‘Impossible’ tile that never repeats has been invented
A geometry problem that has puzzled scientists for 60 years may have been solved by an amateur mathematician with a newly discovered 13-sided geometry.
Called the “hat” for its vague resemblance to a fedora, the elusive shape is “Einstein” (from the German “Einstein” or “one stone”). This means it can completely cover the surface without creating a repeating pattern. This was not yet achieved with a single tile.
“I’m always looking for interesting shapes, but this one was more than that,” its creator, David Smith, a retired printer from the North of England, said in a telephone interview.2022 Shortly after discovering the shape in November 2011, he contacted a mathematics professor and later released a self-published publication with two other academics. scientific paper about it.
“To be honest, I’m not very good at math. I did it in school and I wasn’t very good at it,” Smith said. That’s why I got these other guys involved. I was a little lucky to find the shape, but I was also the one who was persistent. “
from 20,426 to 1
Most wallpapers and tiles in the real world are cyclical. That is, we can identify small clusters that are constantly repeated to cover the entire surface. However, “hats” are non-periodic tiles. That is, it is possible to completely cover the surface without gaps, but it is not possible to identify regularly repeated clusters to do so.
Fascinated by the idea that such an aperiodic set of shapes could exist, mathematicians first pondered the problem in the early 1960s, but initially believed in shapes. it was impossibleIt turned out to be wrong because in a few years they created a set of 20,426 tiles that work in combination. That number soon dwindled him to a little over 100, then to six.
In the 1970s, the work of British physicist and Nobel Prize winner Roger Penrose further reduced the number of shapes from six to two. penrose tilingAnd that’s where things have been stuck for decades.
Smith got interested in this issue when he launched in 2016. blog on the subject. Six years later, in late 2022, when he thought he had beaten Penrose for his discovery of Einstein, he contacted Craig his Kaplan, a professor in his science department at the University of Waterloo in Canada.
“From my point of view, it started with an out-of-the-box email,” Kaplan said in a phone interview. I knew you made the announcement.”
With the help of software, the two realized they were on to something.
How the “hat” works
According to Kaplan, there is nothing inherently magical about a “hat.”
“It’s a polygon that’s very simple to describe. There are no strange or irrational angles. It’s basically what you get by cutting a hexagon.” It may have been ‘discovered’ by other mathematicians who did it, but they didn’t think to check its tiling properties.
The discovery has caused quite a stir since it was announced in late March. artistic performance, knit quilt, cookie cutter, TikTok Explainer Even Jimmy Kimmel jokes opening monologue.
“I think it might open some doors,” Smith said.
the shape is publicly available, 3D printingis not copyrighted.
“We never try to defend it,” Kaplan said. “It belongs to everyone and we hope people will use this in all kinds of decorative, architectural and artistic content.”
What about bathroom tiles? “I hope to see a lot of bathrooms decorated with it, but it’s going to be a little difficult,” he added. , because the rules for how to place them are very simple, and with this comes another challenge: you could potentially start placing it and hack it in the corner of the room.No more We’ve created a space where no hat can be stuffed.”
peer review first
Far from being content to rewrite the history of mathematics, Smith has already discovered a “sequel” to “The Hat.” called “tortoiseThe new shape is also Einstein, but it is larger than the “hat” because it is made of 10 kites (sections) instead of 8.
“It’s a little addicting,” Smith confessed of his constant quest for new forms.
The scientific paper on “The hat,” co-authored with software developer Joseph Myers and mathematician Chaim Goodman-Strauss at the University of Arkansas, has not yet been peer-reviewed. It is a standard validation process by other scientists. in scientific publications—but will in the coming months.
“I’m really excited to see what comes out of that process,” Kaplan said, admitting that it could mean the findings could be challenged. We strongly believe in the importance of peer review as a way to do so, and we agree that there are still reasons to be uncertain until it happens, but based on the evidence we have accumulated, it is It’s hard to imagine how we could be wrong.”
If confirmed, the findings could have important implications for other areas of research, said Rafe Mazzeo, a professor of mathematics at Stanford University. He was not involved in the research.
“Tilings have many uses in physics, chemistry and even crystal studies,” he said in an email. “The discovery of aperiodic tilings many years ago was controversial because their existence was so unexpected.
“This new discovery is a surprisingly simple example. There is no known standard technique for finding new aperiodic tiles, so this really required a new idea. It’s always exciting.” he added.
Mazzeo said he was also happy to hear about a mathematical discovery that was very easy to visualize and explain.