Snowflakes are Dancing!

by Nicholas Mee on November 22, 2017

The father of modern astronomy Johannes Kepler

Johannes Kepler (1571-1630) is the father of modern astronomy and a key figure in the dawn of science. He was a devout and profoundly spiritual man who studied theology and was possessed by a life-long drive to understand the structure of Creation. He believed that geometry and symmetry lay at the heart of the architecture of the universe.

Kepler is most famous for deducing that the orbits of the planets around the Sun are shaped like ellipses. His advances in astronomy laid the foundation for the scientific revolution brought about by Isaac Newton. Kepler also wrote the first modern treatise on the science of optics and was the first to understand how the eye’s lens projects an image on the retina.

Science was in its infancy in Kepler’s time and Kepler’s books are a mixture of deep and valuable insights mingled with ideas that today seem fanciful and very strange indeed. But in many ways Kepler was the ideal modern scientist. He never hid his mistakes and modestly discussed all the blind alleys that he explored before reaching his final destination.

Falling Starflakes

In 1611, Kepler received an invitation to join the New Year’s celebrations of his friend Johannes Matthäus Wackher von Wackenfels, privy counsellor to Rudolf II, the Holy Roman Emperor. Kepler later recalled how, one winter’s day while he was musing over a suitable New Year’s gift for his friend,

“by a happy chance water-vapour was condensed by the cold into snow, and specks of down fell here and there on my coat, all with six corners and feathered radii. Upon 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 since it comes down from heaven and looks like a star.”

Snowflake by Ken Libbrecht

Unable to preserve a beautiful and intricate snowflake as his present, Kepler decided to write a little booklet about the snowflake and its symmetry in honour of his friend. This booklet is called De Niva Sexangular (‘The Six-Cornered Snowflake’). Kepler realised that although the exact shape of the icy filigree of each snowflake is different, they all display the same hexagonal symmetry. It was the origin of this symmetry that intrigued him. In the booklet, he looked for clues in the geometrical structure of other familiar symmetrical objects such as crystals, bees’ honeycombs and the seed cases of pomegranates. Remarkably Kepler came to the correct conclusion that the symmetrical shape of crystals is due to the regular arrangement of the atoms from which they are formed. This was almost 300 years before the existence of atoms became established.

Star Polyhedra

In his quest for the secrets of the universe Kepler investigated the properties of regular geometrical shapes, such as the stellated polyhedra. There are four regular stellated polyhedra and they are known today as the Kepler-Poinsot polyhedra. (Louis Poinsot was a 19th century French mathematician.)  These polyhedra are shown in the animation below.

The four regular stellated polyhedra, also known as the Kepler-Poinsot polyhedra.

Divine Symmetry

Kepler’s belief in divine harmony has proved to be an invaluable insight into the laws of the universe. Symmetry is at the core of all modern theories of fundamental physics. This is true of Einstein’s theories of relativity and it is true of the Standard Model – our best theory of particle physics and the structure of matter. The deeper physicists look the more profound the symmetries appear to be. Indeed, all approaches to developing a more complete theory of the universe are based on the incorporation of even grander and more subtle symmetries. But where the symmetry of a snowflake or regular polyhedron is obvious to the eye, these grand new theories are constructed around abstract higher dimensional jewels.

The best candidate for an ultimate theory describing all the fundamental particles and all the forces that act on them is string theory. It is an approach to physics based on fundamental one-dimensional entities known as strings, rather than point particles. If string theory is correct, then all the fundamental particles are different modes of vibration of a single type of object – the string. For instance, when a string vibrates in one way we might see it as an electron and when it vibrates another way we might see it as a quark or a photon.

Higher Dimensional Jewels

One of the curious features of string theory is that it only works if there are ten dimensions of space and time. In order to describe the physics of the real universe, string theorists have devised cunning ways in which six of the nine spatial dimensions are curled up so tightly that we are not aware of them. Even so, the shape of these extra dimensions determines the properties of the particles and forces that we do see in particle accelerator experiments. According to string theorists the six extra dimensions form a shape known as a Calabi-Yau manifold. The animation below shows a computer generated image of a three-dimensional projection of one of the Calabi-Yau manifolds that they have studied.

Projection of a six-dimensional Calabi-Yau manifold known as a quintic hypersurface.

As yet string theory remains an intriguing, but unproven, approach to physics that has captivated many of the world’s best mathematicians and physicists for several decades. So far there is no experimental evidence to show that it genuinely describes the way our universe is constructed. No-one knows whether it falls into the same class as the brilliant insights of Johannes Kepler or whether it will end up in the same category as his other curious and whacky ideas.

More Information

A video version of this post is now available on The Cosmic Mystery Tour YouTube Channel here: The Cosmic Mystery Tour. Please don’t forget to click the subscribe button.

For further information about Kepler and his search for symmetry within the structure of matter see my book Higgs Force: Cosmic Symmetry Shattered.

Arthur Koestler’s classic book The Sleepwalkers includes a wonderful account Kepler’s life and research into astronomy.

The animation of the stellated polyhedra was originally published many years ago on my POLYTOPIA multimedia CD-ROM. More information is available about POLYTOPIA at the following link:

I produced the animation of the Calabi-Yau manifold for the Henry Moore and Stringed Surfaces Exhibition held at the Royal Society from April to July 2012:

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