Honeycomb conjecture

Let Γ be a locally finite graph in R², consisting of smooth curves, and such that R² \ Γ has infinitely many bounded connected components, all of unit area. Let C be the union of these bounded components.[1]

${\displaystyle \limsup _{r\to \infty }\ {\frac {\operatorname {perimeter} (C\cap B(0,r))}{\operatorname {area} (C\cap B(0,r))}}\geq {\sqrt[{4}]{12}}.}$

Equality is attained for the regular hexagonal tile.

The first record of the conjecture dates back to 36 BC, from Marcus Terentius Varro, but is often attributed to Pappus of Alexandria (c. 290 – c. 350).[2] The conjecture was proven in 1999 by mathematician Thomas C. Hales, who mentions in his work that there is reason to believe that the conjecture may have been present in the minds of mathematicians before Varro.[1][2]

It is also related to the densest circle packing of the plane, in which every circle is tangent to six other circles, which fill just over 90% of the area of the plane.

• Weaire–Phelan structure, a counter-example to the Kelvin conjecture on the solution of the similar problem in 3D.