AI Helps Discover 3D 'Einstein Tile' That Never Repeats

Imagine building with blocks that could fill an entire universe without ever making the same pattern twice.

That's exactly what researchers just accomplished with a shape called Chair44, the first strong example of a three-dimensional "einstein tile." The name comes from German (ein Stein means "one shape"), and it represents a mathematical breakthrough decades in the making.

Ioannis Tsiokos, director of AI research lab Automorph, led the discovery using advanced AI tools to help find and verify the special shape. He partnered with AI models to search through countless possibilities and prove that Chair44 meets the strict requirements for this rare type of tile.

The shape itself is surprisingly simple to visualize. Picture a large cube made of eight smaller cubes stacked together, then remove just one cube. What remains is a seven-cube chair shape that can connect with other chairs to build bigger and bigger versions of itself, infinitely, without the pattern ever repeating.

This discovery builds on excitement from 2023, when amateur mathematician David Smith found the first two-dimensional einstein tile, nicknamed "the hat." That discovery sparked so much joy in the math community that researchers held a two-day celebration called Hatfest at the University of Oxford.

While a 3D einstein tile technically existed before (the Schmitt-Conway-Danzer biprism), mathematicians considered it "weakly aperiodic" because it didn't fully capture the spirit of what they were seeking. Chair44 solves what computer scientist Craig Kaplan calls "a nagging open problem in tiling theory."

Why This Inspires

The discovery shows how human curiosity and artificial intelligence can work together to crack puzzles that have challenged brilliant minds for generations. Tsiokos isn't even a mathematician by training, proving that passion and creative problem-solving can come from anywhere.

The finding also opens doors for young people exploring geometry and mathematics. Kaplan hopes Chair44 will "generate renewed excitement in the mathematics of tilings and encourage more young people to play around with geometry."

The reaction hasn't been purely celebratory, as some mathematicians question the heavy use of AI in the research process. Yet the mathematics community has rallied to verify the result, with multiple researchers now presenting their own proofs confirming that Chair44 works exactly as claimed.

Sometimes the most beautiful discoveries come from asking simple questions about shapes and space, then refusing to give up until we find answers that expand what we thought was possible.