This floor is built from exactly two shapes: a fat diamond and a thin one. Press deflate and every diamond splits into smaller diamonds by a single fixed rule. You can do it forever. Drag to wander, scroll to lean closer. And watch the counter under the floor: it is quietly proving that no matter how far this pattern grows, it will never, ever repeat itself.
For the curious kid.
A checkerboard repeats: cut out a little square of it, stamp it over and over, and you rebuild the whole floor. This floor has no such stamp: no patch of it, however large, tiles the rest by repetition. How could anyone know that, about a floor that goes on forever? Count the diamonds. Every time you deflate, the counter shows more fat diamonds than thin ones, and the ratio creeps toward the golden ratio, 1.61803…, a number that is famously not a fraction. A repeating floor would be made of copies of one stamp, so its ratio of fat to thin would have to be a fraction: this many fat, that many thin, repeated. The ratio here refuses to be a fraction. So the floor refuses to repeat. An infinite fact, caught by counting.
Deeper.
This is Penrose's P3 tiling, drawn by substitution: each rhombus is two mirrored triangles (fat = a pair of 36°-36°-108° triangles, thin = a pair of 36°-72°-72°), and one deflation step replaces every triangle with two or three smaller ones, shrunk by the golden ratio φ. The triangle counts evolve by a fixed matrix whose leading eigenvector has slope exactly φ. That is what the counter is converging to, and since φ is irrational, no periodic tiling can match it: a period would make the fat-to-thin ratio rational. Aperiodicity, from linear algebra.
Penrose found this pair of tiles in 1974, whittled down from a set of 20,426 tiles that logicians had needed for the first aperiodic tiling in 1966. For decades it looked like a mathematician's curiosity, until 1982, when Dan Shechtman saw a "forbidden" tenfold symmetry in the diffraction pattern of an aluminium alloy: matter itself, tiled without repetition. He was told to leave his research group; he got the Nobel Prize in 2011. (And the story is not over: in 2023 a single 13-sided tile, the "hat", was found to do alone what Penrose's pair does together. It may get its own room here one day.)
A rule small enough to hold in one hand, repeated without mercy, can build a thing that never once repeats itself.