‘Nasty’ geometry breaks decades-old tiling conjecture | Hacker News Reader

‘Nasty’ geometry breaks decades-old tiling conjecture

quantamagazine.org

81 pointsisaacfrond24 comments

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.

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.

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’.

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.

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.

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.

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.

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.

Is that something GPT would do?

Not really. Terrence and Terence are too visually similar and tbh I've seen more former than latter.

I hadn't realised I'd misspelled his name until I saw your comment. For some reason, it looks better with two 'r's.

It's not too weird -- "Terry" is always spelled with two r's, and yes English is weird but if you didn't have any more specific knowledge you'd also assume the name Terry is derived from is also always spelled with two r's.

Context matters.

Commenter was replying to someone who said

> 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.

and I highlight

> one of those people who just seem to be in a totally different league.

Hasn't your bet already been beat in concept by https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Polg%C3%A1r ?

He is a chess teacher who decided to train his daughters in chess from a very young age. What do you know, two of them became the first and second best players, with the best being considered the best woman chess player of all time. Unlikely that they somehow just all got chess genius "genes" from him.

It really does seem that heavy investment from a young age by a good teacher can work wonders.

I don't agree that becoming a chess grandmaster is on the same level or even near the same level as becoming a Fields medalist or being in the shortlist of the best mathematicians alive.

> It really does seem that heavy investment from a young age by a good teacher can work wonders.

This is not the bet. It is not any arbitrary skill.

I specifically said

> 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.

> Unlikely that they somehow just all got chess genius "genes" from him.

But likely that they inherited high IQ.

So perhaps when you start with sufficiently intelligence kids, you can train them to become virtually anything. But this does not apply to everyone.

I'd say that Terence got the investment from world renowned mathematicians because he was unusually gifted even at such a young age. It appears that some people are innately adept at mathematical thinking but as you say, people can still learn and contribute even when they don't have such a spectacular start.

> 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. =)

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.

Yes, Perelman was by far the hardest to come up with, but I really wanted at least one contemporary name in there.

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.

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.

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.

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)

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