Take a cube built from 2 × 2 × 2 unit cubes and delete one from a corner. Seven unit cubes remain: the three-dimensional L-tile, or “chair”. Mark its faces in three colours and it fills all of space, never repeating. That is Chair44, the first strongly aperiodic monotile in three dimensions. Chaim Goodman-Strauss (National Museum of Mathematics) has redrawn its proof in 8 pages (arXiv:2609.24779), working in the line Robert Berger opened in 1966 (Memoirs of the AMS 66): matching rules, like jigsaw bumps, that permit a hierarchy and nothing else.

Why it cannot repeat

The bare chair is a rep-tile: 8 copies pack into a chair twice the size, 56 unit cubes with one central tile wrapped by seven outer ones — a supertile. Unmarked, the chair also tiles periodically or at random. The marks are short: blue meets blue, red meets only green, and at the seven corners and the concave “front socket” the colours share one cyclic order.

Now take one corner of a supertile: it is the back corner of an outer tile. Its marks match that tile’s own socket with red and green swapped; the socket matches the central tile’s corner with them swapped again. Two swaps cancel. So a supertile’s corners carry exactly a single tile’s corner marks: the supertile is a big tile, meeting neighbours only the way tiles do. By induction this holds at every level, so every tile sits in one unique infinite tower of supertiles — and a unique tower cannot slide onto a shifted copy of itself.

The 44 counts the ways chairs meet face to face inside the supertiles, catalogued in Goodman-Strauss’s 1999 European Journal of Combinatorics paper. “Strongly” matters too: the Schmitt–Conway–Danzer biprism (Peter Schmitt, Vienna, 1988) could not tile periodically either, but allowed a screw symmetry; Chair44 allows no period at all.

Readers of our 18 September piece, “One shape, no repeat”, will remember the hat — found by David Smith, co-authored by Goodman-Strauss — and its mirror-twin tax.

61 pages, then 8

Ioannis Tsiokos, a software developer in Athens, found the tile by prompting an LLM alongside his own “Six Birds” framework; Quanta Magazine names the model GPT-6 Astra. His 16 September announcement (arXiv:2609.19214) runs 61 pages by Quanta’s count, and he says plainly: “The 3D Einstein was found by Astra, not me.”

“Hats off to Tsiokos for this discovery,” Goodman-Strauss begins, crediting “the human inquiry” behind it. His criticism targets the format, not the person: needless length and “irrelevant puff”, “hallmarks of an LLM-driven research paper”. “We must, absolutely, insist on a higher standard for scientific discourse.” His remedy is the redrawn proof plus a paper net you can cut out (his Figure 5), play-tested by 7 named people. Within 4 days of the announcement, Felix Flicker (University of Bristol) posted 5 pages of simpler rules (arXiv:2609.23783), sparse enough that “physical systems — geometric or otherwise — can encode them.”

Goodman-Strauss closes hopefully: “New tools in human hands together with human insight produce new mathematics of interest to humans.” A machine search can find an einstein; it cannot by itself write the page that lets a colleague believe it, and the page outlasts the file.

←TODAY: A seven-cube chair with three colours of marks is proven strongly aperiodic in 8 pages.
→3012: Zurich-3012 stocks one part, one rule; the joint tells the hand where the next block goes.
Fulcrum: The machine found the form; people made it checkable, and only checkable forms get passed on.

PAZ’s reading: one mould

This reading is ours; no paper mentions buildings. Chair44 tilings are homochiral — no tile fits its own reflection — so a Chair44 kit would need one part and one mould, closing the loop the hat left open. The assembly rule would live in the joint: Goodman-Strauss notes the colours can become bumps and nicks, giving an unmarked solid whose shape alone forces aperiodicity, closer to a mortise than a drawing. The kit nests: 8 parts make the shape at 2×, 64 at 4×. No source reports one built or load-tested.

Atelier: A Swiss studio using AI for form-finding faces the same split: the tool finds the form, the Büro must understand it. Monday move: add one line to your Grasshopper handover checklist — every generated geometry ships with a one-page derivation (objective, method, assumptions) a colleague can check by hand.

Hack: Grow the chair into its supertile from unit cubes and let the count check the rep-tile arithmetic before you trust a picture. Run it in a Rhino 8 Python 3 script component:

from itertools import product
chair = set(product((0,1), repeat=3)) - {(1,1,1)}
grown = {(2*x+i,2*y+j,2*z+k) for x,y,z in chair for i,j,k in product((0,1), repeat=3)}
print(len(chair), len(grown), len(grown) // len(chair))  # 7 56 8

56 unit cubes is exactly 8 chairs of volume. That is necessary, not sufficient: only the dissection proves the pieces fit.

Then print Figure 5, cut out and fold two or more tiles, and try to make them sit wrong.