> This is lovely!

Thanks.

> I went to a talk by Dr Alan Norton (I think that was his name) almost 20
> years ago where he showed pictures similar to this he'd rendered at IBM
> Bell Labs in USA. Ever since then I've wanted to attempt something like
> that but unfortunately my mathematical ability is pathetic. He used 3d
> "cross-sections" thru a 4d mandelbrot object calculated using quaternions.

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.

> By the way, did you intend to send the source with this? If so only the
> render text output came. Do you intend to post the source? I would *love*
> to look at it.

To draw it - simple. (Thanks again to Tor for pointing out this more 
efficient method.) To make it go fast - well, that's much harder! Take a 
look at my original post - 15 hours!!!

(I wonder how long it took the poor bloke from IBM! Admittedly he was 
probably using a custom-writted tool highly optimised just for this 
particular task. But even so, it can't have been fast...)

I will post the exact source if enough people are interested. It's not 
complex.

Andrew @ home.

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