The central ideas in Hales's proof of the Honeycomb conjecture.

Question

I have been trying to understand Thomas Hales's proof of the honeycomb conjecture: https://arxiv.org/pdf/math/9906042.pdf

However, despite the fact that the paper is clearly written, I find myself lost. I am wondering if anybody who knows this proof would be so kind to give a brief ordered list of the central ideas of the proof, so that I am able to understand what is a central idea and what is a technical detail and enable me to follow the proof better.

Thank you very much,

Maithreya

Answer 1

Too long for a comment.

I started to read the paper. I don’t know whether I’ll finish it, but Hales have already answered your question and it seems that its farther detalization is not simple but technical:

“To solve the problem, we replace the planar cluster with a cluster on a flat torus. The torus has the advantages of compactness and a vanishing Euler characteristic. This part of the proof is reminiscent of [FT43], which transports the planar cluster to a sphere. The key inequality, called the hexagonal isoperimetric inequality, appears in Theorem 4. It asserts that a certain functional is uniquely minimized by a regular hexagon of area 1. The isoperimetric properties of the functional force the minimizing figure to be convex. A penalty term prevents the solution from becoming too “round.” The optimality of the hexagonal honeycomb results”.

Even the rigorous formulations of Honeycomb conjecture which he used is very technical, for instance:

“Theorem 1-A (Honeycomb conjecture). Let $\Gamma$ be a locally finite graph in $\Bbb R^2$, consisting of smooth curves, and such that $\Bbb R^2\setminus\Gamma$ has infinitely many bounded connected components, all of unit area. Let $C$ be the union of these bounded components. Then $$\operatorname{limsup}_{r\mapsto\infty}\frac{\operatorname{perim}(C\cap B(0, r))} {\operatorname{area}(C ∩ B(0, r))}\ge\sqrt[4]{12}.$$

Equality is attained for the regular hexagonal tile”.