Chair44, a 3D tile found with AI help, tiles space without ever repeating
Ioannis Tsiokos of Automorph announced Chair44, a 3D monotile built from seven-cube chairs that tiles space without ever repeating. Reaction is mixed because he made significant use of generative AI in the proof.
The Scientist · Science desk
What happened
- Tsiokos says he used Anthropic's Claude Fable 5.1 to write the paper and OpenAI's ChatGPT-6 Astra to search for a 3D monotile and to verify that Chair44 met the criteria for an einstein.
- Craig Kaplan said an aperiodic 3D monotile has been known for decades, the Schmitt-Conway-Danzer biprism, a kind of convex polyhedron.
- Starting from the seven-cube chair, four copies join into a larger chair, and the step repeats without end, according to Tsiokos.
- The 2D precedent is the hat tile of 2023, found by amateur mathematician David Smith and proved by him and three others to be the first known 2D aperiodic monotile.
Why it matters
- What's now possible capability Tiling theory now has a candidate strongly aperiodic monotile in 3D, so the open question shifts from finding one to confirming this one.
- Who's exposed exposure Tsiokos names ChatGPT-6 Astra as both the searcher and the verifier, so a check independent of that model is still needed before the strong label is settled.
- What it sets up precedent The hat's proof carried four authors, and Chair44's preprint has one human name, so more of the confirming falls to readers outside the author list.
Two strengths of aperiodicity are in play, and Kaplan keeps them apart. He said the Schmitt-Conway-Danzer biprism "obeys the letter of the law of aperiodicity but not the spirit," and he calls it weakly aperiodic [7]. Scientific American reports that Chair44 fulfills the decades-long quest for a strong example [8]. Kaplan, who co-wrote the hat proof [3], said: "I'm pleased that we have the solution to what was, until recently, a nagging open problem in tiling theory." [9]
The construction is recursive. Tsiokos said the trick to forcing aperiodicity is that "you need to grow the chair in such a way that the behavior of the chair grows with it" [13]. At each level the smaller chairs have to be arranged in a nonrepeating pattern, or the tiling would not be aperiodic [16]. Tsiokos described the mechanism as "some sort of key locks between the chairs" [17]. The thing this outline doesn't tell you is what those locks are.
Tsiokos is open about his background. "I'm not a mathematician myself," he said [12]. He stressed the human input: "There was a lot of human work that went into what went into Astra to do this." [14]
The hat has a precedent for outsiders, since David Smith is an amateur [3]. Its discovery was celebrated with a two-day Hatfest at the University of Oxford, organized by five researchers [15].
What to watch
- A human-written or independently checked proof of Chair44 from tiling theorists, which would replace Astra's verification with a check the field can inspect.
- Whether other researchers can build the tiling from the preprint's rules for how smaller chairs lock together.
Clarity's read
What the record supports and how the coverage leans. The claims behind it follow.
Reality
- Evidence72
- Adoption
- Not scored
- Hype gap+8
- Incentives
- Not scored
- Confidence70
Claim ledger
Ranked by verification strength, evidence, and original report placement.
- [1]
Ioannis Tsiokos, director of Automorph, which he describes as an AI research lab, announced the Chair44 monotile.
- [2]
Chair44 can infinitely tile three-dimensional space without gaps or overlaps, and the tiling is aperiodic, meaning the pattern never repeats.
- [3]
In 2023 David Smith, an amateur mathematician, discovered the hat tile; in an article later published in Combinatorial Theory, he and three other researchers (Joseph Samuel Myers, Craig Kaplan and Chaim Goodman-Strauss) proved it was the first known two-dimensional aperiodic monotile.
- [4]
Tsiokos is the sole human author of the preprint announcing Chair44.
- [5]
Tsiokos says he used Anthropic's Claude Fable 5.1 to write the paper and OpenAI's ChatGPT-6 Astra to search for a 3D monotile, including to verify that Chair44 met the criteria for an einstein.
- [6]
Craig Kaplan said an aperiodic monotile in 3D has been known for decades, the Schmitt-Conway-Danzer biprism, a kind of convex polyhedron.
- [7]
Kaplan said the biprism "obeys the letter of the law of aperiodicity but not the spirit," making it weakly aperiodic.
- [8]
Scientific American reports that the Chair44 discovery fulfills a decades-long quest to find a strong example of a 3D aperiodic monotile.
- [9]
Kaplan said: "I'm pleased that we have the solution to what was, until recently, a nagging open problem in tiling theory."
- [10]
Tsiokos describes the base shape as a large cube composed of eight tiny cubes with one cube removed, leaving a seven-cube chair.
- [11]
A larger chair is built by connecting four of the smaller chairs, and the process is repeated over and over to form a full tiling.
- [12]
Tsiokos said: "I'm not a mathematician myself."
- [13]
Tsiokos said the trick to ensuring an aperiodic tiling is that "you need to grow the chair in such a way that the behavior of the chair grows with it."
- [14]
Tsiokos said: "There was a lot of human work that went into what went into Astra to do this."
- [15]
When the hat monotile was discovered, five researchers organized a two-day Hatfest celebration at the University of Oxford.
- [16]
As bigger chairs are built, the smaller chairs must be arranged in a nonrepeating pattern so that the resulting tiling is aperiodic.
- [17]
Tsiokos described the recursive building of chairs as requiring "some sort of key locks between the chairs" so that they form bigger and bigger chairs.
- [18]
The hat paper had four named authors.
- [19]
Reception to the Chair44 finding has been mixed, largely because of Tsiokos's significant use of generative AI in the proof.
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Sources
2 independent publishers whose own reporting we read for this story.
- phys.orgNewly discovered 3D 'chair' shape can fill all space without repeating patterns
1 article · October 13, 2026
- scientificamerican.comA unique shape, a 3D einstein tile, has been discovered
1 article · October 12, 2026
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Topics
- Formal VerificationFollow
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