Period Orbits on a 120-Isosceles Triangular Billiards Table

1. Millersville University Period Orbits on a 120-Isosceles Triangular Billiards Table By David Brown, Ben Baer, FaheemGilani Sponsored by Drs. Ron Umble and ZhigangHan

2. Introduction • Consider a frictionless 120-isosceles billiards table with a ball released from the base at an initial angle

3. 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

4. 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

5. 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

6. 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.

7. Definitions (cont.) • A periodic orbit is stable if its period is independent of initial position • Otherwise it is unstable

8. The Problem Find and classify the periodic orbits on a 120 isosceles triangular billiards table.

9. 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.

10. 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)

11. 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]

12. 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.

13. 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

14. 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}.

15. Facts About Orbits (cont.) Periodic Orbits

16. Thank You!