Aperiodic tilings! Just a couple of years ago someone discovered a single tile (down from the set of ~20000 that was first used to prove that aperiodic tiling was even possible) that can completely cover an infinite plane without ever falling into a repeating pattern.
The use of “aperiodic” is somewhat loose here compared to what I would expect. Like… I can instantly see several places with the same pattern just on that small sample…
Aperiodic, in this sense, doesn’t mean that there aren’t any bits that repeat. In fact, if you pick any patch of tiles of any arbitrary size, that patch will be repeated infinitely many times. What it means to be aperiodic is that if you slide the whole tiling over so that one of the patches aligns with the repeated bit, there will still be something outside the patch that doesn’t align. Compare that with, say, a repeating grid of squares, where if you slide one square onto a different square then everything lines up, all the way to infinity; it’s impossible to tell that it’s been slid over.
Aperiodic tilings! Just a couple of years ago someone discovered a single tile (down from the set of ~20000 that was first used to prove that aperiodic tiling was even possible) that can completely cover an infinite plane without ever falling into a repeating pattern.
The use of “aperiodic” is somewhat loose here compared to what I would expect. Like… I can instantly see several places with the same pattern just on that small sample…
Aperiodic, in this sense, doesn’t mean that there aren’t any bits that repeat. In fact, if you pick any patch of tiles of any arbitrary size, that patch will be repeated infinitely many times. What it means to be aperiodic is that if you slide the whole tiling over so that one of the patches aligns with the repeated bit, there will still be something outside the patch that doesn’t align. Compare that with, say, a repeating grid of squares, where if you slide one square onto a different square then everything lines up, all the way to infinity; it’s impossible to tell that it’s been slid over.
Neat!