Ioannis Tsiokos, director of the AI research lab Automorph and the sole human author of a new preprint, set an old geometry question before OpenAI's ChatGPT-6 Astra. He had no mathematical training, as he put it: 'I'm not a mathematician myself.'
The model's proposed answer is Chair44, a three-dimensional shape that can fill space without producing a repeating pattern. Tsiokos gave the model the aperiodic-monotile problem, alongside the Six Birds emergence calculus framework and a construction record.
The question had been posed in 2012 by Joshua E. S. Socolar and Joan M. Taylor: could one simply connected three-dimensional shape force nonperiodicity through its geometry alone? It stayed open for 14 years.
There was already a three-dimensional aperiodic monotile, the Schmitt-Conway-Danzer biprism, known for decades. Craig Kaplan, a mathematician who worked on the two-dimensional version of the problem, said it “obeys the letter of the law of aperiodicity but not the spirit,” because it tiles only through screw motions.
The two-dimensional problem had fallen in 2023. Amateur mathematician David Smith, working with Joseph Samuel Myers, Craig Kaplan and Chaim Goodman-Strauss, found the hat, a shape resembling a fedora that tiles a flat surface without repetition.
Chair44 takes a different route. It is built from a seven-cube chair, with 24 exposed unit panels. Tiny square pyramids of differing heights and matching recesses sit on those faces, creating 44 legal contacts between neighbouring tiles.
Those features rule out every tiling except non-periodic ones. Chair44 is a strongly aperiodic monotile: congruent copies can tile three-dimensional space, with reflections allowed, while every such tiling has no translational period and a symmetry group of order at most 24.
The model produced six hash-chained packets covering the periodic control, frame change, the 44-contact atlas and a written alignment proof. Tsiokos used Anthropic's Claude Fable 5.1 to help write the paper.
The verification was then checked in Lean, a proof-assistant programming language. It comprises 85 Lean files and 29,815 lines on Lean 4.31.0.
The paper, titled 'A Strongly Aperiodic Monotile in Three Dimensions,' was posted to arXiv on 16 September 2026 under Tsiokos's name alone. The AI-based approach is described as diametrically opposed to the method used to discover the two-dimensional hat.
Kaplan welcomed the claimed resolution. “I'm pleased that we have the solution to what was, until recently, a nagging open problem in tiling theory,” he said.
The discovery by a researcher without an academic background in the area has also angered professional mathematicians.
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