  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.
