Win a chance to name one of the new mathematical symmetry groups mentioned in this feature.
OSLO, May 2008. King Harald of Norway presents mathematicians John Thompson and Jacques Tits with the Abel prize, one of the highest accolades in mathematics. There is a pleasing symmetry at the heart of this year’s award. The winners are being honoured for ground-breaking work that led to the completion of a project started by Niels Abel, the 19th-century Norwegian mathematician after whom the prize is named. Appropriately enough, that project concerns mathematicians’ attempts to answer the question: what is symmetry?
Most people’s response is to point to the left-right reflectional symmetry of the human face. Or a flower, or a snowflake. But a snowflake has additional symmetries to that of a human face: as well as looking at its two halves, you can also turn a snowflake 60 degrees to match up its shape again. This begins to get at the essence of what symmetry is – a transformation or move that you can do to a structure which somehow makes it look like it did before you moved it. So how many other types of symmetry are there?
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Remarkably, we now have a definitive answer. Thompson, of the University of Florida in Gainesville, and Tits, of the Collège de France in Paris, are responsible for ideas that have culminated in what is essentially a “periodic table” of symmetry. It has been as influential in the world of symmetry as the periodic table of elements has been to chemistry, allowing anyone exploring the complicated mathematical symmetries of an object to reduce it to something far simpler.
“The periodic table of symmetry is as influential in mathematics as its namesake in chemistry”
That matters because the symmetries of a structure often reveal secrets about how it behaves. For chemists, symmetry is key to classifying the possible crystals that can exist; in biology the mechanism of a virus owes much to its symmetrical shape; even the menagerie of fundamental particles revealed by physicists’ super-colliders only make sense when you start to see them as facets of some strange, higher-dimensional symmetrical shape. And much of the technology we take for granted, such as mobile phones and the internet, depends on codes that exploit symmetry to preserve data as it is transmitted around the world.
Symmetry has fascinated civilisations since ancient times too, but it wasn’t until the 19th century that we developed the language to understand its mathematics. This language allowed us to pull symmetry apart and discover its basic building blocks.
Just as molecules can be broken down into atoms like sodium and carbon, or numbers can be built out of the indivisible primes such as 3, 5 and 7, the mathematicians of Abel’s generation discovered that symmetrical objects can be decomposed into indivisible symmetrical objects. Christened “simple groups”, they are the atoms of symmetry.
Abel’s contemporaries discovered that prime numbers are behind some of the first simple groups. Take a 15-sided polygon, for example. Its symmetries can be built from the symmetries of a pentagon and a triangle sitting inside the shape. To see how this works, imagine rotating the polygon by one-15th of a turn. Another way to do this is to first rotate the pentagon by two-fifths of a turn; then pull back in the opposite direction, rotating the triangle by one-third of a turn (see Diagram). The reason this works is because 1/15 = 2/5 – 1/3.
In fact the symmetries of any flat, regular polygon can be broken down into the symmetries of the prime-sided shapes which fit inside the larger shape. For example, since 105 = 3 × 5 × 7, the symmetries of a 105-sided figure are built from the symmetries of a triangle, a pentagon and a heptagon.
The Abel prize rewards Thompson for a stunning theorem he proved with the late mathematician Walter Feit, showing that many more symmetrical objects in the mathematical world can be built out of prime-sided shapes. The beauty of their proof is that it applies to a plethora of shapes beyond simple 2D polygons. However complicated the shape, just knowing that it has an odd number of symmetries is enough to show that it could be pulled apart.
Thompson and Feit’s theorem was impressive not just because it was a massive stepping stone to understanding the world of symmetry but also because the proof itself was massive. Called the odd order theorem, the 1963 paper describing it ran to 255 pages and took up an entire issue of the Pacific Journal of Mathematics (vol 13, p 775). At the time, it was possibly the longest proof that had ever been published.
So prime-sided polygons are the first indivisible symmetrical objects in the mathematician’s periodic table of symmetry, but they are not the only ones. More exotic shapes were uncovered by 19th-century mathematicians when they tried to crack one of the other big problems of the day.
They knew of formulae that allowed them to work out solutions to equations involving x2, x3 or x4. But they could not find a formula for solving “quintic” equations that include x raised to the fifth power, such as x5 + 6x + 3 = 0.
Abel discovered the reason why a formula was so hard to pin down was that there wasn’t one. A young French mathematician called Évariste Galois was then able to push Abel’s ideas a step further and give some basis as to why this could be. His revolutionary realisation was that behind every equation there is a symmetrical object.
The first hint of symmetry at work is evident in simple equations such as x2 = 4. Its two solutions, x = 2 and x = -2, are in some ways a mirror image of each other. A cubic equation has three solutions, and these are connected by the symmetries of a triangle. Once you get to quartic equations like x4 – 5x3 – 2x2 – 3x – 1 = 0, the four solutions are connected by the symmetries of a tetrahedron, the 3D shape made from piecing together four equilateral triangles.
What Galois discovered is that if the symmetries of the object behind the equation can be broken down into prime-sided shapes, then a formula for finding the equation’s solutions does exist. This unexpected connection with solving equations was the first indication that symmetry could be the key to unlocking many questions that did not at first sight seem to have anything to do with the concept.
When Galois began to examine quintic equations, he discovered that the symmetrical object at their heart is the dodecahedron, the 3D shape built from 12 pentagons. And the reason there is no formula for solving quintic equations is because the rotational symmetries of the dodecahedron cannot be broken up into prime-sided shapes.
There are 60 different ways to spin a dodecahedron so that all the pentagonal faces line up as they did before it was spun: in the language of mathematics, a dodecahedron has 60 different rotational symmetries. Even though 60 is a highly divisible number, Galois proved that the group of 60 rotations of the dodecahedron are as indivisible as if the shape were prime-sided.
Try to break the group of symmetries of the dodecahedron apart by using the rotations of one of its pentagonal faces and the result makes no sense. You can turn the object by one-fifth of a turn about a face, but there are no other shapes whose symmetries can be combined with those of the pentagon to build the symmetries of the dodecahedron.
Having discovered that the dodecahedron has so much in common with prime-sided shapes such as triangles and heptagons, the hunt was then on to find all indivisible shapes.
Mathematicians began to move away from physical objects and turned instead to more abstract structures. Remarkably, they found that shuffling a deck of cards behaves very much like the rotations or reflections of a physical shape. To understand how this works, start off by imagining a tetrahedron. Its rotational symmetry means that there are 12 ways of placing it on its triangular base so that it looks the same, as well as 12 reflectional symmetries. Now imagine sticking a jack, queen, king and ace of spades on the faces. When you rotate the tetrahedron, the movements are equivalent to shuffling the deck of cards. In fact, the symmetries of a tetrahedron can be modelled by the shuffles of a deck of four cards, which can take any one of 24 possible combinations. Similarly, the 60 symmetries of a dodecahedron are intimately related to the shuffles of five cards. What is powerful about this approach is that even though there is a limited number of three-dimensional shapes, we can keep on exploring symmetries by adding cards to the deck.
By changing their perspective and moving from 3D shapes to packs of cards, mathematicians discovered that the dodecahedron is not an isolated shape but the beginning of a new infinite family of indivisible symmetrical structures to add to the periodic table of symmetry alongside the prime-sided shapes.
Monster of the deep
So far, so good. But even greater rewards lie in navigating beyond the third dimension and into hyperspace. The symmetries of these higher-dimensional shapes are the key to unlocking the behaviour of fundamental particles and building the standard model of particle physics. Symmetries are also behind many of the fundamental conservation laws in physics, as the German mathematician Emmy Noether discovered in 1915.
The reason mathematicians can manipulate objects in hyperspace is because of Descartes’s dictionary, which turns geometry into numbers. Just as every position on the globe can be specified by two map coordinates, we can translate shapes into numbers. A square, for example, can be described by the coordinates of its corners: (0,0), (1,0), (0,1) and (1,1). Add an extra coordinate and you add a dimension, so you can then specify the eight corners of a cube as (0,0,0), (0,0,1) and so on.
So what about a four-dimensional cube? Although the pictures run out the numbers don’t, and this allows us to explore the geometry and symmetry of this shape. A four-dimensional cube is known as a tesseract and has 16 corners, 32 edges, 24 square faces and is constructed out of eight three-dimensional cubes. Its symmetries turn out to be related to another family of indivisible symmetries called simple groups of Lie type, after another Norwegian mathematician, Sophus Lie.
The symmetries of “hypercubes” are behind one of 16 new families of Lie groups. And it is unlocking the secrets of these groups for which the Belgian-born mathematician Tits is being recognised with the award of the Abel prize. Tits constructed geometrical settings in higher dimensions which help explain the symmetries of these families.
There are more indivisible symmetries to add to the periodic table, but the other groups aren’t as well behaved as the Lie groups, shuffles or prime-sided shapes. At the end of the 19th century, a French mathematician called Emile Mathieu had discovered five indivisible symmetries that didn’t seem to fit into any of these patterns, nor did they create a family of their own. They just seemed to be sitting there like orphans. Were these five the only exceptional groups of symmetries, or what mathematicians called sporadic groups?
In 1965, Thompson received a letter from the Croatian mathematician Zvonimir Janko, who claimed to have discovered a sixth sporadic group. At first Thompson was quite dismissive of the claim, but as he analysed Janko’s proposal he realised the Croatian could be onto something.
Janko’s discovery turned out to be the beginning of a crazy period in the story of symmetry when mathematicians discovered a whole range of strange indivisible sporadic groups of symmetry that didn’t seem to fit any of the patterns determined by previous generations. Many of the discoveries depended on using a formula developed by Thompson to predict how many symmetries such a sporadic group might have.
Often the birth of these sporadic groups mirrored the discovery of fundamental particles in physics. By exploiting the symmetries underlying the standard model of particle physics, theorists predicted the existence of particles such as the charm quark several years before experiments found evidence for them. Similarly, mathematicians used Thompson’s formula to predict objects before they were actually constructed.
Thompson and Tits are among those who have their names attached to some of these sporadic groups. The culmination of this period of exploration was the prediction by German mathematician Bernd Fischer of an object that can only be seen from 196,883-dimensional space and has more symmetries than there are atoms in the sun. Robert Griess, a mathematician at the University of Michigan in Ann Arbor, eventually constructed the object in 1980.
Called simply “the monster”, it is the largest of the sporadic groups. Far from being some anomalous freak with no relation to reality, we are beginning to realise that the symmetries of the monster might actually underpin some of the deepest ideas of string theory – currently our best hope of uniting relativity and quantum physics.
We are finally coming to the realisation that the monster was the last: there are no more indivisible symmetries to add to the periodic table of symmetry. In what many regard as one of the greatest achievements of mathematics, we now have a complete list of the building blocks of symmetry. It is the power of mathematical proof that we can be so sure that the list is complete, but it is thanks to the work of mathematicians like Thompson and Tits that we are able to produce such a definitive answer. It’s now up to the next generation to explore what symmetrical objects we can build from these atoms of symmetry.
