Thursday, December 22, 2011

400 years of snowflakes

Here is the pre-edited version of my In Retrospect piece for Nature celebrating the 400th anniversary of Kepler’s seminal little treatise on snowflakes.

Did anyone ever receive a more exquisite New Year’s gift than the German scholar Johannes Matthäus Wackher von Wackenfels, four hundred years ago? It was a booklet of just 24 pages, written by his friend Johannes Kepler, court mathematician to the Holy Roman Emperor Rudolf II in Prague. The title was De nive sexangula (On the Six-Cornered Snowflake), and herein Kepler attempted to explain why snowflakes have this striking hexagonal symmetry. Not only is the booklet charming and witty, but it seeded the notion from which all of crystallography blossomed: that the geometric shapes of crystals can be explained in terms of the packing of their constituent particles.

Like Kepler, Wackher was a self-made man of humble origins whose brilliance earned him a position in the imperial court. By 1611 he had risen to the position of privy councillor, and was a man of sufficient means to act as Kepler’s some-time patron. Sharing an interest in science, he was also godfather to Kepler’s son and in fact a distant relative of Kepler himself. It is sometimes said that Kepler’s booklet was in lieu of a regular gift which the straitened author, who frequently had to petition Rudolf’s treasury for his salary, could not afford. In his introduction, Kepler says he had recently noticed a snowflake on the lapel of his coat as he crossed the Charles Bridge in Prague, and had been moved to ponder on its remarkable geometry.

Kepler came to the imperial court in 1600 as an assistant to the Danish astronomer Tycho Brahe. When Tycho died the following year, Kepler became his successor, eagerly seizing the opportunity to use Tycho’s incomparable observational data to deduce the laws of planetary motion that Isaac Newton’s gravitational theory later explained.

Kepler’s analysis of the snowflake comes at an interesting juncture. It unites the older, Neoplatonic idea of a geometrically ordered universe that reflects God’s wisdom and design with the emerging mechanistic philosophy, in which natural phenomena are explained by proximate causes that, while they may be hidden or ‘occult’ (like gravity), are not mystical. In Mysterium Cosmographicum (1596) Kepler famously concocted a model of the cosmos with the planetary orbits arranged on the surfaces of nested polyhedra, which looks now like sheer numerology. But unlike Tycho, he was a Copernican and came close to formulating the mechanistic gravitational model that Newton later developed.

Kepler was not by any means the first to notice that the snowflake is six-sided. This is recorded in Chinese documents dating back to the second century BCE, and in the Western world the snowflake’s ‘star-like’ forms were noted by Albertus Magnus in the thirteenth century. René Descartes included drawings of sixfold stars and ice ‘flowers’ in his meteorological book Les Météores (1637), while Robert Hooke’s microscopic studies recorded in Micrographia (1665) revealed the elaborate, hierarchical branching patterns.

“There must be a cause why snow has the shape of a six-cornered starlet”, Kepler wrote. “It cannot be chance. Why always six? The cause is not to be looked for in the material, for vapour is formless and flows, but in an agent.” This ‘agent’, he suspected, might be mechanical, namely the orderly stacking of frozen ‘globules’ that represent “the smallest natural unit of a liquid like water” – not explicitly atoms, but as good as. Here he was indebted to the English mathematician Thomas Harriot, who acted as navigator for Walter Raleigh’s voyages to the New World in 1584-5. Raleigh sought Harriot’s expert advice on the most efficient way to stack cannonballs on the ship’s deck, prompting the ingenious Harriot to theorize about the close-packing of spheres. Around 1606-8 he communicated his thoughts to Kepler, who returned to the issue in De nive sexangula. Kepler asserted that hexagonal packing “will be the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container.” This assertion about maximal close-packing became known as Kepler’s conjecture, which was proved using computational methods only in 1998 (published in 2005) [1].

Less commonly acknowledged as a source of inspiration is the seventeenth-century enthusiasm for cabinets of curiosities (Wunderkammern), collections of rare and marvelous objects from nature and art that were presented as microcosms of the entire universe. Rudolf II had one of the most extensive cabinets, to which Kepler would have had privileged access. The forerunners of museum collections, the cabinets have rarely been recognized as having any real influence on the nascent experimental science of the age. But Kepler mentions in his booklet having seen in the palace of the Elector of Saxony in Dresden “a panel inlaid with silver ore, from which a dodecahedron, like a small hazelnut in size, projected to half its depth, as if in flower” – a showy example of the metalsmith’s craft which may have stimulated his thinking about how an emergent order gives crystals their facets.

Yet despite his innovative ideas, in the end Kepler is defeated by the snowflake’s ornate form and its flat, plate-like shape. He realizes that although the packing of spheres creates regular patterns, they are not necessarily hexagonal, let alone as ramified and ornamented as that of the snowflake. He is forced to fall back on Neoplatonic occult forces: God, he suggests, has imbued the water vapour with a “formative faculty” that guides its form. There is no apparent purpose to the flake’s shape, he observes: the “formative reason” must be purely aesthetic or frivolous, nature being “in the habit of playing with the passing moment.” That delightful image, which touches on the late Renaissance debate about nature’s autonomy, remains resonant today in questions about the adaptive value (or not) of some complex patterns and forms in biological growth [2]. Towards the end of his inconclusive tract Kepler offers an incomparably beautiful variant of ‘more research is needed’: “As I write it has again begun to snow, and more thickly than a moment ago. I have been busily examining the little flakes.”

Kepler’s failure to explain the baroque regularity of the snowflake is no disgrace, for not until the 1980s was this understood as a consequence of branching growth instabilities biased by the hexagonal crystal symmetry of ice [3]. In the meantime, Kepler’s vision of crystals as stackings of particles informed the eighteenth-century mineralogical theory of René Just Haüy, the basis of all crystallographic understanding today.

But the influence of Kepler’s booklet goes further. It was in homage that crystallographer Alan Mackay called his seminal 1981 paper on quasicrystals ‘De nive quinquanglua’ [4]. Here, three years before the experimental work that won Dan Shechtman this year’s Nobel prize in chemistry, Mackay showed that a Penrose tiling could, if considered the basis of an atomic ‘quasi-lattice’, produce fivefold diffraction patterns. Quasicrystals showed up in metal alloys, not snow. But Mackay has indicated privately that it might indeed be possible to induce water molecules to pack this way, and quasicrystalline ice was recently reported in computer simulations of water confined between plates [5]. Whether it can furnish five-cornered snowflakes remains to be seen.

1. Hales, T. C. Ann. Math. 2nd ser. 162, 1065-1185 (2005).
2. Rothenberg, D. Survival of the Beautiful (Bloomsbury, New York, 2011).
3. Ben-Jacob, E., Goldenfeld, N., Langer, J. S. & Schön, G. Phys. Rev. Lett. 51, 1930-1932 (1983).
4. Mackay, A. L. Kristallografiya 26, 910-919 (1981); in English, Sov. Phys. Crystallogr. 26, 517-522 (1981).
5. Johnston, J. C., Kastelowitz, N. & Molinero, V. J. Chem. Phys. 133, 154516 (2010).

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