IndexWaves, Chaos & MatterTee

Arnold's Cat

Cat Map Chaos Physics

\[\Gamma:\ (x, y) \mapsto (x + y,\ x + 2y) \bmod 1\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

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.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

For scale: size M is 20 in (50.80 cm) wide, about 3.143 × 1034 Planck lengths (lP = 1.616255 × 10−35 m). A tape measure is quicker.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

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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.

Arnold's Cat has been torn to static more times than anyone can count, and always comes back smug.

In the 1960s the mathematician Vladimir Arnold, working with André Avez, drew a cat on a square and applied a simple rule: send the point (x, y) to (x + y, x + 2y), then wrap anything that falls off the edge back onto the square, as if the square were the surface of a doughnut (a torus). That is "mod 1", the first equation on this page. The rule is the matrix A in the second equation.

Two numbers make it chaotic. det A = 1, so every step keeps area exactly: nothing is compressed or lost. But A has eigenvalues λ± = (3 ± √5)/2 ≈ 2.618 and 0.382 (the golden ratio squared and its inverse). Along one direction the picture is stretched by 2.618 every step, along the other squeezed by 0.382. Nearby points separate exponentially, at a Lyapunov exponent ln 2.618 ≈ 0.962 per step. After a few steps the cat is stripes, then noise. Mathematicians call such maps Anosov and mixing.

Pixels change the ending. On an N×N grid the rule becomes arithmetic mod N, and it simply rearranges the N² pixels; it can be undone (A⁻¹ = [[2, −1], [−1, 1]] is also an integer matrix). A rearrangement of a finite set, repeated, must eventually return every element to where it started. Worked number: for N = 124 that happens after exactly Π(124) = 15 steps, which is why frame 15 on the shirt is the same cat as frame 0. The period depends on N in an irregular way: 101 gives 25 steps, 100 gives 150, 5 gives 10 (each checked by direct iteration; Dyson and Falk, 1992, studied these periods). This exact return is a property of the finite grid. On the continuous torus a typical point never comes back exactly; only rational points are periodic.

Arnold's Cat: "Stretch me, cut me, fold me back. Lose a single pixel? Never. Give me fifteen steps and I will be sitting right here, licking a paw."

Equations

\[\Gamma:\ (x, y) \mapsto (x + y,\ x + 2y) \bmod 1\]
\[A = \begin{pmatrix}1 & 1\\ 1 & 2\end{pmatrix},\quad \det A = 1,\quad \lambda_\pm = \frac{3 \pm \sqrt{5}}{2}\]
\[\Pi(124) = 15\]
SymbolMeaningUnit
\(x, y\) coordinates on the unit torus (on an N×N pixel grid: integers, arithmetic mod N) \(\mathrm{1}\)
\(\Gamma\) Arnold's cat map —
\(A\) integer matrix of the map (Arnold–Avez form, acting on the column (x, y)) \(\mathrm{1}\)
\(\det A\) determinant of A; 1 means area is preserved \(\mathrm{1}\)
\(\lambda_\pm\) eigenvalues of A: ≈ 2.618 (stretch) and ≈ 0.382 (squeeze), = φ² and φ⁻² \(\mathrm{1}\)
\(\Pi(N)\) period: number of steps after which every pixel of an N×N grid is back in place \(\mathrm{1}\)
\(N\) grid size in pixels per side \(\mathrm{1}\)

Sources

  1. Arnold, V. I. & Avez, A. (1968) Ergodic Problems of Classical Mechanics, W. A. Benjamin, New York
  2. Dyson, F. J. & Falk, H. (1992) Period of a Discrete Cat Mapping, Amer. Math. Monthly 99, 603
  3. 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)

Source Arnold, V. I.; Avez, A. (1968). Ergodic Problems of Classical Mechanics. New York: W. A. Benjamin.

Field notes

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