With a New, Improved ‘Einstein,’ Puzzlers Settle a Math Problem

In March, a team of mathematical tilers announced their solution to a storied problem: They had discovered an elusive “einstein” — a single shape that tiles a plane, or an infinite two-dimensional flat surface, but only in a nonrepeating pattern. “I’ve always wanted to make a discovery,” David Smith, the shape hobbyist whose original find spurred the research, said at the time.

Mr. Smith and his collaborators named their einstein “the hat.” (The term “einstein” comes from the German “ein stein,” or “one stone” — more loosely, “one tile” or “one shape.”) It has since been fodder for Jimmy Kimmel, a shower curtain, a quilt, a soccer ball and cookie cutters, among other doodads. Hatfest is happening at the University of Oxford in July.

“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.

The researchers might have been satisfied with the discovery and the hullabaloo, and left well enough alone. But Mr. Smith, of Bridlington in East Yorkshire, England, and known as an “imaginative tinkerer,” could not stop tinkering. Now, two months later, the team has one-upped itself with a new-and-improved einstein. (Papers for both results are not yet peer reviewed.)

This tiling pursuit first began in the 1960s, when the mathematician Hao Wang conjectured that it would be impossible to find a set of shapes that could tile a plane only aperiodically. His student Robert Berger, now a retired electrical engineer in Lexington, Mass., proceeded to find a set of 20,426 tiles that did so, followed by a set of 104. By the 1970s, Sir Roger Penrose, a mathematical physicist at Oxford, had brought it down to two.

And then came the monotile hat. But there was a quibble.

Paper cutouts of David Smith’s “hat” tile assembled into a contiguous patch; none of the tiles are reflections, or mirror images of the original.Credit…David Smith

Dr. Berger (among others, including the researchers of the recent papers) noted that the hat tiling uses reflections — it includes both the hat-shaped tile and its mirror image. “If you want to be picky about it, you can say, well, that’s not really a one-tile set, that’s a two-tile set, where the other tile happens to be a reflection of the first,” Dr. Berger said.

“To some extent, this question is about tiles as physical objects rather than mathematical abstractions,” the authors wrote in the new paper. “A hat cut from paper or plastic can easily be turned over in three dimensions to obtain its reflection, but a glazed ceramic tile cannot.”

The new monotile discovery does not use reflections. And the researchers did not have to look far to find it — it is “a close relative of the hat,” they noted.

“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.”

Originally, the team discovered that the hat was part of a morphing continuum — an uncountable infinity of shapes, obtained by increasing and decreasing the edges of the hat — that produce aperiodic tilings using reflections.

But there was an exception, a “rogue member of the continuum,” said Craig Kaplan, a co-author and a computer scientist at the University of Waterloo. This shape, technically known as Tile (1,1), can be regarded as an equilateral version of the hat and as such is not an aperiodic monotile. (It generates a simple periodic tiling.) “It’s kind of ridiculous and amazing that that shape happens to have a hidden superpower,” Dr. Kaplan said — a superpower that unlocked the new discovery.

The equilateral polygon monotile at left; two “Spectre” monotiles at right.Credit…Craig Kaplan
A selection of “Spectre” tiles that prohibit reflections.Credit…David Smith

Inspired by explorations by Yoshiaki Araki, president of the Japan Tessellation Design Association in Tokyo, Mr. Smith began tinkering with Tile (1,1) shortly after the first discovery was posted online in March. “I machine-cut shapes from card, to see what might happen if I were to use only unreflected tiles,” he said in an email. Reflected tiles were forbidden “by fiat,” as the authors put it.

Mr. Smith said, “It wasn’t long before I produced a reasonably large patch” — fitting tiles together like a jigsaw puzzle, with no overlaps or gaps. He knew he was on to something.

Investigating further — with a combination of traditional mathematical reasoning and drawing, plus computational handiwork by Dr. Kaplan and Dr. Myers — the team proved that this tiling was indeed aperiodic.

“We call this a ‘weakly chiral aperiodic monotile,’” Dr. Kaplan explained on social media. “It’s aperiodic in a reflection-free universe, but tiles periodically if you’re allowed to use reflections.”

The adjective “chiral” means “handedness,” from the Greek “kheir,” for “hand.” They called the new aperiodic tiling “chiral” because it is composed exclusively of either left- or right-handed tiles. “You can’t mix the two,” said Chaim Goodman-Strauss, a co-author and outreach mathematician at the National Museum of Mathematics in New York.

The team then went one better: They produced a family of strong or “strictly chiral aperiodic monotiles” through a simple modification of the T(1,1) tile: They replaced the straight edges with curves.

Named “Spectres,” these monotiles, owing to their curvy contours, only allow nonperiodic tilings, and without reflections. “A left-handed Spectre cannot interlock with its right-handed mirror image,” said Dr. Kaplan.

“Now there is no quibbling about whether the aperiodic tile set has one or two tiles,” Dr. Berger said in an email. “It’s satisfying to see a glazed ceramic einstein.”

Doris Schattschneider, a mathematician at Moravian University, said, “This is more what I would have expected of an aperiodic monotile.” On a tiling listserv, she had just seen a playful “Escherization” (after the Dutch artist M.C. Escher) of the Spectre tile by Dr. Araki, who called it a “twinhead pig.”

“It’s not simple like the hat,” Dr. Schattschneider said. “This is a really strange tile. It looks like a mistake of nature.”

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