3d combination of lots of 2d hypercomplex mandelbrots

[qooqoo]

if you've read about hypercomplex numbers from wikipedia, you know that the complex numbers aren't the only geometrically interesting (and algebraically useful) commutative rings that can be expressed by 2x2 matrices.

in fact, i^2=k where k=-1 is just one possibility.

[main idea]
so has anyone tried making a 3d model where the z-coordinate is the value of i^2. i^2 = z
this way, every z = k plane (where k is any real) represents one hypercomplex ring, and any given object is interpreted within the laws of that ring.
if that object is the mandelbrot set, then for z=-1, you have the regular complex version, but for z = 0 you have a plane segment and for z = 1 you have another deformation of the complex number form into the i^2=+1 version.
[/idea]

this object is mathematically interesting because it surveys lots of possible interpretations. since the mandelbrot set doesn't make any specific use of the unique algebraic properties of the complex numbers, why should it belong solely to that ring?
the same extension should be possible for julia sets.

i'm trying to do this in pov-ray. i haven't really learnt to use it properly and it isn't coming out well.
has anyone tried this before? it's very likely yes. can anyone try rendering it now?
does it make a detailed 3d fractal?

[edit]
i'm sorry if this is unclear and you don't get it.
see below for a formula.

should i have posted this in the mandelbulb section?

[JodyVL]

*sigh* .. I truly wish I understood what you meant...


 

[qooqoo]

here's a formula.

(x,y,z)^2 = (x^2 + z*y^2, 2*x*y, z)

|(x,y,z)|^2 = x^2 - z*y^2

iteration Z = Z^2 + c
using above.
same distance^2 limit of 4 where distance is |(x,y,z)| (i guess)

[DarkBeam]

I really think that z can't stay unchanged if you are doing a 3D squaring (x,y,z)^2 = (x^2 + z*y^2, 2*x*y, z)
Anyway welcome

[qooqoo]

I really think that z can't stay unchanged if you are doing a 3D squaring (x,y,z)^2 = (x^2 + z*y^2, 2*x*y, z)

z stays unchanged.

Anyway welcome

:)thanks

[DarkBeam]

I really think that z can't stay unchanged if you are doing a 3D squaring (x,y,z)^2 = (x^2 + z*y^2, 2*x*y, z)
Anyway welcome

z stays unchanged.

Most likely you have expanded something in the wrong way, every known fractal formula (2d, 3d, .........) actually changes all axis values at each iteration. Anyway good luck

[fractower]

This method seems to use z as a parameter rather than a vector element.

[David Makin]

I think all these have been tried (in 3D and 4D) where all rows/columns are rings with a to h being appropriate unit vectors (R,I ,J,K).

Code:

*   R   I   J   K
R   R   I   J   K
I   I  -R   a   b
J   J   c   d   e
K   K   f   g   h

Break the ring rule and you have a form that's virtually impossible to use in a sensible manner (even the standard Mandelbulb triplex is pretty bad from that point of view).
Change R*x = x but maintaining the ring rule is an interesting idea but obviously violates the rules of both real and complex numbers since a pure real squared no longer results in a real.

[Syntopia]

This method seems to use z as a parameter rather than a vector element.

Yes, it seems to be a parametrized family of 2D number systems, and not really 3D numbers.

For instance, you also have addition in the Mandelbrot formula - and if you want z=-1 to be the standard Mandelbrot plane, you need to define addition such that only the first two components are added.

You can of course still visualize the different 2D families layered on the z-axis. But if you try visualizing the different z-planes in 2D (which is easy), you see that the z-parameter just seem to stretch the Mandelbrot set (at least until it changes sign). So the 3D model will look stretched in the z-direction - I don't think it will inhibit any interesting 3D fractal structure.

[qooqoo]

so for i^2<0 it's just a stretched complex plane.
for i^2=0 it's all (x,y) where -2<=x<=0.25
for i^2>0 it's a stretched split-complex plane.
is that a summary of it and there's nothing else going on? is it always just a stretch of one of three forms parallel to one axis?
is the only escape if the f in i^2=f(z) is discontinuous?

can anyone still show how the split-complex mandelbrot set evolves from the dual mandelbrot. in other words, letting i^2 be part of the whole region [0..1]

[qooqoo]

for i^2=z
it's a stretch from the complex number form by -1/z as long as z<0
looking at it sideways would show half a rectangular hyperbola.
sucks