With new, improved ‘Einstein’, puzzlers settle a math problem

NEW YORK: 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 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. 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 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-andimproved 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 (Papers for both results are not yet peer reviewed.)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.

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 shape happens to have a hidden superpower,” Dr. Kaplan said — a superpower that unlocked the new discovery.