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The E8 root system consists of 240 vectors in an eight-dimensional space. These vectors are the corners of an eight-dimensional object called the Gosset polytope 4 21 represented here in two dimensions (Image: John Stembridge, based on a drawing by Peter McMullen)Related Articles

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Mapping E8

Jeffrey Adams, University of Maryland

Lie group, Wikipedia

A fiendishly complicated mathematical challenge has finally been conquered by mathematicians.

The team has exhaustively explored an esoteric 248-dimension structure called E8 and the results take up 60 gigabytes of data. If written out in tiny print, the results would cover an area the size of Manhattan.

“E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood,” says the team leader Jeffrey Adams from the University of Maryland in College Park, US.

E8 (pronounced E-eight) is an example of a so-called Lie group. A Norwegian mathematician invented Lie groups in the 19th century to study symmetry. A Lie group underlies objects like balls, cylinders or cones that are symmetrical when rotated by small amounts.

Tough unpacking

Mathematicians take these descriptions to wild extremes by imagining the 3D objects in myriad dimensions. The group E8 encapsulates the symmetries of a geometric object like a sphere, cylinder or cone, but in 57 dimensions. E8 itself has 248 dimensions.

The tough job for mathematicians was to explore this structure, effectively unpacking all the information about E8 – the catalogue of objects it can act on and how it acts.

“It’s a mathematical entity that we know exists, but we had to explore its inner structure,” says Hermann Nicolai, a mathematical physicist at the Albert Einstein Institute in Potsdam, Germany, who was not involved in the work. “It’s a bit like making a plan of a complicated building, or exploring an ancient pyramid to see how it was built.”

Unified theory

Adams and 17 other researchers solved the problem in a four-year project using a supercomputer at the University of Washington in Seattle. Their resulting map of E8 contains 60 gigabytes of information (see more on their website).

“This is an impressive achievement,” said Nicolai. He adds that the unique structure of E8 might help in the quest for a unified theory of gravity and the other forces in nature. This is because the underlying symmetries of the unified theory, if it really exists, will have to be complicated and unique.

“It will require an extremely special structure, mathematically speaking,” says Nicolai, “and E8 is an example of a symmetry that might fit the bill.”

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# 248 dimension object

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