A unique shape, a 3D einstein tile, has been discovered

It may look like a child’s Lego creation, but it’s a mathematical marvel. Researchers have just discovered a three-dimensional “einstein” tile—a special shape with unique capabilities.

The finding follows a major announcement from 2023, when a new hat-shaped tile called an “einstein” rocked the popular mathematics world. David Smith, an amateur mathematician, discovered that shape. In an article later published in Combinatorial Theory, he and three other researchers—Joseph Samuel Myers, Craig Kaplan and Chaim Goodman-Strauss—proved that “the hat” was the first known example of a two-dimensional aperiodic monotile, also called an “einstein” (ein Stein roughly translates to “one shape” in German). In other words, the hat is a shape that can infinitely tile, or cover, a 2D surface without ever creating a repeating pattern. These building-block hats can be placed next to one another to create bigger and bigger shapes that also look like the hat. But each individual hat will be situated in a nonrepeating pattern that continues to change no matter how large the overall tiling is.

Now a researcher has taken this concept into the third dimension with a 3D shape dubbed the Chair44 monotile. Chair44 can infinitely tile a three-dimensional space by covering it without gaps or overlapping shapes. Moreover, that tiling is aperiodic, meaning that the pattern is never repeated. The finding was announced by Ioannis Tsiokos, director of Automorph, which he describes as an AI research lab.

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“I’m not a mathematician myself,” says Tsiokos, who is the sole human author of the preprint announcing Chair44. Tsiokos says he used Anthropic’s Claude Fable 5.1 artificial intelligence model to write the paper and the OpenAI model ChatGPT-6 Astra to search for a 3D monotile, including to verify that the Chair44 monotile met the criteria for an einstein.

Chair44 isn’t exactly the first example of such a monotile in three dimensions, according to Kaplan, a computer scientist whose research focuses on the intersection of mathematics and art. “For decades, we’ve known of an aperiodic monotile in 3D: a shape known as the Schmitt-Conway-Danzer biprism,” Kaplan says. That biprism, however—a kind of convex polyhedron—“obeys the letter of the law of aperiodicity but not the spirit,” making it weakly aperiodic, he notes. The Chair44 discovery, on the other hand, fulfills a decades-long quest to find a strong example.

“I’m pleased that we have the solution to what was, until recently, a nagging open problem in tiling theory,” Kaplan says. “We can also hope that this discovery will generate renewed excitement in the mathematics of tilings and encourage more young people to play around with geometry.”

Propagating the Chair

To understand the geometric intuition behind the Chair44 shape, Tsiokos says, imagine a large cube composed of eight tiny cubes that are joined together. Next, remove just one cube. “Now you have a seven-cube chair,” he says.

The next goal on this journey is to create a bigger “chair” shape. That larger shape can be constructed by connecting four of these smaller chairs. To create a full tiling, just repeat this process over and over again, ad infinitum.

As you build bigger and bigger chairs, however, you must arrange the smaller chairs in a nonrepeating pattern so that the resulting tiling will be aperiodic.

“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,” Tsiokos says. “To have this iterative, recursive building of chairs” requires making sure “that there are some sort of key locks between the chairs so that they form bigger and bigger chairs” that are made from a nonrepeating tiling pattern.

Experts Respond

When the hat monotile was discovered, it was so lauded that five researchers organized a two-day Hatfest celebration at the University of Oxford. Time will tell if Chair44 inspires its own celebration. For now, the reception to the finding has been mixed, largely because of Tsiokos’s significant use of generative AI in the proof.

“There was a lot of human work that went into what went into Astra to do this,” Tsiokos says.

Chaim Goodman-Strauss, one of the mathematicians behind the hat proof and an outreach mathematician at the National Museum of Mathematics, wrote a preprint reacting to the Chair44 results. In it, he described Tsiokos’s paper as having “hallmarks of an LLM-driven research paper,” including “so much irrelevant puff to push through.” Goodman-Strauss is one of at least two researchers who have now presented their own human-written proofs demonstrating that Chair44 is an aperiodic monotile.

Tiling problems have been “a big part of my career,” Goodman-Strauss says. He has studied the shape that Tsiokos calls Chair44 for more than 30 years, he says, and is used to calling it “the L.”

To explain how tiling with the shape works, Goodman-Strauss begins with an L-shaped building block. “You can make a bigger version of it out of eight little copies,” he says. “The problem is, that shape can also do all kinds of other crazy stuff,” he says. But because you can build a tiling from increasingly larger L shapes made up of smaller ones that are organized without repeating a pattern, each tile exists within what he calls “a supertile” that’s arranged in a unique configuration out of the smaller tiles.

Hence, Chair44 or the L is an aperiodic monotile in 3D.

Felix Flicker, a theoretical physicist at the University of Bristol in England, was one of Hatfest’s organizers. “I know a lot of mathematicians are unhappy with how the tile was discovered,” Flicker says. “The positive take is that we have a new important result.” The announcement comes after a summer in which AI models have repeatedly conquered many of math’s longstanding open challenges, including one of the seven Millennium Prize Problems. “The risk is that our jobs get reduced to sifting through AI slop in search of nuggets of gold,” Flicker says.

Flicker also wrote a Chair44 preprint, which he says aims to “provide the first human-written, human-readable, and therefore referee-able and defensible proof that Chair44 is a strongly aperiodic monotile.” (Even Flicker noted in his paper that he used Claude for coding and discussions in his study.)

Tsiokos’s Chair44 paper “is just the first example of something that we’re going to be encountering a lot of” in terms of generative AI usage, Goodman-Strauss says. “One of the big lessons here in the age of LLMs is that we should remember all the [math] questions we asked before that were difficult to deal with.” Perhaps some of those problems can now be solved through the use of LLMs as a mathematician’s tool—although those technologies aren’t a substitute for human reasoning, he notes.

The use of generative AI to find new potential math research results isn’t going away. “It’s just that we have got to come to terms with what’s expected in terms of communication” of those results, Goodman-Strauss says.