New Geometric Perspective Cracks Old Problem About Rectangles 59 points by exanimo_sai 80 days ago | hide | past | favorite | 12 comments

 Here's a nice 3blue1brown video about the easier version of the inscribed rectangle problem, that some inscribed rectangle exists: https://www.youtube.com/watch?v=AmgkSdhK4K8And I remember thinking this was a nice overview of some aspects of the "square peg problem" (the Toeplitz conjecture): https://www.ams.org/notices/201404/rnoti-p346.pdf
 > It starts with a closed loop — any kind of curvy path that ends where it starts. The problem Greene and Lobb worked on predicts, basically, that every such path contains sets of four points that form the vertices of rectangles of any desired proportion.> Their final proof — showing the predicted rectangles do indeed exist — transports the problem into an entirely new geometric setting. There, the stubborn question yields easily.Except that the article then contradicts itself by saying how they haven't actually proved this. They proved it for smooth closed curves, not for any closed curves.This is a really bizarre article. They seem like they want to describe the math. But they can't bring themselves to do it in a way that might be helpful. They're just waving words around.And then there's this:> it’s possible to rotate the Möbius strip in four-dimensional space so that you only change one of the coordinates in each point’s four-coordinate address — like changing the street numbers of all the houses on a block, but leaving the street name, city and state unchanged. (For a more geometric example, think about how holding a block in front of you and shifting it to the right only changes its x coordinates, not the y and z coordinates.)You can certainly translate a space along an axis without affecting its coordinates along other axes. But that's not a rotation.
 > This is a really bizarre article. They seem like they want to describe the math. But they can't bring themselves to do it in a way that might be helpful. They're just waving words around.Welcome to another issue of Quanta!
 I think it depends on the author. It's hard to distill math and science down to something that the science-interested layman can understand and enjoy. I find Natalie Wolchover's[0] writing to be pretty consistently good, for example.
 I don’t think that’s really a contradiction. Indeed I think that sort of language would be quite a normal thing for a mathematician to colloquially say when describing the problem. Probably they would say smooth rather than “curvy” but they mightn’t say anything at all. In plenty of contexts, “loop” means “map from S1 to your space” and “map” means “continuous function” (or maybe a pointed continuous function in some contexts). Usually I would expect such problems to be introduced with a brief description and some examples and nonexamples. If the proof had some extra reasonable requirements on the curve, one might just say that the curve has to be “well-behaved in some sense” and omit details which aren’t particularly interesting.
 > While locked down due to COVID-19, Joshua Greene and Andrew Lobb figured out how to prove a version of the “rectangular peg problem.”Emphasis mine.
 The text you quote is not even part of the article.But it is related to text that precedes my quote:> One of the problems the two friends looked at was a version of a century-old unsolved question in geometry.> “The problem is so easy to state and so easy to understand, but it’s really hard,” said Elizabeth Denne of Washington and Lee University.> It starts with a closed loop — any kind of curvy path that ends where it starts. The problem Greene and Lobb worked on predicts, basically, that every such path contains sets of four points that form the vertices of rectangles of any desired proportion.Emphasis mine. The article explicitly describes the problem they solved, only to later point out that they actually solved a different problem.
 Oh my god here, is the same qualification near the top of the article.> One of the problems the two friends looked at was a version of a century-old unsolved question in geometry.On every Quanta article there's always someone welching about some gotcha they think they've found that just demonstrates how trash Quanta's popularizations of mathematical topics is. But whenever I read the article in question it always turns out that the writers and editors over there somehow manage to thread the needle in making their material accessible without being mathematically inaccurate. Vague, yes, but that's why they link to the research in question because it's a pop article not a journal publication.You edited your comment to add another objection. It is equally insubstantial unless you dug through the paper yourself and demonstrated that the transformation applied could not reasonably be called a rotation.
 > Oh my god here, is the same qualification near the top of the article.>> One of the problems the two friends looked at was a version of a century-old unsolved question in geometry.Did you read past the first sentence of my comment?> You edited your comment to add another objection. It is equally insubstantial unless you dug through the paper yourself and demonstrated that the transformation applied could not reasonably be called a rotation.I disagree. The article strongly implies that the transformation applied could be called a rotation, and I see no particular reason to doubt that. ("The Möbius strip can be rotated by any angle between 0 and 360 degrees, and he proved that one-third of those rotations yield an intersection between the original and the rotated copy.")But I very much object to the article's idea that I should help myself think about a rotation that only changes point values along a single dimension by visualizing an entirely unrelated transformation. How is that supposed to help?
 >welchingDid you mean whinging?
 My mind was in between whinging and kvetching.
 I've seen the trick with describing rectangles as middlepoint, diagonal length and angle used in solving an interview questions, it was great. https://youtu.be/EuPSibuIKIg

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