> Yes, POV-Ray has the ability to slice 3D cross sections of various 4D 
> Julia sets - but not Mandelbrot sets.
> 
> Quaternions (and hypercomplex numbers) are ways of taking any fractal 
> defined in ordinary complex numbers, and doubling the number of 
> dimensions. The usual quadratic Mandelbrot set is 2D, so if you compute 
> it using quaternions or hypercomplexes, you get a 4D figure. 
> Unfortunately, IIRC it's just a surface of (double?) rotation...
> 
> My image is basic on the *cubic* Mandelbrot set. Without launching into 
> a huge maths lecture... your generic quadratic equation is something 
> like Ax^2 + Bx + C, where A, B and C are constants. However, more any 
> values of A, B and C, you can an image that is a rotated and/or scaled 
> version of something you could compute with z^2 + c. Thus, this 
> simplified formula is general enough to demonstrate anything worth seeing.
> 
> Since there is 1 (complex number) parameter, that means the 
> parameter-space (in which the Mandelbrot set lives) is 2D.
> 
> Similarly, your generic cubis is something like Ax^3 + Bx^2 + Cx + D. 
> However, anything this can draw can also be drawn (maybe rotated and/or 
> scaled) by z^3 - 2(A^2)z + B, so that's what they use. Notice that the 
> dynamic space still consists of 1 variable - z - so the Julia sets are 
> still 2D. However, since there are now *two* parameters, both 
> complex-valued, the Mandelbrot set is 4D. And it's not just a surface of 
> revolution; there are REAL DETAILS in all 4 axis.

OK, *how many* typing mistakes are there in there?! Oh dear...

Most significantly, it's z^3 - 3(A^2)z + B. (i.e., the linear 
coefficient is multiplied by *3*, not *2*!)

Also forgot to mention... to compute the quadratic Mandelbrot set, 
iterate the critical point 0 and see what it does. However, to compute 
the cubic Mandelbrot set, you must example *two* orbits - these are +A 
and -A. (Don't use 0; it renders wrong.) Some authors define a set 
called M+ = {all (A, B) where the orbit of +A is bounded} and another 
set called M- = {all (A, B) where the orbit of -A is bounded}. The usual 
Mandelbrot set M is then the intersection of these two sets.

The set I pictured is just M+; I have some neat pictures showing M+ and 
M- on the same plot...

Andrew @ home.

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