Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry

TL;DR

This paper introduces symplectic tiling billiards, proves a periodic orbit result, and connects it to hyperbolic geometry, revealing a hyperbolic structure on the configuration space of a hexagonal linkage.

Contribution

It unites symplectic and tiling billiards, establishes a periodic orbit result, and constructs hyperbolic structures on moduli spaces of planar polygons.

Findings

Periodic orbits in symplectic tiling billiards are characterized in a special case.

The configuration space of a hexagonal linkage forms a 10-cusped hyperbolic 3-manifold.

The manifold is tiled by 15 regular ideal octahedra.

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.