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This infinite tiling pattern could end 60 years of mathematical exploration
A shape (highlighted in green) was discovered that allowed the area to be infinitely tiled without creating a repeating pattern.Credit: D. Smith other./ar Xiv
After 60 years of searching, mathematicians may finally have discovered a true single ‘aperiodic’ tile, a shape that can cover an infinite plane but never repeats in a pattern.
Periodic tilings have translational symmetry. For example, a honeycomb pattern can repeat indefinitely, and any number of cells will make him look identical even after shifting in any of the six directions. However, such shifts are not possible with aperiodic tiling.
Breakthrough in shape
In March, a team announced significant progress in searching for aperiodic tiles. David Smith, a hobby mathematician based in Bridlington, England, discovered a shape he suspected might be an aperiodic tile, and collaborated with three professional mathematicians to determine if his tile was a mirror image, In other words, I wrote up proof that it was the tile, along with the inverted image.Can be used to construct infinite aperiodic tilings of the plane1. (The proof has not yet been peer-reviewed, but mathematicians have reportedly said it appears to be rigorous.)
A pattern consists of a single tile like this wavy tile.Credit: D. Smith other./ar Xiv
Smith’s geometry was not a single aperiodic tile. This is because Smith and its mirror image are effectively two separate tiles. Also, I needed both versions to tile the entire plane. Now, however, the same group of mathematicians have reported a modified version of the original tile that can build aperiodic tiles without flipping.2This proof has been posted to the preprint server arXiv and has not yet been peer reviewed.
The first aperiodic tiles were discovered in the 1960s and contained 20,426 tiles. After various refinements, UK mathematician Roger Penrose of the University of Oxford, who won the Nobel Prize in Physics in 2020 for his fundamental work on black hole theory, was the first to consist of only two tile types. found an aperiodic tiling of . They are not just mirror images of each other. Penrose tiles now adorn the courtyard of the Mathematics Department at Oxford University.
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Sources 2/ https://www.nature.com/articles/d41586-023-01801-8 The mention sources can contact us to remove/changing this article |
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