Forget the 'Einstein' tile, meet the true aperiodic tile 'Spectre'

Hot on the heels of the discovery of the "one-in-a-million" shape "Einstein" in March, a new shape called "Spectre" might be the true owner of the title.
Christopher McFadden
An image of the new shape making a non-repeating pattern.

David Smith et al, 2023.

Back in March of this year, a team of computer scientists found a way to completely cover a flat surface using only one shape, which they call "Einstein" or "the hat." The former has nothing to do with the famous physicist but is a play on words from the German ein stein ("one stone"). Considered to be a "one-in-a-million" shape at the time, it was believed to be an exceptionally special discovery in mathematics as this shape can cover a surface without repeating a specific pattern. This kind of tiling is called "aperiodic tiling," it's been a tricky problem in mathematics for quite a long time.

Back in the 1960s, mathematician Hao Wang believed that it was impossible to find a collection of shapes that could tile a plane without repeating. However, his student Robert Berger, who is now a retired electrical engineer in Lexington, Massachusetts, proved him wrong by discovering a set of 20,426 tiles that could do just that. Later, he found another set of 104 tiles. By the 1970s, mathematical physicist Sir Roger Penrose from Oxford had narrowed it down to just two sets, until the invention of the "Einstein," of course.

While this was a significant finding, there was a slight problem: the original "hat" shape needed to be flipped over (like a mirror image) to cover some parts of the surface. This made some people argue that they were technically using two different shapes (the original and the mirror image) instead of just one. “Who would believe that a little polygon could kick up such a fuss,” said Marjorie Senechal, a mathematician at Smith College who is on the roster of speakers for the event.

Not wanting to leave this problem unsolved, the team kept working. They found a different shape closely related to the hat but could cover the surface without needing to be flipped over. It still did the aperiodic tiling, but now with only a single shape, no mirror images were needed.

“I wasn’t surprised that such a tile existed,” said the co-author Joseph Myers, a software developer in Cambridge, England. “That one existed so closely related to the hat was surprising,” he added.

The new shape, which they called "Spectre," was discovered by tweaking an "equilateral version" of the hat, a shape that didn't initially seem to have the aperiodic tiling ability. By modifying this shape a bit, they found it could do the non-repeating tiling without mirror images.

This new shape is considered a big step forward in studying tiling in mathematics. It could also lead to some interesting applications, like new designs for soccer balls and shower curtains. It and its predecessor are even being celebrated at a special event at the University of Oxford.

You can read more about this mysterious tile in the journal Arxiv.

Study abstract:

The recently discovered “hat” aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat—the equilateral member of the continuum to which it belongs—is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon’s edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.

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