Fifty New Invariants of N-Periodics in the Elliptic Billiard
Dan Reznik, Ronaldo Garcia, Jair Koiller
Abstract
We introduce 50+ new invariants manifested by the dynamic geometry of N-periodics in the Elliptic Billiard, detected with an experimental/interactive toolbox. These involve sums, products and ratios of distances, areas, angles, etc. Though curious in their manifestation, said invariants do all depend upon the two fundamental conserved quantities in the Elliptic Billiard: perimeter and Joachimsthal’s constant. Several proofs have already been contributed (references are provided); these have mainly relied on algebraic geometry. We very much welcome new proofs and contributions.
Topics & Concepts
Dynamical billiardsMathematical proofMathematicsPure mathematicsAlgebraic numberElliptic curveAlgebraic geometryAlgebra over a fieldInvariant (physics)DiffeomorphismMathematics and ApplicationsMathematical Dynamics and FractalsAnalytic and geometric function theory