The plaid model and outer billiards on kites

TL;DR

This paper demonstrates a coarse isomorphism between the plaid model and the arithmetic graph for outer billiards on kites, showing the model predicts orbit dynamics and confirming unbounded orbits for irrational kites.

Contribution

It proves a coarse isomorphism linking the plaid model to outer billiards on kites, advancing understanding of orbit behavior in polygonal billiards.

Findings

Plaid model predicts symbolic dynamics of outer billiards.

Establishes unbounded orbits for irrational kites.

Provides a new proof of unbounded orbits in polygonal billiards.

Abstract

This paper is the third in a series which explores a combinatorial method for generating lattice polygons in the plane. I call this method the plaid model. In this paper I prove the main result I had been aiming for since the beginning, which is to show that there is a coarse isomorphism between the plaid model and the so-called arithmetic graph for outer billiards on kites. The content of the theorem is that the plaid model predicts the symbolic dynamics of the outer billiards orbits, up to an error of one unit. This result combines with the work in the other papers to give a second proof that outer billiards has unbounded orbits for every irrational kite. So far, these are the only known polygonal examples with this property.