Scientists have solved one of the toughest problems in mathematics, performing a calculation to figure out the symmetry of a complicated 248-dimensional object known as the Lie group E8. The solution is so large that it would take days to download over a standard Internet connection.

All Things Considered
March 20, 2007

Lie groups were invented in the 19th century by the Norwegian mathematician Sophus Lie, to express the symmetry of three-dimensional objects like spheres, cones and cylinders.

The final result of the E8 calculation is a matrix containing 453,060 rows and columns. There are 205,263,363,600 entries in the matrix, each of which is a polynomial.

If each entry of the matrix were written in a one-inch square, the resulting grid would measure more than 7 miles on each side. That means that the 120-year-old math problem yields an answer so large that it would cover a piece of paper the size of Manhattan.

The research into figuring out how the shape known as E-8 works was funded by the American Institute of Mathematics, based in Palo Alto, Calif.

"E-8 is a giant, mysterious, very symmetrical object, maybe the most symmetrical object in the entire universe," says Brian Conrey, the institute's director.

"It has 248 dimensions, which sounds pretty frightening at first. But really, you think of the dimensions as being degrees of freedom."

The dimensions, he says, can be viewed as being variable attributes.

Conrey says that mathematicians are thrilled that his institute has cracked E8. They expect the solution to propel future advances in science and technology — though Conley admits that he's not sure how that will happen.

"I don't know if it'll make a smaller hard drive, or make your cell phone have a clearer signal or show up in electronic gadget anytime soon," he says. "But I wouldn't be surprised if it did, at some point."

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