Billiards with Bombs

ABSTRACT

In this article, we define a variant of billiards in which the ball bounces around a square grid erasing walls as it goes. We prove that there exist periodic tunnels with arbitrarily large period from any possible starting point, that there exist nonperiodic tunnels from any possible starting point, and that there are versions of the problem for which the same starting point and initial direction result in periodic tunnels of arbitrarily large period. We conjecture that there exist starting conditions which do not lead to tunnels, justify the conjecture with simulation evidence, and discuss the difficulty of proving it.

Keywords:

• billiards
• cutting sequence
• discrete dynamical system

2000 AMS Subject Classification:

• 37D50

Acknowledgments

The author thanks Xavier Bressaud for beginning the mathematical study of Breakout and Nicolas Bedaride for suggesting the 1-skeleton variant from which this article grew. The author also thanks Richard Schwartz for his advice and encouragement throughout this process.

Funding

This work was done as part of a doctoral dissertation under the supervision of Richard Schwartz at Brown University; the author thanks the Brown math department as a whole for funding and a very congenial working environment.