A thirteen-sided shape known as a “hat” can give mathematicians their all. When laying tiles, most people tend to use simple and repetitive geometric shapes, such as squares and triangles, but some people prefer not to repeat patterns. A recent study found a new shape that never produces repeating patterns when tiling (combining) flat surfaces.
In the mathematical world, an aperiodic monotile is a shape that tiles the plane and never repeats, and mathematicians have been looking for such a shape, and it is not even clear that it is possible to exist. In 1974, British mathematician Roger Penrose (Roger Penrose) created Penrose tiles, which consist of two rhombuses of different shapes, which can be paved on an infinite plane in a non-repetitive manner. The tiles of this kind cover the global plane, and the pattern will not repeat.
But is it possible that there is a “single shape” that never repeats a pattern when covering a plane? After a long search, four mathematicians from York University, Cambridge University, University of Waterloo, and University of Arkansas have now discovered this special single shape, which only needs 13 sides, and can be neatly combined to extend a non-periodic plane: forever Does not form repetitive patterns.
The new aperiodic monotile discovered by Dave Smith, Joseph Myers, Craig Kaplan, and Chaim Goodman-Strauss, rendered as shirts and hats. The hat tiles are mirrored relative to the shirt tiles. pic.twitter.com/BwuLUPVT5a
— Robert Fathauer (@RobFathauerArt) March 21, 2023
The team demonstrated this shape property through computer simulations. What is fascinating is that even if the side length changes, the final shape does not lose its “aperiodic” characteristics.
While the paper has not yet been peer-reviewed, experts agree that the results stand up to scrutiny and that the new aperiodic tiles could spark further research by the materials science community. Time to revamp our bathroom tile styles.
(First image source: Science News video screenshot)
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