The dynamics of digits: Calculating pi with Galperin's billiards

TL;DR

This paper provides a comprehensive analysis of Galperin billiards, a classical system that calculates digits of pi through particle collisions, including explicit solutions, invariants, and implications for digit calculation accuracy.

Contribution

It offers a complete explicit solution for the system's dynamics, identifies new invariants, and explores the system's integrability and superintegrability, advancing understanding of its mathematical properties.

Findings

Explicit solutions for particle positions and velocities as functions of collision number and time.

Identification of a second invariant providing integrability for any mass ratio.

Discovery of a third invariant for specific mass ratios, indicating superintegrability.

Abstract

In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number $\pi$. This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of $\pi$ in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be $\pi$ itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls' positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of $\pi$ with Galperin billiards, including curious cases with irrational number bases.