Periodic Orbits for Billiards on an Equilateral Triangle
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.
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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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