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The quaternion/octonion extend the complex numbers, which extend the real numbers. So we go:

  • 1-tuple: Real numbers.
  • 2-tuple: Complex numbers.
  • 4-tuple: Quaternions.
  • 8-tuple: Octonions.

The Wikipedia link describes this doubling process:

In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one.

But if these are just vectors in the end, I wonder if there are vectors in the odd dimensions like 3, 5, etc., or other non-power-of-two dimensions like 10, 12, etc.. This way there would be a potentially more general construction describing the vector, and the power-of-two case would be a special case, sort of thing.

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    $\begingroup$ You can of course define a vector (a collection of numbers) in any dimension that you want. But the point is, for example in the case of quaternions, to construct some kind of algebra within the context of that kind of mathematical element. For example, it's possible to define a quaternion multiplication and addition, so that they can be used in meaningful ways. It turns out that the definition of quaternion product gives us also an inverse quaternion (!). In three dimensions, this is not possible to do. So to summarise: It all depends on what you want from the system. $\endgroup$ – Matti P. Mar 6 at 8:43
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    $\begingroup$ By Hurwitz's theorem, these are the only finite-dimensional unital real algebras endowed with a positive-definite quadratic form. $\endgroup$ – Fabio Lucchini Mar 6 at 8:48
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    $\begingroup$ We should remember this question: I'm pretty sure it generalizes a lot of "why isn't there stuff besides the Cayley-Dickson algebras" that we have seen before. It seems the scope is unlikely to become any broader. $\endgroup$ – rschwieb Mar 6 at 16:53
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    $\begingroup$ What about $0$nions? $\endgroup$ – Xander Henderson Mar 6 at 17:24
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    $\begingroup$ @EricDuminil - Hamilton himself spent about 20 years (IIRC) looking for a 3D equivalent to the complex numbers before he famously realized that 4D worked while on a walk and was so excited he carved the formulas defining quaternions into a bridge he was crossing. But he died in 1865 while Frobenius proved his theorem in 1877. $\endgroup$ – Paul Sinclair Mar 6 at 17:54
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The Frobenius theorem says that the only finite-dimensional associative division algebras over $\mathbb R$ are exactly $\mathbb R$, $\mathbb C$ and $\mathbb H$, up to isomorphism.

The octonions are not on this list, but they are not very well-behaved either; their multiplication is not even associative. And it only gets worse as you move further into the Cayley-Dickson sequence.

(As Fabio Lucchini points out, Hurwitz's theorem states that if you don't require associativity, but still want the algebra to have inverses and a norm that agrees with the multiplication, you get $\mathbb R$, $\mathbb C$, $\mathbb H$, and also $\mathbb O$, but nothing more).

At some point things get so bad that one may wonder if you would not rather consider, for example, $\mathbb R^3$ with the cross product. Or just declare any random bilinear map $V\times V\to V$ to be your multiplication.

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    $\begingroup$ While I'm sure you answer is correct, I don't know how broad of an audience can understand At some point things get so bad that one may wonder if you would not rather consider, for example, R3 with the cross product. I mean, maybe you can unpack that a bit? $\endgroup$ – ArtB Mar 6 at 14:46
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    $\begingroup$ @Art, Henning is asking you and other readers to consider the algebra of $3$-component vectors, where addition is as usual, and multiplication is done through the usual cross product. (The dot product produces scalars, so that's already not for consideration.) $\endgroup$ – J. M. is a poor mathematician Mar 6 at 15:06
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Henning Makholm's answer explains this well. But here's another way to think of it, if you want some intuition on why it works that way.

The real numbers, complex numbers, and quaternions are all related to geometric algebra. In particular, think about how complex numbers represent rotations and scaling in two dimensions: $i$ is a ninety-degree rotation in the plane. Similarly, quaternions represent rotations and scaling in three dimensions, with $i, j, k$ acting as ninety-degree rotations in three orthogonal planes ($xy, yz, zx$). And if you extend this, you can think of the real numbers as representing "rotations and scaling" in one dimension, where there's no plane to rotate in, so it's really just scaling.

What happens when you extend this to four dimensions and above? Well, rotations in four dimensions get complicated. In three dimensions and below, every rotation is "simple": it can be represented as a plane of rotation and an angle within that plane. In four dimensions, that stops working. Because in four dimensions, you can have two planes that aren't parallel and also don't intersect (such as the $xy$-plane and the $zw$-plane). If you rotate in both of those planes at the same time, there's no single "plane of rotation" any more.

The rotations-and-scaling representation in four dimensions would have eight elements $\langle 1, xy, xz, xw, yz, yw, zw, xyzw\rangle$, where the last one is what you get when you multiply $xy$ by $zw$ and can't really be visualized except as a "directed hyperspace". (Formally, it's a quadvector, aka a four-blade, but neither of those words really helps.) And because rotations are no longer simple, the math stops working quite as nicely, so this one doesn't get its own name: it's related to the octonions, but not quite the same.

You can keep going further, but the math just gets messier and messier and less and less elegant. If you try this in five dimensions, you get a relative of the "sedenions", with sixteen elements. And so on and so forth. The sedenions are useful if you want to work with rotations in five dimensions, but we don't generally need to do that, and they're really ugly and not that exciting in general. So most people just don't care about them.

If you want to look into this further, the "scalings and rotations" are formally called the even Clifford sub-algebra in $n$ dimensions, written as $Cl^{+}_{n}$ or $Cl^{[0]}_{[0,n]}(\mathbb{R})$. (It sounds a whole lot more complicated than it is.)

TL;DR: the reals, complexes, and quaternions have the number of elements they do because they correspond to rotations and scalings in a particular space. Other numbers of elements don't have that correspondence, and higher-dimensional spaces get messy and become less elegant.

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  • $\begingroup$ The even 4D Clifford algebra is $\mathbb{H} \times \mathbb{H}$, not $\mathbb{O}$ (in fact no Clifford algebra can be isomorphic to the octonions since they are all associative by construction). However, it is true that in 7D the even Clifford algebra is related to the linear transformations induced by octonion multiplication in $\mathbb{R}^8$ (this is described in detail e.g. here), so the octonions are still relevant to geometric algebra, although in a slightly more convoluted way. $\endgroup$ – pregunton Mar 6 at 20:03
  • $\begingroup$ @pregunton Oops, my mistake! I'll edit this later, or you're free to add a suggested edit. $\endgroup$ – Draconis Mar 6 at 20:11
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Of course, one can define real vector spaces $\Bbb R^n$ for any positive integer $n$. What distinguishes the real numbers ($\Bbb R$), the complex numbers ($\Bbb C$), quaternions ($\Bbb H$), and octonions $(\Bbb O)$ is additional structure, including a multiplication operation $\Bbb A \times \Bbb A \to \Bbb A$.

Hurwitz' Theorem states that (up to isomorphism) these are precisely all of the normed division algebras over $\Bbb R$. These are algebras over $\Bbb R$ with (1) the property that $x y = 0$ implies $x = 0$ or $y = 0$ and (2) a multiplicative norm $||\,\cdot\,||$ (so, satisfying $||x y|| \leq ||x||\,||y||$). (For a truly enjoyable introduction to this rich circle of ideas, I highly recommend John Baez' classic article The Octonions.)

One can still define, however, other canonical algebraic operations $\Bbb A \times \Bbb A \to \Bbb A$ on real vector spaces $\Bbb A = \Bbb R^n$ for various small $n$ that still have other interesting properties, and the normed division algebras and the Cayley-Dickson construction lead naturally to many of these.

For example, each of the normed division algebras is equipped with an (involutive) linear conjugation operation $\bar\cdot : \Bbb A \to \Bbb A$ compatible with the addition, multiplication, and norm operations (for $\Bbb R$ this is just the trivial map, for $\Bbb C, \Bbb H, \Bbb O$ these are just the usual conjugation maps). In each case, the $+1$-eigenspace of the conjugation map is just the copy of $\Bbb R$, and so the $-1$-eigenspace has dimension $\dim \Bbb A - 1$. We denote the latter space $\operatorname{Im} \Bbb A$ (so, as a vector space, it is isomorphic to $\Bbb R^{\dim \Bbb A - 1}$), denote the projection $\Bbb A \to \operatorname{Im} \Bbb A$ onto it by $\operatorname{Im}$, and call its elements imaginary. Now, $\Bbb A \times \Bbb A \to \operatorname{Im} \Bbb A$, $(x, y) \mapsto \operatorname{Im} (x \bar y)$, restricts to an new, skew-symmetric binary operation, $$\times:\operatorname{Im} \Bbb A \times \operatorname{Im} \Bbb A \to \operatorname{Im} \Bbb A .$$ In particular, these operations do not have identities (after all, the identity $1 \in \Bbb A$ is not imaginary).

For $\Bbb A = \Bbb R$ and $\Bbb A = \Bbb C$, this construction just leads to the zero map on $\Bbb R^0$ and $\Bbb R^1$---so, not very interesting.

For $\Bbb A = \Bbb H$, $\times : \Bbb R^3 \times \Bbb R^3 \to \Bbb R^3$ is nothing other than the familiar cross product, recovering Henning Makholm's suggestion. Tracing through definitions (and doing some easy calculations) shows that associativity of $\Bbb H$ implies the triple cross product identity $({\bf x} \times {\bf y}) \times {\bf z} = \langle {\bf y}, {\bf z} \rangle {\bf x} - \langle {\bf x}, {\bf z} \rangle {\bf y}$ on $\Bbb R^3$, and forming the cyclic sum of this identity in ${\bf x}, {\bf y}, {\bf z}$ gives the Jacobi identity, $$({\bf x} \times {\bf y}) \times {\bf z} + ({\bf y} \times {\bf z}) \times {\bf x} + ({\bf z} \times {\bf x}) \times {\bf y} = {\bf 0} ,$$ so $\times$ is actually the bracket operation of a real Lie algebra, namely $\mathfrak{so}(3, \Bbb R)$. (NB that $\times$ itself is not associative.) More generally, any property of the multiplication on $\Bbb A$ induces a property of $\times$ on $\operatorname{Im} \Bbb A$.

For $\Bbb A = \Bbb O$, $\times : \Bbb R^7 \times \Bbb R^7 \to \Bbb R^7$ is the somewhat exotic $7$-dimensional cross product. It does not satisfy the Jacobi identity, so it is not the bracket of some Lie algebra, but it is intimately related to (the compact real form) of the exceptional Lie algebra $\mathfrak{g}_2$.

We can generate more examples of canonical algebraic structures on $\Bbb R^n$ if we tweak the Cayley-Dickson construction by defining for a suitable algebra $\Bbb A$ the multiplication rule $$(a, b)(c, d) := (ac \color{red}{+} \bar d, da + b \bar c)$$ on $\Bbb A \times \Bbb A$ (replacing the red $\color{red}{+}$ with $-$ gives the usual Cayley-Dickson construction). If we successively apply this again to $\Bbb R$, we produce the split-complex numbers ($\widetilde {\Bbb C}$), the split-quaternions ($\widetilde {\Bbb H}$), and the split-octonions ($\widetilde {\Bbb O}$), again respectively of dimension $2, 4, 8$. These are not normed division algebras---instead of a norm, these are equipped with an indefinite quadratic form, making them composition algebras (and together with the four normed division algebras, this accounts for all real composition algebras up to isomorphism). These algebras, as the name and notation suggest, share many algebraic features with their nonsplit counterparts. Two are actually familiar: As $\Bbb R$-algebras, $\widetilde{\Bbb C} \cong \Bbb R \oplus \Bbb R$ and $\widetilde{\Bbb H} \cong M(2, \Bbb R)$ (the space of $2 \times 2$ real matrices). In particular, unlike the normed division algebras, these algebras have zero divisors.

The same construction as before now leads to a "split" cross product on $\Bbb R^3$ that can be realized as the Lie bracket on $\mathfrak{sl}(2, \Bbb R) \cong \mathfrak{so}(1, 2)$, as well as a "split" cross product on $\Bbb R^7$. In both cases these are inequivalent to the cross products described earlier.

Finally, something special happens in the split setting that has no analogue in the normed division algebra setting: Up to isomorphism, there is a $6$-dimensional algebra $\Bbb S \cong \Bbb R^6$, $\widetilde{\Bbb H} \subset \Bbb S \subset \widetilde{\Bbb O}$, unique up to isomorphism called, of course, the sextonions. It turns out (but is not immediately obvious) that the split $7$-dimensional cross product restricts to a binary operation on $\operatorname{Im} \Bbb S \cong \Bbb R^5$.

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    $\begingroup$ Nice answer! Since you're mentioning the sextonions, there is also the 3-dimensional subalgebra of "ternions" $\widetilde{\Bbb C} \subset \Bbb T \subset \widetilde{\Bbb H}$ arising from a similar split-null extension (which turns out to be isomorphic to the upper triangular $2\times 2$ real matrices) and a couple other subalgebras in 2D and 5D (I believe these are all but I don't know for sure, I asked a question about it here). $\endgroup$ – pregunton Mar 7 at 7:56
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    $\begingroup$ Thank you! I hope your question gets more attention---it's certainly an interesting one! I'll think about it and at least see whether I can come up with any examples not on your list. $\endgroup$ – Travis Mar 7 at 8:14

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