“Octonions were largely neglected since their discovery in 1843, but in the past few decades they have assumed a curious importance in string theory. And indeed, if string theory is a correct representation of the universe, they may explain why the universe has the number of dimensions it does.” John C. Baez & and John Huerta, “The Strangest Numbers in String Theory,” *Scientific American*, May 2011.

“The set of all real numbers forms a line, so we say that the collection of real numbers is one-dimensional. We could also turn this idea on its head: the line is one-dimensional because specifying a point on it requires one real number. The set of all complex numbers of the form *a*+*b i*, where *i*²=–1, and *a* and *b* are ordinary real numbers, describe points on the plane and their basic operations —addition, subtraction, multiplication and division— describe geometric manipulations in the plane. Almost everything we can do with real numbers can also be done with complex numbers.

On October 16, 1843, William Rowan Hamilton was walking with his wife along the Royal Canal to a meeting of the Royal Irish Academy in Dublin when he had a sudden revelation: quaternions *a* + *b i* + *c j* + *d k*, with *i*²=*j*²=*k*²=–1. Quaternions provide an efficient way to represent threedimensional rotations. Hamilton’s college friend, John Graves, discovered on December 26 a new eight-dimensional number system that he called the octaves and that are now called octonions. In 1845 the young genius Arthur Cayley rediscovered the octonions. For this reason, the octonions are also sometimes known as Cayley numbers.”

In the generalization of numbers as tuples, division is the hard part: a number system where we can divide is called a division algebra. Not until 1958 did three mathematicians prove an amazing fact that had been suspected for decades: any division algebra must have dimension one (which is just the real numbers), two (the complex numbers), four (the quaternions) or eight (the octonions).

Hamilton didn’t like the octonions because they break some cherished laws of arithmetic. Real an complex numbers are commutatitve, but quaternions are noncommutative. The octonions are much stranger. Not only are they noncommutative, they also break another familiar law of arithmetic: the associative law (xy)z = x(yz). What the octonions would be good for? They are closely related to the geometry of seven and eight dimensions, and we can describe rotations in those dimensions using the multiplication of octonions.

In the 1970s and 1980 s theoretical physicists developed a strikingly beautiful idea called supersymmetry, a symmetry between matter and the forces of nature. Every matter particle (such as an electron) has a partner particle that carries a force, and vice versa. Supersymmetry also encompasses the idea that the laws of physics would remain unchanged if we exchanged all the matter and force particles. Even though physicists have not yet found any concrete experimental evidence in support of supersymmetry, the theory is so seductively beautiful and has led to so much enchanting mathematics that many physicists hope and expect that it is real.

In the standard three-dimensional version of quantum mechanics that physicists use every day, spinors describes the wave motion of matter particles and vectors describes that of force particles. Particle interactions require the combination of spinors and vectors by a simulacrum of multiplication. As an alternative, imagine a strange universe with no time, only space. If this universe has dimension one, two, four or eight, both matter and force particles would be waves described by a single type of vectorial object (vectors and spinors coincide), just real numbers, complex numbers, quaternions or octonions, respectively. Supersymmetry emerges naturally, providing a unified description of matter and forces.

In string theory, every object corresponds to a little string with one dimension in space and another one in time, hence two dimensions have to be added to every point in sapce. Instead of supersymmetry in dimension one, two, four or eight, we get supersymmetry in dimension three, four, six or 10. Coincidentally string theorists have for years been saying that only 10-dimensional versions of the theory are self-consistent: anomalies appear in anything other than 10 dimensions, breaking down string theory. But 10-dimensional string theory is, as we have just seen, the version of the theory that uses octonions. So if string theory is right, the octonions are not a useless curiosity, on the contrary, they provide the deep reason why the universe must have 10 dimensions: in 10 dimensions, matter and force particles are embodied in the same type of numbers—the octonions.

Recently physicists have started to go beyond strings to consider membranes. In string theory we had to add two dimensions to our standard collection of one, two, four and eight, now we must add three. Supersymmetric membranes naturally emerge in dimensions four, five, seven and 11. Researchers tell us that M-theory (the “M” typically stands for “membrane”) requires 11 dimensions—implying that it should naturally make use of octonions.

Neither string theory nor M-theory have as of yet made no experimentally testable predictions. They are beautiful dreams—but so far only dreams. The universe we live in does not look 10- or 11-dimensional, and we have not seen any symmetry between matter and force particles. Only time will tell if the strange octonions are of fundamental importance in understanding the world we see around us or merely a piece of beautiful mathematics.”

More information about these especulations in Peter Woit, “This Week’s Hype,” Not Even Wrong, April 28th, 2011, where the expository article about octonions by John Baez that appeared in the AMS Bulletin (copy here, a web-site here) is recommended. In the comments, Thomas Larsson recalls that “octonions is the last division algebra, but if you relax your axioms a little the Cayley-Dickson construction gives an infinite tower of increasingly uninteresting algebras: n=1, Reals; n=2, Complex numbers; n=4, Quaternions, not commutative; n=8, Octonions, not associative; n=16: Sedenions, not alternative but power associative, n=32: 32-ions?; …

See also Philip Gibbs, “Octonions in String Theory,” viXra log, April 29, 2011, and Lubos Motl, “John Baez, octonions, and string theory,” The Reference Frame, April 29, 2011.

The key paragraph from Lubos is “In the slow comments under the 2009 blog entry, Robert Helling argued that there is a lot of interesting fog about the closure of the supersymmetry algebra etc. I find this whole approach to these issues irrational.” Actually, Helling refers to the papers that show that there are more structure than the naive match argued in his blog entry.

Hamilton in his letter of 17 October 1843 to John Graves, seems very confused about the relationships between i. j, +1 and -1. He asks what are we to do with ij when i and j are the unequal roots of a common square. In fact there is no law of arithmetic which makes ij equal to anything but +1. It is these doubts of Hamilton which are the source of his fallacious theory of the non-commutative properties of the multiplication of imaginary numbers. All multiplication whether of real or imaginary numbers is commutative.

Hamilton’s quaternions equation i^2=j^2=k^2=ijk=-1 is incorrect because -1 cannot have more than two square roots, in the same way that any real, imaginary or complex number cannot have more than two square roots, more than three cube roots, more than four fourth roots, more than five fifth roots etc. This means that k^2=-1 is incorrect unless k can equal either i or j.

Further to my previous comments, -1 does not have more than two square roots, nevertheless -1 does have three cube roots which are cos60+isin60, cos180+isin180 which equals -1, and cos300+isin300. Hamilton seemed to have no understanding of these matters.