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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.

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22
votes
2answers
3k views

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}...
30
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2answers
2k views

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 ...
28
votes
2answers
2k views

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 $2^n$ ...
7
votes
2answers
269 views

Why do division algebras always have a number of dimensions which is a power of $2$?

Why do number systems always have a number of dimensions which is a power of $2$? Real numbers: $2^0 = 1$ dimension. Complex numbers: $2^1 = 2$ dimensions. Quaternions: $2^2 = 4$ dimensions. ...
12
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1answer
2k views

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 ...
5
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1answer
780 views

Can octonions be used to rotate 7-dimensional vectors?

A friend told me that, the same way you can represent a 3-vector as an imaginary quaternion then conjugate it by a unit quaternion with real part $\cos(\frac{\theta}{2})$ to rotate it by $\theta$ ...
4
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3answers
525 views

Math beyond Quaternions

Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
25
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5answers
7k views

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 ...
11
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0answers
223 views

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 $\...
9
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1answer
212 views

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 $v_1,\cdots,v_r,w$ ...
8
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0answers
387 views

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 ...
4
votes
1answer
467 views

Octonionic Hopf fibration and $\mathbb HP^3$

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations $S^1\to\...
3
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0answers
62 views

The order of elements in finite octonions

After great success of this question I would like to continue this thread. Choose any element $x$ in algebra $\mathbb O_q$ of octonions over finite field $\mathbb F_q$. There is following equation ...
2
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1answer
213 views

Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
7
votes
1answer
128 views

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^...
2
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1answer
69 views

Classification of subalgebras of composition algebras

Let $F$ be an algebraically closed field. It is known that the only composition algebras over $F$ are $F$ itself, the direct sum $F\oplus F$ (also called split-complexes), the algebra of $2\times 2$ ...
0
votes
1answer
78 views

Polarization identity $2(a,b)(c,d)=(ac,bd)+(ad,bc)$

I am interested in following along this Wikipedia article's derivation of properties of composition algebras (in particular, Euclidean Hurwitz algebras). Let $A$ be a unital, not necessarily ...