3 ms·

‘Nasty’ geometry breaks decades-old tiling conjecture

81 points by isaacfrond 4y ago

isaacfrond 4y ago
It's a paper by Terrence Tao, and Rachel Greenfeld. Terrence Tao features here often. They disproved a longstanding conjecture in geometry known as the periodic tiling conjecture. This conjecture stated that it was impossible to devise a tile that could cover a higher-dimensional space aperiodically, but not periodically. The two mathematicians were able to construct a tile that could do just this, breaking the conjecture. This finding has implications for the understanding of patterns and symmetry in higher-dimensional spaces.
- ndsipa_pomu 4y ago
  I feel that Terrence Tao doesn't get much recognition outside of maths circles but he's one of those people who just seem to be in a totally different league. Maybe it's just that maths doesn't capture the public's imagination as it's incomprehensible to people not involved with the field.
  - evolve2k 4y ago
    We’ll this article is maybe a good case in point. Surely as they were talking of tiled geometry there could have been space for a few visual examples, even if they hit a practical limit and said ‘It’s theoretical from here but this gives the idea’.
    - klyrs 4y ago
      I was too. I mean, they show the aperiodic Penrose tiles, which is nice because that's approachable. But the punchline of the story quashes the hope of a nice illustration of the result: the proof works in 2^(100^100)-dimensional space. We always joke that one must only visualize n-space and set n to 2^(100^100)... but that isn't much of a help. It really is a nasty geometry.
  - voxl 4y ago
    Terrence Tao is just another human, who had a lot of investment at a young age from world renowned mathematicians. The guy also isn't a huge fan of "genius"-type hype around his work. Don't let the existence of someone be the excuse you use to tell yourself you couldn't also do something cool or great.
    - thereare5lights 4y ago
      I'd bet any amount of money you want that you could just do the same thing with any arbitrary human and create a mathematician of his caliber. The idea that any kid who gets a lot of investment at a young age from world renowned mathematicians will be another Terrence Tao or even 10% of Tao is not plausible. He may not like the genius label, but in his case, it is absolutely true.
      - deleted 4y ago
        [deleted]
      - pgayed 4y ago
        Commenter doesn't say you would produce another Terrence Tao: > Don't let the existence of someone be the excuse you use to tell yourself you couldn't also do something cool or great.
        robinhouston 4y ago
        Meta: I’m fascinated by the way that a string of five comments, by different users, have all copied the misspelling of Terence Tao’s name from the original comment by isaacfrond.
        pyinstallwoes 4y ago
        Is that something GPT would do?
        Ygg2 4y ago
  - _0ffh 4y ago
    > doesn't get much recognition outside of maths circles I don't know. I'm definitely not "maths circles" and I don't know a whole lot of mathematicians by name. Those I do know are either personal acquaintances or people I consider to be more or less famous (people like Gauss, Fermat, Laplace, Hilbert, von Neumann, or Perelman). I definitely know Tao's name. He's not an acquaintances of mine, so I reason he must be more or less famous. =)
    - ndsipa_pomu 4y ago
      Well, I wouldn't include myself in 'maths circles' either, but I enjoy reading popularised maths articles and occasional maths videos (e.g. 3Blue1Brown, Mathologer). It just seems far more niche than other topics - I wonder how many people are familiar with Perelman as well.
      - _0ffh 4y ago
        Yes, Perelman was by far the hardest to come up with, but I really wanted at least one contemporary name in there.
- deleted 4y ago
  [deleted]
- tromp 4y ago
  What I found interesting about the construction is that they used dimensions like lines of code in a program, each trying to accomplish one small task. PS: Terence is with one r.
  - User23 4y ago
    Yeah, it’s really neat seeing mainstream mathematicians finally applying mathematical methods computing scientists discovered half a century ago. The isomorphism between programs and proofs implies we’ll see this or similar techniques developed for constructing programs that can be readily mathematically reasoned about applied more and more often in different areas. Good stuff.
rurban 4y ago
Pretty intuitive result. In 2D it was disproven, Penrose found two. But extending this to 3D it would be much easier to find such a tiling. Now they found one of a single shape.
- ginnungagap 4y ago
  The periodic tiling conjecture is true in dimension 2: if you have a single tile that tiles Z^2 aperiodically by translations, then it also tiles Z^2 by translations periodically, this is a result of Bhattacharya (and the reason why the expectation was for the conjecture to be true in general)

81 points by isaacfrond 4y ago

It's a paper by Terrence Tao, and Rachel Greenfeld. Terrence Tao features here often. They disproved a longstanding conjecture in geometry known as the periodic tiling conjecture. This conjecture stated that it was impossible to devise a tile that could cover a higher-dimensional space aperiodically, but not periodically. The two mathematicians were able to construct a tile that could do just this, breaking the conjecture. This finding has implications for the understanding of patterns and symmetry in higher-dimensional spaces.

ndsipa_pomu 4y ago
I feel that Terrence Tao doesn't get much recognition outside of maths circles but he's one of those people who just seem to be in a totally different league. Maybe it's just that maths doesn't capture the public's imagination as it's incomprehensible to people not involved with the field.
- evolve2k 4y ago
  We’ll this article is maybe a good case in point. Surely as they were talking of tiled geometry there could have been space for a few visual examples, even if they hit a practical limit and said ‘It’s theoretical from here but this gives the idea’.
  - klyrs 4y ago
    I was too. I mean, they show the aperiodic Penrose tiles, which is nice because that's approachable. But the punchline of the story quashes the hope of a nice illustration of the result: the proof works in 2^(100^100)-dimensional space. We always joke that one must only visualize n-space and set n to 2^(100^100)... but that isn't much of a help. It really is a nasty geometry.
- voxl 4y ago
  Terrence Tao is just another human, who had a lot of investment at a young age from world renowned mathematicians. The guy also isn't a huge fan of "genius"-type hype around his work. Don't let the existence of someone be the excuse you use to tell yourself you couldn't also do something cool or great.
  - thereare5lights 4y ago
    I'd bet any amount of money you want that you could just do the same thing with any arbitrary human and create a mathematician of his caliber. The idea that any kid who gets a lot of investment at a young age from world renowned mathematicians will be another Terrence Tao or even 10% of Tao is not plausible. He may not like the genius label, but in his case, it is absolutely true.
    - deleted 4y ago
      [deleted]
    - pgayed 4y ago
      Commenter doesn't say you would produce another Terrence Tao: > Don't let the existence of someone be the excuse you use to tell yourself you couldn't also do something cool or great.
      - robinhouston 4y ago
        Meta: I’m fascinated by the way that a string of five comments, by different users, have all copied the misspelling of Terence Tao’s name from the original comment by isaacfrond.
        pyinstallwoes 4y ago
        Is that something GPT would do?
        Ygg2 4y ago
- _0ffh 4y ago
  > doesn't get much recognition outside of maths circles I don't know. I'm definitely not "maths circles" and I don't know a whole lot of mathematicians by name. Those I do know are either personal acquaintances or people I consider to be more or less famous (people like Gauss, Fermat, Laplace, Hilbert, von Neumann, or Perelman). I definitely know Tao's name. He's not an acquaintances of mine, so I reason he must be more or less famous. =)
  - ndsipa_pomu 4y ago
    Well, I wouldn't include myself in 'maths circles' either, but I enjoy reading popularised maths articles and occasional maths videos (e.g. 3Blue1Brown, Mathologer). It just seems far more niche than other topics - I wonder how many people are familiar with Perelman as well.
    - _0ffh 4y ago
      Yes, Perelman was by far the hardest to come up with, but I really wanted at least one contemporary name in there.
deleted 4y ago
[deleted]
tromp 4y ago
What I found interesting about the construction is that they used dimensions like lines of code in a program, each trying to accomplish one small task. PS: Terence is with one r.
- User23 4y ago
  Yeah, it’s really neat seeing mainstream mathematicians finally applying mathematical methods computing scientists discovered half a century ago. The isomorphism between programs and proofs implies we’ll see this or similar techniques developed for constructing programs that can be readily mathematically reasoned about applied more and more often in different areas. Good stuff.

rurban 4y ago

Pretty intuitive result. In 2D it was disproven, Penrose found two. But extending this to 3D it would be much easier to find such a tiling. Now they found one of a single shape.

ginnungagap 4y ago
The periodic tiling conjecture is true in dimension 2: if you have a single tile that tiles Z^2 aperiodically by translations, then it also tiles Z^2 by translations periodically, this is a result of Bhattacharya (and the reason why the expectation was for the conjecture to be true in general)