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In 2023, after decades of searching, mathematicians discovered a shape called 鈥渢he hat鈥� 鈥� so named for its passing resemblance to a fedora 鈥� that can tile a 2D surface without gaps and without ever creating a repeating pattern. Now, a researcher has used AI to go one dimension better and discover a 3D shape that can do the same thing.

Simple shapes like squares can tile a 2D surface, but will form repeating patterns. So-called aperiodic tiles achieve the former without the latter 鈥� you can scroll in any direction, infinitely, and not find a regularly repeating pattern.

That sounds like a nifty but ultimately useless mathematical curiosity, but since the 2023 publication of the 2D tile, it has been used in scientific papers in fields from engineering to chemistry. Some have explored the likely physical properties of a . Others have found that structures built using hat-shaped building blocks could be than those built using famously strong honeycomb-like building blocks.

Now, , a software developer with no background in mathematics, has used AI to discover a 3D monotile he calls Chair44. He did so by describing the 2D problem to OpenAI鈥檚 GPT-6 Astra model聽and asking it to find a similar result in three dimensions.

鈥淎stra found this, I didn鈥檛 find the tile,鈥� says Tsiokos. 鈥淚 gave Astra the theory, I gave it the laws, I gave it the theorems, the Lean code, the papers, and I said: 鈥楬ere鈥檚 the problem.鈥� It鈥檚 kind of cheeky: I鈥檝e been in forums with other mathematicians, and [of] course they鈥檙e all angry with me鈥� because I don鈥檛 have the background to land this result.鈥�

Tsiokos says he also used AI to write the paper describing the finding, and has published around 40 papers in the same way, with another 20 that he is yet to release. But, as he suggests, not all mathematicians are impressed by the way the discovery has been disseminated.

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at the University of Arkansas聽described the paper as 鈥渂asically a piece of slop鈥�, and says that while the result is valid, the paper does little to clarify or explain it. 鈥淚ncontrovertibly, there鈥檚 a very, very pretty result that emerges here,鈥� he says. Goodman-Strauss has since .

at the University of Bristol in the UK has also showing that Chair44 can be simplified and retain its aperiodicity.

at the University of Waterloo, Canada, was one of the researchers behind the 2D aperiodic tile. He says the new paper has 鈥渁ll the usual deficiencies of AI-generated mathematics鈥� in terms of readability and clarity, but that, despite this, the actual shape appears to be a valid solution.

鈥淚t goes without saying that many of us have been contemplating the search for a 3D aperiodic monotile, and I can say that it鈥檚 a natural question people ask when I give talks on the hat,鈥� says Kaplan. 鈥淏ut I was always daunted by the simple prospect of visualising the problem or candidate solutions. Even if I had some shape that looked promising, how should I go about understanding large 3D assemblies of that shape?聽 Evidently, a computer doesn鈥檛 have to struggle as hard.鈥�