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The honeycomb conjecture states that dividing a plane into regions of equal area has a perimeter at least equal to that of the regular hexagonal honeycomb tiling. That is to say when dividing a surface into equal area shapes (leaving no interstitial space), hexagons tile with the smallest perimeter possible. Therefore, a hexagonal structure is the most efficient (uses the least amount of material) to fill a plane with equal area cells.
In 1999, the American mathematician Thomas Hales published a proof of the honeycomb conjecture.
Around 36 B.C., Roman Scholar Marcus Terentius Varro wrote about the hexagonal structure of the bee's honeycomb in his book on agriculture. Varro wrote:
Does not the chamber in the comb have six angles . . . The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space.
In the 4th century AD, Greek mathematician Pappus of Alexandria presented a proof to the problem in the preface of his fifth book. However, Pappus's argument is incomplete and is only a comparison of three suggestive cases (triangle, square, and hexagon). It was known to the Pythagoreans that only these three regular polygons can tile the plane. Pappus stated that with the same quantity of material for the construction of these shapes, the hexagon would be able to hold the most honey. His reasons for restricting to only these three regular polygons were not mathematical but based on bees avoiding dissimilar figures. Pappus also excluded gaps between cells of the honeycomb without mathematical argument, stating:
foreign matter could enter the interstices between them and so defile the purity of their produce.
The isoperimetric property of honeycombs means there is a vast amount of literature throughout the centuries referring to the bee as a geometer. Darwin explained the honeycomb structure through natural selection:
That motive power of the process of natural selection having been economy of wax; that individual swarm that wasted least honey in the secretion of wax, having succeeded best.
The honeycomb conjecture had been solved under special hypotheses, including by Hungarian mathematician László Fejes Tóth. In 1943, Tóth proved the honeycomb conjecture under the hypothesis that the cells are convex. Under this hypothesis, the boundaries of the cells are forced to be polygons, removing potential counterexamples of regions bounded by circular arcs.
In June 1999, American mathematician Thomas Hales, while working at the University of Michigan, proved the hexagonal honeycomb conjecture without any special hypotheses.
Allowing for curved sides complicates the problem as sides can bulge out increasing the area. However, one side bulging out causes the neighboring region to bulge in reducing the area. Hales proved the disadvantages of bulging in outweighs the advantage of bulging out, and therefore polygons are better (and regular hexagons the best of all) than cells bound by circular arcs. The proof begins with a reduction to finite clusters, which requires assuming that the regions are connected.
A form of the theorem can be expressed by letting be the isoperimetric constant for a regular N-gon (polygon with N sides). is the ratio of the circumference squared to the area of a regular N-gon, then the perimeter of a regular hexagon with unit area is .
Let:
be a disk of radius r at the origin.
be a locally finite graph in consisting of smooth curves, and such that has infinitely many bounded connected components, all of unit area.
be the union of these bounded components
Then:
Equality is attained only for a regular hexagonal tile.
The isoperimetric property of hexagons that allows them to most efficiently tile identical cells together in a flat plane means they are present in numerous examples across nature. These include:
- Honeycombs within beehives
- Layers of bubbles on the surface of water tend towards hexagonal shapes
- Snowflakes are a hexagonal crystalline form of ice
- Basalt columns
- Insect eyes
