Mathematical billiards are dynamical systems in which a point bounces around the inside of a convex shape on the plane (or, more generally, on some surface). Outer billiards is the dual situation. Choose a convex shape K on the plane (say, a convex polygon). Choose a starting point x0 outside the shape. Draw a ray from x0 to x1 such that
- the line segment x0x1 is tangent to K and the tangency point is the midpoint of the segment, and
- moving from x0 to x1 the shape K is on the right.
Iterating this process gives a (affinely invariant) dynamical system. The basic questions are the expected ones: one wants to know whether orbits are periodic, whether they are bounded, etc.
The idea of outer billiards seems to be due to B. H. Neumann. They were popularized in the 1970s by J. Moser, who related them to celestial mechanics. This research monograph deals with outer billiards in which the shape K is a kite, i.e., a convex quadrilateral having a diagonal which is a line of symmetry. The main result is that if the ratio of the diagonals in the kite is irrational, then there are uncountably many orbits that are unbounded (both forward and backward).
The author clearly has a knack for inventing names (kites, erratic orbits, the comet theorem, the hexagrid theorem, the room lemma…), which helps make the book a pleasant read. There are lots of pictures. The exposition is a bit breathless, at times seeming to rush forward in its enthusiasm without filling in all the details. That is to be expected, I guess, from a book in the Annals of Mathematics Studies series.
Most of the results, the author tells us, were first discovered experimentally, using a program called Billiard King, available from the author's web site. Also on the site is an interactive guide to the book that uses Java applets to explain what is going on. The online materials make the book much more accessible than it might otherwise have been.
Not many books combine serious mathematics, neat software, and an element of play. This one does. It would be a great source for independent projects with students.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.