How to hear the shape of a billiard table

TL;DR

This paper investigates how the bounce spectrum of a polygonal billiard table encodes geometric information, providing methods for reconstructing edge orderings and angles, but showing finite data cannot determine the exact shape.

Contribution

It introduces methods to recover edge orderings and angles from the bounce spectrum and proves the impossibility of exact shape reconstruction from finite data.

Findings

Reconstruction of edge cyclic orderings from bounce spectrum

Determination of polygon angles from bounce spectrum

Finite bounce data cannot uniquely determine the exact shape

Abstract

The bounce spectrum of a polygonal billiard table is the collection of all bi-infinite sequences of edge labels corresponding to billiard trajectories on the table. We give methods for reconstructing from the bounce spectrum of a polygonal billiard table both the cyclic ordering of its edge labels and the sizes of its angles. We also show that it is impossible to reconstruct the exact shape of a polygonal billiard table from any finite collection of finite words from its bounce spectrum.