The E8 pattern: key to the theory of everything? (Image: Peter McMullen / Claudio Rocchini / Wikimedia Commons )
Eight dimensions is a rarefied space that is home to the octonions – “the crazy old uncle nobody lets out of the attic”, as mathematician John Baez of the University of California, Riverside, puts it.
Octonions are indeed odd creatures. They are one of only four number systems in which division is possible, and so allow the full range of algebraic operations to be performed. The way octonions interact, however, is peculiarly exasperating and unlike anything we are familiar with from our conventional number system (see diagram).
So why bother with octonions at all? It’s because, for some problems in theoretical physics, they are an invaluable tool. Matrices filled with octonions are the building blocks of a bizarre mathematical structure known as the “E8 exceptional Lie group“, which sits at the heart of a particular form of string theory.
In 2007, E8 hit the headlines when physicist Garrett Lisi, who has no university affiliation and spends most of his time surfing in Hawaii, used the E8 group to seemingly unify gravity with the three other fundamental forces without using string theory. The publicity surrounding that work ruffled some feathers. “String theorists have been working on [E8] since the late 70s,” says Michael Duff of Imperial College London. “We didn’t need surfer dudes to tell us that it was interesting.”
Duff himself is agnostic about the value of octonions, pointing out that no theory in which they pop up has yet been tested by experiment. “Whether octonions have …