Bees have encouraged mathematical speculation for two millennia, since classical scholars tried to explain the geometrically appealing shape of honeycombs. How do bees tackle complex problems that humans would express mathematically? In this series we’ll explore three situations where understanding the maths could help explain the uncanny instincts of bees.

Honeycomb geometry

Honeybees collect nectar from flowers and use it to produce honey, which they then store in honeycombs made of beeswax (in turn derived from honey). A question that has puzzled many inquiring minds across the ages is: why are honeycombs made of hexagonal cells?

The Roman scholar Varro, in his 1st century BC book-long poem De Agri Cultura (“On Agriculture”), briefly states

“Does not the chamber in the comb have six angles, the same number as the bee has feet? The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space .”

This quote is the earliest known source suggesting a link between the hexagonal shape of the honeycomb and a mathematical property of the hexagon, made more explicit a few centuries later by Pappus of Alexandria (sometimes considered to be the last Ancient Greek mathematician). Writing after the Roman Empire’s glory days, Pappus points out that there are three regular polygons that tile the plane without gaps—triangles, squares and hexagons—and bees, in their wisdom, choose the design that holds the most honey given a set amount of building material .

The idea that bees economically choose the regular tiling polygon with the greatest area for a given perimeter satisfied the ancients. But why should all the honeycomb cells be identical—could bees do any better than using regular polygons?

The amazingly regular honeycombs we are used to seeing are built by domestic honeybees, using perfectly regular wax foundations provided by their beekeepers as a guide. Wild honeybees, however, don’t have the luxury of nestboxes fitted with wax foundations. Their honeycomb cells, though still very regular, aren’t always hexagonal—sometimes pentagons and heptagons creep in. After choosing an irregularly-shaped tree cavity, many worker bees will start independently building the hive’s hexagons at once, meeting in the middle along seams. It is along these seams where the five and seven-sided irregularities and other defects tend to appear .

More intriguingly, what happens if you provide honeybees with the wrong size of foundation as a guide? H. Randall Hepburn tried this with variously-sized hexagonal foundations (with flat bases unlike the standard ridged commercial ones). The largest caused the bees to to build gorgeous rosettes—they filled each large hexagonal foundation with five or six cells (again, typically five or seven-sided) surrounding a central cell. The next size down caused the bees to build in an irregular pattern, while the size smaller than that led to the bees building hexagonal cells on each vertex of the foundational hexagons, and leaving a hexagonal void or ‘false cell’ in the centre.

So bees don’t necessarily need to use regular polygons at all. Allowing for the possibility of a mixture of shapes leads to two subtly different mathematical problems:

(A) Which mixture of shapes that tile the plane, and each have unit area, have the minimum average perimeter?

(P) Which mixture of shapes that tile the plane, and each have unit perimeter, have the maximum average area?

Fejes Tóth, a Hungarian mathematician, had proved by 1963 that the regular hexagonal tiling was the solution to the isoperimetric problem, (P), but only proved that it solved the iso-areal problem, (A), under the assumption that all the shapes were convex. This convexity condition is more restrictive than it may at first seem: a bulging edge on one shape leads to a concave indentation on a neighbouring shape, so the only convex shapes that can tile the plane are polygons.

It wasn’t until 1999 that Thomas Hales proved that hexagons also divide a plane into shapes of equal area with the least perimeter.

A key step in the proof (though not Hales’ breakthrough) is showing that, on average, such shapes would have six sides. Take a planar graph (ie. with no edges crossing) on a sphere. Using the Euler characteristic of a planar graph (which also holds on a sphere), we know that the number of the vertices, edges and faces of the graph satisfies $V-E+F=2$. Nudging the edges slightly, it turns out to be sufficient to consider only vertices of degree 3. It’s quite easy to convince oneself with a diagram (like the one above) that this is plausible. Moving edges by a tiny amount, you can make sure no more than three lines meet at one vertex. You might change the numbers of edges and vertices, but without changing the perimeter or area by much.

If the $i$th face has $e_i$ edges (and vertices), then adding these up for each face, we would count each edge twice and each vertex three times . So, by the Euler characteristic of the whole graph, overall we must have $$V-E+F=\sum_i \left ( \frac{e_i}{3}-\frac{e_i}{2} + 1 \right )=\sum_i \left ( 1 – \frac{e_i}{6} \right )=2.$$As the number of faces increases, $1 – \frac{e_i}{6}$ must become very small, and so the average of $e_i$ over all faces $i$ tends to $6$, which is the number of sides we were hoping for. This step can be used to show a finite version of the theorem on the plane, which is a good start for the infinite version.

So out of all the prisms arranged side-by-side, hexagonal prisms use the least beeswax to build a unit bee-sized volume. But honeycombs are not made up of hexagonal prisms: the hidden end of the honeycomb cell is not flat. There are two layers of cells, back-to-back and offset, and the end-cap between them is a pyramid made of three rhombuses, so the whole shape could also be described as ‘half of an elongated rhombic dodecahedron’. The ridged wax honeycomb foundation pictured near the start mimics these rhombuses. This shape is more efficient than a hexagonal prism with a flat base.

What bees haven’t realised, is that there’s a slightly more efficient way to cap off their hexagonal cells. Keeping the hexagonal footprint of their hive, they could make repeated use of a truncated octahedron, formed by lopping off the vertices of an octahedron. This would lead to a saving of almost 2% in the idealised version with walls of negligible thickness, which isn’t much.

Using a curvy non-polyhedral variant of the truncated octahedron, Lord Kelvin, the 19th century physicist, held the record for ‘least surface area for a fixed volume’ in the three-dimensional version of the honeycomb conjecture for a full century . This was assumed to be the best possible until 1994, when two physicists, Denis Weaire and Robert Phelan, armed with computers, devised an ever-so-slightly more efficient way of dividing space into cells of equal volume, using two different curvy solids (2.2% more efficient than the rhombic dodecahedra). It was after this discovery that the long-standing two-dimensional honeycomb conjecture seemed less obviously true: a suggestion from Weaire is what nudged Hales to attempt his proof .