I am trying to find someone who can render a 3D image for me, since I
don't have the knowledge or tools to do so myself.
The image is a 3D mandelbrot set, created within a 3D space which uses
an extension of the algebra for complex numbers. The third dimension
uses a j component, where j^2 = -1 and ij = 1.
The formula f(n) = f(n - 1)^2 + k, f(0) = 0 is still used, but this time
you have f(n) and k as points in 3 dimensions and not 2.
To implement one iteration, lets say f(x-1) = a + bi + cj and k = x + yi
f(x) = (a + bi + cj)^2 + x + yi + zj
= a^2 + 2abi + 2acj - b^2 + 2bc - c^2 + x + yi + zj
= (a^2 - b^2 - c^2 + 2bc + x) + (2ab + y)i + (2ac + z)j
I suspect either Mathematica or POV-Ray would be able to render such an
image, but neither I nor my friends have been able to work out how. I
have been able to produce cross-sections.
doesn't i == j then? I am curious, could you expand on this?