# The Honeycomb Conjecture

@article{Hales2001TheHC, title={The Honeycomb Conjecture}, author={Thomas C. Hales}, journal={Discrete \& Computational Geometry}, year={2001}, volume={25}, pages={1-22} }

This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.

## 363 Citations

The Honeycomb Problem on the Sphere

- Mathematics
- 2002

The honeycomb problem on the sphere asks for the perimeter-minimizing partition of the sphere into N equal areas. This article solves the problem when N=12. The unique minimizer is a tiling of 12…

The Least-Perimeter Partition of a Sphere into Four Equal Areas

- MathematicsDiscret. Comput. Geom.
- 2010

We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.

On Hamiltonian Properties of Honeycomb Meshes

- Computer Science, Physics
- 2019

In this paper, we investigated Hamiltonian properties of honeycomb meshes which are created in two different ways. We obtained different Hamilton paths for Honeycomb Meshes for any dimension with…

Certain hyperbolic regular polygonal tiles are isoperimetric

- MathematicsGeometriae Dedicata
- 2021

The hexagon is the least-perimeter tile in the Euclidean plane. On hyperbolic surfaces, the isoperimetric problem differs for every given area. Cox conjectured that a regular $k$-gonal tile with…

Least-Perimeter Partitions of the Sphere

- Mathematics
- 2007

We consider generalizations of the honeycomb problem to the sphere S and seek the perimeter-minimizing partition into n regions of equal area. We provide a new proof of Masters’ result that three…

Proof of the honeycomb asymptotics for optimal Cheeger clusters

- MathematicsAdvances in Mathematics
- 2019

Perimeter-minimizing Tilings by Convex and Non-convex Pentagons

- Mathematics
- 2013

We study the presumably unnecessary convexity hypothesis in the theorem of Chung et al. [CFS] on perimeter-minimizing planar tilings by convex pentagons. We prove that the theorem holds without the…

Approximation of Partitions of Least Perimeter by Γ-Convergence: Around Kelvin’s Conjecture

- MathematicsExp. Math.
- 2011

A numerical process to approximate optimal partitions in any dimension is reported to relax the problem into a functional framework based on the famous result of Γ-convergence obtained by Modica and Mortolla.

Planar clusters and perimeter bounds

- Mathematics
- 2005

We provide upper and lower bounds on the least-perimeter way to enclose and separate n regions of equal area in the plane (theorem 3.1). Along the way, inside the hexagonal honeycomb, we provide…

On Generalizing the Honeycomb Theorem to Compact Hyperbolic Manifolds and the Sphere

- Mathematics
- 2006

We provide a possible alternate proof to the Honeycomb Conjecture in the plane. We generalize the proof of the hexagonal isoperimetric inequality to S and H under certain conditions and deduce that a…

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