Honeycomb conjecture

A regular hexagonal grid

This honeycomb forms a circle packing, with circles centered on each hexagon.

The honeycomb conjecture states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 by mathematician Thomas C. Hales.^[1]

Theorem[]

Let ${\displaystyle \Gamma }$ be a locally finite graph in ${\displaystyle \mathbb {R} ^{2}}$, consisting of smooth curves, and such that ${\displaystyle \mathbb {R} ^{2}\backslash \Gamma }$ consists of infinitely many bounded connected components, all of unit area. Let ${\displaystyle B(0,r)}$ be a disk of radius ${\displaystyle r}$ centered at the origin. Let ${\displaystyle C}$ be the union of these bounded components.^[1] The theorem states:

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

Equality is attained for the regular hexagonal tile.

History[]

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] In the 17th century, Jan Brożek used a similar theorem to argue why bees create hexagonal honeycombs. 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.

See also[]

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

References[]

This geometry-related article is a stub. You can help Wikipedia by .

• v
• t