A Dynamical System Using the Voronoï Tessellation

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1. INTRODUCTION. Suppose you know the locations of post offices or cell phone satellites, and you want to know what regions they serve. Or maybe you know the locations of atoms in a crystal, and you want to know what a fundamental region looks like. There are lots of reasons you might want to make a tiling around a given discrete set of points. A natural way to do it is with Voronotessellations—so natural, in fact, that it has been rediscovered numerous times over the years. On the other hand, if you have a tiling, you might want to decorate each tile with a few points to create or destroy symmetry. Or you might look at all the vertices of the tiling—points where three or more tiles meet—to extract combinatorial information. If you have a tiling, there are many ways to obtain a point set from it. So, you can get tilings from point sets and point sets from tilings: doesn't this give you a way to associate point sets to point sets or tilings to tilings? Once you have a map from a class of objects back to itself, you can take a dynamical systems viewpoint to analyze the situation. In this paper we are going to do exactly that, with a new dynamical system based on the vertices of Vorono¨i tessellations. For those uninitiated with the Voronotessellation, we begin with its definition and then give the definition of our dynamical system. From there, the remainder of § 1i s spent exploring the evolution of simple point sets, using these simplified examples to develop both the intuition and vocabulary needed for more interesting cases. In § 2w e give a new proof of a theorem, first proved in (4), quantifying the growth in size of point sets over repeated iteration. Following that we will point out some interesting corollaries and give some estimates on the growth rate. We devote § 3t o discussing what questions interest us most from the dynamical systems viewpoint. For now, let's turn to the definitions. 1.1. Our Dynamical System. We start with a finite point set P ⊂ R 2 , which we call the generating set, the members of which we refer to as the generators .( We relax the assumption that P be finite and/or planar in §3.) The Vorono¨ o polygon of a point p ∈ P, denoted V (p) ,i s given by