The Einstein Stone

Take a floor and a single shape of tile. Almost any shape you choose, if it tiles the floor at all, will let you do it the boring way: lay down a patch, then slide that whole patch sideways by some fixed amount and have it land exactly on more tiles, forever. A pattern with a period — wallpaper, brickwork, a honeycomb. Square tiles do it. Hexagons do it. Even strange shapes do it.

The question that stayed open for fifty years: is there a single tile that refuses? A shape that fills the infinite plane with no gaps and no overlaps, but in a pattern that never repeats — no slide, in any direction, by any distance, ever maps the tiling onto itself? Such a thing is called an aperiodic monotile, or, in a multilingual pun mathematicians could not resist, an einstein — German ein Stein, "one stone."

Roger Penrose came close in the 1970s with a famous pair — his kite and dart force non-repetition, but you need two shapes. For decades after, the count stayed stubbornly at two. Single-tile candidates always cheated: they came with coloured matching rules, or were secretly disconnected, or only worked in three dimensions. A plain, connected, undecorated tile that forced aperiodicity by its outline alone — nobody knew if one could exist.

In November 2022 it turned up in an amateur's cardstock cut-outs. One shape. Thirteen sides. No repeat, ever.

Its discoverer was David Smith, a retired print technician from Bridlington on the Yorkshire coast, who plays with shapes for the pleasure of it. Using a tile-fiddling program he found a 13-sided figure (soon nicknamed the "hat" for its fedora silhouette) that he could not get to repeat. He sent it to the computer scientist Craig Kaplan, who grew enormous patches by machine; they brought in the combinatorialist Joseph Myers and the geometer Chaim Goodman-Strauss, and had a proof about two months after the find (the team formed in January 2023, and Myers proved it in about a week). The hat is real. Here it is.

I · How one tile forbids a pattern

Why can't the hat repeat? The proof is a beautiful piece of bootstrapping. The hats are forced to clump into four kinds of cluster — call them H, T, P, F (the authors call them metatiles). Those clusters are in turn forced to clump into bigger clusters of the same four kinds. And those into bigger ones still. Every tiling of the plane by hats secretly carries this infinite hierarchy of clusters-within-clusters, like Russian dolls running both inward and outward forever.

Each step up the hierarchy magnifies the picture by a fixed amount. Count how many small clusters of each kind sit inside a large one and you get a substitution matrix; its dominant eigenvalue — the factor by which areas grow at each step — is exactly φ⁴ ≈ 6.854, where φ = (1+√5)/2 is the golden ratio. The lengths grow by φ² ≈ 2.618. And here is the hinge of the whole argument: φ is irrational. A repeating pattern has a lattice of whole-number periods, and no lattice can survive being scaled by an irrational factor at every level of an infinite hierarchy. The irrationality of the golden ratio is what makes the repeat impossible.

None of this is mine to assert. The instrument below grows a real patch by running those substitution rules — the published ones — and the live counter at the bottom re-derives the φ⁴ growth in front of you. Drag to pan, scroll or pinch to zoom, and turn up the detail to inflate the hierarchy another level. Look for the repeat. There isn't one.

II · The catch: the hat needs a mirror

Switch the colour mode above to mark the mirrors and you will see the one blemish on the hat's victory. Scattered through every hat tiling, about one tile in eight, are hats that have been flipped over — mirror images of the rest. They are unavoidable. You cannot tile the plane with hats of a single handedness; the pattern demands both the hat and its reflection.

The exact density is gorgeous. In the limit, the fraction of reflected hats is precisely 1 / (φ⁴ + 1) ≈ 0.1273 — one mirror hat for every φ⁴ ≈ 6.854 ordinary ones, about 1 in 7.85. The live census in the instrument counts them as you inflate and watches the ratio home in on that value. (It is the same golden ratio again, surfacing in the bookkeeping of the clusters.)

For most people this is a footnote: the hat is a single connected shape with no markings, and every tiling it makes is non-repeating, so by the textbook definition it solved the einstein problem. But a purist could object. If you are only allowed to rotate and slide tiles — never flip them, as if each tile were printed with a fixed front and back — then the bare hat is not enough. You need a separate supply of mirror tiles. Is there a shape that needs no mirror at all?

Two months after the hat, the same four people answered. Yes.

III · The spectre — a tile that casts no reflection

The hat lives in a whole family of tiles, Tile(a, b), got by stretching its two edge lengths. The hat is Tile(1, √3); its cousin the "turtle" is Tile(√3, 1); the degenerate ends of the family, Tile(1, 0) and Tile(0, 1) — a "comet" and a "chevron" — are the flexible ones that do tile periodically. Sitting exactly in the middle is the equilateral member, Tile(1, 1), a 14-sided shape with all edges the same length.

Tile(1,1) is special. As a bare polygon it is not aperiodic at all: it admits a periodic tiling that uses equal numbers of the tile and its mirror image (Figure 6.2 of the hat paper). Only if reflections are forbidden by fiat does it become aperiodic, which makes it weakly chiral. But replace its straight edges with matching curves (gentle S-bends, an outie answered by an innie) and the mirror image can no longer interlock with the original at all. The result is the spectre: a single curved tile that tiles the infinite plane, never repeats, and does so using only rotations and translations. No flips. No mirror supply. The authors themselves called this the "vampire einstein" problem in the spectre paper, and the press took up the name: the tile that casts no reflection.

Switch the instrument to the spectre and turn on mark the mirrors: the census reports zero. Every tile in the patch has the same handedness. That is the whole point of the second tile, made visible — and, like the hat's patch, this one is built from the published rules and checked for gaps and overlaps before it is drawn.

This continuum is what Joseph Myers saw: the hat and the turtle are not two discoveries but two points on one road. Walk it below. Every shape along the way is the same fourteen-sided tile with its two edge-lengths re-dialled — and the theorem is sharper than "mostly aperiodic": every single member is strongly aperiodic — a genuine einstein — except exactly three points. The two flat polyiamonds at the ends tile periodically; so does the one perfectly balanced tile in the dead centre. Even the ratios that look like they ought to snap to a lattice — a clean 2 : 1 stretch, or 1 : 2 — keep tiling without repeat (the paper dispatches both by hand). Only the 1 : 1 midpoint breaks, and it breaks beautifully: dress its straight edges in curves and it becomes the spectre.

IV · What is actually true here

It would be easy to make a page like this lie — to draw a pretty pattern and call it aperiodic. So nothing here rests on the drawing. The patches are generated by the real substitution systems from the two papers (Craig Kaplan's reference code), reimplemented from scratch and run headless before they were ever put on screen. A verification program then checks the two things a bug would break: that no two tiles overlap (an exact test over every neighbouring pair) and that no interior gaps exist (a fine raster flood-filled from the outside). Both come back clean, for thousands of tiles, for the hat and for the spectre. The substitution matrix's eigenvalue reproduces φ⁴ to fifteen digits, and the reflected-hat density reproduces 1/(φ⁴+1). The numbers in the live readouts are the same numbers the offline checker prints.

The one thing this page does not re-prove is aperiodicity itself: that no periodic tiling exists. That is the theorem of the two papers, and it rests on the irrationality of φ in the inflation, not on any finite picture. What you can verify by hand is everything else: grow a patch as large as your patience allows, slide it against itself, and fail to find the repeat. The full checks can be downloaded and run with Node: verify.mjs (the hat) and spectre-verify.mjs (the spectre), beside the engines they check, engine.mjs and spectre-engine.mjs.