Octonion

"It is possible to form an analogous theory with seven imaginary roots of (-1)."

"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)."

"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."

"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 algebras non-associativity, which must be handled with care."

"Octonions are to physics what the Sirens were to ."

"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."

"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?"

"It is nearly irresistible to ask if the octonions, \mathbb O, the last of the set of four normed division algebras over \mathbb R, 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."

"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."