248 Dimension Maths Puzzle Solved

What? 248 th dimension!! Leave it, where is the 4th dimension?!

An international team of mathematicians has detailed a vast complex numerical "structure" which was invented more than a century ago.

Mapping the 248-dimensional structure, called E8, took four years of work and produced more data than the Human Genome Project, researchers said. "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." E8 is a "Lie group", which is used as a means of describing symmetrical objects.

Part of the E8 matrix. Image: David Vogan / MIT

(The structure is described in the form of a vast matrix)

The Story

The 18-member group of mathematicians and computer scientists was convened by the American Institute of Mathematics in Palo Alto to map a theoretical object known as the "Lie group E8." Familiar structures such as balls and cones have symmetry in three dimensions, and there are Lie groups to describe them. E8 is much bigger.

Lie (pronounced Lee) groups were invented by 19th-century Norwegian mathematician Sophus Lie in his study of symmetrical objects, especially spheres, and differential calculus. The E8 group, which dates to 1887, is the most complicated Lie group, with 248 dimensions, and was long considered impossible to solve.

(Lie groups were invented by the Norwegian Sophus Lie.)

"To say what precisely it is something even many mathematicians can't understand," said Jeffrey Adams, the project's leader and a math professor at the University of Maryland.

The problem's proof, announced at the Massachusetts Institute of Technology, took the researchers four years to find. It involves about 60 times as much data as the Human Genome Project.

When stored in highly compressed form on a computer hard drive, the solution takes up as much space as 45 days of continuous music in MP3 format.

"It's like a Mount Everest of mathematical structures they've climbed now," said Brian Conrey, director of the institute.

"What's attractive about studying E8 is that, it's as complicated as symmetry can get", observed David Vogan from the Massachussetts Institute of Technology (MIT) in the US. Professor Vogan is presenting the results at MIT in a lecture entitled The Character Table for E8, or ‘How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness’.

How they did it?

Conceptualising, designing and running the calculations took a team of 19 mathematicians four years and one more for software writing before the solution could be tackled by computer. The final computation took more than three days' (77 hours to be precise) solid processing time on a Sage supercomputer.

The single calculation required computer power and memory that wasn't available until recently, said the National Science Foundation, which provided funding along with the American Institute of Mathematics.

What came out was a matrix of linked numbers, which together describe the structure of E8. It contains more than 60 times as much data as the human genome sequence.

Each of the 205,263,363,600 (now that is really a big number!!!) entries on the matrix is far more complicated than a straightforward number; some are complex equations.

The team calculated that if all the numbers were written out in small type, they would cover an area the size whole of Manhattan island.

And Applications…

In addition to facilitating further understanding of symmetry and related areas of mathematics, the team hopes its work will contribute to areas of physics, such as string theory, which involve structures possessing more than the conventional four dimensions of space and time.

The calculation does not have any obvious practical applications but could help advance theoretical physics and geometry. "While mathematicians have known for a long time about the beauty and the uniqueness of E8, we physicists have come to appreciate its exceptional role only more recently," commented Hermann Nicolai, director of the Max Planck Institute for Gravitational Physics (the Albert Einstein Institute) in Germany.

"Yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner."

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