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

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= , 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}.

Facts About Orbits (cont.)

Periodic Orbits

Thank You!