Did you know? (accessible math stuff 1)
(This post was originally posted on the now-closed website cohost in October 2023. It is part of what I ultimately hope to be a very occasional series of math tidbits that I strive to make accessible and interesting for most readers.)
If you draw a continuous closed curve on a sheet of paper, there are four points on that curve that form the four vertices of a square.
This is true of any curve you could draw.
Mathematically, most curves aren't as nice as the ones you can draw: it can be that no matter how much you zoom in on a point, there are irregularities around it - you cannot represent their surrounding as a continuous function's graph (an accessible example of that, if I'm not mistaken, is the famous fractal that is the Koch snowflake).
And so, in the general case, we don't know if the same property holds.
This fascinating thing (go on! Test it with curves you draw! Find four points in them that form a square!) is called the Inscribed Square Problem, and is in my opinion an absolutely fascinating math problem that was stated by Toeplitz in 1911, and that is still open to this day.
(Note that "Inscribed" can be misleading: just as with the figure above, the square doesn't have to be "inside" the curve, and basically its edges don't matter, we just require its four vertices to be on the curve.)
Of course, progress has been made on the question: mathematicians proved that you could find any kind of triangle (notably, equilateral triangles) in absolutely any curve; and that that was false for regular pentagons (no such pentagon vertices can be found in a square-shaped closed curve, for instance). Also, the square case has been cracked for wider and wider classes of curves, with still recent results by Tao (2017 - there is an inscribed square in any union of two contracting Lipschitz curves) and Chambers (2022 - this also holds for any curve which is "close" to a C² curve).
I just wanted to share this because I first heard about it a month ago and I still haven't recovered from the following:
- this property is true for any curve you can draw;
- and it remains unclear whether it is true in general;
- and it's so easy to state! I mean, NOBODY TOLD ME YOU COULD ALWAYS FIND A SQUARE IN A CLOSED CURVE!
Math research is very much like that: discovering what seems to be a behavior of some math objects and trying to prove it, building a dozen variants of the question on the way toward achieving that, and unearthing just as many riddles, some of them easy, some of them absurdly hard. And the base question doesn't have to be high-level stuff at all.
Of course, this has nothing to do with what I do in mathematics - I will probably never have tools to chip away at this specific problem. But I feel so glad it exists. Like many other easy-to-state, hard-to-prove problems (the Collatz conjecture, the Ein Stein problem that was solved very recently...), I think it crystallizes something about why math is fascinating.
If you want the gist of the problem, several main results, and rather accessible proofs (with drawings) of both warmup exercises and some big theorems, this page is the one that really sold me on this! It's also the one in which I found the drawing above.
Apart from that, the Wikipedia page of the problem was my other source for this post, notably history-wise.
(edit: when I shared this on cohost, joXn also commented by adding this video/article by 3Blue1Brown, that talks about this problem and the slightly weaker Inscribed Rectangle Problem - so I'm adding the ressource here if you're interested!)
Thank you for reading this far!