Image: https://www.shutterstock.com/g/Bobkov+Evgeniy

Back in 1611, Johannes Kepler, court mathematician to the Holy Roman Emperor Rudolf II, found himself flat broke in the lead-up to the holiday season. (Apparently Rudolf didn’t pay his court all that regularly.) As he wandered across the Charles Bridge in Prague, he noticed a delicate snowflake attached to the lapel of his coat and marvelled at its geometric beauty. Why does the snowflake always appear ‘six cornered’?, he wondered. Never five or eight or twelve. The theory he then developed became the foundation of the field of crystallography, proposing that the geometric shapes of crystals is determined by the most efficient way for molecules to arrange themselves.

Stop and think about that for a second …. Kepler came up with a theory about how molecules arrange themselves almost 200 years before the notion of atoms forming molecules was even formulated! He wrote his musings down in the 24-page booklet “*De nive sexangula*”, which he presented to his friend Johannes Matthäus Wackher von Wackenfels as a New Year’s gift. (Anyone feeling strapped for cash right now? Now that’s a priceless homemade gift!)

“Cannonballs, Whitepoint Gardens, Charleston, SC” by Martin LaBar is licensed under CC BY-NC 2.0

The crux of Kepler’s argument was that the six-sided hexagonal shape arises because of how the ‘smallest natural units of a liquid like water’ (i.e. molecules before we called them that) arrange themselves to fill the space most efficiently. This idea of how molecules arrange themselves in close-packed configurations was inspired by earlier work from an English mathematician Thomas Harriot, who had advised Raleigh on how to stack cannonballs on the ship’s deck in the most efficient manner: hexagonal, like the snowflakes.

Harriot shared his cannonball-stacking analysis with Kepler, who immediately saw the connection to the symmetric crystals decorating his coat, suggesting that the six-sided structure comes about naturally since “in no other arrangement could more pellets be stuffed into the same container”. Now here is another truly mind-blowing part to this story: Kepler’s conjecture from 1611 was not formally proved until almost 400 years later by mathematician Thomas Hales and collaborators, with the formal proof eventually published by Forum of Mathematics, Pi, in 2017!

Hexagons are found just about everywhere you care to look in nature, not just in wintertime on your lapel, like honeycombs made by bees, a collection (“raft”) of bubbles on a frothy pond surface, or the compound eyes of insects. If you want to cover a flat surface as efficiently as possible with objects that all have the same shape and size, you have three options: equilateral triangles, squares, and hexagons. Hexagons often come out as the winner because they require the least amount of side length to contain a given area.

For example, let’s think about an equilateral triangle, a square, and a hexagon, each with a total area of 100 cm^{2}. For a square, that means that each side is 10 cm long, for a total perimeter/wall length of 10 + 10 + 10 + 10 = 40 cm. For an equilateral triangle, the area is calculated as the side length squared multiplied by the square root of 3, all divided by 4. This tells us that an equilateral triangle of area 100 cm^{2} has a side length of 15.2 cm. The total perimeter here is then 15.2 + 15.2 + 15.2 = 45.6 cm. Lastly, the naturally-preferred hexagon. The area for such a shape is calculated as the side length squared multiplied by the factor 3/2 times the square root of 3. (This is the same thing as six-times the area of an equilateral triangle, since a hexagon is the same thing as six close-packed equilateral triangles.) For a total area of 100 cm^{2}, the hexagon has a side length of only 6.2 cm. Adding these up for the perimeter gives us a total wall length of 37.2 cm, the smallest of these three shapes.

It takes energy to build a honeycomb structure. None other than Charles Darwin noted that the observed shape made sense from the evolutionary perspective, declaring that the hexagonal honeycomb is “absolutely perfect in economizing labor and wax.” Once again, Mother Nature, you are brilliant!

## References

In retrospect: On the Six-Cornered Snowflake, Philip Ball *Nature* volume 480, page 455 (2011)

On the Six-Cornered Snowflake, Kepler’s Discovery

THE SIX-CORNERED SNOWFLAKE (*De nive sexangula*) By Johannes Kepler, Translated by Jacques Bromberg, March 11, 2010 Paul Dry Books ISBN – 10:1589880536

A proof of the Kepler conjecture, Thomas C. Hales et al, Forum of Mathematics, Pi , Volume 5 , 2017 , e2

How Physics Gives Structure to Nature by Philip Ball, April 2016, for Nautilus