Periodic Orbits for Billiards on an Equilateral Triangle

Authors: Baxter, Andrew M.; Umble, Ronald

Source: American Mathematical Monthly, Volume 115, Number 6, June-July 2008 , pp. 479-491(13)

Publisher: Mathematical Association of America

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Abstract:

How many ways can one set a billiard ball in motion on a frictionless triangular equilateral table so that the ball retraces the same path after n bounces? Such a path is called a periodic orbit of period n. When n is odd there is at most one such orbit, but when n is even there are uncountably many. Fortunately there is a natural equivalence relation on orbits of even period. Using techniques from plane geometry, number theory, and combinatorics we construct a bijection between equivalence classes of these orbits and a new type of integer partition. This allows us to count equivalence classes containing orbits of a given period by counting partitions.

Document Type: Research article

Publication date: 2008-06-01

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American Mathematical Monthly publishes articles, notes, and other features about mathematics and the profession. AMM readers span a broad spectrum of mathematical interests, and include professional mathematicians as well as students of mathematics at all collegiate levels.
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