Snowflakes and Symmetries Strena seu de niue sexangula

Symmetry should clearly be at the heart of new classification schemes in the near-infrared (Figure 171). Near-infrared observations show an ubiquity of one and two-armed spirals; three armed spirals are rare; we have not yet observed any spiral galaxy with four, five or six spiral arms of old stars.

Symmetry was much on Kepler's mind, in his Latin monograph Strena seu de niue sexangula published by Godfrey Tampach in 1611. The English title is: A New Year's Gift, or On the six-cornered snowflake. Kepler's introduction recounts that on crossing the Karlsbrucke bridge (built in 1352) over the Vltava river, he noticed a star from heaven, a snowflake, fall on the lapel of his coat. Kepler decided to give it (or rather, his speculations on it) to his patron, Counsellor Wacker of Regensburg, as a gift. He asks the question: Why are snow-flakes six sided? Why do they possess a six-fold symmetry? This led Kepler to focus on the facultas formatrix (formative faculty) or morphogenetic field.

A slight diversion here, but a most interesting one. Imagine filling a large container with small equal-sized spheres. The density of the arrangement is the proportion of the volume of the container which is taken up by the spheres. In order to maximize the number of spheres

in the container, one needs to find an arrangement with the highest possible density, so that the spheres are packed together as closely as possible. Experiment shows that dropping the spheres in randomly will achieve a density of approximately sixty-five percent. However, a higher density can be achieved by carefully arranging the spheres as follows:

Let us start with a layer of spheres in a hexagonal lattice, then place the next layer of spheres in the lowest points one can find above the first layer, and so on - exactly as oranges, for example, are stacked in a shop. This natural method of stacking the spheres creates one of two similar patterns called cubic close packing and hexagonal close packing. Each of these two arrangements has an average density just over seventy-four percent. The Kepler conjecture says that this is the best that can be done - no other arrangement of spheres has a higher average density than this:

Coaptatio fiet arctissima, ut nullo praeterea ordine pluresglobuli in idem vas compingi queant.

which translates as follows:

This packing is the closest possible, and with no other stacking more spheres can be brought together in the same room.

In terms of this "packing problem," Kepler was over 350 years ahead of his time; extensive and exhaustive computer calculations by Professor Thomas Hales showed that Kepler was correct; the Kepler conjecture is now very close to becoming a theorem.

Writing in the British journal Nature, Professor Ian Stewart at the University ofWarwick comments: "Kepler thought his way to a version of the atomic lattice of a crystal: a tightly packed assembly of identical units ... Only in 1998 did Thomas Hales prove that Kepler was right."

The marvel of one snowflake, with its six-sided symmetry! What lessons to be learnt from the wondrous world of the microcosm. We need to understand why Nature so largely favors two-armed spirals of old stars. Why two-fold symmetries?

Each snowflake so minute - each, a star from heaven in its own right. Lessons from the microcosm, the small.

The words of Francis Bacon ring in our ears:

The eye of the understanding is like the eye of the sense; for as you may see great objects through small crannies or holes, so you may see great axioms of nature through small and contemptible instances.

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