Einstein Hat Tiling

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One tile, copied, turned and flipped, covers an endless floor, and the pattern never repeats.

What’s going on

Squares, triangles and hexagons tile a floor in patterns that repeat forever. For sixty years mathematicians asked whether a single shape could tile a floor only in patterns that never repeat: an “einstein”, German for “one stone”. In 2023 the answer arrived: the “hat”, a 13-sided shape made of eight kites, found by retired print technician and amateur mathematician David Smith and proved with Craig Kaplan, Joseph Samuel Myers and Chaim Goodman-Strauss.

How the pattern grows

Hats are grouped into four larger “metatiles”, H, T, P and F. Each metatile can be replaced by a bigger arrangement of the same four, again and again, so the tiling grows outward without end. The colors show which metatile each hat came from (see the legend); the reflected hats, about one in seven, form their own sparse lattice.

The Shape slider

The hat is one member of a family of shapes, Tile(a, b), whose edges come in two lengths. Sliding between them morphs it: Tile(1,1) at the middle is a special case that can tile periodically, the hat sits near one end, and the “turtle” near the other. All the others tile the plane only aperiodically.

How to use it

▶ Start grows the tiling; ↺ Reset starts over.
Palette changes the colors; ✦ Random colors each hat randomly; ◈ Cycle and ◎ Spiral animate the colors once it has grown.
Size sets how big each tile is; Shape morphs it through the family; Speed sets the color animation speed.
Drag to pan; pinch or scroll to zoom.
PNG / JPG / WEBP save a picture (SD/HD picks the resolution); ⏺ Rec video films the animation.

On phones the controls are under ⚙.

Press the share button (top right). It makes a link-preview picture of the tiling as it is now. Add a title if you like, then post to Facebook, copy the link, or send it another way. The share page’s button reopens the tiling with the same palette, coloring, size, shape and position.

Videos: press ⏺ Rec video (it’s available once the tiling has grown and is cycling). The video is saved to your device and also appears in the share dialog, where it can go on your share page or be posted as a Facebook reel.

Credits

The hat: David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss, “An aperiodic monotile” (2023). The metatile construction follows Craig Kaplan’s published method.

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Made by Jason Buchheim for odd.place.

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