Arnold's Cat Map - Chaos That Returns Home

Original Image

After 0 iterations

Animation — watch chaos return to order!

Iteration: 0

Grid Size (N×N)

50×50

Pattern

Grid Size 50×50

Period —

Max Period (3N) 150

Current Iteration 0

The Cat Map

(x', y') = (2x+y, x+y) mod N

Matrix: [[2,1],[1,1]]

🔄 The Paradox

Completely chaotic mixing...
yet ALWAYS returns to start!
Period ≤ 3N iterations

🐱 Why "Cat Map"?

Vladimir Arnold illustrated this transformation in the 1960s using a picture of a cat. The map stretches and shears the image, then wraps it back onto the torus (mod N). After enough iterations, the scrambled mess magically reconstructs the original!

🌀 Chaotic Yet Periodic

The map is genuinely chaotic: nearby points separate exponentially (Lyapunov exponent = ln(φ) where φ is the golden ratio!). But on a discrete N×N grid, there are only N² states—so it must eventually return to its starting point.

📐 Area Preserving

The matrix [[2,1],[1,1]] has determinant = 1, so area is preserved (it's a symplectic map). Points get scrambled but never created or destroyed. This connects to Hamiltonian mechanics and Liouville's theorem.

🔐 Cryptography

Send a scrambled image; the recipient applies more iterations until the original appears! The period depends on N (often much less than 3N), so knowing N lets you decrypt. Arnold's map inspired real encryption schemes.