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

Buy & download fulltext article:


Price: $20.00 plus tax (Refund Policy)


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: June 1, 2008

More about this publication?
Related content



Free Content
Free content
New Content
New content
Open Access Content
Open access content
Subscribed Content
Subscribed content
Free Trial Content
Free trial content

Text size:

A | A | A | A
Share this item with others: These icons link to social bookmarking sites where readers can share and discover new web pages. print icon Print this page