# 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:

$$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions}$$

or:

"Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions"

With the following "properties":

• From $\mathbb{R}$ to $\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering).
• From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity.
• From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity.
• From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.

The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ?

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One useful generalization starting from $\mathbb{H}$ and extending to all powers of $2$ is Clifford algebras: en.wikipedia.org/wiki/Clifford_algebra . You can also keep applying the Cayley-Dickson construction past $\mathbb{S}$ (en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) although I don't know if this is useful. – Qiaochu Yuan Nov 28 '11 at 17:13
Dragons. Beyond sedenions there are dragons. Beware. – Mariano Suárez-Alvarez Nov 28 '11 at 17:15
If you want to keep with your "throw away" theme, then from $\mathbb{R}$ to $\mathbb{C}$ you lose the ordering – Jason DeVito Nov 28 '11 at 19:23
(somewhat) related: mathoverflow.net/questions/19929/19975#19975 – Grigory M Nov 28 '11 at 23:02
@Jason: Thanks, I have add this to the first action from $\mathbb{R}$ to $\mathbb{C}$) – Willem Noorduin Nov 29 '11 at 9:26

What you are talking about is precisely the Cayley-Dickson construction.

Remark: I am left wondering what is gained by going past Octonions. The the first 4 are very special as they are the unique 4 normed divison algebras over $\mathbb{R}$. Perhaps someone with more knowledge can point out the possible uses of the Sedenions and their higher counterparts.

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Wikipedia states, "The Cayley–Dickson construction can be carried on ad infinitum, at each step producing a power-associative algebra whose dimension is double that of algebra of the preceding step." – ttt Nov 28 '11 at 17:27
As for applications of sedenions, I am quoting Wikipedia: "Moreno (1998) showed that the space of norm $1$ zero-divisors of the sedenions is homeomorphic to the compact form of the exceptional Lie group $G_2$." – darij grinberg Nov 28 '11 at 17:44
@darij: that can't be right. Clifford algebras are associative and the octonions aren't. – Qiaochu Yuan Nov 28 '11 at 19:30
Oh right, I mixed up things there. Comment deleted. – darij grinberg Nov 28 '11 at 23:12
Just for info, after Degen's 8-square, there is the non-bilinear Pfister's 16-square identity. – Tito Piezas III Aug 14 '13 at 4:21

If with "beyond" you mean something like "after" the answer is yes: the Cayley-Dickson's construction has no end and you can always extend your number system to a higher dimension one (always with $2^n$ dimensions with $n\in \Bbb N$), after Sedenions $\Bbb S$ come Trigintaduonions $\Bbb T$ which someone say are useful in electrical and computer engineering (I can't really figure out how) but they don't loose any other properties more than Sedenions do, this set and the others that came after it are not very interesting (with an only mathematical point of view) cause of that, they just have some subgroups that look like the Sedenions.

If with "beyond" you mean something like "other than" the answer is still yes: there are a lot of hypercomplex numbers with 16 dimensions system like the extension of tessarines, duals and hyperbolic number and so on.

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I believe John Baez has answered your question in a series of short articles/blogposts which you could begin for example here.

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