When tiling a floor, most folks want to see a repeating pattern that shows attention to detail and pleasant symmetry. But in geometry, experts explore patterns that do not repeat. This is called aperiodic tiling. The Wikipedia entry on aperiodic tiling is flagged as "too technical for most readers to understand," and they are right. A challenge in geometry has been to find the lowest number of tile shapes that will produce a non-repeating pattern. Up until now, Penrose tiling, discovered by Roger Penrose, has been the lowest, with two tiles, shown above. It looks pretty symmetrical to me, but the caveat is that while these tiles don't have translation symmetry, they may have reflection symmetry and rotational symmetry. Penrose tiles look really nice on the floor of a round room. The search for a single tile shape that produces aperiodic tiling is called the Einstein problem, because the German words "ein stein" mean "one tile."
However, recently a new science paper was submitted claiming an aperiodic tiling shape that uses only one tile! They call the shape of this tile "the hat." It looks more like a shirt to me, but how cool is this floor pattern?
The paper is awaiting peer review before publication. There's a lot to unpack here for geometry geeks, but for the rest of us, can you imagine trying to lay a floor with the hat? It would be like putting a chaotic jigsaw puzzle together with grout.
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