Octonion

The octonion number system is an extension of the quaternions of mathematics. It was first discovered by Irish mathematician John T. Graves in 1843, and was also independently discovered by British mathematician Arthur Cayley in 1845. In modern physics, the octonions have applications in quantum field theory and string theory. Quotes There is still something in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. ... If with your alchemy you can make three pounds of gold, why should you stop there? Letter from John T. Graves addressed to William Rowan Hamilton on October 26, 1843, shortly after Hamilton's discovery of the quaternions It is possible to form an analogous theory with seven imaginary roots of (-1). Arthur Cayley, "P.S. On Quaternions" (1845) The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. John C. Baez https://webhomes.maths.ed.ac.uk/~v1ranick/papers/baezocto.pdf 10.1090/S0273-0979-01-00934-X The octonions Bull. Amer. Math. Soc. 39 145–205 2002 (p. 145) Why have the octonions languished in such obscurity compared to the quaternions? Besides their rather inglorious birth, one reason is that they lacked a tireless defender such as Hamilton. But surely the reason for this is that they lacked any clear application to geometry and physics. John C. Baez https://webhomes.maths.ed.ac.uk/~v1ranick/papers/baezocto.pdf 10.1090/S0273-0979-01-00934-X The octonions Bull. Amer. Math. Soc. 39 145–205 2002 (p. 147) Besides their possible role in physics, the octonions are important because they tie together some algebraic structures that otherwise appear as isolated and inexplicable exceptions. John C. Baez https://webhomes.maths.ed.ac.uk/~v1ranick/papers/baezocto.pdf 10.1090/S0273-0979-01-00934-X The octonions Bull. Amer. Math. Soc. 39 145–205 2002 (p. 147) The octonions are much stranger. Not only are they noncommutative, they are also break another familiar law of arithmetic: the associative law (xy)z = x(yz). Baez John C. Huerta John The Strangest Numbers in String Theory Scientific American 2011-05-01 https://www.scientificamerican.com/article/the-strangest-numbers-in-string-theory/ But mathematicians know that the number system we study in school is but one of many possibilities. And indeed, other kinds of numbers are important for understanding geometry and physics. Among the strangest alternatives is the octonions. Largely neglected since their discovery in 1843, in the last 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. Baez John C. Huerta John The Strangest Numbers in String Theory Scientific American 2011-05-01 https://www.scientificamerican.com/article/the-strangest-numbers-in-string-theory/ But perhaps most important, it wasn’t clear in Hamilton’s time just what the octonions would be good for. They are closely related to the geometry of 7 and 8 dimensions, and we can describe rotations in those dimensions using the multiplication of octonions. But for over a century that was a purely intellectual exercise. It would take the development of modern particle physics—and string theory in particular—to see how the octonions might be useful in the real world. Baez John C. Huerta John The Strangest Numbers in String Theory Scientific American 2011-05-01 https://www.scientificamerican.com/article/the-strangest-numbers-in-string-theory/ Despite its counter-culture status, the octonions have long drawn the curiosity of generations of physicists. The algebra is known to appear without warning in apparently disparate areas of mathematics, within algebra, geometry, and topology. However, despite its ubiquity, its practical uses in physics have remained elusive, due to the algebra's non-associativity, which must be handled with care. Cohl Furey Standard model physics from an algebra? University of Waterloo 2015 https://arxiv.org/abs/1611.09182 (p. 37) (Ph.D. thesis) It is nearly irresistible to ask if the octonions, , the last of the set of four normed division algebras over , have a calling in nature. Certainly several have thought so, but for the most part, the octonions have remained as a well kept secret from mainstream physics. More often than not, the octonions are passed by in haste because they are non-associative, and hence at times temperamental. As we will show, this property is in fact a gift, which will offer a way to streamline some of the standard model’s complex structure. Cohl Furey Towards a unified theory of ideals arXiv arXiv 2018 10.48550/arXiv.1002.1497 https://arxiv.org/abs/1002.1497 The octonions have been found to apply to areas in particle physics and extended to sedenions more recently. It may not be necessary to go to higher dimensions to achieve new representations in particle physics. All Cayley-Dickson algebras of dimension higher than octonions consist of multiple copies of octonions and various quasi-octonions so it is possible that just octonions and the quasi-octonion algebras are all that is necessary. Wilmot G. P. Structure of the Cayley-Dickson algebras 2025 10.48550/ARXIV.2505.11747 https://arxiv.org/abs/2505.11747 Nonassociative things are strongly disliked by mathematicians ... Because while it’s very easy to imagine noncommutative situations — putting on shoes then socks is different from socks then shoes — it’s very difficult to think of a nonassociative situation. John C. Baez quoted in: Wolchover Natalie The Peculiar Math That Could Underlie the Laws of Nature Quanta Magazine 2018-07-20 https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ Accordions are the octonions of the music world — tragically misunderstood. Cohl Furey quoted in: Wolchover Natalie The Peculiar Math That Could Underlie the Laws of Nature Quanta Magazine 2018-07-20 https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ I realized that the eight degrees of freedom of the octonions could correspond to one generation of particles: one neutrino, one electron, three up quarks and three down quarks. Cohl Furey quoted in: Wolchover Natalie The Peculiar Math That Could Underlie the Laws of Nature Quanta Magazine 2018-07-20 https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ The real theory which we would like to get to should include gravity with all these theories in such a way that gravity is seen to be a consequence of the octonions and the exceptional groups. ... It will be hard because we know the octonions are hard, but when you’ve found it, it should be a beautiful theory, and it should be unique. Michael Atiyah in 2010, as quoted in: Wolchover Natalie The Peculiar Math That Could Underlie the Laws of Nature Quanta Magazine 2018-07-20 https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ Octonions are to physics what the Sirens were to Ulysses. Pierre Ramond, particle physicist and string theorist at the University of Florida, quoted in: Wolchover Natalie The Peculiar Math That Could Underlie the Laws of Nature Quanta Magazine 2018-07-20 https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ See also William Rowan Hamilton History of mathematics Mathematics Quaternion Sedenion External links Category:Algebra Category:Numbers Category:Theoretical physics