Poncelet’s Porism is one of my favorite examples of something in mathematics that is simple to state, yet the result feels both surprising and inevitable:
Draw two ellipses, one inside the other. Then there exists an n-gon circumscribed around the inner ellipse and inscribed inside the outer ellipse if and only if there are infinitely many such n-gons, one with a vertex at each point of the outer ellipse.
Many of you have probably grabbed a pencil and a piece of paper and tried to sketch an example to convince yourself what exactly this says. If you haven’t, I recommend checking out this website, where Michael Borcherds has used Geogebra to create some wonderful illustrations of this fact, at least in the case of two circles.
Probably a smaller number of you have tried to figure out why this is true, and even to come up with a proof of it on your own. I would encourage you to keep working on that proof, as there is a reasonable chance that you will come up with one. There are a number of proofs of this in the literature, employing a range of methods ranging from the naive (but ugly) approach of writing down coordinates, to more elegant approaches using algebraic geometry, differential equations, dynamical systems, and even quantum mechanics. The reason I like Poncelet’s Porism is that it can be phrased in a way that any high school geometry student could understand, and while it is certainly far from trivial to prove it is also far from impossible. In fact, I think most mathematicians who were locked in a room with nothing to think about but Poncelet’s Porism could eventually come up with a proof of their own using the tools from their own areas of expertise. While I have not replicated this experiment exactly, my graduate school advisor did give me the problem to solve in preparation for my oral examination saying he would only let me take the exam when I had a solution, and I think that is as close as the IRB at most schools will let one get to a true simulation. (In case it isn’t clear, I did eventually find a solution with only a little nudging).
The other way in which Poncelet’s Porism is a great question is that even a solution leads to many natural other questions — Given two ellipses, how can you figure out which values of n will give n-gons and which won’t? Given one ellipse, how often does another ellipse lead to triangles being inscribed about one and circumscribed about the other? What if we consider shapes other than ellipses? What is the analagous question in higher dimensions? The book Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians, and Pencils of Quadrics by Vladimir Dragović and Milena Radnović looks at a number of these questions, and gives a nice picture of both Poncelet’s Porism and a number of the areas of mathematics that are close by.
The book opens with an introduction to Poncelet’s Porism which describes the problem, gives some historical information, and gives a high-level overview of some of the mathematics that will arise as the authors attempt to analyze the problem. (For those who are wondering, it does also contain a discussion of the word “porism” and how it differs from a theorem, lemma, corollary, or problem.) There are then several chapters dedicated to building up machinery and discussing applications to the question of Poncelet: the authors introduce the theory of billiards in general, billiards on ellipses in particular, and periodic orbits in chapter two. The third chapter looks at hyperelliptic curves and their Jacobians, including discussions of Riemann surfaces, complex tori, and Abel’s Theorem. These are followed by a theorem on projective geometry, starting with conics and quadrics and developing the notion of pencils of quadrics in a variety of ways. All three of these chapters are trying to do a difficult job, explaining the relevant parts of deep subjects to readers with very little background without either turning this into a 1000-page textbook or boring the readers who do know the background material. In general, they do a pretty good job, although I think a reader who has never seen the topics might be running to find a copy of their favorite algebraic geometry textbook to fill in details.
While these earlier chapters give proofs of Poncelet’s Porism as stated above, the fifth chapter introduces the “Full Poncelet Theorem” which says that, given a number of conics C, C1, …, Cn, there is a polygon inscribed in C and circumscribed about all of the Ci if and only if there are infinitely many, and one can choose the order in which the polygon intersects the Ci as well as some more information about the polygon arbitrarily. This theorem was proven by Poncelet in 1822, and the authors follow a presentation of the proof due to Lebesgue that looks at the cubic Cayley curve which gives a parametrization of contact points of tangents from a given point to all conics in a pencil. This allows one to give a condition on pairs of conics satisfying the original Poncelet Porism.
The later chapters in the book describe connections between Poncelet’s results and other areas of mathematics. One chapter looks at connections between Poncelet and Marden’s Theorem (the result that Dan Kalman calls “The Most Marvelous Theorem In Mathematics”). Another goes more in depth into the theory of billiards, and in particular analyzes topological properties of elliptical billiards and integral potential perturbations on them. Many people who know a modern proof of Poncelet’s result know that elliptic curves often play a crucial role, so it is not surprising that when one considers higher dimensional cases it is Jacobians of hyperelliptic curves that appear, and the book discusses this connection in depth. Yet another chapter looks at connections between Poncelet’s Theorem and continued fractions of polynomials, and in the final chapter the author brings in the Quantum Yang-Baxter Equation and (2-2) correspondences.
If the above doesn’t make it clear, much of this book gets quite technical, and I have to admit that at times I wish that the authors had given more background information and more detail to keep the book accessible. In fact, one failing of the book might be that the authors wanted to cover their own research results from recent years but lack perspective on the background they require or how it fits into a bigger picture. That said, Dragović and Radnović do a good job of covering a wide range of material, including both citations and descriptions of classical material as well as extremely cutting-edge results. (How often do you read a book whose bibliography contains items published in both 1766 and 2010?) The exposition in the book is generally good, and my guess is that any mathematician who picks up this book will learn something from it.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College. His research interests are predominantly in arithmetic geometry, Galois theory, and cryptography. He can be reached at firstname.lastname@example.org.