Given a closed, convex curve C in the plane, a billiard path on C is a polygon P inscribed in C such that, at each vertex v of P, the two edges of P incident with v make equal angles with the tangent line to C at v. (Intuitively, this is the path a billiard ball would follow on a frictionless pool table bounded by C.) For most convex curves C, there are only a finite number of triangular billiard paths on C, a finite number of quadrilateral billiard paths, and so on. Obviously, when C is a circle, there are infinitely many closed billiard n-gons inscribed in C, but, surprisingly, the same is true when C is an ellipse. (This is a famous theorem due to Chasles.) The interesting question is whether there are other convex curves, besides ellipses, for which there are infinitely many closed billiard n-gons for some n. In this talk, I’ll discuss the above-mentioned phenomenon and show how it is related to the geometry of non-holonomic plane fields (which will be defined and described). This leads to some surprisingly beautiful geometry, which will require nothing beyond multivariable calculus from the audience.


Robert Bryant

Research Area

Duke University


Thu, 10/10/2013 - 2:00pm to 3:00pm


Chemistry Lecture Theatre 4, University of Sydney