Shred a picture completely. It puts itself back.

One rule, applied over and over: slide every row sideways, then slide every column up, and wrap around the edges. Within a few steps the picture is gone. Keep going and it returns — not approximately, but exactly, every pixel home.

What you're seeing

The rule is about as simple as a rule can be. Take the pixel at position x across and y down. Move it to x+y across and x+2y down. Anything that falls off an edge wraps around to the other side. That is the whole thing: a shear, another shear, and a wrap.

Watch what it does. The face stretches along one diagonal and squashes along the other, and the wrap chops the stretched version into stripes and lays them back down side by side. Two pixels that started as neighbours end up more than a picture-width apart within five steps. By step eight there is nothing left to recognise — just an even amber haze.

And then it comes back. Not a blurry approximation of the cat, not a cat with soft edges: the original, pixel for pixel. The reason is that the rule never loses anything. Its determinant is exactly 1, so no two pixels are ever sent to the same place and no destination is ever left empty. Each step is a reshuffle of a finite deck. Keep shuffling the same way and you must eventually deal the original order again.

The genuinely strange part is when. Drag the size dial. At 72 pixels across the picture is home in 12 steps. Shrink it to 64 — a smaller picture, less to scramble — and it takes 48. At 128 it takes 96, at 48 it takes 12 again. There is no trend to find. Making the picture bigger is as likely to shorten the journey as lengthen it.

Press break the rule to see what the invertibility was doing. That swaps one number in the recipe, sending y to x+3y instead of x+2y. The scrambling looks much the same at first. But this version has determinant 2, so pairs of pixels now land on the same square and one of them is simply lost. Dark holes open up, the picture thins out, and it never returns. Scrambling was never the hard part — coming back is what needs the rule to throw nothing away.

The rule, exactly. Each step sends the pixel at (x, y) to (x + y, x + 2y) mod N whose matrix has determinant 1, making it a bijection of the N×N grid. The return period is the multiplicative order of that matrix mod N, computed live on this page rather than looked up. Its eigenvalue (3 + √5) / 2 ≈ 2.618 is the per-step stretch. Verified in node (improve/verify/92-cat-map.js, 49 checks). The map is confirmed a bijection at eight grid sizes, and the period is confirmed two independent ways that must agree — the matrix order, and brute-force iteration of every lattice point — at seventeen sizes. The period is also confirmed minimal, since one step fewer leaves essentially every pixel displaced. Negative controls: the determinant-2 rule collides hundreds of pixels and never restores the grid in 20,000 steps; a determinant-0 rule collapses the grid onto a line; and a plain shear returns too, but only in exactly N steps and with no exponential stretch — so returning is not the rare part, returning after genuine mixing is. The gate also caught an error in this page's own first proof: estimating the stretch as a geometric mean from step zero is biased by the starting vector and reported 2.590, not 2.618.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 92.