The game of 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).
One can also consider billiard paths on polygonal billiard tables. The only closed billiard path of a single circuit in an acute triangle
is the pedal triangle. There are an infinite number
of multiple-circuit paths, but all segments are parallel to the sides of the pedal
triangle. There exists a closed billiard path inside a cyclic
quadrilateral if its circumcenter lies inside
the quadrilateral (Wells 1991).
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).
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