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This hat was the first true example of an “einstein.” That’s the name for a special type of shape that can tile a plane. Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps. It can even tile a plane that’s infinitely big. But an einstein tile does so with a pattern that never repeats.

“Everybody is astonished and is delighted, both, ” says Marjorie Senechal. She’s a mathematician at Smith College in Northampton, Mass. She was not involved with the discovery. This ends a 50-year search for such a shape. “It wasn’t even clear that such a thing could exist, ” Senechal says of the einstein.

Means “one stone.” That refers to using a single tile shape. The hat sits weirdly between order and disorder. The tiles fit neatly together and can cover an infinite plane. But they are aperiodic (AY-peer-ee-AH-dik). That means the hat can’t form a pattern that repeats.

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Think about a tiled floor. The simplest ones are made with one shape that fits together neatly with others like itself. If you use the right shape, the tiles fit together with no gaps and no overlaps. Squares or triangles work well. You could cover an infinitely large floor with them. Hexagons also show up on many floors.

Floor tiles are typically arranged in a periodic, or repeating, pattern. You could shift the tiles over by one row and your bathroom floor would look exactly the same.

The hat could also cover an infinitely large floor. But it won’t form a pattern that repeats, no matter how hard you try.

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David Smith identified the hat. He does math as a hobby, not as his job. He describes himself as an “imaginative tinkerer of shapes.” He was part of a team of researchers that reported the hat in a paper posted online March 20 at arXiv.org.

The hat is a polygon — a 2-D shape with straight edges. It’s surprisingly simple, says Chaim Goodman-Strauss. Before this work, if you’d asked him what an einstein would look like, says, “I would’ve drawn some crazy, squiggly, nasty thing.” Goodman-Strauss is a mathematician. He works at the National Museum of Mathematics in New York City. He teamed up with Smith and other mathematicians and computer scientists to study the hat.

Mathematicians previously knew of tilings that couldn’t repeat. But all used two shapes or more. “It was natural to wonder, could there be a single tile that does this?” says Casey Mann. He’s a mathematician at the University of Washington Bothell. He was not involved with the discovery. “It’s huge, ” he says of the hat finding.

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Mathematicians found the first true “einstein.” That’s a shape that can be tiled to cover an infinite plane, never repeating its pattern. The hat is one of a family of related tiles. In this video, the hats morph into these different shapes. At the extremes of this family are tiles shaped like a chevron and a comet. By comparing these shapes, researchers showed that the hat couldn’t form a pattern that repeats.

The researchers proved that the hat was an einstein in two ways. One came from noticing that the hats arrange themselves into larger clusters. Those clusters are called metatiles.

Metatiles then arrange into even larger supertiles, and so on. This approach revealed that the hat tiling could fill an entire infinite plane. And it showed that its pattern would never repeat.

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The second proof relied on the fact that the hat is part of a family of shapes that are also einsteins. You can gradually change the relative lengths of the sides of the hat. If you do that, you can find other tiles that can take on the same nonrepeating pattern. The scientists studied the relative sizes and shapes of the tiles at the ends of that family. At one end there was a tile shaped like a chevron. At the other end was a shape that looked a bit like a comet. Comparing those shapes showed that the hat couldn’t be arranged in a periodic pattern.

The work has yet to be peer-reviewed. That’s the process in which other experts in a field read and critique the work. But the experts interviewed for this article think that the result will likely hold up.

Similar tilings have inspired artwork. The hat appears to be no exception. Already the tiles have been made to look like smiling turtlesand ajumble of shirts and hats.

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An aperiodic turtle tessellation based on new aperiodic monotile Tile (1, 1.1). In the tiling, it is said that around 12.7% of tiles are reflected. The green one is an instance. One more reflected turtle is hidden in the tiling. Who is the reflected? pic.twitter.com/GZJRP35RIC— Yoshiaki Araki 荒木義明 (@alytile) March 22, 2023

The new aperiodic monotile discovered by Dave Smith, Joseph Myers, Craig Kaplan, and Chaim Goodman-Strauss, rendered as shirts and hats. The hat tiles are mirrored relative to the shirt tiles. pic.twitter.com/BwuLUPVT5a— Robert Fathauer (@RobFathauerArt) March 21, 2023

And the hat wasn’t the end. In May, the same team made another announcement. They found a new type of einstein shape. This one is even more special. The researchers shared it May 28 in a paper at arXiv.org.

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The first einstein made a pattern that involved both the tile and its mirror image. The new tile also makes a pattern that never repeats, but without its reflection. Because the shape isn’t paired with its reflection, you might call it a “vampire einstein, ” the researchers say. They found a whole family of vampire einsteins that they’re calling “spectres.”

“I would never have predicted that we’d stumble upon a shape that solves this [vampire-einstein problem] so quickly, ” says team member Craig Kaplan. He’s a computer scientist at the University of Waterloo in Canada.

Researchers should continue the hunt for einsteins, he says. “Now that we’ve unlocked the door, hopefully other new shapes will come along.”

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A shape called a spectre covers an infinite plane but only with a pattern that doesn’t repeat (small section shown) and which requires no mirror images of the shape. Although certain clustered arrangements of the tiles can reappear, the entire pattern doesn’t repeat indefinitely, as a checkerboard pattern does, for example.D. SMITH, J.S. MYERS, C.S. KAPLAN AND C. GOODMAN-STRAUSS (CC BY 4.0)Soon, a set of obscure notes by foundational physicist Sir Isaac Newton will be auctioned at Sotheby’s. Yes, Newton “discovered gravity” in the popular imagination, but it’s his obsessive interest in pyramidology that shows in the previously undiscovered notes.

Like many people throughout history, Newton believed the pyramids in Egypt had spiritual and metaphysical significance in some way, and he studied them for years to try to reveal their “secrets.”

Newton was trying to uncover the unit of measurement used by those constructing the pyramids. He thought it was likely that the ancient Egyptians had been able to measure the Earth and that, by unlocking the cubit of the Great Pyramid, he too would be able to measure the circumference of the Earth.He hoped that would lead him to other ancient measures, allowing him to uncover the architecture and dimensions of the Temple of Solomon—the setting of the apocalypse—and interpret the Bible’s hidden meanings.

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It’s easy to reverse engineer a history where people like Newton were so-called pure scientists—untroubled by religious politics, pseudoscience, and other areas we don’t consider part of science today. But for thousands of years, the most rational scientists were still grappling with complicated problems that made it hard to rule out these more extreme theories. Without cell or atomic theory, for example, how could they conclude that miasma theory and the classical elements weren’t accurate?

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All this is to say that even Newton was deep in his pseudoscience cups, studying alchemy and searching for esoteric meaning in different places. The pyramids of ancient Egypt, specifically the most photogenic and iconic batch at Giza, have always been a magnet for esoterica. In fact, the field of pseudoscience speculation about the pyramids is so large that, like ufology, it has its own name: pyramidology.

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For Newton, pyramidology related to his interest in alchemy and even his underlying Christian beliefs. He thought the pyramids held secrets to the details of the specific Christian apocalypse detailed in the book of Revelations. And during Newton’s lifetime, mixing the default Christian beliefs with science

Pseudoscience was verboten. Even though his research was directly related to how fervently he believed in the Bible, he had to hide these studies from everyone.

The book of Revelations is a drastic tonal and subject matter departure from the rest of the Bible books that precede it, and it’s stuffed with symbolism and numbers that have intrigued scholars for centuries. They’ve considered the puzzling book to be

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Why are the pyramids involved in this story? Well, for a similar reason. The pyramids are such a staggering human achievement, surrounded by the rich cultural history of the ancient Egyptians who built them, that they became a

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