Updated: Mar 30 2007, 05:30am hrs

FOR more than a century mathematicians have known about Lie groups. These are families of shapes named after Sophus Lie, a Norwegian mathematician who discovered them. There are four simple families of Lie groups and five othersthis being mathematicsthat are not quite so simple.

The simplest member of the simplest Lie group is the circle, which looks the same however it is rotated. Its higher-dimensional cousin, the sphere, has the same properties, only more so, and is thus the second-simplest of the same family. The five non-simple groupsdubbed exceptional in their complexity and symmetryare harder to envisage and, for almost 120 years, the details of the most intricate of these have lain beyond reach. This week a group of mathematicians led by Jeffrey Adams of the University of Maryland announced that they had completed a map of the largest and most complicated one, a structure known to mathematicians as E{-8}.

Lie groups have two defining features: surface and symmetry. A sphere has two surface dimensions. In other words, any place on its surface is defined by just two numbers, the longitude and the latitude. But it has three dimensions when it comes to symmetries. A sphere can spin on an axis that runs, say, from north to south, or on each of two axes placed at right angles to this. E{-8} is rather more difficult to visualise. Its surface has 57 dimensionsthat is, it takes 57 co-ordinates to define a point on it, and it has 248 axes of symmetry.

Grappling with such a structure is as tricky as it sounds. But Dr Adamss team decided to have a go. They want to create an atlas of maps of the Lie groups.

This involves making a description in the form of a matrix for each structure. (A matrix is a multi-dimensional array of numbers, such as that found in a sudoku puzzle.)

Dr Adams and his colleagues began by writing a computer program that would generate such matrices, a task that took them more than three years. It transpired that they needed 453,060 points to describe E{-8} but that they also needed to express the relationship between each of these points. That meant they had to devise a matrix with 453,060 rows and the same number of columns. In total this gives 205 billion entries. To complicate things further, many of these entries were not merely numbers but polynomialssequences in which a given number is raised to a series of different powers, for example its square and its cube.

Processing such a vast quantity of data was beyond the capacity of even modern supercomputers, so the team were forced to tinker with the problem to make it tractable. This tinkering led them to a piece of ancient maths known as the Chinese remainder theorem.

This theorem is contained in a book written in the late third-century AD by a mathematician called Sun Tzu (not to be confused with the military strategist of the same name). It is used to simplify large calculations by breaking them down into many smaller ones, the results of which can then be recombined to generate the answer to the original question.

One problem addressed in the original book concerns counting soldiers. Sun Tzus solution was that the soldiers should first split into groups of three, then groups of five, then groups of seven, with the number unable to join a group (in other words, the remainder) being noted each time. The three remainders can then be used to calculate how many soldiers are present.

For example, if two were left over from the groups of three, three left over from the groups of five and two left over from the groups of seven, there would have been 23 soldiers in the unit (or possibly 233, but the difference should be obvious to even the stupidest commanding officer).

The researchers worked out how to use the remainder theorem to bring their calculation within the capacity of a supercomputer called Sage, which spent more than three days crunching the numbers to generate the map of E{-8}. Not content with letting the supercomputer do all the arithmetic, the mathematicians simultaneously jotted down some calculations of their own on the back of an envelope. They worked out that if each entry in the matrix were written on paper that was one inch square, the answer would cover an area the size of Manhattan.

**And the point is**

Apart from the satisfaction of mapping E{-8} at long last, mathematicians are pleased because the structure keeps popping up in another branch of intellectual endeavour: string theory. This purports to be the best explanation of the universe beyond the Standard Model of physics that describes all known particles and forces, but which is generally acknowledged to be incomplete. String theory requires that the universe has many more dimensions than those that are obvious, but that most of these extra dimensions are too small to be discerned with todays equipment. One of the ways in which they can be hidden involves E{-8}, so having a mathematical map of its structure could be handy. Cheaper, too, than building a particle accelerator the size of the solar system.

* The Economist Newspaper Limited 2007*