Vladimir I. Arnold (1937–2010), a giant of Soviet mathematics, drew a cat on the unit square and applied one linear map: (x,y) → (2x+y, x+y) mod 1. Multiply by the matrix M=[[2,1],[1,1]], wrap around the torus, repeat.
The cat shears, wraps, and dissolves into noise. It looks destroyed. It is not: the map is a bijection — nothing is lost, only rearranged. HISTORY
The determinant of M is det = 2·1 − 1·1 = 1. A linear map with det = 1 preserves area: the unit square maps onto the torus with no squeezing, no gap, no overlap.
Because area is conserved and the entries are integers, on any N×N grid M is a permutation of the cells. This is the seed of everything below. LIT
Arnold's hyperbolic toral map became a textbook exemplar of a chaotic yet reversible system, feeding KAM theory (Kolmogorov–Arnold–Moser), catastrophe theory, and modern ergodic theory.
Today the same permutation is used to scramble images in watermarking and encryption — because it is guaranteed to unscramble. LINEAGE
A recognisable picture on an N×N grid, and the map to apply. Small N so the period computes instantly.
Each step re-positions every pixel by M. Watch it scramble — then, at k = T(N), snap exactly back to the start.
| quantity | value |
|---|---|
| det M | 1 |
| bijection on grid | yes |
| period T(N) | — |
| identical to start? | yes (k=0) |
The eigenvalues of M multiply to 1 and sum to 3, so they are the roots of λ²−3λ+1: λ = (3±√5)/2. One is below 1 (contracting), the larger is above 1 (stretching).
| λ₋ (contract) | λ₊ (stretch) | λ₊ = φ²? | Lyapunov lnλ₊ |
|---|---|---|---|
| — | — | — | — |
The stretch factor is exactly the golden ratio squared, φ² ≈ 2.618. Nearby points fly apart by that factor each step — positive Lyapunov exponent, genuine chaos.
Because the grid is finite and M is a permutation, iterating Mk mod N must eventually return to the identity. That smallest k is the discrete period T(N) — Poincaré recurrence, made exact and computed, not assumed.
| N | T(N) | MT mod N |
|---|
"Chaos means the pattern never repeats."
The stretching (chaos) and the return (recurrence) are not in tension — the finiteness of the grid forces both to be true at once.
Swap M for D = [[2,0],[0,1]] — det = 2, not area-preserving. Watch the witness go red: on an even grid D collapses two cells onto one (mass collides), so it is not a permutation and never cleanly returns.