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Of all the videos I've made, one of my favorites topics to have covered was a proof of the inscribed rectangle problem by H. Vaughan using a Mobius strip.

But now there's a new result!
@QuantaMagazine recently did a great article about recent work by Greene and Lobb using a beefed-up version of the same idea, letting a Mobius strip encode geometric properties of the curve to solve a more general result, check it out:…
From recent comments on the video above, it looks there's a little confusion where some people thought this means the inscribed square problem (i.e. the Toeplitz conjecture) has been solved, but that's not quite the case.
We've known since 1916, thanks to A. Emch, that inscribed squares always exist in smooth curves. The challenge is curves that aren't piecewise smooth (think fractals), where things like the limiting behavior on the slope of two nearby points is not in your control.
What Greene and Lobb proved was a more general result in the case of smooth (Jordan) curves saying that not only are there squares (which was known), but there are rectangles of any particular shape.
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