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I was toying with possible approaches for finding inscribed squares in Jordan curves, and I know that every arbitrary Jordan curve ($\gamma $) must admit some amount of inscribed rectangles, because the self-intersections of a 3-dimensional surface with $x$ and $y$-values given by pairs of points on $\gamma$ and $z$-values given by the distance between these pairs of points must always intersect when being mapped onto a Mobius Strip (or vice-versa, with the Mobius Strip being mapped onto the output space) in such a way that the edges of the output plane are mapped onto the edges of the Mobius Strip.

A great visual representation of this phenomena, before my question is asked, is given by 3blue1brown's video on inscribed rectangles in a Jordan curve. The link is provided below. Here is an image in his video upon which this 3D surface is about to be mapped to a Mobius Strip:

Figure 1: Mobius Strip mapped onto Output Space

I highly recommend watching the actual mapping on 3b1b's video (14:26 - 14:37) by clicking on the link below:

Figure 2: Mapping Animation by 3blue1brown

Now, my question: if there always exists inscribed rectangles on any Jordan curve, what is the minimum amount of these rectangles that can exist on any Jordan curve? 1? 2? Is it the same for any curve, some constant? Perhaps it's different depending on the curve itself, given by some formula?

Here's another way to word my question: what is the minimum amount of times the Mobius Strip intersects itself at a point when mapped onto that 3D output space given by the distances between pairs of points on $\gamma$?

I appreciate your help.

Ben Myers
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  • What do you mean by an "inscribed square in a curve"? You seem to be very mixed up about basic geometric facts and about the question you want to ask: the Moebius strip can be imbedded in Euclidean space without any self-intersections, but your question about inscribed rectangles has nothing to do with this. – Rob Arthan Aug 22 '18 at 01:34
  • Thanks for your response, and sorry for the confusion. By "inscribed square in a curve", I mean 4 distinct points on a Jordan curve that form a square. And, yes, of course the Möbius strip doesn't have to intersect itself when embedded in Euclidean space, but in the particular case where it is mapped onto an output plane given by distances between pairs of points on the Jordan curve, it always does. Each self-intersection from this mapping implies the existence of one inscribed rectangle in a Jordan curve. My question inquires about the minimum amount of these self-intersections. – Ben Myers Aug 22 '18 at 02:42
  • https://math.stackexchange.com/questions/2218214/where-can-i-find-the-inscribed-rectangle-problem-proof?rq=1 – Ben Myers Aug 22 '18 at 12:02

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