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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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 $n<4$…
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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…
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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: $2^4 = 16$…
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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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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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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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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…
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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,…
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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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