Why is every cell in this honeycomb a hexagon?

More than 2,000 years ago, **Marcus Terentius Varro**, a roman citizen, proposed an answer, which ever since has been called “The Honeybee Conjecture.” He thought that if we better understood, there would be an elegant reason for what we see.

“The Honeybee Conjecture” is an example of mathematics unlocking a mystery of nature. And luckily, NPR, with the help of physicist/writer **Alan Lightman**, (who wrote The Accidental Universe: The World You Thought You Knew) helps explains Varro’s hunch.

**Why the preference for hexagons? Is there something special about a six-sided shape?**

“It is a mathematical truth,” Lightman writes, “that there are only three geometrical figures with equal sides that can fit together on a flat surface without leaving gaps: equilateral triangles, squares and hexagons.”

So which to choose? The triangle? The square? Or the hexagon? Which one is best? Here’s where our Roman, Marcus Terentius Varro made his great contribution. His “conjecture” — and that’s what it was, a mathematical guess — proposed that a structure built from hexagons is probably a wee bit more compact than a structure built from squares or triangles. A hexagonal honeycomb, he thought, would have “the smallest total perimeter.” He couldn’t prove it mathematically, but that’s what he thought.

Compactness matters. The more compact your structure, the less wax you need to construct the honeycomb. Wax is expensive. A bee must consume about eight ounces of honey to produce a single ounce of wax. So if you are watching your wax bill, you want the most compact building plan you can find.

In 1999 Thomas Hales produced a mathematical proof, confirming that Varro was right.

**Why are they all the same size?**

For bees to assemble a honeycomb the way bees actually do it, it’s simpler for each cell to be exactly the same. If the sides are all equal — “perfectly” hexagonal — every cell fits tight with every other cell. Everybody can pitch in. That way, a honeycomb is basically an easy jigsaw puzzle. All the parts fit.

**Update:** I ran across this interesting paper, which argues the honeybee comb have a circular shape at first and then transform into the hexagon.

We report that the cells in a natural honeybee comb have a circular shape at ‘birth’ but quickly transform into the familiar rounded hexagonal shape, while the comb is being built. The mechanism for this transformation is the flow of molten visco-elastic wax near the triple junction between the neighbouring circular cells. The flow may be unconstrained or constrained by the unmolten wax away from the junction. The heat for melting the wax is provided by the ‘hot’ worker bees.

**Still Curious?** Learn more about The Honeycomb Conjecture.