At Long Last, Mathematicians Have Found a Shape With a Pattern That Never Repeats

Experts have searched for decades for a polygon that only makes non-repeating patterns. But no one knew it was possible until now

The same 13-sided shape forms a non-repeating pattern
Infinitely many copies of a 13-sided shape can be arranged with no overlaps or gaps in a pattern that never repeats. David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss (CC BY 4.0)

From bathroom floors to honeycombs or even groups of cells, tilings surround us. These patterns cover a space without overlapping or leaving any gaps. Like a rug filled with diamond shapes, where each section looks the same as the one next to it, every tiling ever recorded has eventually repeated itself—until now.

After decades of searching for what mathematicians call an “einstein tile”—an elusive shape that would never repeat—researchers say they have finally identified one. The 13-sided figure is the first that can fill an infinite surface with a pattern that is always original.

Repeating patterns have translational symmetry, meaning you can shift one part of the pattern and it will overlap perfectly with another part, without being rotated or reflected. The shape described in a new paper does not have translational symmetry—each section of its tiling looks different from every part that comes before it.

A number of rugs with repeating patterns
The designs on these rugs have translational symmetry—the patterns on the rugs repeat themselves. Juli Kosolapova via Unsplash

Sarah Hart, a mathematician at Birkbeck, University of London, who didn’t contribute to the finding, tells New Scientist’s Matthew Sparkes that she had thought finding an “einstein” (named for the German words for “one stone,” or one tile) could not be done. “There are infinitely many possible candidate tiles, and even the existence of a solution feels quite counterintuitive,” she says to the publication.

“Everybody is astonished and is delighted, both,” Marjorie Senechal, a mathematician at Smith College who did not participate in the research, tells Science News’ Emily Conover. “It wasn’t even clear that such a thing could exist.”

David Smith, a retired printing technician and nonprofessional mathematician, was the first to come up with the shape that could be a solution to the long-standing “einstein problem.” He shared his ideas with scientists who took on the challenge of trying to mathematically prove his conjecture, per the New York Times’ Siobhan Roberts.

The team published a preprint paper detailing the findings on the site arXiv last week, and it has not been peer-reviewed yet. But experts say the work is expected to be supported with further investigation, per Science News.

“This appears to be a remarkable discovery,” Joshua Socolar, a physicist at Duke University who did not contribute to the finding, tells the Times. “The most significant aspect for me is that the tiling does not clearly fall into any of the familiar classes of structures that we understand.”

a tiling of the 13-sided shape
Each "einstein" tile has eight kite shapes inside of it. David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss (CC BY 4.0)

The “einstein” tile is made up of eight kites, or four-sided polygons with two pairs of adjacent, equal-length sides. Researchers call it “the hat” because of its resemblance to a fedora.

The shape is simpler than some experts expected it to be. Chaim Goodman-Strauss, a mathematician at the University of Arkansas and one of the authors of the paper, tells Science News that if he’d been asked to guess what the shape might look like before the finding, “I would’ve drawn some crazy, squiggly, nasty thing.”

In the 1970s, mathematician Roger Penrose discovered that two shapes could form a non-repeating tiling pattern together, prompting hopes that a single shape may be found to do this one day. Researchers have been able to make other non-repeating patterns in the past, but the challenge has been finding a shape that can only make a non-repeating pattern, Goodman-Strauss tells the Times.

The shape of “the hat” can also be morphed to form additional tile shapes that make non-repeating patterns, as shown in the video above.

This new finding could lead to materials science investigations—for example, shapes that form non-repeating tilings could help design stronger materials, Hart tells New Scientist. The elusive shape might also spark creative inspiration for new decorative designs or art.

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