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Octonians and 4D vector products

As you probably know, octonians (https://en.wikipedia.org/wiki/Octonion) were invented by Graves in 1843 – not long after quaternions were invented by Hamilton. They’re not associative. But they’re also not exactly the same as the product of two 4D vectors. I find that puzzling. Complex numbers are exactly the product of two 2D vectors and quaternions are exactly the product of two 3D vectors. Why do octonians break this pattern?

4D vector products seem quite similar to octonians, but the eighth basis, e1e2e3e4, squares to positive one instead of negative one. That seems to be the main difference, but maybe I’m missing others?

What is special about 4D in this regard? And is the difference related to the reason there are no other division algebras? I note that 4D is the first place where non-blade bivectors are possible (example: e1e2 + e3e4). Is that part of the answer? Do we need Galois Theory to untangle this?

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I investigated a similar topic involving the similarities and differences between octonions and 3D multivectors (as opposed to the 4D rotors you are describing) a year or two ago which is how I stumbled onto the fact that 3D multivectors are isomorphic to complex quaternions, with imaginary quaternions living in the odd-grade elements and performing hyperbolic rotations instead of the usual elliptic rotations that quaternions induce. (Bonus fact: they are also isomorphic to complex 2x2 matrices!)

The most important takeaway for me was that unit octonions topologically live on a high dimensional sphere, with indistinguishable basis elements that all square to -1 and which all perform symmetrical actions. In contrast the unit 3D multivectors live on a high dimensional torus, sweeping the hypersphere of unit quaternions around a 2D complex circle similar to how a normal 3D torus differs from a 3D sphere by sweeping a circle around another circle.

The topological hole in the torus creates a split between the actions - you can’t continuously rotate an action that orbits the hole into an action that rotates orthogonal to the hole. In much the same way, the quaternion 3D rotation action and the complex action that rotates odd graded elements into even graded elements and vice versa are split and act independently.

You noted that with the 4D even subalgebra there is an element that squares to +1 instead of -1. This is the critical feature that makes it not a division algebra, because it introduces the ability to construct elements at 45 degrees between squaring to +1 and squaring to -1. It introduces a lightcone of nilpotent elements that divide your space, a topological hole that splits your algebra.

You also noted that vector products cannot produce all 4D bivectors, and non-blade bivectors are possible. Consider that you also can’t construct your 8th basis vector. The vector product just gives you six bivector components and a scalar component - a seven dimensional space of outputs that is not closed.

As the dimensionality of your space increases, the number of grade-1 elements grows linearly but the number of higher grade elements grows exponentially. 3D is the inflection point, the highest dimension where the space is still small enough to describe everything in terms of linear components. The space is just the right size to accommodate the notion of an ‘axis of rotation’ that conflates the planes of rotation with directions of translation. It’s exactly the right size to conflate 3D volume elements with scalars, and have no other elements left over to describe or express.

Once you hit 4D and go beyond, the number of ways to transform the space explodes compared to the dimensionality of the space. Describing operations in terms of their linear components simply becomes inadequate. The isomorphism between translation and rotation breaks down. The isomorphism between higher grade elements and scaling breaks down. Products between vectors are no longer sufficient to completely describe or understand your space.

Nonetheless, it seems fair to say that vector products form the roots from which the rest of geometric algebra stems. The grade 1 elements and the reflections they describe are the foundation, the atoms out of which the fractal complexity of the higher dimensional algebra emerges. Clifford algebras explore linear transformations. They explore actions you can describe with matrices, actions that preserve the overall structure and adjacency of your geometry. While the Cayley–Dickson construction has its own fractal complexity, there’s a good reason that it diverges from the Clifford algebras when it hits octonions. It is exploring something fundamentally different.

At first glance the foundation of the Cayley-Dickson construction seems to be about rotation since it builds algebras recursively from the complex unit circle, but that’s not really right. Planes of rotation are still fundamentally flat things. In GA when you perform a rotation the shape is maintained, but some points move more than others. Components that are not part of the rotating subspace don’t even see the action happen. To any point on the z axis, any rotation in the xy plane is indistinguishable from doing nothing. Even for points that do move, knowing where they started and where they ended is not enough to know what rotation took them there. A point moving from the x axis to the y axis could have gotten there by rotating 90 degrees about the z-axis, but it just as easily could have rotated 180 degrees through a perpendicular axis on the x-y diagonal.

What the Cayley-Dickson construction is instead exploring, what it is really about, is geodesic flow. It looks at actions that rotate every point on the surface of a sphere at the same rate. Actions that don’t look like the identity to any of the points on the surface. Actions that can be uniquely determined from the vantage of any point that knows where it started and where it ended up. Unit quaternions look like rotations when applied using the sandwich product common in GA, but if you just look at how quaternions transform each other with a single multiplication then it is more natural to look at the motion as a turbulent vortex. It’s a twisting rotation of the fibers of the Hopf fibration.

There’s still room with the quaternions to project down to 3D and see their action on each other as a kind of screw transformation. You can pretend one of the circles is rotating through a point at infinity so it looks like translation along a straight line, and the other circle is rotation that you are performing orthogonal to your translation. There’s still psychological room with projective geometry to see the action as something flat. But all of that ends when you step up a dimension and finally reach the octonions. No amount of mental gymnastics is sufficient to untwist the geometry of that algebra and stretch the flow of the octonions out into something linear. The geodesic fibers are just woven together too tightly!

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