Puzzlers solve a mathematical problem with a new, improved ‘Einstein’

In March, a team of mathematician tilers announced their solution to a legendary problem: they had discovered an elusive “einstein” – a single shape tiling a plane, or an infinite two-dimensional flat surface, but only in a non-repeating 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 associates called their einstein “the hat.” (The term ‘einstein’ comes from the German ‘ein stein’ or ‘one stone’ – more loosely ‘one tile’ or ‘one shape’.) Since then it has been fodder for Jimmy Kimmel, a shower curtain, a quilt, a football and cookie cutters, among other doodads. Hat party happens in July at the University of Oxford.

“Who would believe that a small polygon could cause such a stir,” said Marjorie Senechal, a mathematician at Smith College who is on the list of speakers for the event.

The researchers would have been satisfied with that the discovery and the noise, and left well enough alone. But Mr. Smith of Bridlington in East Yorkshire, England, and known as an ‘imaginative tinkerer’, couldn’t stop tinkering. Now, two months later, the team has boosted itself with a new and improved einstein. (Papers for both results have not yet been peer-reviewed.)

This search for tiles first began in the 1960s, when mathematician Hao Wang conjectured that it would be impossible to find a set of shapes that could only occasionally tile a face. His student Robert Berger, now a retired electrical engineer in Lexington, Massachusetts, then found a set of 20,426 tiles that did just that, followed by a set of 104. In the 1970s, Sir Roger Penrose, a mathematical physicist at Oxford, had brought it down two.

And then came the monotile hat. But there was one complaint.

Dr. Berger (among others, the researchers of the recent papers) noted that the hat tiles use 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 is a reflection of the first,” said Dr. Berger. .

“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 out of paper or plastic can easily be flipped three-dimensionally to get its reflection, but a glazed ceramic tile can’t.”

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

“I wasn’t surprised that such a tile existed,” says study 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 changing continuum — a myriad infinity of shapes, obtained by enlarging and shrinking the edges of the hat — that produce aperiodic tiling using reflections.

But there was an exception, a “rogue member of the continuum,” said Craig Kaplan, a co-author and computer scientist at the University of Waterloo. This shape, technically known as Tile(1,1), can be thought of 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,” said Dr. Kaplan – a super power that unlocked the new discovery.

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 initial discovery was posted online in March. “I machine cut shapes out of cardboard to see what would happen if I just used non-reflective tiles,” he said in an email. Reflected tiles were banned “by fiat”, as the authors put it.

Mr. Smith said, “It didn’t take me long to produce a reasonably sized patch” — fitting tiles together like a jigsaw puzzle, with no overlaps or gaps. He knew he was onto something.

Further research – using a combination of traditional mathematical reasoning and drawing, plus computational handwork by Dr. Kaplan and Dr. Myers – the team proved that these tiles were indeed aperiodic.

“We call this a ‘weakly chiral aperiodic monotile,'” Dr. Kaplan 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 ‘dexterity’, from the Greek ‘kheir’ for ‘hand’. They called the new aperiodic tiling “chiral” because it is composed solely of left- or right-handed tiling. “You can’t combine 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 step further: they produced a family of strong or “strictly chiral aperiodic monotiles” by a simple modification of the T(1,1) tile: they replaced the straight edges with curves.

These monotiles, called “Spectres”, allow only non-periodic tilings due to their rounded contours, and no reflections. “A left-handed Specter cannot interlock with its right-handed mirror image,” said Dr. Kaplan.

“Now there’s no more debate about whether the aperiodic tile set has one or two tiles,” said Dr. Berger 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 expected from an aperiodic monotile.” On a tile frame, she had just seen a playful “Escherization” (after Dutch artist MC Escher) of the Specter tile by Dr. Araki, who called it a”double-headed pig.”

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