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 $2^n – ion$ product. I tried it out for 3 dimensional cross product (as imaginary part of quaternion product) […]

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 $S^1\to\mathbb RP^{2n+1}\to\mathbb CP^n$ and $S^2\to\mathbb CP^{2n+1}\to\mathbb HP^n$. Question. Does octonionic Hopf fibration $S^7\to S^{15}\to\mathbb OP^1$ give rise to a fibration $\mathbb HP^3\to\mathbb OP^1$? (On one hand, constructing a map $\mathbb HP^3\to\mathbb OP^1$ […]

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 complex order derivative valid without restrictions for all complex number z. I mean: Is a well-defined definition […]

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 numbers?

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 this for the exceptional algebras $\mathfrak{f}_4, \mathfrak{e}_6, \mathfrak{e}_7$ and $\mathfrak{e}_8$ as isometry groups of “projective planes” […]

… 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 real-world uses for them? For example, quaterions have a nice connection to computer […]

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 $n<4$ instances: As we move from $\mathbb{R}$ to $\mathbb{C}$ we lose ordering From $\mathbb{C}$ to $\mathbb{H}$ […]

Motivating question: What lies beyond the Sedenions? I’m aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$ “Reals” $\subset$ “Complex” $\subset$ “Quaternions” $\subset$ “Octonions” $\subset$ “Sedenions” $\subset$ $\ldots$ and that at each step you’re given a multiplication table […]

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, the only composition algebras are of 1, 2, 4, and 8 dimensions. What’s so special about 8?

Intereting Posts

Show there is a continuous isomorphism $l_{\infty}\rightarrow \left(l_1\right)^* $
Does there exist a bijection between empty sets?
Can two analytic functions that agree on the boundary of a domain, each from a different direction, can be extending into one function?
Can I use the “Secretary Problem” to find the worst candidate, too?
Why is the error function defined as it is?
Question on using sandwich rule with trig and abs function to show that a limit exists.
I need help with Differential equations.
Positive definite matrix must be Hermitian
Coupon Problem generalized, or Birthday problem backward.
What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?
Groups such that any nontrivial normal subgroup intersects the center nontrivially
For $k<n/2$ construct a bijection $f$ from $k$-subsets of $$ to $(n-k)$-subsets s.t. $x\subseteq f(x)$
Confusion related to the definition of NP problems
Reflection principle for harmonic functions
If $X$ is an infinite set and there exists an injection $X \to \mathbb{N}$, is there also a bijection?