Team Solves Mammoth, Century-Old Math Problem

The E8 root system, from a drawing made in the 1960s by Peter McMullen. i i

The E8 root system, from a drawing made in the 1960s by Peter McMullen. This computer-generated image was created by John Stembridge, based on McMullen's drawing. John Stembridge hide caption

itoggle caption John Stembridge
The E8 root system, from a drawing made in the 1960s by Peter McMullen.

The E8 root system, from a drawing made in the 1960s by Peter McMullen. This computer-generated image was created by John Stembridge, based on McMullen's drawing.

John Stembridge

Scientists have solved one of the toughest problems in mathematics, performing a calculation to figure out the symmetry of a 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.

Lie groups were invented in the 19th century by the Norwegian mathematician Sophus Lie [pronounced LEE], 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 — or, an expression of variables and exponents.

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 math problem first put forth 120 years ago yields an answer so large that it could 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.

E8 "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 should think of the dimensions as being degrees of freedom."

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

"So, if you were studying the weather, you might look at temperature, and humidity and pressure and wind speed and all those things.

"And pretty soon, you might have a list of 10 different attributes you're looking at. Well, we would call that 10-dimensional. So, if you just think of the 248 as being 248 variables, then you're off to a good start."

Conrey says that mathematicians expect the solution to propel future advances in science and technology — though he admits that he's not sure how that will happen.

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

For now, Conrey says, his institute is on to its next big challenge, a hypothesis that has to do with prime numbers. It's been a mystery since it was first contemplated in 1859.

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