Professor Marcus du Sautoy on Thompson and Tits

Thompson and Tits win the Abel Prize 2008

In the early hours of the morning of the 30th of May 1832 a gunshot was heard ringing out across the fields in the 13th arrondissement in Paris. A peasant on his way to market, hearing the shot, ran towards the scene of the shooting. On the ground he discovered a young man writhing around in agony. It was obviously a duelling wound. The young man's name was Evariste Galois, a well-known revolutionary. He was taken to the local Cochin hospital where he died the next day in the arms of his brother. "Do not cry" he pleaded "it takes all my courage to die at the age of 20."

But it isn't for his contributions to revolutionary politics that Galois is now remembered. The young revolutionary had stayed up the whole night before the duel trying to explain a revolutionary mathematical breakthrough he'd made. Maybe the lack of sleep contributed to his bad aim that morning. But contained in the package he left behind him was the beginnings of a new language called group theory that would finally help mathematicians to articulate one of the most important concepts of nature: symmetry.

And it is for helping to complete an epic saga written in this language of symmetry that John Thompson and Jacques Tits are being rewarded with this year's Abel prize.

Symmetry is a central concept in both science and the arts. Symmetry is key to our evolutionary survival. It is an indicator of good genes. It controls the behaviour of molecules, crystals and viruses. It has unlocked the secrets of the fundamental particles that make up the material world. It is also central to many of the codes that are used to preserve the integrity of data as it is transmitted around the world. Such error-correcting codes use symmetry to reveal when errors have crept into the data. For artists too symmetry is a central theme. From architecture to music, from poetry to painting, symmetry underpins many of the structures used in the creative world. But it wasn't until the nineteenth century that we finally had the language to answer the question: what is symmetry?

That saga begins with the extraordinary revelation contained in Galois's manuscripts that just as molecules can be broken down into atoms like sodium and chlorine, or numbers can be built out of the indivisible primes, symmetrical objects too can also be decomposed into indivisible symmetrical objects. Christened simple groups, these symmetrical objects are the atoms of the world of symmetry. Galois's breakthrough meant that it might be possible to create a Periodic Table of Symmetry containing a list of all these simple groups of symmetry. Such a classification had the prospect to be as influential as the Periodic Table of elements has been to Chemistry. Prime numbers, the building blocks of all numbers, are behind some of the first simple groups to be feature in the classification.

For example, take a flat 15 sided shape or polygon. For mathematicians the symmetries of a shape are all the ways I can rearrange the shape and place it back down inside an outline of the shape so that the shape looks like it did before I moved it. Each symmetry of a shape is bit like a magic trick move: look away and I move the shape but when you look back it's as if the shape hasn't moved. For example I can rotate the 15-sided polygon by a 15 of a turn and it sits back down inside an outline of the shape.

Galois understood that the symmetries of this 15 sided polygon can be built out of the symmetries of two smaller shapes sitting inside the large shape, namely a pentagon and a triangle. How can you rotate the 15-sided polygon through a fifteenth of a turn using the rotations of the pentagon and triangle? First rotate the pentagon by two fifths of a turn; then pull back in the opposite direction by rotating the triangle by a third of a turn. The combined effect is a rotation of a fifteenth of turn. The reason this works is because: 1/15=2/5-1/3.

So the group of symmetries of the 15 sided polygon are built out of the symmetries of a pentagon and a triangle. But the rotations of these prime sided shapes cannot be broken down. So just as prime numbers are the building blocks of all numbers it turns out that prime sided shapes are some of the first building blocks of the world of symmetry.

One of the achievements that John Thompson is being recognized for with the award of the Abel prize is a stunning theorem he proved with the late Walter Feit. Together they proved that many symmetrical objects in the mathematical world can be built out of the symmetries of these prime sided shapes. Called the Odd Order Theorem, it states that if an object or structure has an odd number of symmetries then its symmetries can be broken down into the symmetries of these prime sided shapes.

It was an extremely influential paper. For the first time it gave mathematicians a belief that they could really construct a complete Periodic Table of Symmetry. The proof was impressive for other reasons. Published in 1963 it ran to 255 pages and took up the whole issue of The Pacific Journal of Mathematics. It was, at the time, possibly the longest proof that had ever been published.

Although Thompson's work revealed that these prime sided shapes were at the heart of many symmetrical shapes, there turned out to be other shapes that couldn't be broken down so easily. For example, take the classic football made up of pentagons and hexagons. This shape has 60 rotational symmetries or 60 magic trick moves which rearrange the shape so that all the pentagons and hexagons line up again. 60 is an even number so Thompson's theorem can't be used to break it down into prime sided shapes. But is there another way to break this shape down?

60 is a very divisible number. It is one of the reasons that the Babylonians used it as the base for their number system and why we have 60 minutes in the hour. But despite the high divisibility of the number, Galois proved that the 60 rotations of the football are as indivisible as if it were a prime sided shape. Sure, there are rotations of a pentagon sitting as subset of the symmetries of the football. But try to divide by the symmetries of one of the faces and the result makes no sense. There are no shapes whose symmetries can be combined with those of the pentagon to realise the symmetries of the football.

The symmetries of this shape turned out to be the tip of the ice-berg. There are many other shapes whose symmetries were indivisible but to create these shapes one has to move away from the physical world of 3 dimensions and enter the abstract world of hyperspace. For example the symmetries of a hypercubes give rise to a new family of indivisible shapes.

What do mathematicians mean by a cube in 4 dimensions? To play with such shapes, mathematicians needed to create a new language. In the seventeenth century, Descartes produced a dictionary which changed geometry into numbers. This dictionary is used by everyone who employs a map or negotiates a route with their SAT NAV. Every location on the surface of the earth can be translated into a pair of numbers which denote the distance east-west and north-south from the origin of this map located at Greenwich. So for example the GPS location of the Norwegian Academy of Sciences is (10.7, 55.9). A geometric position translated into numbers. If I wanted to locate my place in space rather than on a two dimensional surface I would need to use three numbers.

Using these 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). Mark these locations on a piece of graph paper and you've got the corners of a square. The corners of a cube are got by adding an extra dimension. So the eight corners of the cube can be described by the eight coordinates of numbers starting at (0,0,0), (1,0,0), (0,1,0) ... continuing to the extremal point at (1,1,1).

So what about a four dimensional cube? Although the pictures run out, the numbers don't. One side of this dictionary carries on. So a mathematician will describe a four dimensional cube as the object whose corners are given by the coordinates with 4 numbers starting at (0,0,0,0), (1,0,0,0) and stretching out to the furthest point at (1,1,1,1). Using the numbers I can explore the geometry and symmetry of this shape. So for example a 4 dimensional cube or what is known as a tesseract has 16 corners, 32 edges, 24 square faces and is constructed out of 8 cubes.

One can see shadows of these shapes in our 3 dimensional world. The Arch at La Defense in Paris is in fact a shadow of a 4 dimensional cube in 3 dimensions. Just as an artist will represent a 3D cube on the 2D canvas by drawing a square inside a square, the architect at La Defense has constructed a cube inside a cube to create a shadow of the hypercube. The symmetries of this shape turn out to be related to a new infinite family of indivisible symmetries to add to the prime sided shapes. They are called simple groups of Lie type after another Norwegian mathematician, Sophus Lie, who began his investigation of these groups while in prison in France after he'd been mistakenly arrested as a spy during the Franco-Prussian war.

The symmetries of hypercubes in higher dimensions turned out to be behind one of 16 new families of Lie groups. And it is unlocking the secrets of these groups for which the Belgian mathematician Jacques 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 new families.

The Periodic Table of symmetry was shaping up nicely ... except there were 5 symmetrical shapes discovered by a French mathematician called Emile Mathieu that didn't seem to fit into any of these nice families of groups but just seemed to sit there like orphans. One of these symmetrical objects can actually be heard as its symmetries are at the heart of Messiaen's piano piece Ile du Feu. (Although Messiaen threaded many mathematical themes through his work it is unlikely he was aware of just how mathematical sophisticated this piece of music is.) But were these 5 shapes the only exceptional shapes or were there more lurking out there in the shady corners of the mathematical world?

In 1965 Thompson received a letter from a Croatian mathematician called Zvonimir Janko claiming to have discovered a new indivisible symmetry, a sixth sporadic group. At first Thompson was quite dismissive of the claim but as he analysed Janko's proposal he realised that the Croatian could be on to something.

Janko's discovery turned out to be the beginning of a crazy period in the story of symmetry where 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 the fundamental particles in physics. A prediction about the existence of a new fundamental particle generally proceeds its first observation. Similarly mathematicians would often use Thompson's formula to predict for example an object with 604,800 symmetries before it was actually constructed. The construction would often depend on finding the right geometric setting to realise that number of symmetries.

Both Thompson and Tits are amongst those who have their names attached to some these new sporadic groups of symmetries that appeared over the decades since Janko's discovery. The culmination of this period of exploration was the discovery of a 26th object that could only be seen once you'd got to 196,883-dimensional space and has more symmetries than there are atoms in the sun. Called simply the Monster, it turned out to be largest of the sporadic groups. But we are finally coming to the realisation that there are no more indivisible symmetries to add to the periodic table of symmetry, the details of which are recorded in the Atlas of Finite Groups. It is thanks to the work of mathematicians like Thompson and Tits that we believe we now have a complete list of the building blocks of symmetry, one of the greatest achievement in the history of mathematics. It's now up to the next generation to explore what symmetrical objects we can build from these atoms of symmetry.