Billiards

Billiards is played on a rectangular table (known as a billiard table) upon which balls are placed. One ball (the "cue ball") is then struck with the end of a "cue" stick, causing it to bounce into other balls and reflect off the sides of the table. Real billiards can involve spinning the ball so that it does not travel in a straight line, but the mathematical study of billiards generally consists of reflections in which the reflection and incidence angles are the same. However, strange table shapes such as circles and ellipses are often considered. The popular 1959 animated short film Donald in Mathmagic Land features a tutorial by Donald Duck on how to win at billiards using the diamonds normally inscribed around the edge of a real billiard table.

Many interesting problems can arise in the detailed study of billiards trajectories. For example, any smooth plane convex set has at least two double normals, so there are always two distinct "to and fro" paths for any smoothly curved table. More amazingly, there are always distinct -gonal periodic orbits on smooth billiard table, where is the totient function (Croft et al. 1991, p. 16). This gives Steinhaus's result that there are always two distinct periodic triangular orbits (Croft and Swinnerton 1963) as a special case. Analysis of billiards path can involve sophisticated use of ergodic theory and dynamical systems.

Given a rectangular billiard table with only corner pockets and sides of integer lengths and (with and relatively prime), a ball sent at a angle from a corner will be pocketed in another corner after bounces (Steinhaus 1999, p. 63; Gardner 1984, pp. 211-214). Steinhaus (1999, p. 64) also gives a method for determining how to hit a billiard ball such that it caroms off all four sides before hitting a second ball (Knaster and Steinhaus 1946, Steinhaus 1948).

Alhazen's billiard problem seeks to find the point at the edge of a circular "billiards" table at which a cue ball at a given point must be aimed in order to carom once off the edge of the table and strike another ball at a second given point. This problem is insoluble using a compass and ruler construction (Elkin 1965, Riede 1989, Neumann 1998).

On an elliptical billiard table, the envelope of a trajectory is a smaller ellipse, a hyperbola, a line through the foci of the ellipse, or a closed polygon (Steinhaus 1999, pp. 239 and 241; Wagon 1991). The closed polygon case is related to Poncelet's porism.

Polygonal billiard tables also admit rich trajectory dynamics. The only closed billiard path of a single circuit in an acute triangle is the orthic triangle, the pedal triangle of the orthocenter. There are infinitely many multiple-circuit paths with segments parallel to the sides of the orthic triangle. There exists a closed billiard path inside a cyclic quadrilateral if its circumcenter lies inside the quadrilateral (Wells 1991).

Rational polygonal billiards restricts the table angles to rational multiples of . In this setting, a direction containing some periodic trajectory need not have every regular trajectory periodic. This distinction is central to Boshernitzan's questions about rational periodic directions (Boshernitzan 1992).

There are four identical closed billiard paths inside and touching each face of a cube such that each leg on the path has the same length (Hayward 1962; Steinhaus 1979, 1999; Gardner 1984, pp. 33-35; Wells 1991). This path is in the form of a chair-shaped hexagon, and each leg has length . For a unit cube, one such path has vertices (0, 2/3, 2/3), (1/3, 1, 1/3), (2/3, 2/3, 0), (1, 1/3, 1/3), (2/3, 0, 2/3), (1/3, 1/3, 1). Lewis Carroll (Charles Dodgson) also considered this problem (Weaver 1954).

There are three identical closed billiard paths inside and touching each face of a tetrahedron such that each leg of the path has the same length (Gardner 1984, pp. 35-36; Wells 1991). These were discovered by J. H. Conway and independently by Hayward (1962). The vertices of the path are appropriately chosen vertices of equilateral triangles in each facial plane which are scaled by a factor of 1/10. For a tetrahedron with unit side lengths, each leg has length . For a tetrahedron with vertices (0, 0, 0), (0, , ), (, 0, ), (, , 0), the vertices of one such path are (, , ), (, , ), (, , ), (, , ).

Conway has shown that period orbits exist in all tetrahedra, but it is not known if there are periodic orbits in every polyhedron (Croft et al. 1991, p. 16).

Strict convexity does not by itself rule out ergodic dynamics. Fan and Hasselblatt (2026) proved that sufficiently asymmetric lemon billiards have the Bernoulli property, which implies mixing and hence ergodicity. Their tables are intersections of two disks and have corners where their circular boundary arcs meet. Thus the result does not supply a smooth strictly convex ergodic table.