### Abstract

Original language | English |
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Journal | arXiv.org e-Print archive |

Publication status | In preparation - 23 Feb 2016 |

### Fingerprint

### Keywords

- math.MG

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

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

Research output: Contribution to journal › Article › Research

*arXiv.org e-Print archive*.

}

TY - JOUR

T1 - Packings of Regular Pentagons in the Plane

AU - Hales, Thomas

AU - Kusner, Wöden

PY - 2016/2/23

Y1 - 2016/2/23

N2 - 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.

AB - 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.

KW - math.MG

M3 - Article

JO - arXiv.org e-Print archive

JF - arXiv.org e-Print archive

ER -