Arithmetic Billiards-Reference-Cited by-同舟云学术

Arithmetic Billiards

Published:2022-06-01 Issue:16 Volume:9 Page:43-54
ISSN:2182-1976
Container-title:Recreational Mathematics Magazine
language:en
Short-container-title:

Author:

Perucca Antonella¹,De Sousa Joe Reguengo¹,Tronto Sebastiano¹

Affiliation:

1. Department of Mathematics , University of Luxembourg , 6 av. de la Fonte, 4364 Esch-sur- Alzette , Luxembourg

Abstract

Abstract Arithmetic billiards show a nice interplay between arithmetics and geometry. The billiard table is a rectangle with integer side lengths. A pointwise ball moves with constant speed along segments making a 45° angle with the sides and bounces on these. In the classical setting, the ball is shooted from a corner and lands in a corner. We allow the ball to start at any point with integer distances from the sides: either the ball lands in a corner or the trajectory is periodic. The length of the path and of certain segments in the path are precisely (up to the factor √2 or 2√2) the least common multiple and the greatest common divisor of the side lengths.

Publisher

Walter de Gruyter GmbH

Subject

General Chemical Engineering

Link

https://www.sciendo.com/pdf/10.2478/rmm-2022-0003

Reference10 articles.

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3. [Lap04] Glenda Lappan, James T. Fey, William M. Fitzgerald, Susan N. Friel, and Elizabeth Difanis Phillips, Comparing and Scaling: Ratio, Proportion, and Percent in Connected Mathematics Project, Upper Saddle River NJ, Pearson Prentice Hall, 2004. ISBN: 0131656352. (See pp. 64–66.)

4. [Pap] Paper Pool Game, NCTM (National Council of Teachers of Mathematics) Illuminations, https://illuminations.nctm.org/Unit.aspx?id=6527.

5. [Per18] Antonella Perucca, Arithmetic billiards, Plus Magazine, 24 April 2018, University of Cambridge, https://plus.maths.org/content/arithmetic-billiards-0.

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