
New Geometric Perspective Cracks Old Problem About Rectangles - exanimo_sai
https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/
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kmill
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=AmgkSdhK4K8](https://www.youtube.com/watch?v=AmgkSdhK4K8)

And 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](https://www.ams.org/notices/201404/rnoti-p346.pdf)

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thaumasiotes
> 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.

~~~
twic
> 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!

~~~
mabbo
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.

[0][https://www.quantamagazine.org/authors/natalie/](https://www.quantamagazine.org/authors/natalie/)

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thaumasiotes
Wait a minute.

> 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.

> The rectangular peg problem is a close offshoot of a question posed by the
> German mathematician Otto Toeplitz in 1911. He predicted that any closed
> curve contains four points that can be connected to form a square. His
> “square peg problem” remains unsolved.

How can the square problem possibly remain unsolved if the article is
celebrating a proof that you can find the corners of a rectangle _of any
proportion_ in any closed curve. What if I choose the proportion "length = 1
times width"?

------
ajuc
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](https://youtu.be/EuPSibuIKIg)

