Packings of Regular Pentagons in the Plane

TL;DR

This paper proves the pentagonal ice-ray conjecture, establishing the maximum packing density of regular pentagons in the plane as approximately 0.92, using a computer-assisted geometric analysis.

Contribution

It confirms that optimal packings are double lattices and reduces the conjecture to finite area minimization problems solved computationally.

Findings

Maximum packing density is about 0.92.

Optimal packings are double lattices with aligned columns.

The proof involves computer-assisted area minimization of Delaunay triangles.

Abstract

We show that every packing of congruent regular pentagons in the Euclidean plane has density at most $(5-\sqrt5)/3$, which is about 0.92. More specifically, this article proves the pentagonal ice-ray conjecture of Henley (1986), and Kuperberg and Kuperberg (1990), which asserts that an optimal packing of congruent regular pentagons in the plane is a double lattice, formed by aligned vertical columns of upward pointing pentagons alternating with aligned vertical columns of downward pointing pentagons. The strategy is based on estimates of the areas of Delaunay triangles. Our strategy reduces the pentagonal ice-ray conjecture to area minimization problems that involve at most four Delaunay triangles. These minimization problems are solved by computer. The computer-assisted portions of the proof use techniques such as interval arithmetic, automatic differentiation, and a meet-in-the-middle algorithm.