In Matt Parker’s recent video he looks into some interesting properties of the shapes commonly found in nature in bee’s hives:
He brings up an interesting point in that the hexagons that bees make are not necessarily built to be that way, but that when crunched together circles may make the hexagon or hexagonal angles, the proven most-efficient 2d packing. Remember that a circle (a water droplet or bubble for example) has the greatest area enclosed for the least perimeter – some interesting proofs shown here. However in 3d-space there is a more effective packing he shows, and an odd foam 3d space is more effective but non workable for this problem. And as seen in experiments the bees are not calculating it but pushing it together at off angles such that it happens to go to that shape.
It makes you wonder why the 2d shapes are naturally pushed toward the optimal, though the 3d shape when left to itself or pushed together does not go to the optimal shape packing? For that matter, a large droplet of water doesn’t grow to a sphere, but when water is added goes from a circle to an off-circle shape and runs off rather than grow in the third dimension. An interesting thought.