I recently learned that Elizabeth Denne of Smith has made some recent progress on the inscribed square problem in joint work with Jason Cantarella (Georgia) and John McCleary (Vassar). Since she knows this problem, its history, its variants, and her own contributions much better than I do, I asked her if she’d be willing to make a guest post: and here it is!
A) Given a simple closed curve in R^2 (Jordan curve) are there four points on the curve which are the vertices of a square?
B) Given a closed curve in R^2 are there four points on the curve which are the vertices of a square?
Currently I am collaborating with Jason Cantarella (UGA) and John McCleary (Vassar College) on this problem.
First off, both problems are still open for a continuous map of S^1 into R^2. Secondly, I know of no counterexamples to the problem and believe the answer to problem A is yes. It appears (as John Cowan (7/25/07) commented) that problem B is not interesting, as parts of the curve could be “cut off”. So I’m not sure of the literature on B. I do think the problem is interesting for the following reasons. Don’t “cut off” parts of the loop and still try and solve the problem. In the Jordan curve case (A) when considering vertices ABCD in order about the square , they will match their order along the curve. In the case of problem B, this won’t necessarily be the case. (Just think about the Figure 8 example that John Cowan suggests.)
So from now on I’ll just consider problem A. All bloggers went straight to the heart of the issue. Namely, once you know that there is an inscribed square for some kind of Jordan curve (polygonal, smooth) you can’t necessarily show that every Jordan curve has an inscribed square. The reason that inscribed squares in the approximating curves may shrink to zero in the limit.
I’ll now describe what the known results are, repeating some of what others have written.