Period Orbits on a 120-Isosceles Triangular Billiards Table
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Transcript Period Orbits on a 120-Isosceles Triangular Billiards Table
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:
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!