Mathematicians have finally found the elusive Einstein tile


The 13-sided shape known as the “hat” attracts the attention of mathematicians.

This is the first real example of an “einstein”, a single shape that forms a special tiling of a plane: like a bathroom floor tile, it can cover the entire surface without gaps or overlaps, but only in a pattern that never repeats.

“Everyone is surprised and happy, both,” says mathematician Marjorie Senechal of Smith College in Northampton, Massachusetts, who was not involved in the discovery. Mathematicians have been looking for such a form for half a century. “It was not even clear that such a thing could exist,” says Seneschal.

Although the name “Einstein” is reminiscent of the iconic physicist, it is of German origin ein Stein , which means “one stone” and refers to one tile. Einstein sits in a strange purgatory between order and disorder. Although the tiles fit neatly together and can span an infinite plane, they are aperiodic, meaning they cannot form a repeating pattern.

Thanks to the periodic pattern, the tiles can be moved to perfectly match the previous placement. An infinite chessboard, for example, looks the same if you move the rows by two. While it is possible to arrange other unit tiles in patterns that are not periodic, the hat is special because it cannot form a periodic pattern.

The “hat” (one highlighted) is a polygon made up of eight smaller kite shapes (dark gray lines).D. SMITH AND OTHER / ARXIV.ORG 2023

Defined by David Smith, a lay mathematician who describes himself as “fantasy with imagination,” and reported in a paper published March 20 online at, the hat is polycystic — a bunch of smaller kite shapes stuck together. It’s a type of shape that hasn’t been studied closely in the search for Einstein, says Chaim Goodman-Strauss of the National Museum of Mathematics in New York, one of a group of skilled mathematicians and computer scientists with whom Smith has teamed up to study the hats.

This is a surprisingly simple polygon. Before this work, if you asked what Einstein would look like, Goodman-Strauss replied, “I would draw some crazy, twisted, hideous thing.”

Mathematicians used to know about unique tilings, which included several tiles of different shapes. In the 1970s, mathematician Roger Penrose discovered that only two different shapes form a mosaic that is not periodic. This begs the question, “Could there be a single tile that does this?” says mathematician Casey Mann of the University of Washington Bothell, who was not involved in the study. He was finally found, “he is huge.”

An image of a Taylor-Socolar tile that looks like a hexagonal shape with polygons sticking out from the edges and pairs of smaller rectangles surrounding it.
Taylor-Sokolar tiles are the closest mathematicians have come to an “einstein,” a single tile that forms a pattern that never repeats. But Taylor-Socolar tiles have disconnected parts (illustrated), stretching the definition of a tile. PARCLY TAXEL/WIKIMEDIA COMMONS ( CC BY-SA 4.0 )

Other forms approached. Taylor-Socolar tilings are non-periodic, but they are a set of several disconnected pieces—not what most people think of as a single tiling. “This is the first solution without stars,” says mathematician Michael Rao of the CNRS and École Normale Supérieure de Lyon in France.

Smith and his colleagues proved that the tile was Einstein in two ways. One came from noticing that the caps clustered into larger clusters called metatiles. These meta-records are then ordered into even larger super-records, and so on ad infinitum, in the type of hierarchical structure common to non-periodic cuts. This approach showed that the hat tile could fill the entire infinite plane and that its pattern would not repeat itself.

The second proof was based on the fact that the hat is part of a continuum of shapes: by gradually changing the relative lengths of the hat’s sides, the mathematicians were able to form a family of tiles that could have the same non-repeating pattern. By looking at the relative sizes and shapes of the tiles at the extremes of this family—one shaped like a chevron and the other like a comet—the team was able to show that the hat cannot be arranged in a periodic pattern.

Mathematicians have found the first true “Einstein,” a hat-like shape that can be stretched to cover an infinite plane, but with a pattern that cannot repeat itself. The hat is one of a family of related tiles with many different shapes. In this video, the hats transform into different shapes. By comparing the extreme shapes of this family, one chevron-shaped and the other resembling a comet, the researchers were able to show that the hat cannot form a repeating pattern.

Although the paper has not yet been peer-reviewed, experts interviewed for this article agree that the result is likely to stand up to close scrutiny.

Non-repeating patterns may have connections to the real world. Materials scientist Dan Schechtman won the 2011 Nobel Prize in Chemistry for the discovery of quasicrystals, materials with atoms arranged in an ordered structure that never repeats, often described as analogous to Penrose mosaics. The new aperiodic tile could spark further research in materials science, says Seneschal.

Such tiles have inspired artists, and the hat seems to be no exception. The tile has already been artistically reproduced in its form smiling turtles and piles of shirts and hats . It’s probably only a matter of time before someone puts a hat tile on top of a hat.

And the hat is not the end. Researchers should continue to hunt for new Einsteins, says computer scientist Craig Kaplan of the University of Waterloo in Canada, a co-author of the study. “Now that we’ve opened the door, hopefully other new forms will emerge.”

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