4 ms·

> and divide each square in two in a non-periodic way. That doesn't actually sound simple. Can you provide an example of how you would do that?

by mabbo 3y ago

enriquto 3y ago
You traverse the squares in a spiral pattern starting from a central one. Then each square is divided vertically or horizontally according to the binary digits of pi (or whatever).
- zuminator 3y ago
  It doesn't even need to be (pseudo-) random. You can trace a spiral pattern out from the origin like so: 1 vertical up, turn right, 1 horizontal right, turn down, 2 vertical down, turn left, 2 horizontal left, turn up, 3 vertical up, turn right, 3 horizontal right, turn down, etc. The resulting spiral pattern will never repeat.
BurningFrog 3y ago
For each 2x2 block, decide randomly/arbitrarily to split it horizontally or vertically.
- cvoss 3y ago
  This solution won't be accepted without a lot more work. Flipping a coin is not guaranteed to produce an aperiodic tiling, since, in the trivial case where the coin comes up heads indefinitely, you'll be producing a periodic tiling. (You will argue it's not likely to do this, and I'd agree. In fact, I'd conjecture you'll only produce a periodic tiling with measure zero probability.) But what's required is to actually show an aperiodic tiling. If you have a proof that coin flipping will produce such a tiling with non-vanishing probability, then, presumably, embedded in your proof, you will likely find a decision procedure for producing an aperiodic tiling deterministically.
  - JadeNB 3y ago
    > If you have a proof that coin flipping will produce such a tiling with non-vanishing probability, then, presumably, embedded in your proof, you will likely find a decision procedure for producing an aperiodic tiling deterministically. I'm not sure it's obvious to expect a proof that something happens with positive probability to embed a constructive proof of existence (which seems to be what you mean). The difference between these two ideas in combinatorics is exploited with the probabilistic method (https://en.wikipedia.org/wiki/Probabilistic_method https://en.wikipedia.org/wiki/Probabilistic_method).

> and divide each square in two in a non-periodic way. That doesn't actually sound simple. Can you provide an example of how you would do that?

by mabbo 3y ago

> and divide each square in two in a non-periodic way. That doesn't actually sound simple. Can you provide an example of how you would do that?

enriquto 3y ago

You traverse the squares in a spiral pattern starting from a central one. Then each square is divided vertically or horizontally according to the binary digits of pi (or whatever).

zuminator 3y ago
It doesn't even need to be (pseudo-) random. You can trace a spiral pattern out from the origin like so: 1 vertical up, turn right, 1 horizontal right, turn down, 2 vertical down, turn left, 2 horizontal left, turn up, 3 vertical up, turn right, 3 horizontal right, turn down, etc. The resulting spiral pattern will never repeat.

BurningFrog 3y ago

For each 2x2 block, decide randomly/arbitrarily to split it horizontally or vertically.

cvoss 3y ago
This solution won't be accepted without a lot more work. Flipping a coin is not guaranteed to produce an aperiodic tiling, since, in the trivial case where the coin comes up heads indefinitely, you'll be producing a periodic tiling. (You will argue it's not likely to do this, and I'd agree. In fact, I'd conjecture you'll only produce a periodic tiling with measure zero probability.) But what's required is to actually show an aperiodic tiling. If you have a proof that coin flipping will produce such a tiling with non-vanishing probability, then, presumably, embedded in your proof, you will likely find a decision procedure for producing an aperiodic tiling deterministically.
- JadeNB 3y ago
  > If you have a proof that coin flipping will produce such a tiling with non-vanishing probability, then, presumably, embedded in your proof, you will likely find a decision procedure for producing an aperiodic tiling deterministically. I'm not sure it's obvious to expect a proof that something happens with positive probability to embed a constructive proof of existence (which seems to be what you mean). The difference between these two ideas in combinatorics is exploited with the probabilistic method (https://en.wikipedia.org/wiki/Probabilistic_method https://en.wikipedia.org/wiki/Probabilistic_method).