Post ATz0D33ranKNt2j0IC by OscarCunningham@mathstodon.xyz
 (DIR) More posts by OscarCunningham@mathstodon.xyz
 (DIR) Post #ATz0CzyR4WCgJLA7uq by divbyzero@mathstodon.xyz
       2023-03-21T14:56:35Z
       
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       Super exciting news about Smith, @jsm28, @csk, Goodman-Strauss's discovery of an "einstein"—a single shape that tiles the plane aperiodically! I decided to create a 3D-printable version of it. You can find it here: https://www.thingiverse.com/thing:5923307(I now see that others did this as well—probably not surprising since the geometry of the shape is so simple, relatively speaking.)Here's their preprint: https://arxiv.org/abs/2303.10798
       
 (DIR) Post #ATz0D0trd08nBS1zQe by divbyzero@mathstodon.xyz
       2023-03-22T00:44:02Z
       
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       Here are 30 tiles that I printed today and the beginning of an aperiodic tiling. The red ones are mirror images of the others.
       
 (DIR) Post #ATz0D1WrI1Kh8OH8D2 by OscarCunningham@mathstodon.xyz
       2023-03-22T02:28:24Z
       
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       @divbyzero I wonder if it's possible to tile the plane and then three-colour the tiling so that no adjacent regions have the same colour?
       
 (DIR) Post #ATz0D2MyAH15k0ekQy by divbyzero@mathstodon.xyz
       2023-03-22T02:42:33Z
       
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       @OscarCunningham The center blue tile has five neighbors. So those six tiles require four colors. But I had a similar thought—about colorability. I was wondering if you color all the “reversed” tiles one color (red in this picture), can you still four-color the tiles?
       
 (DIR) Post #ATz0D33ranKNt2j0IC by OscarCunningham@mathstodon.xyz
       2023-03-25T08:58:25Z
       
       0 likes, 1 repeats
       
       @divbyzero The aperiodic 'hat' tilings seem to have a four-colouring where one of the colours consists of precisely the 'flipped' tiles.Even more amazing, when I was trying it I never had to make any choices after the first four colours. My moves were always forced. So it seems like this is a natural unique colouring. Wonderful!I just tried this by hand, I don't have a proof that it extends forever.
       
 (DIR) Post #ATz0DEigFNVu2ouFwO by OscarCunningham@mathstodon.xyz
       2023-03-25T09:22:52Z
       
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       @divbyzero Jesse Clark (@myhf) points out that when four tiles meet at a point they still have all different colours. https://twitter.com/myhf/status/1639053012399439872?s=20
       
 (DIR) Post #ATz0DErXiQbQUJ3Kkq by OscarCunningham@mathstodon.xyz
       2023-03-25T08:59:30Z
       
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       @divbyzero This has practical applications for mathematics departments undergoing renovation.