Last November, after a decade of failed attempts, David Smith, a self-proclaimed form hobbyist from Bridlington in East Yorkshire, England, surmised that he might finally have solved an open problem in the mathematics of tiling: that is, he thought he might have one “Einstein” discovered.
In less poetic terms, an Einstein is an “aperiodic monotile,” a shape that tiles a plane or infinite two-dimensional flat surface, but only in a non-repeating pattern. (The term “Einstein” comes from the German “ein stein” or “ein Stein” – looser, “a Fliese” or “a shape”.) Your typical wallpaper or tiled floor is part of an infinite pattern that repeats itself periodically; If it is shifted or “translated”, the pattern can be placed exactly on itself. Aperiodic tiling exhibits no such “translational symmetry” and mathematicians have long sought a single shape that could tile the plane in this way. This is known as the Einstein problem.
“I am always tinker and experiment with molds,” said Mr. Smith, 64, who worked among other things as a printing technician and took early retirement. Although he liked math in high school, he wasn’t very good at it, he said. But he has long been “obsessively fascinated” by the Einstein problem.
And now one new paper – by Mr. Smith and three co-authors with mathematical and computational expertise – proves that Mr. Smith’s discovery is true. The researchers dubbed their Einstein “the hat” because it resembles a fedora. (Mr. Smith often wears a bandana around his head.) The paper has not yet been peer-reviewed.
“This appears to be a remarkable discovery!” Joshua Socolar, a physicist at Duke University who read an early copy of the article provided by The New York Times, said in an email. “The most important aspect for me is that the tiles do not clearly fall into any of the known classes of structures that we understand.”
“The mathematical result raises some interesting physics questions,” he added. “One could imagine encountering or creating a material with such an internal structure.” Socolar and Joan Taylor, an independent researcher in Burnie, Tasmania, previously found a hexagonal monotile of separate pieces, which some felt stretched the rules. (They also found a connected 3-D version of the Socolar-Taylor tile.)
From 20,426 to one
Initially, mathematical tiling was motivated by a broad question: was there a set of shapes that could only irregularly tile the plane? 1961 became the mathematician guessed Hao Wang that such sets were impossible, but his student Robert Berger soon proved this conjecture wrong. dr Berger discovered an aperiodic set of 20,426 tiles, and then a set of 104.
Then the game became: how few tiles would be enough? In the 1970s, Sir Roger Penrose, a mathematical physicist at Oxford University who received the 2020 Nobel Prize in Physics for his research on black holes, brought the number down two.
Others have since found forms for two tiles. “I have a pair or two mine,” said Chaim Goodman-Strauss, another author of the paper, a professor at the University of Arkansas who also holds the title of Outreach Mathematician at the University of Arkansas National Museum of Mathematics in NYC.
He found that black and white squares can create strange non-periodic patterns in addition to the familiar periodic checkerboard pattern. “It’s really quite trivial to be able to create weird and interesting patterns,” he said. The magic of the two Penrose tiles is that they only produce non-periodic patterns – that’s all they can do.
“But then the Holy Grail was, could you get by with one – one tile?” said Dr. Goodman-Strauss.
Just a few years ago, Sir Roger was in search of an Einstein, but he put that exploration aside. “I reduced the number to two, and now we have reduced it to one!” he said about the hat. “It’s a feat. I see no reason not to believe that.”
The paper provided two proofs, both performed by Joseph Myers, a co-author and software engineer in Cambridge, England. One was a traditional proof based on a previous method, plus custom code; another used a new, non-computerized technique developed by Dr. Myers was developed.
Sir Roger found the evidence “very complicated”. Nevertheless, he was “extremely fascinated” by the Einstein, he said: “It’s a really good shape, remarkably simple.”
Imaginative crafts
The simplicity came honestly. Mr. Smith’s investigations were mostly done by hand; one of his co-authors described him as an “imaginative inventor”.
At the beginning he “fiddled” with the computer screen PolyForm puzzle solverSoftware developed by Jaap Scherphuis, a Tiler and puzzle theorist in Delft, Netherlands. But when a shape had potential, Mr. Smith would use a Silhouette cutting machine to create an initial batch of 32 cardstock copies. He then fitted the tiles together like a jigsaw puzzle with no gaps or overlaps, flipping and rotating the tiles as needed.
“It’s always nice to get involved,” said Mr. Smith. “It can be quite meditative. And it provides a better understanding of how a shape does or does not tessellate.”
In November, when he found a tile that appeared to fill the plane without a repeating pattern, he emailed Craig Kaplan, a co-author and computer scientist at the University of Waterloo.
“Could this shape be an answer to the so-called ‘Einstein problem’ – now wouldn’t that be a thing?” Mr. Smith wrote.
“It was clear that something unusual was happening with this shape,” said Dr. chaplain. Using a computational approach that builds on previous research, his algorithm created larger and larger swathes of hat tiles. “There seemed to be no limit to how big a blob of tiles the software could construct,” he said.
With this raw data, Mr. Smith and Dr. Kaplan the hierarchical structure of the tiles with the eye. dr Kaplan discovered and decoded telltale behaviors that opened up a traditional proof of aperiodicity – The method mathematicians “pull out the drawer every time you have a candidate set of aperiodic tiles,” he said.
The first step, said Dr. Kaplan, was to “define a set of four ‘metatiles,’ simple shapes representing small groupings of one, two, or four hats.” The metatiles combine into four larger shapes that behave similarly. This gathering of metatiles to supertiles to supersupertiles, ad infinitum, covered “bigger and bigger mathematical ‘floors’ with copies of the hat,” said Dr. chaplain. “We then show that this kind of hierarchical arrangement is essentially the only way to tile the plane with hats, which turns out to be sufficient to show that it can never tile periodically.”
“It’s very smart,” said Dr. Berger, a retired electrical engineer in Lexington, Mass., in an interview. At the risk of sounding finicky, he pointed out that since hat tiling uses reflections – the hat-shaped tile and its mirror image, some might wonder if this is a two-tile aperiodic monotile set rather than a single tile .
dr Goodman-Strauss had addressed this subtlety on a tiling listserv: “Is there a hat or two?” The consensus was that a monotile counts as such even when using its reflection. That leaves an open question, said Dr. Berger: Is there an Einstein that does the work without reflection?
Hide in the hexagons
dr Kaplan clarified that “the hat” is not a new geometric invention. It is a poly dragons — it consists of eight dragons. (Take a hexagon and draw three lines connecting the center of each side to the center of the opposite side; the six resulting shapes are kites.)
“It’s likely that others have considered this hat shape in the past, just not in a context where they were studying its tile properties,” said Dr. chaplain. “I like to think it was hiding in plain sight.”
Marjorie Senechal, a mathematician at Smith College, said, “In a way, it’s been sitting there waiting for someone to find it.” Senechal’s research explores the neighboring realm of Mathematical crystallographyand connections with quasicrystals.
“What impresses me the most is that these aperiodic tiles are arranged on a hexagonal lattice that is as periodic as possible,” said Doris Schattschneider, a mathematician at Moravia University whose research focuses on this mathematical analysis of periodic tilesespecially that of the Dutch artist MC Escher.
dr Senechal agreed. “It sits right in the hexes,” she said. “How many people are going to kick the world and wonder why I didn’t see that?”
The Einstein family
Incredibly, Mr. Smith later found a second Einstein. He called it “the tortoise” – a polydrake made up of ten rather than eight dragons. It was “spooky,” said Dr. chaplain. He remembered panicking; he was already “neck-deep in his hat”.
But dr Myers who did it similar calculationsShe promptly discovered a deep connection between the hat and the turtle. And he realized that there was, in fact, a whole family of related Einsteins – an unbroken, uncountable infinity of forms that morph into one another.
Mr. Smith wasn’t as impressed with some of the other family members. “They looked a bit like imposters or mutants,” he said.
But this Einstein family motivated the second proof, which offers a new tool for proving aperiodicity. The math seemed “too good to be true,” said Dr. Myers in an email. “I didn’t expect such a different approach to proving aperiodicity – but everything seemed to come together as I wrote down the details.”
dr Goodman-Strauss regards the new technique as a crucial aspect of the discovery; To date, there have only been a handful of aperiodicity proofs. He conceded it was “strong cheese,” perhaps only something for die-hard connoisseurs. It took him a few days to process. “Then I was blown away,” he said.
Mr. Smith was amazed to see the research work come together. “I wasn’t any help, to be honest.” Appreciating the illustrations, he said, “I’m more of an image person.”
