Johannes Kepler Illustration of Johannes Kepler and the geometry of snow

What a mineral shows the observer of snow

Johannes Kepler (1571–1630), a German mathematician and astronomer, wanted a New Year’s gift that would weigh next to nothing. A snowflake suited him perfectly. Its outline presented a puzzle that could be inspected again on the very next flake. Why did six directions keep recurring when the length and branching of the rays could change? At one point in his search for an answer Kepler turned to a piece of ore.

An objection from Dresden

But the earth’s shaping power does not confine itself to one figure. It knows the whole of geometry and is practised in it.

– Johannes Kepler, Strena seu de nive sexangula, 1611, p. 23.[1]

Kepler recalls something he had seen with his own eyes in Dresden. A solid with twelve faces projected from a piece of silver-bearing ore. It was about the size of a small hazelnut. The comparison is easy to pass over. Yet it turns a figure that one might draw on paper into a particular thing that once lay before an observer. On that piece of ore nature had made a different figure from the one that could be picked out on a winter coat.

What a counterexample can do

Kepler considers several possible explanations for the shape of snow. It is an attractive idea that hexagons can cover a surface without leaving gaps. A snow star with thin, widely separated arms, however, hardly looks like a filled surface. Nor can a general shaping power of the earth by itself explain the choice of six. The twelve-faced solid from Dresden suffices to show why. Kepler’s delight in the variety of figures comes through particularly clearly at the point where an elegant idea runs up against that unassuming piece of ore.