Numerical exploration of a hexagonal string billiard

TL;DR

This paper numerically investigates a hexagonal string billiard, revealing similarities and differences with elliptical billiards, and provides evidence against the Birkhoff-Poritsky conjecture.

Contribution

It introduces a new class of billiards derived from string constructions and offers numerical evidence challenging a longstanding conjecture.

Findings

Shares properties with elliptical billiards

Exhibits essential differences from elliptical billiards

Provides numerical evidence against the Birkhoff-Poritsky conjecture

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.