Why is honeycomb shaped like hexagons?

This excerpt from “The Accidental Universe: The World You Thought You Knew” by

Alan Lightman answers this question.

“Each cell of a honeycomb is a nearly perfect hexagon, a space with six identical and equally spaced walls. Isn’t that surprising? Wouldn’t it be more plausible to find cells of all kinds of shapes and sizes, fitted together in a haphazard manner? 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: equilateral triangles, squares, and hexagons. Any gaps between cells would be wasted space. Gaps would defeat the principle of economy. Now you might ask why the sides of a cell in a beehive need to be equal in length. It is possible that each cell could have a random shape and unequal sides and the next cell could then be custom made to fit to that cell, without gaps. And so on, one cell after another, each fit to the one before it. But this method of constructing a honeycomb would require that the worker bees work sequentially, one at a time, first making one cell, then fitting the next cell to that, and so on. This procedure would be a waste of time for the bees. Each insect would have to wait in line for the guy in front to finish his cell. If you’ve ever seen bees building a beehive (or watched a video of bees on YouTube), they don’t wait for one another. They work simultaneously. So the bees need to have a game plan in advance, knowing that all the cells will fit together automatically. Only equilateral triangles, squares, and hexagons will do. But why hexagons? Here unfolds another fascinating story. More than two thousand years ago, in 36 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. And the smallest total perimeter, or smallest total length of sides, means the smallest amount of wax needed by the bees to construct their honeycomb. 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 had made only a conjecture. Astoundingly, Varro’s conjecture, known by mathematicians as the Honeycomb Conjecture, was proven only recently, in 1999, by the American mathematician Thomas Hales. The bees knew it was true all along. There’s more to the bee story. Bees are related to the question of why flowers have so much symmetry. Bees need flowers for their food and for making wax, and flowers need bees for pollination. Experiments published in 2004 by researchers at the Freie Universität in Berlin and the CNRS Université Paul-Sabatier in Toulouse show that bees are more attracted to flowers with symmetry. And why are bees attracted to flowers with more symmetry? The same researchers propose that symmetrical stimuli from the flowers are more easily processed by the visual system in the bee brain—that is, they require less neurological apparatus. Again, the principle of economy at work."