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URL: Mathematicians have discovered an elusive ‘einstein’ tile
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https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile
Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.
Related Information
Magic tile shape grows organically, never creating a pattern
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This is the most complex shape known that can fit together organically. It was just discovered by mathematicians in 2023. It is easy to cookie-cut these out of clay (notice the cutter I made at the top). Placing the tiles is tricky because it is only logical to seek a pattern, but that does not work. Starting with a center tile and moving outward in a spiral around it seems to be the best way. Mathematicians are seeking to prove that placement can grow infinitely without ever repeating a pattern. Making the cookie cutter in Fusion 360 was easier than expected because the shape is built from the pie-slices that result from cutting a regular hexagon into six pieces midway across the straight sides. Because of the complexity of the shape I have found that it is best to print multiple cutters (I can do eight at a time), and stamp the shapes without using stretch wrap (letting them dry overnight in the cutters). The randomness seems confirmed in that when I piece together a few dozen tiles it is very difficult to do a count (because they are not in rows). In addition, to piece together 28 tiles requires turning eleven of them over - if there was a pattern I would expect to turn over exactly half of them. One issue: To create a setting with straight sides it appears I will need a dozen shapes.
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