IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i4p509-d340670.html
   My bibliography  Save this article

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

Author

Listed:
  • Xabier M. Aretxabaleta

    (Departament de Física, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain)

  • Marina Gonchenko

    (Departament de Matemàtiques, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain)

  • Nathan L. Harshman

    (Department of Physics, American University, 4400 Massachusetts Ave. NW, Washington, DC 20016, USA)

  • Steven Glenn Jackson

    (Department of Mathematics, University of Massachusetts Boston, Boston, MA 02125, USA)

  • Maxim Olshanii

    (Department of Physics, University of Massachusetts Boston, Boston, MA 02125, USA)

  • Grigory E. Astrakharchik

    (Departament de Física, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain)

Abstract

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.

Suggested Citation

  • Xabier M. Aretxabaleta & Marina Gonchenko & Nathan L. Harshman & Steven Glenn Jackson & Maxim Olshanii & Grigory E. Astrakharchik, 2020. "The Dynamics of Digits: Calculating Pi with Galperin’s Billiards," Mathematics, MDPI, vol. 8(4), pages 1-31, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:509-:d:340670
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/4/509/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/4/509/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Shu-Ju Tu & Ephraim Fischbach, 2005. "A Study On The Randomness Of The Digits Of Π," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 16(02), pages 281-294.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Alexander Maye & Chih-hao Hsieh & George Sugihara & Björn Brembs, 2007. "Order in Spontaneous Behavior," PLOS ONE, Public Library of Science, vol. 2(5), pages 1-14, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:509-:d:340670. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.