Inscribed squares: Denne speaks

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!

The problem:
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.

The problem was initially posed by Toeplitz in 1911.

First efforts were made by A. Emch [see Math reviews 1506193] who considered convex Jordan curves.

In 1929 (published 1944) Schnirelmann proved that Jordan curves of bounded continuous curvature have inscribed squares. [Schnirelmann, L.G., On certain geometric properties of closed curves (Russian) Uspekhi Mat. Nauk 10, (1944), pp 34–44.] However Schnirelmann’s proof contained an error which was corrected by Guggenheimer in 1965 [Guggenheimer, H., Finite sets on curves and surfaces, Israel J.~Math. 3 (1965), pp 104–112].

I did not know about the 1950’s Danish paper by C.M. Christensen – thank you JSE.

In 1959, Jerrard \cite{Jerrard} proved that all analytic curves must have an odd or infinitely many inscribed squares. [Jerrard, R.P., Inscribed squares in plane curves, Trans. Amer. Math. Soc. 98, (1961), pp 234–241.] While at Univ. Illinois, Urbana-Champaign, I spoke to Jerrard about this problem. The problem of squares shrinking to zero in the limit was the problem he came up against in looking for a complete solution to the problem.

In 1989 Stromquist extend the result to piecewise C^1 curves. [Stromquist, W., Inscribed squares and square-like quadrilaterals in closed curves, Mathematika 36, (1989), pp 187–197.

In 1991 H.B. Griffiths, proves that certain curves in R^n has an inscribed square-like quadrilateral. Namely one with equal sides and equal diagonals. [Griffiths, H.B., The topology of square pegs in round holes, Proc. London Math. Soc. 62, (1991), pp 647–672.] When n=2 of course, this result reduces to the case of squares inscribed in Jordan curves. He proved this result for a class of curves which don’t allow for infinitely small squares, roughly speaking as follows. Take any point p on the curve. It has a neighborhood in which no tangent vector is parallel to a vector perpendicular to the curve at p.

Finally Nielsen and Wright show that that curves with a certain kind of symmetry have inscribed squares (indeed rectangles). This allows for some fractal-like curves which had previously not been considered.

Thurston’s comments on the problem (about quasifuchsian groups) is new to me as well – thanks Richard. However this leads us to the natural question of what other kinds of polygons may be inscribed in Jordan curves. A straightforward dimension count indicates that generically there is a 2-dimensional family of secant segments inscribed in a curve, a 1-dimensional family of triangles, a 0-dimensional family of quadrilaterals (a finite number). For n>4 the family of n-gons has negative (4-n) codimension.

Meyerson showed that any Jordan curve contains the vertices of an equilateral triangle. This can be extended to show that a triangle similar to any given triangle can be inscribed in the curve. [Meyerson, M.D., Equilateral triangles and continuous curves, Fund. Math. 110, (1980), pp 1–9.]

Jerrard commented to me that if you adjust the arguments in his paper, you can prove that there is a 1-dimensional family of (any similarity type) of triangles inscribed in (analytic) Jordan curves. In his paper he remarks that it is easy to find a Jordan curve which does not have an inscribed regular n-gon for n>4. Consider a semi-circle with the endpoints joined by a diameter. The circular segment must contain at least 3 vertices of the n-gon, but then all other vertices must lie on the same circle.

In the joint work (in progress) with Jason Cantarella (UGA) and John McCleary (Vassar College) we have proved that two classes of simple closed curves in R^n have four points which are the vertices of square-like quadrilaterals. (Those with equal sides and equal diagonals.) The first class is C^1 curves (continuously differentiable), the second class is curves of finite total curvature without cusps.

Some remarks:
1) Our proofs are leaner than previous results due to our use of configuration spaces.
2) Curves of finite total curvature are certainly contained in the class of curves proved by Griffiths.
3) There are C^1 curves not of finite total curvature and curves of finite total curvature which are not C^1 (take any polygon).

We have also proved the following:
Given a Jordan curve which is C^1 at a point p, then p is a vertex of an inscribed n-gon of any metric similarity class. Two n-gons p_1, … , p_n and q_1, … , q_n in metric spaces P and Q are metrically similar if there exists some k for which
d_P(p_i,p_{i+1}) = k d_Q(q_i,q_{i+1})
for i = 1, … , n. (As usual, indices for points on n-gons are interpreted mod n, so q_{n+1} = q_1.)

In response to JSE’s questions: our results do not reprove Jerrard’s results that the number of inscribed squares is odd, nor do our results directly handle the question posed by Richard about quasifuchsian groups.

Readers might also be interested in papers by V.V Makeev which also look at inscribed polygons.

Elizabeth Denne (edenne@email.smith.edu)

Update (5 Sep 2007):  Jason Cantarella points out that “Emch did a
lot more than convex curves. In fact, he proves the result for
piecewise analytic curves (with finitely many pieces) in his paper “On
Some Properties of the Medians of Closed Continuous Curves Formed by
Analytic Arcs” (MR1506274) from 1916.”