Questions tagged [sedenions]

The sedenions are a 16 dimensional nonassociative algebra over the reals.

Filter by
Sorted by
Tagged with
2 votes
0 answers
56 views

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 ...
Efesto's user avatar
  • 1
79 votes
0 answers
946 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 ...
pregunton's user avatar
  • 5,791
3 votes
1 answer
186 views

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 ...
a196884's user avatar
  • 283
2 votes
0 answers
71 views

Multiplication identities of Cayley-Dickson algebras after the sedenions

Consider the multiplication reducts $\{*\}$ of Cayley-Dickson algebras $(X;+,-,*,0,1)$ over the real numbers $\mathbb{R}$. In both the real numbers and the complex numbers, multiplication satisfies ...
user107952's user avatar
  • 20.2k
1 vote
0 answers
56 views

What comes in the next several K-D steps after the sedenions, and what is lost?

Wikipedia and elsewhere seem to say that one can keep on extrapolating forever in hypercomplexification, but that you progressively lose operation-equative symetries or whatever you call, e.g. ...
Ayer AGG'TDd'E-A's user avatar
2 votes
1 answer
102 views

Is division by a null sedenion a valid operation?

So octonion set provides the largest normed division algebra, and starting with sedenions, Cayley-Dickson construction provides algebras with zero divisors. From what I understand, it means there are ...
psychoslave's user avatar
2 votes
2 answers
268 views

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 ...
Mr. J. Larios's user avatar
3 votes
0 answers
131 views

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) ...
Hector's user avatar
  • 219
7 votes
2 answers
513 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. ...
Omega Force's user avatar
6 votes
0 answers
200 views

Are complex split-octonions isomorphic to a more easily-defined algebra?

I write fiction and nonfiction, both which are mathy. My fiction is not usually supermathy but I'm working on a fictional story that has some math in it, and I prefer accuracy to mathbabble. I'm ...
Trixie Wolf's user avatar
33 votes
2 answers
4k 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 $2^n$ ...
Pat Muchmore's user avatar
4 votes
3 answers
853 views

Math beyond Quaternions

Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
BAR's user avatar
  • 179
27 votes
2 answers
4k 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}...
Hooked's user avatar
  • 6,617
66 votes
4 answers
13k views

What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: $$2^0 \mathrm{-ions} \subset ...
Willem Noorduin's user avatar