z' = z² + c produces Mandelbrot or Julia sets, depending on whether inputs are used for c or for the initial z. What matters is *orbit stability* - values that don’t slingshot toward infinity when iterated. Iterations are discrete, can we make these orbits continuous? (1/3) This seems like a job for Lie algebras, which use the exponential map to turn a discrete transform into a geodesic direction to follow for a unit of distance - this distance can then be scaled. Take the log of a transform, scale the resulting tangent, then re-exponentiate. (2/3)

I’ve been working on geometric reasoning for complex quaternions. Here’s where I’ve landed.

The real quaternion part describes an ordinary 3D rotation. The imaginary quaternion also performs an ordinary 3D rotation but subsequently turns the result inside out. You can use complex phase to blend between a unit real and a unit imaginary quaternion transformation. The real part of transformed vectors will squish through a mirror from the right handed orientation to the left handed orientation as the phase goes from real to imaginary.

In 4D the 24-cell arises by taking three copies of the tesseract and aligning each set of 8 cell centers with half the vertices of the other 2 copies.

5 copies of the 24-cell then yield the 120-cell in direct analogy to the way that vertices from 5 cubes can make a dodecahedron. In searching for geometry representing the space of octonions we quickly encounter the fano plane and its group of 168 automorphisms. This looks like a hyperbolic 2D tiling of 24 heptagons, which seems suggestive. Do 7 copies of the 24-cell weave together to form an 8D polytope?

The surfaces of 4D shapes are 3D, so we can visualize them nicely by either flattening them or unfolding them. But in higher dimensions even the surfaces of objects have too many dimensions to visualize clearly. Is there a good way to stay sane and go higher? (1/17) Let’s look at an analogous problem: understanding a 4D hypercube surface in 2D. Is there a way we can make a 2D structure to represent our 8 cubic cells and the connections each has at its boundary to six adjacent neighboring cells? Maybe! (2/17)

Today I noticed that a change of basis of a 3D multivector (i.e. complex quaternion) which obtains idempotent/nilpotent ‘light cone variables’ (elements squaring to themselves or zero) is equivalent to a choice of 2x2 complex matrix representation! (1/6) The 2x2 matrix is a place to measure spin up and spin down as your two orthogonal basis elements, and your choice of axis determines which direction D lands on the main diagonal as (1+D)/2 and (1-D)/2 whilst the orthogonal components get mixed into the off diagonal elements.

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.