Questions tagged [octonions]

For questions on the octonions, a normed division algebra over the real numbers. It is a non-associative higher-dimensional analogue in the hierarchy of real, complex, and quaternionic numbers.

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter ${\mathbb O}$. There are only four such algebras, the other three being the real numbers $\Bbb R$, the complex numbers $\Bbb C$, and the quaternions $\Bbb H$. The octonions are the largest such algebra, with eight dimensions, double the number of the quaternions from which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, power associativity.

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What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ dimensions. I think I understand the first $n<4$…
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What do we lose passing from the reals to the complex numbers?

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex…
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What are some real-world uses of Octonions?

... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Cayley-Dickson process is a numerical beast, but has anybody found any…
Hooked
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Why are properties lost in the Cayley–Dickson construction?

Motivating question: What lies beyond the Sedenions? I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset…
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Is the seven-dimensional cross product unique?

I'm confused about how many different 7D cross products there are. I'm defining a 7D cross product to be any bilinear map $V \times V \to V$ (where $V$ is the inner product space $\mathbb{R}^7$ endowed with the Euclidean inner product) such that for…
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If there are any 3nion, 5nion, 7nion, 9nion, 10nion, etc.

The quaternion/octonion extend the complex numbers, which extend the real numbers. So we go: 1-tuple: Real numbers. 2-tuple: Complex numbers. 4-tuple: Quaternions. 8-tuple: Octonions. The Wikipedia link describes this doubling process: In…
user10869858
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Can octonions be represented by infinite matrices?

It is sometimes possible to multiply matrices of countably-infinite dimension. (Matrix multiplication is defined in the usual way, with rows and columns multiplied termwise and summed.) However, it turns out the associative property fails in general…
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The sixth number system

In high school, I learned there are 5 number systems, namely: Natural numbers ($\Bbb N$) Integers ($\Bbb Z$) Rational numbers ($\Bbb Q$) Real numbers ($\Bbb R$) Complex numbers ($\Bbb C$) I remember one time our teacher told us that there is…
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Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, and that there is a way to generalize…
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Why is 8 so special?

I have been reading about multi-dimensional numbers, and found out that it's been proven that the Octonions are the composition algebra of the largest dimension. I was wondering why, despite having infinitely many different dimensions of numbers,…
InterestedGuest
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Are roots of unity in hypercomplex algebras well defined?

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the quaternions, octonions, and other hypercomplex…
dezakin
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Trinonions, Quaternions, Quinonions, Sextonions, Septonions, Octonions

There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions and septonions? The term sextonions is used…
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Do the octonions contain infinitely many copies of the quaternions?

Note: by "infinitely many", I'm confident I always mean $\beth_1$ many herein. We can easily show the quaternions contain infinitely many copies of $\Bbb C$ because, given any unit vector $\in\Bbb R^3$ of components $b,\,c,\,d$, $\Bbb R[h]$ is…
J.G.
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Octonionic formula for the ternary eight-dimensional cross product

A cross product is a multilinear map $X(v_1,\cdots,v_r)$ on a $d$-dimensional oriented inner product space $V$ for which (i) $\langle X(v_1,\cdots,v_r),w\rangle$ is alternating in $v_1,\cdots,v_r,w$ and (ii) the magnitude $\|X(v_1,\cdots,v_r)\|$…
arctic tern
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Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would like to know if some mathematician has defined a…
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