The E8 root system consists of 240 vectors in an 8-dimensional space - vectors are the vertices (or corners) of an 8-dimensional object called the Gosset polytope 421. In the 1960s, Peter McMullen drew by hand a 2-dimensional representation of the Gosset polytope 421 - this is a computer generated version of that drawing.

Credit: John Stembridge/American Institute of Mathematics

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WASHINGTON: A theoretical structure in 248 dimensions has been successfully constructed by mathematicians, resolving a 120-year puzzle that can be used to test theories about the structure of the cosmos.

Mathematicians and computer scientists from the United States and Europe said they had mapped 'E8', a problem that was first proposed in 1887 but has had to wait until the era of supercomputers and Internet-linked research groups to resolve.

E8 is the mother of all so-called Lie groups - a category of problems invented by a 19th-century Norwegian mathematician, Sophus Lie (pronounced 'Lee'), to explore symmetry.

Spheres, cylinders or cones are familiar examples of simple, symmetrical objects in three dimensions. But E8 is a piece of geometric origami that comes in 248 dimensions.

"(E8) is as complicated as symmetry can get," said David Vogan, a mathematics professor at the Massachusetts Institute of Technology (MIT), who took part in the massive calculation. "Mathematics can almost always offer another example that's harder than the one you're looking at now, but for Lie groups, E8 is the hardest one."

Resolving E8 was a gigantic undertaking, the scientists said. They compared it to the Human Genome Project, which patiently unraveled the human genetic code. The human genome is less than a gigabyte in size, but the E8 calculation is 60 gigabytes, enough to store 45 days of continuous music in MP3 format. The paper print-out of the calculations would cover an area the size of Manhattan island, New York.

The reason was that the input - the E8 equations themselves - was comparatively small but the answer itself was enormous.

"This groundbreaking achievement is significant both as an advance in basic knowledge, as well as a major advance in the use of large-scale computing to solve complicated mathematical problems," said Jeffrey Adams, project leader and mathematics professor at the University of Maryland.

A low-definition picture of E8, released by MIT, showed something like a multicoloured circus tent made, like a child's constructor set, of densely-packed, connected rungs.

"We can never hope to represent the structure in its entirety, it's a mathematical abstraction," said Dutch researcher Marc van Leeuwen, of France's University of Poitiers. "You can make some nice pictures with it, but a sheet of paper has only two dimensions, so you will never see the real object."

It took four years to produce the E8 calculation. Unusually - since mathematics is often a solitary activity - the achievement came through close collaboration of a number of international groups, mixing theoretical mathematics and intricate computer programming on both sides of the Atlantic.

"The literature on this subject is very dense and very difficult to understand," explained Vogan. "Even after we understood the underlying mathematics, it still took more than two years to implement it on a computer."

One of the biggest headaches was finding a computer big enough to crunch the calculation. For a whole year, the team tried to slim down the calculation a thousand-fold, trying to make it more efficient so that it could fit on existing supercomputers. Even so, the calculation was still beyond the current generation of computing power.

Just as the team was despairing of seeing their work realised, one of them came up with a way to break up the calculation in such a way that parts of it could be digested in separate batches.

The results from each batch were then assembled to give the ultimate solution, which took 77 hours to run on a U.S. supercomputer called Sage. A press statement said the E8 breakthrough was "a very
important advance" in physics, for it could be used to test a key theory about the fundamental symmetries in nature.

Among the mooted symmetries it may help unravel is the structure of the cosmos, created by the Big Bang some 13.7 billion years ago, and basic particles themselves, said Hermann Nicolai, director of the Albert Einstein Institute in Potsdam, Germany.

The 18-member research team, called Atlas, included mathematicians from France's universities of Poitiers and Lyon. A paper presenting the research results was tabled on Monday; it was entitled "The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness".