Numerical exploration of a hexagonal string billiard

Abstract

In this paper we are interested in the motion of a ball inside a billiard table bounded by a particular smooth curve. This table belongs to a family of billiards which can all be drawn by a common process: the so-called gardener's string construction. The classical elliptical billiard is, of course, the foremost member of this family. So it should come as no surprise that our hexagonal string billiard shares many basic properties with the latter, but, on the other hand, also exhibits some essential differences with it. We have gathered numerical evidence against the Birkhoff-Poritsky conjecture.