# 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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### 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 ...
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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 $2^n$ ...
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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 ...
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### 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. ...
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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 ...
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### What is a good book on general octonion algebras and the Cayley-Dickson construction?

I want a good reference that discusses properties of octonion algebras, especially over number fields. I'd like to know more about how this generalizes when applying the Cayley-Dickson construction ...
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### Math beyond Quaternions

Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
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### Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
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### Using the Fano plane for octonion multiplication

The Fano plane is the projective plane over the field $\mathbf Z/2$. It can be used to remember octonion multiplication, as nicely explianed in John Baez's article on octonions (see http://math.ucr....
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### Is there a way to introduce quaternions and octonions in a similar way to how we are typically introduced to complex numbers?

So I've been reading a little bit into ideas around quaternions and octonions. I just read the following explanation that introduces them as what happens when you have complex numbers and you then ask ...
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### Nonzero Octonions as a 7-sphere

While reading about Moufang loops in the book "An introduction to Quasigroups and their Representations" by Smith, I've encountered the following statement: The set $S^7$ of nonzero octonions of ...
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### Understanding terminology of, fibers, clutchings and Hopf.

I have some questions regarding the terminology of fiber bundles as used in section 3 of this paper; http://www.sciencedirect.com/science/article/pii/S0723086907000151 The section starts off by ...
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### Do the octonions form a field?

The octonions are a noncommutative nonassociative normed division algebra over $\mathbb{R}$. Multiplication distributes over addition. Somehow, the existence of a norm implies the existence of ...
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### Unit or non-zero octonions form an $A_\infty$-space?

If I have a Moufang loop, can it have a classifying space? I'm thinking of the unit octonions, if that's too general, so perhaps a better question is: are the unit (or non-zero) octonions an $A_\infty$...
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### Pfister's 16-Square Identity and the norm of sedenions

Consider the sequence of numbers: complex numbers $\Bbb C$, quaternions $\Bbb H$, octonions $\Bbb O$, and sedenions $\Bbb S$. The Brahmagupta-Fibonacci 2-Square identity implies that the norm of the ...
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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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### Mirror symmetries of the Albert algebra

A simple Euclidean Jordan algebra (i.e. a factor) is either a spin-factor, the matrices over the reals/complex-numbers/quaternions or the exceptional Albert algebra of 3x3 octonian matrices. My ...