Multi-dimmensional: A 2D drawing of an 8D object that stands as a root of the 57D object called E8, and its 248D symmetries.American Institute of Mathematics/Peter McMullen

A map of one of the strangest and most complex entities in mathematics should be a powerful new tool for both mathematicians and physicists pursuing a unified theory of space, time and matter.

The strange 'thing' that has been mapped is a 'Lie group' called E8 — a set of maths that describes the symmetry of an (unimaginable to most) 57-dimensional object.

The creation of this map, which took 77 hours on a supercomputer, resulted in a matrix of 453,060 ? 453,060 cells, containing more than 205 billion entries — "all related in intricate and complex ways", says Jeffrey Adams, the project leader and a mathematician at the University of Maryland.

This represents 60 gigabytes of data, enough data to store 45 days of MP3 music files, or fill a piece of paper the size of Manhattan (about 60 square kilometres). The human genome takes up 1 gigabyte.

The finished product is essentially a database of information, which should come in very handy to theoretical physicists tackling grand unified theories of everything. "Now that it's done, mathematicians and physicists can use the results very easily," says Ian Stewart of the University of Warwick, UK. Adams agrees: "It's going to be a fabulous tool."

Weird exception

A Lie group is a collection of mathematical descriptors that help to illustrate the symmetry of a smooth object. The Lie group for a sphere, for example, describes all the mathematical operations that can be performed on the sphere without changing its appearance. There are an infinite number of straightforward Lie groups. But there are also five 'exceptional groups': weird one-offs of which E8, discovered in 1887, is one.

It gets stranger: E8, which represents the symmetries of a particular 57-dimensional object, has 248 dimensions itself.

"It's perhaps the most beautiful structure in all of mathematics, but it's very complex," says physicist Hermann Nicolai of the Max Planck Institute for Gravitational Physics in Potsdam, Germany.

Adams's team spent two years working out how the problem could be rendered in a form that wouldn't overwhelm the memory of a computer. The rest of the time was taken writing the code and testing the map, probing the mathematical properties of different regions to see if they provided the expected answer.

"The calculation was known to be possible in principle, but it was thought to be hopeless in practice," says Adams. "But four years ago a group of us said let's really try to do it. We're pretty sure we've got it right, but it's hard to be 100% sure."

"It's probably one of the most complicated pure mathematical calculations anyone's ever done," says Stewart. "Each entry is difficult to calculate — it's amazing they managed to do this."

Balls and string

Besides pure mathematicians, the people most familiar with E8 are physicists, and they might get the most out of the new map.

The mathematics of symmetry lies at the heart of both relativity and quantum physics. String theorists trying to unify these two areas are casting around for a type of symmetry that will let them deal with the troublesome extra dimensions thrown up by their models.

"A unified theory needs unique mathematics," says Nicolai. "What we'd like is a structure with very special properties. E8 has a flavour of this, although we don't know how the symmetry is realized in physical theory — we have to study it in more detail."

"Nobody knows what pieces of mathematics string theorists are going to need, but this will be an important piece of the toolkit," agrees Stewart. "It gives a better chance of making new and unexpected predictions."

The map will be included in the Atlas of Lie Groups and Representations, and will be available online at www.liegroups.org. The researchers also plan to publish their methodologies in a scientific journal.

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