An aperiodic monotile is a single shape, like a puzzle piece, that can make a non-repeating pattern forever.

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This shape, called the HAT was discovered early in 2023.

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This is a huge discovery in math – the last discovery, a 2 tile version, was discovered in the 70’s.

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Here's an article about it:
https://www.theguardian.com/science/2023/apr/03/new-einstein-shape-aperiodic-monotile

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Some very interesting discussion about this is here, including the about-to-be published paper:
https://discourse.mcneel.com/t/an-aperiodic-monotile/156830

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Here's a generator here to make your own:
https://cs.uwaterloo.ca/~csk/hat/app.h