The honeycomb – why hexagons
Only triangles, squares and hexagons fill the plane without gaps, and the hexagon needs the least wall. Suspected since antiquity, proven in 1999.
Bees build their combs from wax, and wax is expensive. For one gram of wax they use up many times as much honey. A comb should therefore create as much room for honey and brood as possible with as little wall as possible. The bees solve this task with the hexagon.
An old conjecture
As early as the 1st century BC the Roman Marcus Terentius Varro wrote that geometers could show the hexagon holds the most space. In the 4th century Pappus of Alexandria devoted a reflection of its own to the “wisdom of the bees”: only three regular polygons fill the plane without gaps, the triangle, the square and the hexagon (see Circle, triangle and square). Of these three, the hexagon encloses the largest area for the same perimeter, and the bees, Pappus said, knew this.
Three tilings with the same cell area. The display gives the wall length per cell, each wall counted half because two cells share it. Hexagons need almost 7 per cent less wall than squares and over 18 per cent less than triangles.
Pappus compared only the regular polygons. Whether some entirely different division, with curved walls or unequal cells, might be even more economical remained open. This honeycomb conjecture was proven only in 1999 by Thomas Hales: every division of the plane into cells of equal size needs at least as much wall as the regular hexagonal grid. Shortly before, the same Hales had also proven Kepler’s conjecture on the densest packing of spheres (see Kissing numbers, sphere packings and 24).
Circles become hexagons
Pack circles as densely as possible and each sits in a ring of six neighbours. They then fill just over 90 per cent of the area. If the circles keep growing, they press against each other and flatten into straight walls. At the end stand hexagons that fill the area completely. The same can be seen in soap foam between two glass plates and in crowded cells in living tissue.
Circles in densest packing grow until they flatten into hexagons. The slider shows the filled share rising from 90.7 per cent to 100 per cent.
How the bees shape their hexagons is disputed among researchers. One idea is that they build round tubes and the warm wax flows by itself into straight walls where the tubes touch. Other observations show that the bees shape the walls deliberately. What is certain is that the result is the most economical shape geometry allows.
The floor of the cell
A comb has two layers of cells, back to back. The floor of each cell is formed by three equal rhombi meeting in a point. In 1712 the astronomer Giacomo Maraldi measured their angles: about 109½ and 70½ degrees. These are exactly the angles of the rhombic dodecahedron, the solid that fills space without gaps and forms the cells of the face-centred cubic lattice (see Filling space without gaps). In 1611, in his treatise on the six-cornered snowflake, Johannes Kepler had already compared the bee cell with this solid.
In 1964 the Hungarian mathematician László Fejes Tóth showed that an even more economical cell floor would exist. The saving is below one per cent, though, and for a wall of wax that needs a certain thickness it makes no difference.
What it means
Six is the first perfect number, 1 + 2 + 3, and the third triangular number. The hexagon consists of six equilateral triangles, and six circles close without gaps around a seventh, as in the Flower of Life. The hexagon is thus the form in which the triangle of the tetractys continues across the plane. That precisely this form is the most economical division of the plane was found by nature long before mathematics could prove it.