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Geometry and Billiards

Serge Tabachnikov
American Mathematical Society/ Mathematics Advanced Study Semesters
Publication Date: 
Number of Pages: 
Student Mathematical Library 30
[Reviewed by
William J. Satzer
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Geometry and Billiards starts a new collection — part of the Student Mathematical Library series — published jointly by the American Mathematical Society and the Mathematical Advanced Study Semesters (MASS) program. Each book in the collection is planned to be based on lecture notes for advanced undergraduate topics courses for MASS or the REU (Research Experiences for Undergraduates) program at Penn State. All are intended to be self-contained and to address non-standard mathematical topics accessible to undergraduates with two years of college mathematics.

Mathematical billiards is the study of the motion of a point mass on a domain with elastic reflections from the boundary (with the natural requirement that the angle of incidence equals the angle of reflection). It is not a single mathematical theory, instead more a kind of playground where a variety of methods and approaches can be tried out. The motivation for mathematical billiards comes from dynamical systems, geometry and physics, and especially geometrical optics.

Key questions arising in mathematical billiards typically involve the relationship between the geometry of the domain — in particular, the shape of its boundary — and the nature of the dynamics that results. For an elliptical region, the dynamics is well-behaved in the sense that the billiard map T is integrable: there is a smooth function (called an integral or integral-of-the-motion) on the phase space of the system that is constant on each trajectory of T. At the opposite extreme is chaotic billiards arising, for example, in a square region with a circular obstacle in the center. More common are mixed examples like mushroom billiards where motion is integrable in the mushroom cap and chaotic in the stem with a continuous transition in between.

A more geometrical question is the existence of caustics. A caustic is the envelope of reflected rays for a given billiard trajectory. Caustics for elliptical billiards are confocal ellipses. In general, what other plane convex billiards with smooth boundary have caustics? That turns out to be a subtle question answerable via the powerful methods of the Kolmogorov-Arnold-Moser theorem on perturbations of integrable systems.

Geometry and Billiards addresses circular and elliptical billiards as well as billiards in conics, quadrics and polygons. There are also sections on billiards and integral geometry, periodic billiard trajectories, caustics and dual billiards. There are a great many digressions. While digressions can be instructive or amusing, there are far too many of them here. Some chapters seem to be completely dominated by them to the extent that it is hard to follow the main content of the chapter. While the intent of the book seems to be to introduce a lot of mathematical ideas in the context of billiards, the narrative bounces around so much that it is hard to make out the theme. Most topics are treated so quickly that the reader scarcely gets a notion of the concept being described.

The book does, however, include a plenitude of fascinating ideas. If a student has the inclination to dip in for an idea that he or she might develop at greater length, then this would be a terrific source.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

  • Motivation: Mechanics and optics
  • Billiard in the circle and the square
  • Billiard ball map and integral geometry
  • Billiards inside conics and quadrics
  • Existence and non-existence of caustics
  • Periodic trajectories
  • Billiards in polygons
  • Chaotic billiards
  • Dual billiards
  • Bibliography
  • Index