After decades of searching, mathematicians discovered a single shape that can cover a surface without forming repeating patterns, launching a small industry of “aperiodic monotile” merchandise.
The “hat” shape can tile an infinite plane without creating repeating patterns
It is rare for a shape to make a splash, but this year one did just that with the announcement of the first ever single tile that can cover a surface without forming repeating patterns. The discovery of this “aperiodic monotile” in March has since inspired everything from jigsaw puzzles to serious research papers.
“It’s more than I can keep up with in terms of the amount and even, to some extent, the level and depth of the material, because I’m not really a practising mathematician, I’m more of a computer scientist,” says Craig Kaplan at the University of Waterloo, Canada. He is on the team that found the shape, which it called the “hat”. Mathematicians had sought such an object for decades.
After revealing the tile in March, the team unveiled a second shape in May, the “spectre”, which improved on the hat by not requiring its mirror image to tile fully, making it more useful for real surfaces.
The hat has since appeared on T-shirts, badges, bags and as cutters that allow you to make your own ceramic versions.
It has also sparked more than a dozen papers in fields from engineering to chemistry. Researchers have investigated how the structure maps into six-dimensional spaces and the likely physical properties of a material with hat-shaped crystals. Others have found that structures built with repeating hat shapes could be more resistant to fracturing than a honeycomb pattern, which is renowned for its strength.
Kaplan says a scientific instrument company has also expressed an interest in using a mesh with hat-shaped gaps to collect atmospheric samples on Mars, as it believes that the pattern may be less susceptible to problems than squares.
“It’s a bit bittersweet,” says Kaplan. “We’ve set these ideas free into the world and they’ve taken off, which is wonderful, but leaves me a little bit melancholy because it’s not mine any more.”
However, the team has no desire to commercialise the hat, he says. “The four of us agreed early on that we’d much rather let this be free and see what wonderful things people do with it, rather than trying to protect it in any way. Patents are something that, as mathematicians, we find distasteful.”
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*Credit for article given to Matthew Sparkes*