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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Projective plane over the octonions (Cayley plane)
You can use octonion algebra's $\mathbb{O}$ over a field to coordinatize projective planes. They are called Cayley planes as far as I know.
You can't use the usual approach with homogeneous ...
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Are all 12 possibilities for products of three octonions in fact possible?
The octonions are a non-associative, non-commutative, $8$-dimensional algebra over the reals. Consider a triple $(x,y,z)$ of three distinct octonions. There are $12$ possible products of those, such ...
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Sequence and Limit Definition at Quaternions
A sequence at $\mathbb{C}$ is a map $z_n: \mathbb{N} \to \mathbb{C}$, where $z_n$ converges to a $z_0 \in \mathbb{C}$ if:
$$\forall \epsilon > 0, \text{ } \exists N \in \mathbb{N}; \text{ } \forall ...
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Square Root of a Arbitrary Octonion
Let $q = a + bi + cj + dk = a + Q \in \mathbb{H}$ a quaternion. So, we have:
$$\sqrt{q} = \sqrt{\frac{|q| + a}{2}} + \frac{Q}{|Q|} \sqrt{\frac{|q| - a}{2}}$$
I have a question: this formula works for ...
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Closed form of product of sedenions
I'm a math student and I'm taking an algebra course. The professor introduced us to the field of quaternions ($\mathbb{Q}$), I became very curious about the topic and I saw that in addition to ...
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Some basics on octonions and quaternions
There is an $\mathbb{R}$-bilinear operation on the octonions $\mathbb{O} \otimes_{\mathbb{R}} \mathbb{O} \rightarrow \mathbb{O}$, which is not associative. My question is instead about the algebra ...
user900250
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Have generalisations of dual numbers/complex numbers/quaternions/octonions... been studied?
Can anyone point me to any generalisations of the notions in the title?
For example say you have:
$$
(a_1, a_2, a_3, ...,a_n) \in \mathbb{R}^n
$$
and
$$
\gamma_1, \gamma_2, \gamma_3, ..., \gamma_n
$$
...
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Is there any practical use for octonions? [closed]
Quaternions are useful for describing orientation/ rotations in 3- dimensions, however is there much practical use for an 8-dimensional base hyper complex number id est Octonions?
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Where the tensor used in the definition of the octonion product come from?
The octonion multiplication table is hard to remember.
It's also not uniquely defined, but I'm assuming the definition that Wikipedia chose is fairly standard.
One of the presentations uses an ...
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Are there an endless number of imaginary square roots for negative one?
When i learned about imaginary numbers, I learned that i represents the square root of negative one, as does i's counterpart below zero, "-i."
But then I learned the same is true of the ...
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How do you find the 3rd roots of hypercomplex imaginary units?
In my instance I want to find the 3rd root of the octonion imaginary unit $e_4$.
I am working on simplifying the octonion
$$
o=5+2e_1 \sqrt[3]{e_4}+3e_2 \sqrt[3]{e_4}+2e_3 \sqrt[3]{e_4}
$$
$$
=5+...
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When is operator conformality preserved under addition and subtraction?
An operator $A$ on an $n$-dimensional real vector space is conformal in case
$$
A^T A = \alpha I \qquad \text{for some} \qquad \alpha \ge 0 ,
$$
and
$$
\mathrm{det} A \ge 0 .
$$
Let $A$ and $B$ be two ...
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Does the $32$-inator exist?
Background
It is common popular-math knowledge that as we extend the real numbers to complex numbers, quaternions, octonions, sedenions, $32$-nions, etc. using the Cayley-Dickson construction, we lose ...
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Extension of Hopf Fibrations
The only hopf fibrations that exist are the first three listed here.
$\require{AMScd}$
\begin{CD}
S^3 @>S^1>> S^2
\end{CD}
$\require{AMScd}$
\begin{CD}
S^7 @>S^3>> S^4
\end{CD}
$\...
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Norm of the Sedenions
Let the Cayley-Dickson doubling of the octonions be called the sedenions. The sedenions are not a division algebra, because they contain zero divisors. The presence of zero divisors means that the ...
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