AI Helps Find 3D Shape That Never Repeats

Imagine building with blocks that somehow never repeat the same pattern, no matter how long you keep going. That's exactly what researchers just discovered with a remarkable 3D shape called Chair44.

The breakthrough follows up on 2023's "hat" tile sensation, when an amateur mathematician named David Smith discovered a flat shape that could cover an endless surface without patterns repeating. Now researcher Ioannis Tsiokos has taken that concept into three dimensions, using AI to find and verify a shape that does the same thing in 3D space.

Here's how Chair44 works. Picture eight tiny cubes joined together with one corner cube removed, creating a chair-like shape. You can connect four of these chairs to make a bigger chair, then repeat that process forever. The key is that as you build bigger and bigger chairs, they lock together in ways that never create the same arrangement twice.

Tsiokos, who runs an AI research lab called Automorph, used Claude and ChatGPT to search for the shape and write the mathematical proof. "I'm not a mathematician myself," he says, but the AI tools helped verify that Chair44 met all the criteria for this special kind of shape, called an aperiodic monotile.

The discovery solves what mathematician Craig Kaplan calls "a nagging open problem in tiling theory." While a similar 3D shape existed before (the Schmitt-Conway-Danzer biprism), it only weakly met the requirements. Chair44 is the strong example researchers have been hunting for decades.

Why This Inspires

This discovery shows how human creativity and AI can team up to crack problems that have stumped brilliant minds for years. Kaplan hopes Chair44 will "generate renewed excitement in the mathematics of tilings and encourage more young people to play around with geometry."

Some mathematicians have questioned the heavy use of AI in the research. But other experts, including mathematician Chaim Goodman-Strauss (who worked on the original hat tile), quickly wrote their own proofs confirming that Chair44 really works. Goodman-Strauss actually studied this shape for over 30 years, calling it "the L."

The math community hasn't thrown a "Chairfest" yet like they did for the hat tile, but the finding represents real progress in understanding how shapes can fill space in endlessly creative ways. Sometimes the patterns we've been searching for are right there, waiting for the right tools to help us see them.