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- A tile shape has been identified that can tile a surface aperiodically.
A geometry problem that has been puzzling scientists for 60 years has likely just been solved by an amateur mathematician with a newly discovered 13-sided shape.
“I’m always looking for an interesting shape, and this one was more than that,” said David Smith, its creator and a retired printing technician from northern England, in a phone interview. Soon after discovering the shape in November 2022, he contacted a math professor and later, with two other academics, they released a self-published scientific paper about it.
“I’m not really into math, to be honest — I did it at school, but I didn’t excel in it,” Smith said. That’s why I got these other guys involved, because there’s no way I could have done this without them. I discovered the shape, which was a bit of luck, but it was also me being persistent.”
Smith became interested in the problem in 2016, when he launched a blog on the subject. Six years later, in late 2022, he thought he had bested Penrose in finding the einstein, so he got in touch with Craig Kaplan, a professor in the School of Computer Science at the University of Waterloo in Canada.
In mid-November of last year, David Smith, a retired print technician and an aficionado of jigsaw puzzles, fractals and road maps, was doing one of his favorite things: playing with shapes. Using a software package called the PolyForm Puzzle Solver, he had constructed a humble-looking hat-shaped tile. Now he was experimenting to see how much of the screen he could fill with copies of that tile, without overlaps or gaps.
https://www.jaapsch.net/puzzles/polysolver.htm
But this is far from the first time a hobbyist has made a serious breakthrough in tiling geometry. Robert Ammann, who worked as a mail sorter, discovered one set of Penrose’s tiles independently in the 1970s. Marjorie Rice, a California housewife, found a new family of pentagonal tilings in 1975. And then there was Joan Taylor’s discovery of the Socolar-Taylor tile. Perhaps hobbyists, unlike mathematicians, are “not burdened with knowing how hard this is,” Senechal said.
Aperiodic tiling with a single tile shape is a type of tiling where a single tile shape is used to cover a surface without any gaps or overlaps, but the pattern does not repeat itself in any direction. This means that the tiling does not have a periodic pattern and can continue infinitely without repeating.
Aperiodic tiling is different from regular tiling in that it does not have a repeating pattern. Regular tiling, also known as periodic tiling, has a pattern that repeats itself in all directions, creating a grid-like structure. Aperiodic tiling, on the other hand, does not have a repeating pattern and can have a more complex and irregular structure.
Some well-known examples of aperiodic tiling with a single tile shape include the Penrose tiles, the Ammann-Beenker tiling, and the Wang tiles. These tilings have been studied extensively by mathematicians and have been found to have interesting properties and applications in various fields.
Aperiodic tiling with a single tile shape has various practical applications in fields such as architecture, cryptography, and materials science. For example, the Penrose tiles have been used in the design of buildings, while the Ammann-Beenker tiling has been used to create quasicrystals, which have unique properties and are being studied for potential use in electronic devices.
Aperiodic tiling with a single tile shape is relevant to the study of mathematics as it involves the exploration of complex geometric patterns and their properties. It also involves the use of mathematical concepts such as symmetry, tessellations, and group theory. Studying aperiodic tiling can also lead to new discoveries and insights in the field of mathematics.