Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry

TL;DR

This work introduces symplectic tiling billiards, a unification of symplectic billiards and tiling billiards, and develops a framework to study periodic orbits via sunbursts and phase-modified weaves. It constructs a concrete, computable correspondence between similarity classes of strictly convex equilateral N-gons and strictly convex equiangular N-gons, denoted $[L]\mapsto[P_L]$, by embedding the polygons into paired sunburst tilings and leveraging a woven periodic orbit. The correspondence is shown to be bijective and algebraic, enabling transfer of Thurston’s hyperbolic structures from equiangular to equilateral polygon moduli spaces; in particular, ${\cal A}_5$ is a hyperbolic surface tiled by 12 pentagons and ${\cal A}_6$ is a hyperbolic 3-manifold tiled by 15 regular ideal octahedra with 10 cusps. This provides a new algebraic hyperbolic perspective on polygon spaces and their degenerations, unifying dynamical, combinatorial, and geometric viewpoints with explicit computational methods.

Abstract

In this paper I will unite two games, symplectic billiards and tiling billiards. The new game is called symplectic tiling billiards. I will prove a result about periodic orbits of symplectic tiling billiards in a very special case and then show how this result combines with the construction in Thurston's paper {\it Shapes of Polyhedra\/} to give hyperbolic structures on moduli spaces of planar equilateral polygons. One corollary is that the configuration space of the hexagonal planar linkage with unit-length rods (modulo isometry) has an algebraically defined hyperbolic structure in which it is a $10$-cusped hyperbolic $3$-manifold that is tiled by $15$ regular ideal octahedra. The $10$ cusps correspond to the $10$ maximally degenerate configurations.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 2.1
• Lemma 2.2
• Theorem 3.1
• Theorem 3.2
• Lemma 3.3
• Lemma 3.4