You don't really need much math to see why this is happening. First, the bonds in a benzene ring aren't discrete like we draw them, alternating between single bonds and double bonds. It's also important to realize that although we typically represent benzene in 2D all molecules really have a 3D geometry.
Electron orbitals can overlap in different ways depending on the geometry of the atom and its electronics. See this for a picture: http://en.wikipedia.org/wiki/File:Benzene_Representations.sv...
So, the electrons in a benzene ring really form more of a cloud around the entire ring. You'd expect this to pull the atoms into a perfect circle with the carbon atoms all being equidistance from each other, all else being equal.
However, each carbon atom also has a hydrogen atom attached to it. So now you have a sort of a circle with 6 "strings" attached at points equidistant around the circle all pulling outward, perpendicular to the circle.
Imagine a perfectly circular piece of string with 6 strings attached equidistantly around the circle. You apply an equal force perpendicular to the surface of the circle.
Hopefully you can see how this would result in the original circular string being "deformed" into a hexagon.
It's a far leap from there to say why hexagons are "so common in nature." Are they? Relative to what? I don't know that any of this has anything to do with the shape of that storm you linked to.
> It's a far leap from there to say why hexagons are "so common in nature."
I was thinking of things (compared to other geometric shapes) like the storm, honey bee cells (honeycombs), basalt columns [1], turtle shells (although irregular), and a common snowflake shape.
There are three regular polygons you can use to tile a plane: triangles, squares, and hexagons. The regular hexagonal packing is the densest sphere packing in the plane, so any time you have objects constrained to a plane which for the sake of maximizing or minimizing some force want to be equidistant from each other you'll get something close to a hexagon.
I'll add that sometimes a shape like that might result from a more evolutionary process. In a 2D plane a circle is the structure which most equally distributes force, so it's the shape most able to hold up under pressure.
But a tile of circles isn't so fortunate. Of all the possible tilings, the hexagonal tiling holds up the best precisely because it's the densest sphere packing in the plane.
Other arrangements might appear, but over the course of time you're more likely to see hexagonal tilings since those are the ones that best survive external forces.
It has to do with the bond angles in water molecules. The bond angles are, in turn, determined by quantum mechanical wave functions. These quantum mechanical wave functions apply to all of chemistry, including benzene rings. So the shapes of snowflakes and the shapes of benzene rings are not totally independent events.
The short answer for why benzene rings are common is aromaticity, which makes it a very stable structure. Rings with fewer than 6 members are uncommon in chemistry, because the angles are not what the bonds naturally want to be and so they are increasingly unstable.
As for nature in general, you could probably come up with a convincing argument that boils down to: 6 is a nice round number. It has 2 and 3 as factors.
>Rings with fewer than 6 members are uncommon in chemistry
In chemistry, or in nature? five membered rings show up all over the place, both aromatic and otherwise. Granted, cyclobutyl (4 member, square) and cyclopropyl (3 membered, triangle) suffer from ring strain and are uncommon, but 5, 6, 7, (or higher) rings show up all over the place.
Examples off the top of my head are the cyclopentadienyl ion pervasive in inorganic chemistry (see ferrocene, et al) and amino acids tryptophan, tyrosine, histidine, and phynylalanine all feature cyclic aromatics 5 and 6 membered, as well as proline with a non-aromatic 5 member ring.
The takeaway point is that although ring-strain (having non ideal angles (120 or 109.5 degrees)) increases the internal energy of the molecule (destabilizing it), other factors, such as aromaticity, which decrease internal energy (stabilizing it) may balance or exceed the ring-strain, still giving a stabilized, if non-ideal geometry.
(But yeah, 3, 4 membered rings, ugh. Look up platonic alkanes for some really crazy strain angles.)
I probably learned it in some chem course, and later forgot as all my math and science got applied to business :(