Old Octonions may rule the World

This week’s That’s Maths column in The Irish Times (TM055, or search for “thatsmaths” at irishtimes.com) is about octonions, new numbers discovered by John T Graves, a friend of William Rowan Hamilton.

Multiplication table for octonions, of the form z=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]

On this day in 1843, the great Irish mathematician William Rowan Hamilton discovered a new kind of numbers called quaternions. Each quaternion has four parts, like the coordinates of a point in four dimensional space. Physical space has three dimensions, with coordinates ( x, y, z ) giving the East-West, North-South and Up-Down positions.

The quaternions could be added, subtracted, multiplied and divided like ordinary numbers, but there was a catch. Two quaternions multiplied together give different answers depending on the order: A times B is not equal to B times A. They break the algebraic rule called the commutative law of multiplication.

Eight dimensions

When Hamilton told his friend John T. Graves about the new numbers, Graves wondered “Why stop at four dimensions?” He soon came up with a system of numbers, now called octonions, each of which has eight components: they are points in 8-dimensional space. But there was a price to pay: the octonians break another algebraic rule, the associative law.

Hamilton promised Graves that he would speak about the octonions at the Royal Irish Academy, but he was so excited by his quaternions that he forgot, and the English algebraist Arthur Cayley got credit when he re-discovered the eight-component numbers a few years later. Only then did Hamilton bring news of Graves’ work to the Academy.

You might wonder — just as Graves did — whether more new numbers can be found. A number system in which we can add, subtract, multiply and divide is called a division algebra. It turns out that there are division algebras only for dimensions 1, 2, 4 and 8. This was long believed to be true but it was not proved until 1958.

Quaternions fell out of favour after Hamilton’s death and were supplanted by vector calculus, which was more efficient and less complicated. They have resurfaced recently in computer graphics and astronautics, and have applications in theoretical mechanics.

String Theory

Octonions fared even worse than quaternions, being regarded as little more than interesting curiosities. But now things may be changing: for decades, physicists have been struggling to reconcile quantum mechanics and general relativity. The goal is to produce a unified description of nature, sometimes called a Theory of Everything. The foremost candidate is called string theory: elementary constituents like electrons are not point particles, but one-dimensional oscillating strings that trace out sheets or tubes as they move through time.

String theory depends on the idea of supersymmetry: every matter particle has a “twin” force particle, and the laws of physics remain unchanged if we swap the matter and force particles. Currently, quantum mechanics represents the two species by different kinds of numbers, spinors and vectors. If we could represent both species by the same kind of numbers, we would have a symmetric description of matter and force, that is, supersymmetry. This is possible only in 1, 2, 4 or 8 dimensions.

Adding two more components, one for time and one for the string dimension, we get spaces of 3, 4, 6 and 10 dimensions. At present, 10-dimensional space based on octonions is considered by string theorists as a likely candidate for describing the universe. This is speculative and controversial but if it turns out to be correct, matter and force particles will be modelled by those curious eight-component octonions discovered by Graves 171 years ago.

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Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to http://www.ramblingroundireland.com/