Aperiodic Monotile Explorer

A bathroom floor repeats. Slide the whole pattern one tile to the left and every joint lands on a joint. For sixty years mathematicians asked whether a single shape could exist that covers the plane perfectly and yet makes that slide impossible, in every arrangement, forever. In 2023 the answer arrived, and it was shaped like a shirt.

The claim of this page is narrow on purpose. The hat tile is a theorem about the infinite plane, proved by four named people in one paper. A picture of a few thousand hats, however large, proves nothing about periodicity. What a picture can show, and what is worth checking exactly, is the humbler fact underneath: that the pieces really do fit, with no overlap and no gap.

From twenty thousand tiles to two

The story starts with logic, not decoration. In 1961 Hao Wang asked whether a procedure could decide if a given set of square tiles with colored edges, placed side by side without turning, can cover the plane. He conjectured that any set which tiles at all can also tile periodically. If that were true, the question would be decidable.

It was false. Wang’s student Robert Berger built a set of 20,426 square tiles that covers the plane but only without translational symmetry, and with it showed the tiling problem undecidable. A set like that is called aperiodic. The word describes the tiles, not one picture: every tiling the set allows must lack a repeat.

Then came the race downward. The survey by Bruneau and Whittaker lists the stations: Berger himself reached 104, Donald Knuth 92, and a line of others brought the count of Wang tiles to 16. A set of 13 followed. In a paper published in 2021, Emmanuel Jeandel and Michael Rao found a set of 11 and proved by exhaustive computer search that no set of fewer than 11 Wang tiles is aperiodic. For squares with colored edges, the race is over.

Free the shapes and allow turning, and the numbers fall faster. MacTutor’s biography of Roger Penrose records the descent in his own work: many tiles at first, then six, then, in its words, the seemingly impossible, two. Robert Ammann found two-tile sets independently in the same decade. Two was where the count stuck for nearly half a century.

The problem of one stone

The remaining question has a punning name. An einstein is a single tile that is aperiodic by itself; the word plays on the German for one stone. It has nothing to do with relativity.

Partial answers came first, and each had an asterisk. In 2010 Joan Taylor, an amateur in Tasmania, found rules for a marked hexagon, and with the physicist Joshua Socolar proved them aperiodic. But the rules reach past a tile’s neighbors to the tiles beyond them. A version that works by shape alone exists, and its tile is not one connected piece of the plane. It also needs its mirror image.

In November 2022 David Smith, another hobbyist, was playing with shapes made of kites. The kites come from cutting each regular hexagon of a honeycomb into six through the midpoints of its sides. Glue eight of them together in a particular way and the result is a polygon with thirteen sides, of lengths 1, 2 and the square root of 3. The hat is 8 kites and nothing more. Smith could find no way to make it repeat, and no obstacle to making larger and larger patches.

He wrote to Craig Kaplan; Joseph Samuel Myers and Chaim Goodman-Strauss joined. Their preprint appeared in March 2023 and the paper was published in the journal Combinatorial Theory in 2024. Its theorem is short: the hat is an aperiodic monotile. No markings and no rules are needed beyond the outline itself.

Why the proof is two proofs

Showing that a shape tiles the plane is the easier half. The authors group hats into four clusters, which they call metatiles and label H, T, P and F. A handful of these clusters, arranged in a fixed way, makes a larger cluster of each kind, and the larger clusters meet each other by the same rules as the small ones. Those form larger ones again. Repeat without end and the plane is covered. Each round multiplies the area by the fourth power of the golden ratio, about 6.854.

The harder half is showing that nothing else is possible. One proof in the paper is a long, computer-checked case analysis: in any tiling by hats, every tile belongs to exactly one metatile, every metatile to exactly one larger cluster, and so on upward. A tiling that could slide onto itself would break that uniqueness. The other proof is new in kind. It stretches the hat until some sides vanish, compares the two extreme tilings that result, and derives a contradiction from the square root of 2.

That stretching is worth a second look, because it turns one tile into a family. Replace the sides of length 1 by any length a, and the sides of length root 3 by any length b, and call the result Tile(a, b). The hat is Tile(1, root 3). Swap the two lengths and a second shape made of ten kites appears, which the authors call the turtle. Every member with a and b positive and unequal is an aperiodic monotile.

Three members are exceptions, and the paper names them. With a at zero the tile collapses to the chevron, made of four triangles. With b at zero it becomes the comet, made of eight. With a equal to b all sides match. Each of these three also tiles periodically, so none is aperiodic. The same hat tiling can still be squeezed into any of them, which is exactly what the family slider on this page does.

The mirror, and the tile that did without it

Count the hats in a patch and a pattern appears. Baake, Gahler and Sadun put it plainly: a hat tiling uses twelve tiles up to translation, the hat and its mirror image each in six directions. The mirrored hats are sparse. Each one sits inside a shell of three ordinary hats, and those four make the H cluster. No other cluster contains a mirrored tile.

So is it really one tile? By the long convention of the subject, a shape and its reflection count as the same, and the authors defend that reading. But they also concede the practical point in their second paper. A hat cut from paper can be turned over. A glazed ceramic tile cannot.

Their answer came within weeks, in May 2023. The equal-sided member of the family, Tile(1, 1), tiles periodically only when both handednesses are mixed. Forbid the mirror image and every tiling it allows is non-periodic. Bend its edges into suitable curves, so that a tile and its mirror image can no longer sit side by side, and the restriction enforces itself. The authors call these shapes Spectres. A Spectre tiles the plane, never periodically, and never with its reflection.

Here is a worked example with the tool above. Choose the seed cluster H and level 0: four hats, one of them mirrored. Raise the level and the patch holds 23 hats, then 157, then 1,076. The mirrored hats number 1, 3, 20 and 137. At level 3 the share is 137 in 1,076, or 12.73%.

Those counts come from the substitution, as carried out here with whole tiles: an H supertile holds three H, one T, two P and three F clusters. That bookkeeping, and the limit it implies for the mirrored share, about 12.73 percent, are this page’s own arithmetic and not statements from the paper. Seed T makes a tidier list: 1, 8, 55, 377, 2,584. Readers who know the Fibonacci numbers will find every one of them in that sequence, four places apart. The golden ratio in the growth rate is not a coincidence.

For anyone making real tiles, the mirrored share is the number that matters. Cut from plywood or paper, one template serves, since any piece can be flipped. Cast in a material with a face, about one tile in eight must come from a second, mirrored mould. A patch of 1,076 needs 137 of them. Or use Spectres and one mould.

What aperiodic does not mean

It does not mean random. A hat tiling is rigidly ordered; Socolar showed that its structure is quasiperiodic with six-fold symmetry, the planar cousin of the quasicrystals that Dan Shechtman first saw in 1982 during a sabbatical at NIST. Before that, as NIST’s account puts it, scientists thought all crystals had a regularly repeating structure.

It does not mean that nothing recurs. Small arrangements of hats turn up again and again; the paper notes that all hat tilings have the same local structure. What never happens is that the whole pattern slides onto itself.

And it does not follow from a picture. The tool on this page grows a patch by the published rules and then checks it independently with whole-number arithmetic. Every tile is a hat or its mirror image, no kite of the grid is covered twice, the rim is a single closed loop, and the angles at every interior corner add to a full turn. Those checks establish a tiling of a finite region. The by-hand mode refuses overlaps and misfits, but it cannot promise that a patch built by hand could be continued forever. That every hat tiling, out to infinity, refuses to repeat is the theorem, and it belongs to Smith, Myers, Kaplan and Goodman-Strauss.