Note on a Conjecture of Wegner

TL;DR

This paper investigates the number theoretic properties of exceptional packings of unit discs in the plane, providing counterexamples to Wegner's conjecture and characterizing these exceptional numbers.

Contribution

It offers a counterexample to Wegner's conjecture and characterizes the exceptional numbers related to extremal disc packings.

Findings

Counterexample to Wegner's conjecture

Characterization of exceptional numbers

Relation between shape and number theoretic properties

Abstract

The optimal packings of n unit discs in the plane are known for those natural numbers n, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal Groemer packing of n unit discs does not exist for all natural numbers n and in this case, the number n is called exceptional. We are interested in number theoretic characterizations of the exceptional numbers. A counterexample is given to a conjecture of Wegner concerning such a characterization. We further give a characterization of the exceptional numbers, whose shape is closely related to that of Wegner's conjecture.