Five-neighbour packings of centrally symmetric convex discs

TL;DR

This paper proves the minimal density for five-neighbour packings of centrally symmetric convex discs, showing it is 9/14 and achieved by affine regular hexagons, extending previous work on convex disc packings.

Contribution

It provides a detailed proof of the minimal density for five-neighbour packings of centrally symmetric convex discs, confirming the conjecture from earlier work.

Findings

Minimal density for five-neighbour packings is 9/14.

Affine regular hexagons achieve this minimal density.

The proof extends previous results on convex disc packings.

Abstract

In an old paper of the author the thinnest five-neighbour packing of translates of a convex disc (different from a parallelogram) was determined. The minimal density was $3/7$, and was attained for a certain packing of triangles. In that paper it was announced that for centrally symmetric convex plates (different from a parallelogram) the analogous minimal density was $9/14$, and was attained for a certain packing of affine regular hexagons, and a very sketchy idea of the proof was given. In this paper we give details of this proof.