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.
Image: John Stembridge/American Institute of Mathematics
http://www.cosmosmag...e.com/node/1110
248-dimension maths puzzle solved
The structure is described in the form of a vast matrix
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.
E8 is a "Lie group", a means of describing symmetrical objects.
The team said their findings may assist fields of physics which use more than four dimensions, such as string theory.
Lie groups were invented by the 19th Century Norwegian mathematician Sophus Lie (pronounced "Lee").
It's as complicated as symmetry can get
David Vogan
Familar structures such as balls and cones have symmetry in three dimensions, and there are Lie groups to describe them. E8 is much bigger.
"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.
"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."
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.
Fundamental force
Conceptualising, designing and running the calculations took a team of 19 mathematicians four years. The final computation took more than three days' solid processing time on a Sage supercomputer.
Lie groups were invented by the Norwegian Sophus Lie
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 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 of Manhattan.
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.
"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."
Edited by Russ, 21 March 2007 - 11:12 AM.
