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Open AccessArticle

The Dynamics of Digits: Calculating Pi with Galperin’s Billiards

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Departament de Física, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
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Departament de Matemàtiques, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain
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Department of Physics, American University, 4400 Massachusetts Ave. NW, Washington, DC 20016, USA
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Department of Mathematics, University of Massachusetts Boston, Boston, MA 02125, USA
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Department of Physics, University of Massachusetts Boston, Boston, MA 02125, USA
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 509; https://doi.org/10.3390/math8040509
Received: 20 February 2020 / Revised: 23 March 2020 / Accepted: 24 March 2020 / Published: 2 April 2020
(This article belongs to the Special Issue Various Routes towards Few-Body Physics)
In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number π . This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of π 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 π 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 π with Galperin billiards, including curious cases with irrational number bases. View Full-Text
Keywords: Galperin billiards; calculating pi; three-body problem; solvable model; integrability; superintegrability; irrational bases Galperin billiards; calculating pi; three-body problem; solvable model; integrability; superintegrability; irrational bases
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Aretxabaleta, X.M.; Gonchenko, M.; Harshman, N.L.; Jackson, S.G.; Olshanii, M.; Astrakharchik, G.E. The Dynamics of Digits: Calculating Pi with Galperin’s Billiards. Mathematics 2020, 8, 509.

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