2007/01/01 by Richard Evan Schwartz · 1 citation
Mathematics · #Mathematical Dynamics and Fractals #Mathematics and Applications #Kite #Mathematics #Uncountable set #Orbit (dynamics) #Quadrilateral #Regular polygon #Polygon (computer graphics) #Pure mathematics #Geometry #Physics #Computer science
paper · doi:10.3934/jmd.2007.1.371
openalex publication_date 2007/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/04/05
The question of B.H. Neumann, which dates back to the 1950s, asks if there exists an outer billiards system with an unbounded orbit. We prove that outer billiards for the Penrose kite, the convex quadrilateral from the Penrose tiling, has an unbounded orbit. We also analyze some finer properties of the orbit structure, and in particular produce an uncountable family of unbounded orbits. Our methods relate outer billiards on the Penrose kite to polygon exchange maps, arithmetic dynamics, and self-similar tilings.