Is nature a mathematician? Patterns and geometry are everywhere. But nature seems to have a particular thing for the number 6. Beehives. Rocks. Marine skeletons. Insect eyes. It could just be a mathematical coincidence. Or could there be some pattern beneath the pattern, why nature arrives at this geometry? We’re going to figure that out… with some bubbles. And some help from our favorite mathematician: Kelsey, from Infinite Series. Happy to help. [OPEN] A bubble is just some volume of gas, surrounded by liquid. It can be surrounded by a LOT of liquid, like in champagne, or just a thin layer, like in soap bubbles. So why do these bubbles have any shape at all? Liquid molecules are happier wrapped up on the inside, where attraction is balanced, than they are at the edge. This pushes liquids to adopt shapes with the least surface.
In zero g, this attraction pulls water into round blobs. Same with droplets on leaves or a spider’s web. Inside thin soap films, attraction between soap molecules shrinks the bubble until the pull of surface tension is balanced by the air pressure pushing out. It’s physics! Physics is great, but mathematics is truly the universal language. Bubbles are round because if you want to enclose the maximum volume with the least surface area, a sphere is the most efficient shape. Yeah. That’s another way of putting it. What’s cool is if we deform that bubble, the pull of surface tension always evens back out, to the minimal surface shape. This even works when soap films are stretched between complex boundaries, they always cover an area using the least amount of material.
That’s why German architect Frei Otto used soap films to model ideal roof shapes for his exotic constructions. Now let’s see what happens when we start to pack bubbles together. A sphere is a three-dimensional shape, but when when we pack bubbles in a single layer, we really only have to look at the cross-section: a circle. Rigid circles of equal wdiameter can cover, at most, 90% of the area on a plane, but luckily bubbles aren’t rigid. Let’s pretend for a moment these bubbles were free to choose any shape they wanted. If we want to tile a plane with cells of equal size and *no* wasted area, we only have three regular polygons to choose from: triangles, squares, or hexagons. So which is best? We can test this with actual bubbles. Two equal-sized bubbles?
A flat intersection. Three, and we get walls meeting at 120˚. But when we add a fourth… instead of a square intersection, the bubbles will always rearrange themselves so their intersections are 120˚, the same angle that defines a hexagon. If the goal is to minimize the perimeter for a given area, it turns out that hexagonal packing beats triangles and squares. In other words, more filling with fewer edges. In the late 19th century, Belgian physicist Joseph Plateau calculated that junctions of 120˚ are also the most mechanically stable arrangement, where the forces on the films are all in balance. That’s why bubble rafts form hexagon patterns.
Not only does it minimize the perimeter, the pull of surface tension in each direction is most mechanically stable. So let’s review: The air inside a bubble wants to fill the most area possible. But there’s a force, surface tension, that wants to minimize the perimeter. And when bubbles join up, the best balance of fewer edges and mechanical stability is hexagonal packing. Is this enough to explain some of the six-sided patterns we see in nature? Basalt columns like Giant’s Causeway, Devil’s Postpile, and the Plains of Catan form from slowly cooling lava. Cooling pulls the rock to fill less space, just like surface tension pulls on a soap film.
Cracks form to release tension, to reach mechanical stability, and more energy is released per crack if they meet at 120˚. Sounds pretty close to the bubbles. The forces are different, but it’s using similar math to solve a similar problem. What about the facets of insect’s eye? Here, instead of a physical force, like in the bubble or the rock, evolution is the driver. Maximum light-sensing area? That’s good for the insect, but so is minimizing the amount of cell material around the edges. Just like the bubbles, the best shapes are hexagons. What’s even cooler, if you look down at the bottom of each facet?? There’s a cluster of four cone cells, packed just like bubbles are. Bubbles can even help explain honeycomb. It would be nice to imagine number-crunching bees, experimenting with triangles and squares and realizing hexagons are most efficient balance of wax to area… but with a brain the size of a poppy seed? They’re no mathematicians. It turns out honeybees make round wax cells at first.
And as the wax is softened by heat from busy bees, it’s pulled by surface tension into stable hexagonal shapes. Just like our bubbles. You can even recreate this with a bundle of plastic straws and a little heat. So is nature a mathematician? Some scientists might say nature loves efficiency. Or maybe that nature seeks out the lowest energy. And some people might say nature follows the rules of mathematics. However you look at it, nature definitely has a way of using simple rules to create elegant solutions. Stay curious. So that’s how nature arrives at the optimal solution for three-dimensional bees, but you know mathematicians love to take things to the next level.