hr-studioPatterns & Tiling · Bonus
A computational design primer · The Hat

One tile. No repeats.

For half a century, mathematicians chased a single shape that could cover the plane forever and never fall into a repeating pattern — an "einstein," from ein Stein, one stone. In 2023 a retired print technician found it. They call it the hat.

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i.

The fifty-year shape

A single tile that tiles the plane only without ever repeating was long suspected to be impossible. Penrose needed two tiles; every proposed single shape needed extra markings or matching rules to cheat its way to aperiodicity.

The hat needs none of that. It is an ordinary 13-sided polygon — found in 2022 by hobbyist David Smith and proven aperiodic with Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss. Drag to turn it.

The hat · one tile · 13 sides
ii.

Eight kites in a coat

The hat isn't arbitrary. Take a hexagonal grid, cut each hexagon into six kites by joining the midpoints of opposite edges, and the hat is exactly eight of those kites, drawn from three neighbouring hexagons — a "polykite."

That is why every edge is a hexagon edge, a half-edge, or a height, and why every corner is a multiple of 30°. Toggle the kites to see the bones.

Built from 8 kites
8 kites · 1 tile
iii.

It needs its own mirror

Here is the hat's one concession: to tile the plane, it must be joined by its mirror image. Most hats sit one way; a smaller number — about one in seven — are flipped. Both handednesses are essential; neither tiles alone.

That asymmetry is what later drove the search for the "spectre," a relative that needs no reflection at all. Flip the tile.

The hat and its reflection
iv.

Grown, not laid

You cannot place hats one by one and hope for the best — you'd get stuck. Instead they assemble through substitution: small clusters of hats group into four "metatiles," which group into larger metatiles, forever. Reverse the recipe and a patch unfolds.

Step the level up and watch a handful of hats inflate into a legal, ever-larger piece of the tiling.

Substitution · inflation
v.

The hat field

Hundreds of hats, every one identical up to rotation and reflection, locked together with no gaps and no overlaps — and no repeat anywhere, no matter how far you look. The warm tiles are the flipped ones, threading through the rest. Push the depth, recolour, let it turn. Or flip to Build and lay the stones yourself — hover near an open edge and a ghost hat appears wherever one can legally sit; click to set it. If you wall yourself in, that's section iv talking: undo, and grow another way.

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flippedunflipped
A shape simple enough to cut from card, strict enough to forbid every repeating pattern at once. The einstein was never exotic — it was just one stone, waiting to be noticed.