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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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 ...
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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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votes
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answers
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views
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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About the octonions [closed]
Given $a\in \mathbb R^3-\{0\}$, then the map $J_a: q\to q\frac{a}{|a|}$ is a complex structure of the quaternions $\mathbb H$.
Now, what about the octonions $\mathbb O$? Do they similarly obtain a ...
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Orders of Paige Loops over Finite Fields
A Moufang loop $M$ is a loop that satisfies the Moufang identity:
$(zx)(yz) = z((xy)z),\forall x,y,z\in M$.
From here, we get the following statement:
For any field $F$ let $M(F)$ denote the Moufang ...
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Investigating and Generalizing Octonionic Nonassociativity
A possible multiplication convention for the octonions is given by the following 7 sets of integers from 1 to 7.
{1,2,3},{1,4,5},{1,6,7},{2,4,6},{2,5,7},{3,4,7},{3,5,6}.
Notice that each of the seven ...
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Automorphism group of octonions is closed in $O(7)$.
So far I have understood that the group of automorphisms of the octonions is a subgroup of $O(7)$. I want to understand how it is a Lie group using the lemma that closed subgroups of Lie groups again ...
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"Canonical" norm on a real finite-dimensional unital associative division algebra?
Let $\mathcal{C}$ denote the category of unital associative finite-dimensional division $\mathbb{R}$-algebras. (As a full subcategory of that of unital $\mathbb{R}$-algebras.)
For $A\in \mathcal{C}$ ...
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General algebra for rotations in $n$ dimensions
Rotations in $2$ dimensional space can be given a nice representation with complex numbers.
The group $(\mathbb{C}, \times)$ under complex multiplication has two one-parameter subgroups: the real line ...
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votes
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answer
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Quaternion with only two imaginary numbers
Why do quaternions not only use two imaginary numbers. Can we not simplify quaternions
$$q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \tag{1}$$
to the form
$$
\begin{align}
q & = a + b\mathbf{i}...
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vote
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Fundamental theorem of algebra for Quaternions and Octonions -- and the completeness
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex $\mathbb{C}$ coefficients has at least one complex root. This includes polynomials with real ...
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Is there a reasonable limit to how far you can generalise complex numbers? [duplicate]
Real numbers satisfy a(bc) = (ab)c as well as ab = ba. They are also comparable.
Generalising to complex numbers, everything stays the same, except the numbers lose their comparibility.
Generalising ...
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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 ...
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Why this is the octonions? [closed]
The multiplication table is different from that of the octonions what I know. I tried to find suitable basis change for which the multiplication table coincide with wikipedia, but no success.
- 3
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Quaternionic and octonionic analogues of the Basel problem
It is a well-known fact that
$$\sum_{0\neq n\in\mathbb{Z}} \frac{1}{n^k} = r_k (2\pi)^k$$
for any integer $k>1$, where $r_k$ are rational numbers which can be given explicitly in terms of Bernoulli ...
- 5,731
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vote
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answers
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Interpolation Scheme Between a set of Quaternions
My question is similar to Quaternion barycentric interpolation, but since that question is 6+ years old, I figured it could be asked again. I also will eventually be trying to do this with unit ...
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votes
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answer
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Proving that the Octonion Norm Respects Multiplication
I'm working with the following definition of the octonions: $\mathbb{O} = \mathbb{H} \times \mathbb{H}$, endowed with the product $$(p,q)(r,s) = (pr - sq^*, p^*s + rq).$$ Conjugation is given by $(p,q)...
- 2,169
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vote
2
answers
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views
Properties lost when going from real number system to quaternions and octonions
What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?
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Is there a classification of bivector-valued cross products?
Background
Vector-valued cross products
Let $\mathbb{F}$ be an field of characteristic $0$. A $k$-ary cross product in the vector space $V=\mathbb{F}^n$ equipped with an inner product $\langle \cdot, \...
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A doubt from book Spinors and Calibrations
In chapter 14, the author shows that the octonionic projective plane is the quotient of exceptional Lie group $F_{4}$ by the group $Spin(9)$ (Theorem 14.99). When he is leading with an element that ...
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who pioneered the study of the sedenions?
The nature of this question is pure historical curiosity.
I found lots of background information about the discovery of both Imaginary and Complex Numbers, and enough information about the first two ...
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answers
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Normed Division Algebras: Why those dimensions? [duplicate]
I recently read a fascinating article about string theory, which discussed higher-dimensional algebras and their applications to supersymmery. The author mentioned that there were only four algebras ...
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An octonion generalization of the usual quantum angular momentum operators.
I'm working through, developing the formalism for an octonion generalization of angular momentum operators, but the whole time I'm thinking that this has probably been done elsewhere. My searches ...
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Octonion Algebras Over Number Fields
Is there any textbook or paper about Arithmetic of Octonion Algebras or Octonion Algebras constructed over number fields?
I know J. Voight book and K. Martin notes about quaternion algebras but I was ...
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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 $\mathbb{R}^7$ ...
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Real, Complex, Quaternionic and Octonionic Projective spaces
Do Octonionic Projective spaces exist or defined similar to $\Bbb RP^n$, $\Bbb CP^n$, $\Bbb HP^n$? If so, are they symmetric spaces?
I am asking this question because I've never seen Octonionic ...
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answers
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Why do $n$-dimensional numbers follow a geometric sequence?
I'm new here so please forgive my poor formatting and understanding.
I'm trying to understand why $n$-dimensional numbers follow a sequence. For example, octonions are $8D$ numbers that (I assume) ...
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A real example of an Octonion product
Goal: find the general Octonion multiplication product like the Quaternion formula given here:
https://en.wikipedia.org/wiki/Quaternion#Multiplication_of_basis_elements
...
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Subalgebras of the Real Octonions under Multiplication
I am interested to know what the subalgebras of the octonions look like. I tried searching for it but to no avail. What did show up was "Subalgebras of the Split Octonions," which was quite nice. Yet, ...
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Max number of solutions to $\exp\bigg(\frac{\rho}{\log(x)}\bigg)=x$ for complex, quaternion and octonion numbers?
If $\rho$ is a positive real number and $x$ is a real variable, I have verified that the following equation should have exactly two solutions, related to each other by being reciprocals of each other.
...
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7D octonionic cross product versus bivector cross product
Why the octonionic cross product product is not "unique" up to some permutations in the octonionic multiplication table BUT the bivector cross product in 7D is unique and isomorphic to one of those ...
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Construction of the Octonions
I get how to construct number systems: $\Bbb{N,~Z,~Q,~R,~C}$. But what about the hypercomplex? How do we go from $\Bbb C$ to $\Bbb H$ and $\Bbb O$?
I've also heard it stated that the octonions are the ...
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Commutation or Anti-commutation of the corresponding imaginary units of the octonions and split-octonions
Given a general octonion x:
$\mathbb{O}$=$\mathbb{H}$+$\mathbb{H}$$L$ by
x=$x^1$+$x^2$i+$x^3$j+$x^4$k+$x^5$i$L$+$x^6$j$L$+$x^7$k$L$+$x^8$$L$
with $L^2$=-1,
and a general split-octonion x:
$\...
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Java Library that supports Quaternions Octonions, Sedenions?
I would like to experiment with multi dimensional complex numbers such as quaternions octonions, sedenions.
I know Apache Commons Maths supports Quaternions, and I've found (although cannot download) ...
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Hermitian Octonion matrices?
The space of $n\times n$ octionion matrics $A$ such that $AA^*=A^*A=I_n$ is not a Lie group due to lack of associativity. But is it a smooth manifold? What is its dimension?
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Octonion Grassmanian?
I just learnt that the octonion projective space $\mathbf{OP}^n$ exists only for $n=1,2$. How about the "octonion Grassmannian", i.e. the space of $k$-subspace of $\mathbf O^n$? For which values of $(...
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The isotropy subgroup of the action of Spin(7) on the Grassmannian G(3,8)
The group Spin$\left( 7\right) \ $is the universal cover of $SO\left(
7\right) $ and is characterized as the subgroup of $SO\left( 8\right) $
consisting of the automorphisms of the triple cross ...
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votes
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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 ...
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If there can exist a model of the octonions without complex numbers
I know the ordering of complex numbers -> quaternions -> octonions, so the octonions are basically built on top of the complex numbers. But I am wondering, still, if there is a way to reformulate them ...
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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^...
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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 $2\times 2$ ...
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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 ...
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Why is not possible to show that $S^7$ is a Lie Group in the following way?
I am taking a course about smooth manifolds following Elon Lages Lima's "Variedades Diferenciáveis" with Lee's "Introduction to smooth manifolds" (loosely following as a supplement) and I've stumbled ...
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