Packings of Regular Pentagons in the Plane

Thomas Hales, Wöden Kusner

Research output: Contribution to journalArticleResearch

Abstract

We show that every packing of regular pentagons in the Euclidean plane has density less than 0.9611. Our proof is computer-assisted. We also give a detailed strategy for proving the Kuperberg-Kuperberg conjecture, which asserts that the optimal packing of 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 Kuperberg conjecture to area minimization problems that involve at most four acute Delaunay triangles.
Original languageEnglish
JournalarXiv.org e-Print archive
Publication statusIn preparation - 23 Feb 2016

Fingerprint

Pentagon
Packing
Delaunay
Triangle
Vertical
Euclidean plane
Acute
Minimization Problem
Estimate
Strategy

Keywords

  • math.MG

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Hales, T., & Kusner, W. (2016). Packings of Regular Pentagons in the Plane. Manuscript in preparation.

Packings of Regular Pentagons in the Plane. / Hales, Thomas; Kusner, Wöden.

In: arXiv.org e-Print archive, 23.02.2016.

Research output: Contribution to journalArticleResearch

Hales, Thomas ; Kusner, Wöden. / Packings of Regular Pentagons in the Plane. In: arXiv.org e-Print archive. 2016.
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