Nature: The Mathematics of the Beehive

The hexagon – a shape with six identical, equally spaced edges – has inspired mathematicians for the longest time. Turns out that nature too, loves this shape. A near perfect example of a hexagon in nature is the honeycomb. Each cell of a honeycomb is hexagonal. At first sight, this is surprising because we would expect to find cells of all kinds of shapes and sizes, fitted together in a rather haphazard manner.  Moreover, a hexagonal honeycomb would require worker bees to work sequentially, one at a time, first making one cell, then fitting the next cell to that, and so on. This is time-consuming as every bee would have to wait in line for the guy in front to finish his cell before he begins his, and if you’ve ever seen bees building a beehive, you know they don’t work this way. They work simultaneously!

So, why do bees build hexagonal honeycombs? The fact that they have been around for so long tells us that they are wise beyond their size, and that they have a clever game plan when building a honeycomb, one that ensures that despite the apparent chaos, in the end, all honeycomb cells will fit together beautifully. That game plan, it turns out, involve a good understanding of math.

It is a mathematical truth that there are only three geometrical figures with equal sides that can fit together on a flat surface without leaving gaps. Any gaps between the cells would be wasted space. A hexagon satisfies this criterion perfectly. To see why, let me recount to you a story.

More than 2000 years ago, in 35 BC, the Roman scholar Marcus Terentius Varro conjectured that the hexagonal grid is the unique geometrical shape that divides a surface into equal cells with the smallest total perimeter. In bee language, what this means is that a hexagonal honeycomb requires the least amount to construct. For every ounce of wax, a bee must consume about eight ounces of honey. That’s a lot of work. requiring visits to thousands of flowers and much flapping of wings. The hexagon minimizes the effort and expense of energy. But Varro only made an intelligent guess, a conjecture in mathematical parlance. Although outwardly simple, his conjecture was rigorously proven only in 1999, by the American mathematician Thomas Hales.

As for the bees – you could say they knew it all along!