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Questions tagged [sedenions]

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

The sedenions are the 16 dimensional nonassociative normed algebra obtained by applying the Caley-Dickson-Construction to the octonions.

Every sedenion has a multiplicative inverse, but due to the nonassociative nature of the algebra, the algebra also has zero divisors. For this reason, sources disagree on calling it a "division algebra."

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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 algebraic properties at each step such as…
pregunton
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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 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions}…
Willem Noorduin
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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 $2^n$ dimensions. I think I understand the first $n<4$…
Pat Muchmore
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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} \subset \mathbb{O} \subset \mathbb{S} \subset…
Hooked
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Is there an equational identity that holds for the sedenions but not the trigintaduonions?

I asked a more general version of this question before a while ago, here: Do you lose any more equational identities when you go past sedenions?, but it didn't get any answers. So, I am asking a more specific question now. The sedenions $\mathbb{S}$…
user107952
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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. Octonions: $2^3 = 8$ dimensions. Sedenions: $2^4 = 16$…
Omega Force
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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 stepping outside my particular math domain so what I…
Trixie Wolf
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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 quaternions there are also two other fields called…
Efesto
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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 norm of a sedenion $x$, denoted $n(x)=x\bar{x}$, is…
a196884
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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?
BAR
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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 pairs of non-null sedenion $(s_a,s_b)$ for which…
psychoslave
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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) ca.uwaterloo.alumni.dwharder.Numbers.Sedenion Are…
Hector
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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 both the commutative and associative identities. In…
user107952
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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 types of Hyper Complex Numbers; Quaternions and…
Mr. J. Larios
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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. commutativity, associativity, power-associativity,…
Ayer AGG'TDd'E-A
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