## Square pegs, square pegs. Square, square pegs.

Lately I’ve been thinking again about the “square pegs” problem:  proving that any simple closed plane curve has an inscribed square.  (I’ve blogged about this before: here, here, here, here, here.)  This post is just to collect some recent links that are relevant to the problem, some of which contain new results.

Jason Cantarella has a page on the problem with lots of nice pictures of inscribed squares, like the one at the bottom of this post.

Igor Pak wrote a preprint giving two elegant proofs that every simple closed piecewise-linear curve in the plane has an inscribed square.  What’s more, Igor tells me about a nice generalized conjecture:  if Q is a quadrilateral with a circumscribed circle, then every smooth simple closed plane curve has an inscribed quadrilateral similar to Q.  Apparently this is not always true for piecewise-linear curves!

I had a nice generalization of this problem in mind, which has the advantage of being invariant under the whole group of affine-linear transformations and not just the affine-orthogonal ones:  show that every simple closed plane curve has an inscribed hexagon which is an affine-linear transform of a regular hexagon.  This is carried out for smooth curves in a November 2008 preprint of Vrecica and Zivaljevic.  What’s more, the conjecture apparently dates back to 1972 and is due to Branko Grunbaum.  I wonder whether Pak’s methods supply a nice proof in the piecewise linear case.

## Cited!

We’re in a new era of mathematical publishing indeed; a paper posted yesterday on the arXiv cites a post from this blog as a reference.

The paper, by Strashimir Popvassiliev, constructs for every positive integer n a simple closed plane curve with exactly n inscribed squares. (It’s an old conjecture of Toeplitz that every simple closed plane curve contains at least one inscribed square.) This seems to speak against philosophy, mentioned by Denne in her guest post here, that “the reason” every curve has at least one inscribed square is because every curve has an odd number of inscribed squares.

I’m not sure Popvassiliev’s example really contradicts this philosophy. Surely the squares should be counted with multiplicity, in the appropriate sense. With a more naive notion of “counting” you can’t expect parity conditions to hold. For instance, you certainly want to say that a straight line intersects a smooth closed curve an even number of times. Naively, you might complain that a tangent line to a circle intersects the circle only once! But of course, it really crosses the curve twice; it’s just that the two crossings are at the same point.