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Separating dots with circles

James Beyer, Jaewon Min, Greg Muller

TL;DR

This work studies how finite sets of points (dots) on the plane or sphere can be separated by circles of two kinds: incident circles through three dots and avoidant circles through none. It proves configuration-invariant counting formulas for how many such circles yield partitions of the remaining points into fixed sizes, and it ties these counts to the combinatorics of k-th order Voronoi decompositions on the sphere. By analyzing how these decompositions change under continuous movements of the dots, the authors show that the induced cluster algebras are independent of the specific dot configuration and are determined only by the numbers of dots and the Voronoi structure. The work also develops a framework of local moves (Postnikov moves) that connect all configurations and establishes a rich link between spherical/planar Voronoi diagrams and cluster algebra theory, with future directions including mapping class group actions and notable special cases such as the Markov and X7 cluster algebras.

Abstract

Given a finite set of points in general position in the plane or sphere, we count the number of ways to separate those points using two types of circles: circles through three of the points, and circles through none of the points (up to an equivalence). In each case, we show the number of circles which separate the points into subsets of size k and l is independent of the configuration of points, and we provide an explicit formula in each case. We also consider how the circles change as the configuration of dots varies continuously. We show that an associated higher order Voronoi decomposition of the sphere changes by a sequence of local `moves'. As a consequence, an associated cluster algebra is independent of the configuration of dots, and only depends on the number of dots and the order of the Voronoi decomposition.

Separating dots with circles

TL;DR

This work studies how finite sets of points (dots) on the plane or sphere can be separated by circles of two kinds: incident circles through three dots and avoidant circles through none. It proves configuration-invariant counting formulas for how many such circles yield partitions of the remaining points into fixed sizes, and it ties these counts to the combinatorics of k-th order Voronoi decompositions on the sphere. By analyzing how these decompositions change under continuous movements of the dots, the authors show that the induced cluster algebras are independent of the specific dot configuration and are determined only by the numbers of dots and the Voronoi structure. The work also develops a framework of local moves (Postnikov moves) that connect all configurations and establishes a rich link between spherical/planar Voronoi diagrams and cluster algebra theory, with future directions including mapping class group actions and notable special cases such as the Markov and X7 cluster algebras.

Abstract

Given a finite set of points in general position in the plane or sphere, we count the number of ways to separate those points using two types of circles: circles through three of the points, and circles through none of the points (up to an equivalence). In each case, we show the number of circles which separate the points into subsets of size k and l is independent of the configuration of points, and we provide an explicit formula in each case. We also consider how the circles change as the configuration of dots varies continuously. We show that an associated higher order Voronoi decomposition of the sphere changes by a sequence of local `moves'. As a consequence, an associated cluster algebra is independent of the configuration of dots, and only depends on the number of dots and the order of the Voronoi decomposition.

Paper Structure

This paper contains 9 sections, 20 theorems, 35 equations, 6 figures.

Table of Contents

  1. Summary of results
  2. Dots in the sphere or plane
  3. Counting incident circles
  4. Higher order Voronoi decompositions
  5. Counting avoidant circles
  6. Moving the dots
  7. Relations between higher order Voronoi decompositions
  8. Future directions
  9. Connection between spherical and planar Voronoi decompositions

Key Result

Theorem A

Given at least three dots in general position in the plane or sphere, the number of circles through three of the dots which separate the remaining dots into subsets of size $k$ and $\ell$ is

Figures (6)

  • Figure 1: Incident and avoidant circles for two configurations of dots
  • Figure 2: Three local moves between bicolored graphs in the sphere
  • Figure 3: Stereographic projection of 6 dots
  • Figure 4: Two configurations of four dots in general position in the plane
  • Figure 5: The stereographic projection of a $2$nd order Voronoi decomposition of the sphere determined by a configuration of $6$ dots, with bicolored vertices
  • ...and 1 more figures

Theorems & Definitions (46)

  • Theorem A
  • proof : Remark 1
  • Theorem B
  • proof : Remark 2
  • Theorem C
  • proof : Remark 3
  • Proposition 4
  • proof : Warning 5
  • Proposition 6
  • proof
  • ...and 36 more