For what m and n can a regular m-gon be inscribed in a regular n-gon?

One day as an undergraduate I visited my supervisor with a problem that had crossed me in an otherwise unrelated assignment.

I showed him on the whiteboard how an equilateral triangle can be inscribed in a square and how a square can be inscribed in a regular pentagon. Both used the intermediate value theorem with given pairs of triangles or rectangles. You can probably see where I went with that. There may be much nicer ways to show it, but that was what I used at the time.

A square inscribed in a regular pentagon.

We casually discussed for a few minutes whether a regular pentagon could similarly be inscribed in a regular hexagon, a regular hexagon in a regular heptagon, and so on, coming up with very little. He showed me another problem involving inscribing and regular n-gons but his one had circles and a line and a constant that nobody knows. I left, my initial puzzle still a mystery.

The question has needled me over the years and become something I have thought about when waiting in queues, at bus stops, and on station platforms. The formulation has generalised in my mind to the set of values for which a regular m-gon can be inscribed in a regular n-gon, rather than my initial n and n+1.

The obvious results sprang out at me, for example in the case when m divides n, but never a fully comprehensive answer. Frequently I would search the internet, sure that there would be a complete solution with elementary techniques that I had simply overlooked.

Today I was going to post the problem on this newly-born blog, as a nice start to set the tone of the posts, but decided of course to have one more check online first.

Lo and behold: Solved, by an S. J. Dilworth of the University of South Carolina.

Perhaps it is not surprising I had not found this before, it was only published in August 2010.

A diagram from the paper.

I have spent a large portion of today reading the paper.