The Math Book: Milestones in the History of Math

The Quest for Lie Group E8

Marius Sophus Lie (1842 - 1899), Wilhelm Karl Joseph Killing (1847 - 1923) - For more than a century, mathematicians have sought to understand a vast, 248-dimensional entity, known to them only as E8. Finally, in 2007, an international team of mathematicians and computer scientists made use of a supercomputer to tame the intricate beast.

As background, consider the Mysterium Cosmographicum (The Sacred Mystery of the Cosmos) of Johannes Kepler (1571 - 1630), who was so enthralled with symmetry that he suggested the entire solar system and planetary orbits could be modeled by Platonic Solids, such as the cube and dodecahedron, nestled in each other forming layers as if in a gigantic crystalline onion. These kinds of Keplerian symmetries were limited in scope and number; however, symmetries that Kepler could have hardly imagined may indeed rule the universe.

In the late nineteenth century, the Norwegian mathematician Sophus Lie (pronounced "Lee") studied objects with smooth rotational symmetries, like the sphere or doughnut in our ordinary three-dimensional space. In three and higher dimensions, these kinds of symmetries are expressed by Lie groups. The German mathematician Wilhelm Killing suggested the existence of the E8 group in 1887. Simpler Lie groups control the shape of electron orbital and symmetries of subatomic quarks. Larger groups, like E8, may someday hold the key to a unified theory of physics and help scientist understand string theory and gravity.

Fokko du Cloux, a Dutch mathematician and computer scientist who was one of the E8 team members, wrote the software for the supercomputer and pondered the ramifications of E8 while he was dying of amyotrophic lateral sclerosis and breathing with a respirator. He died in November 2006, never living to see the end of the quest for E8.

On January 8, 2007, a supercomputer computed the last entry in the table for E8, which describes the symmetries of a 57-dimensional object that can be imagined as rotating in 248 ways without changing its appearance. The work is significant as an advance in mathematical knowledge and in the use of large-scale computing to solve profound mathematical problems.