Kepler’s New Year’s Gift — On the Six-Cornered Snowflake

Original Booklet: On the Six-Cornered Snowflake: A New Year’s Gift by Johannes Kepler 


Some things never change. In winter 1610, Johannes Kepler was stressing out about holiday gifts — in particular, one for his friend and benefactor, the rather grandly-named Johannes Matthaeus Wacker von Wackenfels. Kepler, at the time employed as Imperial Mathematician at the court of Holy Roman Emperor Rudolph II, records his musings on the problem in the opening pages of his now-famous discourse, The Six-Cornered Snowflake.

Kepler sets a high standard for himself. His gift should be of the intellectual variety: an amusing idea or a clever argument. After all, that’s why Wacker keeps him around. Kepler considers several potential topics, dismissing each in turn as being either too serious or too light. Kepler’s intellectual respect for Wacker —  who was an accomplished scholar and amateur philosopher in his own right — renders other topics off limits. (An example: given the impressive size of his patron’s zoological library, Kepler jokingly complains that writing a treatise about animals would be “like bringing owls to Athens.”) Wandering around town, feeling guilty about his procrastination, Kepler notices some snowflakes landing on his coat, “all with six corners and feathered radii” [1]. Kepler immediately identifies the perfect topic for his essay:

“‘Pon my word, here was something smaller than any drop, yet with a pattern; here was the ideal New Year’s gift… the very thing for a mathematician to give.”

The pattern Kepler alludes to here is the six-fold shape of the snowflake [2]. Considering this shape, Kepler alights on what will become the central puzzle of the piece,

“Our question is, why snowflakes in their first falling, before they are entangled in larger plumes, always fall with six corners and with six rods, tufted like feathers.”

In other words, why should snowflakes be six-sided, rather than five-sided, seven-sided, or anything-else-sided? Kepler’s attempts to answer this question are a treasure trove of condensed matter physics: they include the first observation in print that regular shapes can arise from close-packing of identical objects [3] and the famous conjecture that hexagonal packing is the densest way to fill space with spheres [4]. But perhaps the most interesting thing about The Six-Cornered Snowflake is how Kepler sees nature and how he wants the reader to see it.

Left: Kepler’s sketches of sphere packings. Middle: Thomas Bentley’s classic snowflake photographs, taken around 1902. Right: Pyramidal packing of colloidal spheres gives the beetle P. argus its characteristic iridescence. (Scale bar is 1 micron.)

Intellectually and personally, Kepler straddled the gradual transition away from the medieval era of alchemy and astrology [5], and towards the modern age of empirical observation, mathematical models, and testing ideas by experiment. Despite some asides that strike the modern reader as somewhat mystical in character — for instance his numerological obsession with the properties of the natural numbers — we can immediately recognize The Six-Cornered Snowflake as a scientific work. While the whole chain of reasoning is somewhat convoluted [6], The Snowflake includes the following:

1. Kepler imagines matter as being made of tiny, discrete “pieces” that are all identical to one another;

2. He considers (and draws!) how the arrangement of those pieces might influence the material properties of an object, in particular, its shape. In modern parlance, Kepler defines a crystal: a macroscopic object made out of identical pieces arranged into a regular structure called a lattice [7];

3. He tries to articulate a physical principle  — such as close-packing — that might explain why the small pieces arrange themselves in such a way as to produce a crystal lattice.

By Kepler’s own admission, none of his arguments adequately explain the shape of snowflakes. (For one thing, he can’t figure out how a three-dimensional process could possibly create two-dimensional crystals.) But, despite this failure, Kepler still manages to suggest a productive direction for future research. Noting that different substances crystallize into different 3D shapes [8], Kepler finishes his essay by kicking the problem over to another branch of the natural sciences: “I have knocked at the door of chemistry and see how much remains to be said before we can get hold of our cause.” In other words, Kepler correctly intuits that understanding the form of crystals necessitates understanding their chemistry.

Today, we know that macroscopic objects are indeed made of tiny identical pieces — atoms and molecules — and that those pieces often arrange themselves in structures that are highly reminiscent of Kepler’s sphere packings. We have learned how to accurately describe the forces that bind atoms together or push them apart. In addition to the shape of crystals, we know that many important material properties — most strikingly rigidity, the ability of a solid to resist deformation — arise because of the regular arrangement of atoms on the micro-scale. We understand (some) statistical physics, which explains how, at high enough temperature, thermal motion overcomes the forces holding the atoms in place, destroying the lattice and melting the solid. Our knowledge of the physics and chemistry of solids has allowed us to engineer with precision the technologies — in particular silicon-based semiconductors — that underpin the modern world.

Although Kepler couldn’t have begun to imagine all this, scientifically speaking, the world of The Snowflake is very modern: a world of material cause and material effect, of microscopic bodies in motion and in contact, a world of forces that are invisible yet comprehensible, and where the properties of the whole can be understood by considering its parts. As physics students, we learn that all the fundamental physical laws can be written on the back of a napkin. And yet, the materials in the world around us exhibit an amazing variety of properties: solid and liquid, conductive and insulating, magnetic and not. How can such a zoo of behaviors and properties arise from physical laws that are fundamentally simple? Kepler’s essay gives us a framework to understand the apparent contradiction. Kepler says: look inside. Look at the pieces. Look at the structure and the symmetry.

A pretty good New Year’s gift for a soft matter enthusiast, even in 2018.


[1] In today’s world of snowflake wrapping paper, snowflake ornaments, and snowflake emoji, it’s hard to imagine that there was a time when people didn’t know what snowflakes actually look like, but Kepler was apparently the first European to write about the hexagonal symmetry of snow crystals. (The observation is, however, recorded in much older Chinese documents, from the 2nd century BC.)

[2] In fact it isn’t: there are triangular snowflakes too, and how they form is still an active area of research.

[3] Prior to the publication of The Snowflake, English scientist Thomas Harriot privately communicated his ideas on the efficient stacking of cannon balls to Kepler.

[4] This conjecture was only formally proved in 2015, by a group of Mathematicians led by Thomas Hales.

[5] Kepler, who was at times employed as an astrologer, thought that astrology, as practiced in 17th century Germany, was mostly nonsense. However, he himself cast horoscopes that he claimed were correct and argued with scholars who wanted to dismiss the discipline entirely.

[6] Despite Kepler’s assurances in the introduction that his piece is “next to nothing,” The Snowflake is 21 pages long — evidence, I think, of the modern tyranny of page limits and copy editors.

[7] Interestingly, this argument correctly predicts the form of so-called “complex materials” like opal (colloidal silica), where the pieces really are (relatively) uniform hard spheres. The water molecules in an ice lattice have much more complicated, directional interactions arising from the hydrogen bonds between them, and so their crystal structure is harder to understand or predict. In fact, it seems that Kepler generally has something like colloidal particles in mind throughout The Snowflake, rather than modern atoms or molecules.

[8] “But the formative faculty of the earth does not take to her heart only one shape; she knows and is practiced in the whole of geometry. I have seen… a panel inlaid with silver ore; from it, a dodecahedron, like a small hazelnut in size, projected to half its depth, as if in flower.”

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