After breaks from school spent ski-bumming, bartending abroad and intensely training as a mixed martial artist, Furey later met the division algebras again in an advanced geometry course and learned just how peculiar they become in four strokes. When you double the dimensions with each step as you go from real numbers to complex numbers to quaternions to octonions, she explained, “in every step you lose a property.” Real numbers can be ordered from smallest to largest, for instance, “whereas in the complex plane there’s no such concept.” Next, quaternions lose commutativity; for them, a × b doesn’t equal b × a. This makes sense, since multiplying higher-dimensional numbers involves rotation, and when you switch the order of rotations in more than two dimensions you end up in a different place. Much more bizarrely, the octonions are nonassociative, meaning (a × b) × c doesn’t equal a × (b × c). “Nonassociative things are strongly disliked by mathematicians,” said John Baez, a mathematical physicist at the University of California, Riverside, and a leading expert on the octonions. “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.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.”

The octonions’ seemingly unphysical nonassociativity has crippled many physicists’ efforts to exploit them, but Baez explained that their peculiar math has also always been their chief allure. Nature, with its four forces batting around a few dozen particles and anti-particles, is itself peculiar. The Standard Model is “quirky and idiosyncratic,” he said.

In the Standard Model, elementary particles are manifestations of three “symmetry groups” — essentially, ways of interchanging subsets of the particles that leave the equations unchanged. These three symmetry groups, SU(3), SU(2) and U(1), correspond to the strong, weak and electromagnetic forces, respectively, and they “act” on six types of quarks, two types of leptons, plus their anti-particles, with each type of particle coming in three copies, or “generations,” that are identical except for their masses. (The fourth fundamental force, gravity, is described separately, and incompatibly, by Einstein’s general theory of relativity, which casts it as curves in the geometry of space-time.)

Sets of particles manifest the symmetries of the Standard Model in the same way that four corners of a square must exist in order to realize a symmetry of 90-degree rotations. The question is, why this symmetry group — SU(3) × SU(2) × U(1)? And why this particular particle representation, with the observed particles’ funny assortment of charges, curious handedness and three-generation redundancy? The conventional attitude toward such questions has been to treat the Standard Model as a broken piece of some more complete theoretical structure. But a competing tendency is to try to use the octonions and “get the weirdness from the laws of logic somehow,” Baez said.

Furey began seriously pursuing this possibility in grad school, when she learned that quaternions capture the way particles translate and rotate in 4-D space-time. She wondered about particles’ internal properties, like their charge. “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,” she said — a bit of numerology that had raised eyebrows before. The coincidences have since proliferated. “If this research project were a murder mystery,” she said, “I would say that we are still in the process of collecting clues.”

The Dixon Algebra

To reconstruct particle physics, Furey uses the product of the four division algebras, $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ ($latex \mathbb{R}$ for reals, $latex \mathbb{C}$ for complex numbers, $latex \mathbb{H}$ for quaternions and $latex \mathbb{O}$ for octonions) — sometimes called the Dixon algebra, after Geoffrey Dixon, a physicist who first took this tack in the 1970s and ’80s before failing to get a faculty job and leaving the field. (Dixon forwarded me a passage from his memoirs: “What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”)

Whereas Dixon and others proceeded by mixing the division algebras with extra mathematical machinery, Furey restricts herself; in her scheme, the algebras “act on themselves.” Combined as $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, the four number systems form a 64-dimensional abstract space. Within this space, in Furey’s model, particles are mathematical “ideals”: elements of a subspace that, when multiplied by other elements, stay in that subspace, allowing particles to stay particles even as they move, rotate, interact and transform. The idea is that these mathematical ideals are the particles of nature, and they manifest the symmetries of $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$.

As Dixon knew, the algebra splits cleanly into two parts: $latex \mathbb{C}\otimes\mathbb{H}$ and $latex \mathbb{C}\otimes\mathbb{O}$, the products of complex numbers with quaternions and octonions, respectively (real numbers are trivial). In Furey’s model, the symmetries associated with how particles move and rotate in space-time, together known as the Lorentz group, arise from the quaternionic $latex \mathbb{C}\otimes\mathbb{H}$ part of the algebra. The symmetry group SU(3) × SU(2) × U(1), associated with particles’ internal properties and mutual interactions via the strong, weak and electromagnetic forces, come from the octonionic part, $latex \mathbb{C}\otimes\mathbb{O}$.

Günaydin and Gürsey, in their early work, already found SU(3) inside the octonions. Consider the base set of octonions, 1, e 1 , e 2 , e 3 , e 4 , e 5 , e 6 and e 7 , which are unit distances in eight different orthogonal directions: They respect a group of symmetries called G2, which happens to be one of the rare “exceptional groups” that can’t be mathematically classified into other existing symmetry-group families. The octonions’ intimate connection to all the exceptional groups and other special mathematical objects has compounded the belief in their importance, convincing the eminent Fields medalist and Abel Prize–winning mathematician Michael Atiyah, for example, that the final theory of nature must be octonionic. “The real theory which we would like to get to,” he said in 2010, “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.” He added, “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.”