Five-neighbour packings of centrally symmetric convex discs
Endre Makai Jr

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 , 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 , 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.
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Taxonomy
TopicsPoint processes and geometric inequalities · Quasicrystal Structures and Properties · Mathematics and Applications
