Sixty years of shrinking the record: 20,426 tiles, then 104, then 6, then 2
The name is a pun, not a tribute: ein Stein, one stone. The question is whether a single shape can tile the plane while admitting no periodic tiling at all — and it stayed open from 1961 to 2023, not because nobody cared, but because every route to it kept producing near misses.
This page is about the sixty-two years in between: how far the record fell, where it stuck, and what exactly the near misses were missing.
Twenty thousand four hundred and twenty-six, then a hundred and four, then ninety-two, then six, then two. And then nothing, for forty-nine years.
One connected polygon, no decorations, no matching rules. David Smith found it with cut-out cardboard in November 2022; the proof appeared in March 2023.
Caveat: A hat tiling uses both the tile and its mirror image.
The bars are logarithmic, or the last sixty years would be invisible next to the first. Note how the interesting part of the story is not the drop from twenty thousand to six — that took seven years — but the fifty-two years spent stuck at two.
Try it: Click along the entries. The dramatic part looks like the fall from 20,426 to 6 in seven years, but the interesting part is the flat stretch afterwards. Two tiles was reached in 1974 by a route — matching rules, then bumps and notches — that simply had nothing more to give.
It is worth being precise about which half of the problem is hard, because a single tile that covers the plane is not rare at all. Every triangle tiles the plane. So does every quadrilateral — convex or not, symmetric or not, with a construction that never looks at the shape.
No symmetry, no right angles, no equal sides. It still tiles, and the construction did not have to be told anything about it. The white shape is the tile; the dashed blue outline is the same tile turned a half turn about the midpoint of its first side; the two green arrows are the translations that carry the pair over the whole plane. Those arrows are the quadrilateral’s own diagonals, which is why the lattice cell has room for exactly two tiles — the area of any quadrilateral is half the cross product of its diagonals.
Try it: Step through the four shapes, including the dart, which has an interior angle over 180°. Each one tiles, and each tiling is thoroughly periodic — the two green arrows are exactly the periods. The difficulty was never covering the plane. It was covering it in a way that cannot be made to repeat.
“A single aperiodic tile” is four requirements wearing one coat, and every near miss failed a different one. The most instructive is Joshua Socolar and Joan Taylor’s tile of 2010: one connected hexagon that genuinely forces aperiodicity, and not an answer, because the forcing lives in decorations printed on it. Turn the decorations into geometry and the shape falls into disconnected pieces.
| candidate | One tile | Connected and undecorated | Tiles the plane | Never periodically |
|---|---|---|---|---|
| A square | ✓ | ✓ | ✓ | ✗ |
| Any quadrilateral | ✓ | ✓ | ✓ | ✗ |
| Penrose rhombs1974 | ✗ | ✓ | ✓ | ✓ |
| Socolar–Taylor, decorated2010 | ✓ | ✓ | ✓ | ✓ |
| Socolar–Taylor, geometric2010 | ✓ | ✗ | ✓ | ✓ |
| The hat2023 | ✓ | ✓ | ✓ | ✓ |
| The spectre2023 | ✓ | ✓ | ✓ | ✓ |
A connected thirteen-sided polygon with no decorations at all. Its tilings use both the tile and its mirror image, which some argued should count as two tiles.
One tile: A single shape, used in every position.
Connected and undecorated: A connected region of the plane, with no markings, colours or matching rules attached.
Tiles the plane: Copies of it cover the plane with no gaps and no overlaps.
Never periodically: No tiling it admits has a translation symmetry — not merely that some arrangement fails to repeat.
The fourth condition is the one that makes this a hard problem, and it is easy to misread. It is not enough to arrange the tiles so that they fail to repeat — you can do that with squares by sliding one row along. The tile must make repetition impossible, in every arrangement, of which there are infinitely many.
Try it: Compare the two Socolar–Taylor rows. Each satisfies three conditions and fails a different fourth, and the failures are not the same failure. That is the shape of the last thirteen years before 2023: the answer kept being available if you were willing to relax one thing, and nobody was.
In November 2022 David Smith, working by hand with cut-out cardboard, found a thirteen-sided shape he could not make repeat. It satisfies all four conditions. The tiling it makes is below — what you are looking at is the four clusters the proof organises hats into, rather than the hats themselves.
Generations stop at 6, which is 242,962 tiles. The substitution keeps going — generation 7 from an H is 1,667,659 — but building that takes about 0.7 s and 25 MB, so the instrument does not offer it.
Try it: Zoom around and look for a repeat. Then go on to the hat itself, which is where the shape comes from, and why it never repeats, which is where the argument is.