Separating circles on the sphere by polygonal tilings

TL;DR

This paper extends the concept of separating polygonal tilings from Euclidean plane circle packings to spherical and hyperbolic geometries, revealing new geometric properties and limitations.

Contribution

It proves that separating tilings exist for circle packings on the sphere and hyperbolic plane, and identifies Euclidean circles as uniquely separable convex discs.

Findings

Separating tilings exist for circle packings on the sphere and hyperbolic plane.

Euclidean circles are uniquely separable among convex discs in the plane.

The property does not hold for convex discs on the sphere, and its status on the hyperbolic plane remains open.

Abstract

We say that a tiling separates discs of a packing in the Euclidean plane, if each tile contains exactly one member of the packing. It is a known elementary geometric problem to show that for each locally finite packing of circular discs, there exists a separating tiling with convex polygons. In this paper we show that this separating property remains true for circle packings on the sphere and in the hyperbolic plane. Moreover, we show that in the Euclidean plane circles are the only convex discs, whose packings with similar copies can be always separated by polygonal tilings. The analogous statement is not true on the sphere and it is not known in the hyperbolic plane.