In 2023, after decades of searching, mathematicians discovered a shape called “the hat” – so named for its passing resemblance to a fedora – 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 – 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 material with hat-shaped crystals. Others have found that structures built using hat-shaped building blocks could be more resistant to fracturing than those built using famously strong honeycomb-like building blocks.

Now, Ioannis Tsiokos, 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’s GPT-6 Astra model and asking it to find a similar result in three dimensions.

“Astra found this, I didn’t find the tile,” says Tsiokos. “I gave Astra the theory, I gave it the laws, I gave it the theorems, the Lean code, the papers, and I said: ‘Here’s the problem.’ It’s kind of cheeky: I’ve been in forums with other mathematicians, and [of] course they’re all angry with me… because I don’t have the background to land this result.”

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.

Chaim Goodman-Strauss at the University of Arkansas described the paper as “basically a piece of slop”, and says that while the result is valid, the paper does little to clarify or explain it. “Incontrovertibly, there’s a very, very pretty result that emerges here,” he says. Goodman-Strauss has since published a paper clarifying Tsiokos’ work.

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

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

“It goes without saying that many of us have been contemplating the search for a 3D aperiodic monotile, and I can say that it’s a natural question people ask when I give talks on the hat,” says Kaplan. “But 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?  Evidently, a computer doesn’t have to struggle as hard.”