Period Orbits on a 120-Isosceles
Triangular Billiards Table
BY DAVID BROWN, BEN BAER, FAHEEM GILANI
SPONSORED BY DRS. RON UMBLE AND ZHIGANG HAN
Introduction
 Consider a frictionless 120-isosceles billiards table
with a ball released from the base at an initial angle
History
 The 120 isosceles triangle is one of 8 shapes that can
the plane tessellate through edge reflections
 The other shapes are:
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Square/Rectangle
Equilateral Triangle
45 Isosceles Triangle
30-60-90 Triangle
120 Isosceles Triangle
Regular Hexagon
120-90-90 Kite
60-120 Rhombus
History
 Andrew Baxter (working with Dr. Umble) solved the
equilateral case
 Jonathon Eskreis-Winkler and Ethan McCarthy
worked (with Dr Baxter) on the rectangle, 30-60-90
triangle, and 45 isosceles triangle cases
Assumptions
 A billiard ball bounce follows the same rule as a
reflection:
Angle of incidence = Angle of reflection
 A billiard ball stops if it hits a vertex.
θ
θ
Definitions
 The orbit of a billiard ball is the trajectory it follows.
 A singular orbit terminates at a vertex.
 A periodic orbit eventually retraces itself.
 The period of a periodic orbit is the number of
bounces it makes until it starts to retrace itself.
Definitions (cont.)
 A periodic orbit is stable if its period is independent
of initial position
 Otherwise it is unstable
The Problem
Find and classify the periodic orbits on a 120
isosceles triangular billiards table.
Techniques of Exploration
 We found it easier to analyze the path of the billiard
ball by reflecting the triangle about the side of
impact. In the equilateral case we were able to
construct a tessellation, the same can be done with
the 120-isosceles case.
Techniques of Exploration (cont.)
 We used Josh Pavoncello’s Orbit Mapper program
to generate orbits with a given initial angle and
initial point of incidence.
(22 bounce orbit using the Orbit Mapper program)
Results
 There exist at most 2 distinct periodic orbits with a
given initial angle
 Every periodic orbit is represented by exactly one
periodic orbit with incidence angle θ in [60,90]
Facts About Orbits
Theorem 1: If the initial point of a periodic orbit is
on a horizontal edge of the tessellation, so is its
terminal point.
Facts About Orbits (cont.)
 Theorem 2: If θ is the incidence angle of a
periodic orbit, then θ=
0<a≤b with (a,b)=1.
, for integers
a=3
b=5
Facts About Orbits (cont.)
Theorem 3: Given a periodic orbit with initial angle
as before:
(1) The orbit is stable iff 3|b.
(2) If an unstable orbit has periods m<n, then
n  {2m-2,2m+2}.
Facts About Orbits (cont.)
Periodic Orbits
Thank You!