Kepler's seasonal gift

In 1611 Johannes Kepler, the German cosmologist with a passion for geometry, offered his patron, John Matthew Wacker, a highly seasonal “New Year's gift”. Unlike most such gifts, it not only outlasted the festive season but also acquired enduring renown. Kepler's present was his small book, On the Six-Cornered Snowflake. It was a sprightly and speculative expression of delight and perplexity in the face of the marvellous regularity of snow crystals.

Kepler provided no illustration, as his patron well knew what a hexagon looked like. But Kepler was unaware of the astonishing variety within the crystalline structures, which was to be revealed by the microscope and illustrated in Robert Hooke's Micrographia in 1665, as shown here.

The account by Kepler of what triggered his thoughts on the snowflake gives a nice idea of his book's tone: “Specks of snow fell here and there on my coat, all with six corners and feathered radii. ‘Pon my word, here was something smaller than any drop, yet with a pattern; here was the ideal New Year's gift for the devotee of Nothing’.” The jest that his patron was a “devotee of Nothing” may be a philosophical play on the fact that the fundamental unit of pure geometry, the point, is ‘nothing’, as it has no physical dimension.

Kepler tries to identify the cause of the snowflake's six-cornered configuration. But before doing so, he notes the ubiquity of polygonal and polyhedral constructions in nature, from the cosmos to the world in miniature. He is delighted to observe how the polyhedral architecture of the snowflake has analogies across all scales of natural design. He had himself argued that the geometry of the five ‘platonic solids’, or regular polyhedra, provided a kind of invisible armature for the orbits of the planets (see Nature 393, 123; 1998). On a smaller scale, the rule of geometry is manifest in the world of plants, Kepler points out: “in a flower the authentic flag of this faculty is flown, the pentagon”. The classic case is the bee's honeycomb: “The architecture is such that any cell shares not only six walls with the six cells in the same row, but also three plane surfaces on the base with three other cells from the contrary row.”

Such geometrical packing, which Kepler compares to the seeds in a pomegranate, is attributed to a physical process similar to that observed when pellets are systematically compressed in a round vessel. But the regularity of the physical actions must themselves be caused by something, and Kepler concludes that God prescribed to the bee “these canons of its architecture” that resulted in the geometrical constructions.

But what lies behind the form of the snowflake, when no animate architect is involved? Observation and delight are one thing; explanation, beyond the adducing of divine agency, is very much another. Kepler's review of a range of physical explanations for the forms of the snowflake proves frustratingly inconclusive. He is convinced that “the cause of the six-sided shape of a snowflake is none other than that of the ordered shapes of plants and numerical constants”. In the absence of any obvious agent, he adduces the presence of a “formative faculty” — the facultas formatrix — which God has insinuated into matter in terms of “world-building figures”. But, in the final analysis, Kepler senses that something is missing in his explanation, and he concludes by announcing that he has “knocked at the door of chemistry”.