Thread by @JasonHise64

The complex quaternions are more than a 2D phase rotation and a 3D spatial rotation. They are more than the sum of their parts. Today I want to talk about one such feature that you can’t get from complex numbers or quaternions on their own. I want to talk about projectors. (1/15)

A complex quaternion has real scalar and bivector parts and imaginary scalar and bivector parts. Instead of breaking these up into four complex pieces or two quaternion pieces we can instead mix and match, affixing the imaginary bivector to the real scalar and vice versa.

These mixed ‘paravectors’ aren’t quaternions, and complex paravectors are something new. The scalar acts special… almost ‘time-like’. A rotation between the scalar direction and any vector direction has changed from being elliptic to being hyperbolic.

Hyperbolas have asymptotes, and this is where our ‘projectors’ lie. When the scalar part dominates the paravector is ‘time-like’, when the vector magnitude dominates the paravector is ‘space-like’, and when they perfectly balance the paravector is ‘light-like’.

Pure scalars and vectors are unstable equilibria, but light-like paravectors are attractive. If a paravector has any scalar content and any vector content, repeatedly squaring this value will rapidly approach having equal parts scalar and vector as the asymptote is approached.

The interaction between the scalar magnitude and the vector magnitude, when squaring, is identical to the interaction between x and (1-x) in the logistic map.

When scalar and vector are balanced your paravector becomes ‘idempotent’. Squaring it just scales it. This makes it a projector: multiplication by other complex quaternions retains the ‘half’ of them with which the projector is ‘parallel’. Negating the vector gets the other half.

Consider the paravector (1+z)/2, which squares to itself. It will keep half of anything projected by it. Where’s the rest? You can pull it out with the conjugate paravector: (1-z)/2. Your choice of which direction to pair with your scalar determines how you split the algebra.

Now consider a complex paravector. a(1+z) + b(xy+xyz). It represents rotation about some circle. It comes with a conjugate, c(1-z) + d(-xy+xyz), representing rotation about ‘the other, orthogonal circle’. How can we understand this?

Orthogonal circles show up naturally on the 4D sphere where quaternions live. Two parts can rotate into each other, and the other two components can freely perform their own independent rotation. The boundary between where one orbit dominates over the other is a flat torus.

Our orthogonal circles, projected from (1+z)/2 and (1-z)/2, can live in the middle of opposite sides of this boundary. They can rotate at independent rates, and the ratio of rotation rates will trace out a rational knot at the boundary.

Applied repeatedly, values slightly on one side of the boundary or the other will quickly slip downhill until they only orbit the circle initially closer.

We can freely rotate in 3D space to choose a projection direction to use to split the algebra. But we can also take advantage of the split to isolate an arbitrary component whose overall complex phase we want to rotate one way while the complement rotates the other way.

Combined this gives us all of the degrees of freedom required by SU(3). The two complex degrees of freedom at the center map to orthogonal rotations on the torus surface.

Then the 3D spatial rotation with isolated component phase rotation can provide the 6 remaining degrees of freedom. Real and imaginary rotation can be shared rigidly or interleaved twistedly.