Snowflakes, Symmetry and String Theory

by Nicholas Mee on February 12, 2022

There is a YouTube video to accompany this blog article on The Cosmic Mystery Tour channel. (Please don’t forget to click the subscribe button. Subscribing is free.)

This video is about the great German astronomer Johannes Kepler who was a pivotal figure in the birth of science. Kepler is regarded as the father of modern astronomy. He was a devout and profoundly spiritual man who had a lifelong drive to understand the structure of Creation. Kepler believed that geometry and symmetry are built into the fundamental architecture of the universe. Many of his ideas are as relevant today as they were 400 years ago. Kepler’s intuition that the universe has a geometrical blueprint has proved to be very fruitful and remains at the heart of physics even today.

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 scientist. He never hid his mistakes and modestly discussed all the blind alleys he explored before reaching his final conclusions.

Kepler is most famous for his laws of planetary motion. He deduced that the orbit of each planet around the Sun is shaped like an ellipse, and he found rules for how fast the planets travel around their orbits. These advances in astronomy laid the foundation for the Scientific Revolution that came to fruition with the work of Isaac Newton. Kepler also wrote the first modern treatise on the science of optics and was the first to understand how the lens of the eye focuses an image on the retina.

In 1611 Kepler was invited to the New Year celebrations of his friend Johannes Mathaeus Wacker von Wackenfels, privy councillor to Rudolf II, the Holy Roman Emperor. Kepler later recalled, while musing over a suitable New Year’s gift for his friend,

‘by 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 rays. Here was something smaller than any drop yet with a pattern. It 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.’

Unable to preserve a delicate and beautiful snow crystal as a 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.

The Six-Cornered Snowflake

Kepler realised that although the exact shape of each snow crystal is different they all display the same hexagonal symmetry, and 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 rock crystals, the seed cases of pomegranates and bees’ honeycombs. Kepler concluded that the symmetrical shape of snowflakes and other crystals is due to the regular arrangement of the atoms from which they are formed. This was almost 300 years before scientists had even established the existence of atoms. But Kepler’s remarkable insight has proved to be fundamentally correct.

Kepler went on to consider the most compact way to pack collections of equal-sized spheres. In particular, he asked what is the maximum number of such spheres that will fit around a central sphere? Mathematicians refer to this number as the kissing number. It is easy to answer the equivalent question in two dimensions. With seven coins of the same denomination, we can see that six will fit exactly around the seventh, so the maximum kissing number is 6. Coins or circles can be arranged in this way to cover the entire plane. This is the densest possible packing of circles in a plane.

A space-filling honeycomb of octahedra and cuboctahedra.

Kepler believed that the spheres in the densest packings in three-dimensional space have a kissing number of 12. Such packings include those traditionally used by grocers to stack apples and bombardiers to stack cannonballs. This is also how atoms are packed together in many metals and other crystalline solids. So it is rather surprising that a definitive mathematical proof that these packings are indeed the densest did not arrive until 1998 when American mathematician Thomas Hales proved Kepler’s conjecture, as it is known.

Kepler attempted to understand the structure of the universe in terms of elegant mathematics and regular geometrical figures. Kepler’s belief in divine symmetry has proved to be a valuable insight into the laws of the universe. Everywhere that physicists have looked they see symmetry in the structure of physics, from Einstein’s theories of relativity to the standard model which is our best theory of particle physics and the structure of matter. But, whereas the symmetry of a snowflake is obvious to the eye, often these deeper symmetries can only be expressed in terms of abstract higher-dimensional geometries.

And this brings us back to Kepler’s sphere-packing problem. In 2016 the Ukrainian mathematician Maryna Viazovska proved that the E8 lattice gives the densest packing of spheres in 8-dimensional space. Life in eight dimensions can be quite cosy – this packing gives the spheres a kissing number of 240. Viazovska’s proof was then extended to show that the Leech lattice gives the densest possible packing in 24-dimensional space, where the spheres have a kissing number of 196,560.

Although you might not have heard of E8, it turns up in string theory, which is the only serious candidate that we have for an ultimate theory of physics. String theorists speculate that the symmetry group E8 might provide the ultimate description of the fundamental particles and the forces that act on them.

A string wrapped around a torus.

String theory is a theory in which each type of fundamental particle is a different sort of vibration of one fundamental entity – the string – so when the string vibrates in one way we might see it as an electron or if it vibrates in another way we might see it as a quark or a photon. A curious feature of string theory is that it only works in 10-dimensional spacetime. To describe the physics of the real universe string theorists assume that six of the nine spatial dimensions are curled up so tightly we are not aware of them. Even so, these extra dimensions might determine the properties of the particles and forces we see in particle accelerator experiments. According to string theorists the six extra dimensions form a shape known as a Calabi-Yau manifold. This animation is a three-dimensional projection of one of these shapes known as a quintic hypersurface.

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

String theory remains an intriguing but unproven approach to physics so this is all highly speculative. As yet there is no experimental evidence to connect string theory to the structure of the real universe so no one knows whether it is in the same class as some of the brilliant insights of Johannes Kepler or whether it will eventually be dismissed just like some of his other curious and wacky ideas.

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