The whole tile, as published: 24 unit panels, each with eight square pyramids at (±⅛, ±¼) and (±¼, ±⅛) from its centre. The small number is the signed height in units of 1/10000 of a cube edge, + a bump or − a dent; the big grey one is the panel's id. The tinted squares are the three panels inside the notch. Axes are global, so for −x, +y and −z you are looking from inside.
| claim | paper | here |
|---|---|---|
| computing… | ||
Everything in the here column is recomputed in your browser, from the 192 numbers in the recipe tab, when you open this tab. No count is copied from the paper. The same module, unchanged, is held to these numbers by tile.selftest.mjs, which also checks the 44 against the paper's own table one pose at a time.
In September 2026 Ioannis Tsiokos, a software developer in Athens, posted a preprint describing a solid that fills space, but only in patterns that never repeat. He had found it by prompting an AI model, OpenAI's GPT-6 Astra, which also wrote a formal proof in Lean. He calls it Chair44. Within days Felix Flicker reproduced the finite enumerations and found simpler matching rules that force the same tiling. Chaim Goodman-Strauss, one of the four people behind the 2023 “hat”, published short notes that redraw the tile and give a much cleaner proof. He admired the result and was sharp about the preprint's form.
Take a 2×2×2 block of cubes and remove one corner cube. What's left is the chair. Eight chairs fit together into a chair twice the size (the 8 view), eight of those into one four times the size (64), and so on forever. That alone gives a tiling with no period. But plain chairs can also stack periodically, so the shape by itself proves nothing.
Chair44 fixes that with tiny square pyramids, eight on each of the 24 unit squares on its surface, 192 in all. Some stick out and some are dents. There are twelve heights, from 1/10000 to 12/10000 of a cube edge. A bump fits only into a dent of exactly the same height, so where two tiles touch, every pyramid across the shared square must pair up. At true scale you could not see them. The bumps × slider exaggerates them up to 1000 times, and the colours pair each height with its matching dent.
Of the 2,388 ways a second chair can press a face against the first without overlapping, the pyramids allow exactly 44 (the touch view steps through them). Every one of the 44 is a rotation; none is a mirror image. So any tiling uses one handedness only.
The proof then shows that those 44 local fits force every tile into a unique doubled chair (the 8-chair cut, in this orientation). The doubled chairs meet each other in exactly the same 44 ways, at twice the scale. That last step is the check tab's 697 → 116 → 44, which equals the atlas at twice the size. So they form doubled-doubled chairs, and so on at every scale. A tiling with a period v would then also have periods v/2, v/4, … which is impossible unless v = 0. Its symmetries are also capped at the 24 rotations of a cube, so no screw motion survives. That makes it strongly aperiodic, which is what distinguishes it from earlier 3D tiles like the Schmitt–Conway–Danzer biprism.
It recomputes the finite census the proof rests on, from the recipe, with integer arithmetic: 2,388 → 44, the 19 rotations and that they generate all 24, the unique legal cut of the doubled chair, 697 → 116 → 44 for the parents and their halving back to the atlas, no symmetry of the decorated tile but the identity, and every contact inside the 64- and 512-chair patches is one of the 44. It does not check the continuous-geometry part: that pyramids of these exact angles cannot slide into any non-grid position. The paper handles that with its registration lemmas, and Lean checks the rest.
The pyramids are placed as the paper places them. The only liberty taken is the drawing exaggeration, and the mesh is checked to stay a closed surface of volume exactly 7 with it on.