Wednesday, September 27, 2006

Speaking of snowflakes

Why the six-pointed symmetry?

The first person to ask the question seriously was the astronomer Johannes Kepler in a delightful little book called The Six-Cornered Snowflake, published in 1611. All around him Kepler observed beautiful shapes in nature: six-pointed snowflakes, the hexagonal honeycombs of bees, the twelve-sided shape of pomegranate seeds. Why? he asks. Why does nature display such mathematical perfection?

We might add: Why does the stuff of the universe arrange itself into spiral galaxies, planetary ellipses, double-helix DNA, rhomboid crystals, five-petaled flowers, the rainbow's arc? Why the five-fingered, five-toed, bilateral symmetry of the newborn child? Why?

Kepler struggles with the problem, and along the way he does a pretty neat job explaining why pomegranate seeds have twelve flat sides (squeeze spheres into the smallest volume and that's what you get) and why the bee's honeycomb has six sides (because that's the way to make honey containers with the least amount of wax). His book is a tour-de-force of playful mathematics.

In the end, Kepler admits defeat in understanding the snowflake's six points, but he thinks he knows what's behind it all, behind all of nature's beautiful forms: A universal spirit pervading and shaping everything that exists. He calls it nature's facultas formatrix, or "formative capacity."

We would be inclined to say that Kepler was just giving a fancy name to something he couldn't explain. To the modern mind, facultas formatrix sounds like empty words.

We can do rather better with snowflakes than Kepler. We explain the general hexagonal form by invoking the shape of water molecules, and we explain the shape of water molecules with the laws of quantum physics. But the perfect six-fold symmetry? As a snowflake grows, adding water molecules essentially at random, how does one point know what is going on at another point? On the scale of molecules, the faces of the growing crystal are light-years apart. You will find theories out there -- forced "tiling", exquisitely sensitive vibrations, that sort of thing -- but I haven't seen anything yet that is totally convincing.

And we are probably no closer than Kepler to answering the ultimate questions: What is the reason for the curious connection between nature and mathematics? Why are the mathematical laws of nature one thing rather than another? Why do natural forms exist at all?

Maybe facultas formatrix is as good a name as any to cover our ignorance.