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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!