Aperiodic Tiling with a single tile shape

[.Scott]

TL;DR Summary

A tile shape has been identified that can tile a surface aperiodically.

A tile shape has been identified that can tile a surface aperiodically.
Phys Org article

 

[scottdave]

I am reminded of this Veritasium video on Penrose tiles (takes 2 different shapes)

 

[phyzguy]

I just saw this today! I want to tile a floor with these tiles. I wonder how long it will take before someone starts manufacturing them?

 

[robphy]

You can 3-D print them
https://github.com/christianp/aperiodic-monotile

 

[robphy]

I just made a crude rendering in desmos
https://www.desmos.com/calculator/5zxbhw3nqr

I just posted here:

 

[robphy]

According to the CNN article quoted below,
the tiling was discovered by David Smith, an amateur mathematician,
who started a blog on the subject in 2016: https://hedraweb.wordpress.com/ .

https://hedraweb.wordpress.com/2023/03/23/its-a-shape-jim-but-not-as-we-know-it/

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https://www.cnn.com/2023/04/06/world/the-hat-einstein-shape-tile-discovery-scn/index.html

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.

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https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/

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.