period orbits on a 120-isosceles triangular billiards table
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BY DAVID BROWN, BEN BAER, FAHEEM GILANISPONSORED BY DRS. RON UMBLE AND ZHIGANG HAN
Period Orbits on a 120-Isosceles Triangular Billiards Table
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
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= , for integers 0<a≤b with (a,b)=1.
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}.
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