Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry
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 -cusped hyperbolic -manifold that is tiled by regular ideal octahedra. The cusps correspond to the maximally degenerate configurations.
Cite
@article{arxiv.2307.12259,
title = {Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry},
author = {Richard Evan Schwartz},
journal= {arXiv preprint arXiv:2307.12259},
year = {2025}
}
Comments
This paper is a considerable revision. I revised the paper according to two referee reports. The new version is more formally written and has many more results. The core content is the same