:\ (x, y) \mapsto (x + y,\ x + 2y) \bmod 1 \\ A = \begin{pmatrix}1 & 1\\ 1 & 2\end{pmatrix},\quad \det A = 1,\quad \lambda_\pm = \frac{3 \pm \sqrt{5}}{2} \\ \Pi(124) = 15 \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $x, y$ | coordinates on the unit torus (on an N×N pixel grid: integers, arithmetic mod N) | 1 |
| $\Gamma$ | Arnold's cat map | — |
| $A$ | integer matrix of the map (Arnold–Avez form, acting on the column (x, y)) | 1 |
| $\det A$ | determinant of A; 1 means area is preserved | 1 |
| $\lambda_\pm$ | eigenvalues of A: ≈ 2.618 (stretch) and ≈ 0.382 (squeeze), = φ² and φ⁻² | 1 |
| $\Pi(N)$ | period: number of steps after which every pixel of an N×N grid is back in place | 1 |
| $N$ | grid size in pixels per side | 1 |
Sources
- Arnold, V. I. & Avez, A. (1968) Ergodic Problems of Classical Mechanics, W. A. Benjamin, New York
- Dyson, F. J. & Falk, H. (1992) Period of a Discrete Cat Mapping, Amer. Math. Monthly 99, 603
- Periods Π(124)=15, Π(101)=25, Π(100)=150, Π(5)=10: computed by QUILL by direct iteration of the full grid (2026-11-01)
Crossovers
Arnold's Cat meets Helia (Hénon–Heiles System). Arnold's Cat stretches and folds a square until it is scrambled, the model of strong chaos. Helia's galaxy is mixed: at low energy her orbits stay regular, and at high energy they turn chaotic.
Arnold's Cat meets Reversa (Loschmidt's Reversal Paradox). Arnold's Cat snaps back to her starting picture after a finite number of steps on a finite grid. Reversa sees the same lesson as Poincaré recurrence: reversible dynamics can return, but for a real gas the wait is far longer than the age of the universe.
Arnold's Cat meets Chuanna (Chua's Double Scroll). Chuanna's circuit and Arnold's Cat are both chaotic, but in different ways. The cat is a map that preserves area; Chuanna's flow is dissipative and settles onto a strange attractor.
Wear the lore
Concept keywords: Cat Map Chaos, Arnold's Cat, Mechanics, Chaos & Fluids, physics lore, Pixelated Physics