Minimal Area of a Voronoi Cell in a Packing of Unit Circles

TL;DR

This paper provides a concise, self-contained proof that the smallest Voronoi cell area in a packing of unit circles is 2√3, achieved uniquely by a perfect hexagon, clarifying a classical geometric fact.

Contribution

It offers a new, short, and instructive proof of the minimal Voronoi cell area in unit circle packings, emphasizing the uniqueness of the hexagonal configuration.

Findings

Minimal Voronoi cell area is 2√3.

The minimal area configuration is a perfect hexagon.

The proof is concise and self-contained.

Abstract

We present a new self-contained proof of the well-known fact that the minimal area of a Voronoi cell in a unit circle packing is equal to $2\sqrt{3}$, and the minimum is achieved only on a perfect hexagon. The proof is short and, in our opinion, instructive.