Christmas Trilogy 2013 Part III: A New Year’s Offering.

Johannes Kepler wrote more than eighty books and pamphlets covering a wide range of mathematical and scientific topics. One of the most fascinating is the pamphlet he wrote as a New Year’s offering at the beginning of 1611. We’ll let Kepler introduce it for us:

The highly regarded Court Councillor of His Imperial Majesty, Herr Johannes Matthäus Wacker von Wackenfels, Golden Knight …, Supporter of the Sciences and Philosophy, my Gracious Benefactor.

Yes, I know that especial you love nothingness; however certainly not because of its slight value but rather much more because of the joyful and charming games, which one can, with lively jest, play with this word. It is easy for me to fancy that a present for you is all the more desirable and welcome the closer it approaches nothingness.

Wacker von Wackenfels
Aegidius Sadeler

Johannes Matthäus Wacker von Wackenfels (1550 – 1619) was lawyer, diplomat, humanist scholar and courtier, who having worked his way up the greasy pole of Renaissance absolutist court politics had, since 1597, been a member of one of the highest legal councils at the imperial court of Rudolph II, the Holy Roman German Emperor and Kepler’s employer. Wacker was an intelligent and well educated and widely read humanist scholar and Kepler’s closest friend at Rudolph’s court and the two Johanneses very much enjoyed chewing the intellectual cud with each other. It was Johannes Wacker, for example, who first brought Kepler the news of Galileo’s telescopic discoveries. It was good manners in those time for friends to give presents to each other at New Year and the pamphlet of which I have quoted the very flowery opening paragraphs was Kepler’s New Year’s offering to his friend Wacker in 1611.

This opening is followed by two pages of the various forms of nothingness that Kepler knows his friend to already possess. We then arrive at the core of Kepler’s offering to his friend:

As I went over the bridge deep in thought and full of worry and annoyed about my poverty, that is to come to you without a New Year’s offering, always following the same thoughts, to present this nothingness, or to find something that come closest to it, and exercised the astuteness of my thoughts on it, by chance the water vapour thickened through the cold to snow, and single small snowflakes fell on my coat, all were six-cornered with feathered spokes. Yes, by Heracles, that’s it, yes a phenomenon, smaller than a drop, and thereto of regular form. Yes, that is the wished for New Year’s offering for a friend of nothingness! Just as snow falls from the heavens and looks like the stars, so it is also suitable as the present of a mathematician who has nothing and receives nothing. Now quickly bring the present to my benefactor, as long as it exists and hasn’t through body warmth disappeared into nothingness.

Snowflake photo
Alexey Kljatov
The Atlantic Dec 4 2013

Here we have the subject of the pamphlet already expressed in its title, Strena seu de Nive sexangula, in English, New Year’s Offering or The Six-Cornered Snowflake. From here Kepler sets out to investigate the question, why are snowflakes six-cornered?

What follows is a rambling, at time fascinating, at others delightful discourse not just on six-cornered snowflakes but also the hexagonal cells of a honeycomb, the shape of pomegranate seeds, the arrangement of peas in a pod, the regular Platonic solids, the semi-regular Archimedean solids, three and six petaled flowers and various other things. Kepler discusses the tiling of planes and the filling of spaces. As one aspect of the latter he considers the best way to stack canon balls to occupy the least space. He had discussed this subject in his correspondence with Thomas Harriot who had been presented with this highly practical problem by his patron and employer, Sir Walter Raleigh. Kepler’s suggested solution, for which he could offer no proof, entered the history of mathematics as Kepler’s conjecture. Hilbert included it as problem eighteen in his famous list of twenty-three unsolved mathematical problems in 1900. The American mathematician Thomas Hales finally produced a generally accepted proof of Kepler’s conjecture that relies on a computer in 1998. Hales started on a more formal version of his proof, which he estimates will take twenty years, in 2003. Great oaks do truly from little acorns grow!

The Kepler conjecture makes The Six-Cornered Snowflake an important document in the history of mathematics. This is however not its only claim to scientific fame. Although comparatively primitive it is considered the first published scientific work in the discipline of crystallography.

But what of Kepler’s question, why is the snowflake six-cornered? In the end after all his considerations and diversions he is forced to admit defeat and acknowledge that he is unable to produce a satisfactory answer to his own question.

In the seventeenth century Kepler was not the only natural philosopher to consider the snowflake. René Descartes turned his attention to them in his Discourse on the Method

Sketch of snow crystal
René Descartes

As did Robert Hooke in his microscopical investigations, which you can read about here.

Hooke’s snowflakes

The first person to successfully photograph snowflakes was Wilson Alwyn “Snowflake” Bentley (February 9, 1865 – December 23, 1931).

Snowflakes
Wilson Bentley