The more general class of 3d fractals

[cortexman]

I was trying to think of an alternate approach to this issue of creating a 3d mandelbrot, and one thing I am curious about is the general class of 3d fractals.

I was imagining a sphere packing algorithm that worked in 3d, squeezing in the largest spheres that it could at a given digit of precision.

This turns out to be the 3d form of what's known as the Apollonian Gasket. Link: http://mathworld.wolfram.com/ApollonianGasket.html

There is also an Apollonian fractal generator that creates this object: http://thomasbonner.heliohost.org/apolfrac.htm

If you could gain some insight into 3d fractals of this type, maybe you could apply them to the study of the 3d mandelbrot.

This raises the question of what is a fractal? The Apollonian Sphere Fractal is a somewhat complex algorithm which is in stark contrast to z = z^2 + c.

But still, it is a rule-based fractal and Mandelbrot even measured its fractal dimension.

My question is, are there any nice closed formulas that give rise to 3d fractals, and if not, could trying to come up with the equations that generate an Apollonian fractal be easier than continuing with the 3d mandelbrot? And if not the Appolonian, perhaps another type?

[Tglad]

[Incidentally the 3d appollonian gasket is an example of a 'fat cluster' if you're looking at the spheres, otherwise it is a 'thin foam' if you're looking at the border/surface of the spheres... using this classification:http://www.fractalforums.com/new-theories-and-research/categorising-fractals/msg22874/#msg22874 ]
p.s. there is no 3d mandelbrot, no equivalent of complex numbers in 3d, proven a century or two ago I think.

[hobold]

Tglad seems to be too modest ... there exists one type of fractal, discovered by him, that finally has all the necessary theoretical qualities in 3D which the classical Mandelbrot set has in 2D. It is the Mandelbox. It doesn't look at all like the cardioids and minibrots of yore. But just like the Mandelbrot set, the Mandelbox has detail everywhere, and the nature of shapes varies a lot as you move around the box or zoom in.

The Apollonian Gasket, in contrast, is more like a Julia set. It does display some new variety when you zoom in, but not really when you move around.

[Paolo Bonzini]

no equivalent of complex numbers in 3d, proven a century or two ago I think.

It depends on what you mean. Purely imaginary quaternions certainly provide a numeric representation of 3D space and you can operate on them (scale, rotate, translate) using operations on quaternions. You can also use them to operate on 2D space, thought it's a bit more clumsy than complex numbers. For example, if you transform a complex number a+bi to a quaternion q(z) = ai+bj, you have where , and .  Both of these identities have very nice geometric explanations.

[Tglad]

Thanks hobold, the mandelbox definitely keeps surprising us with new shapes and with huge variety. I guess you could class this as a sort of indirect extension of a mandelbrot; trying to copy the essense of it.
And I think this is the interesting search, for 3d fractals that are mandelbrot-like in their variety, and perhaps in their beauty if we're lucky.

On the other hand, I don't see the point in looking for 'the real 3d version of the mandelbrot' because z^2 simply has no equivalent in 3d.
The complex numbers are a complete algebra. They have no 3d equivalent that share their properties. It is proven that the only 'complete' extensions of complex numbers are quaternions and octonions. Neither are conformal under squaring. And being conformal is what makes the mandelbrot what it is.. i.e. smaller scaled (but not stretched) copies of itself.
In fact you can't even rotate space in 3d as the +C causes stretch (this is why the mandelbox still shows stretch in some areas).

[Paolo Bonzini]

On the other hand, I don't see the point in looking for 'the real 3d version of the mandelbrot' because z^2 simply has no equivalent in 3d.
The complex numbers are a complete algebra. They have no 3d equivalent that share their properties. It is proven that the only 'complete' extensions of complex numbers are quaternions and octonions. Neither are conformal under squaring.

But why squaring?  Instead of using the complex map , you can define the Mandelbrot set on quaternions of the form xi+bj as , which is just as beautiful a formula (in 3D it simply gives the lathed Mandelbrot set, though).

It is however a real problem that the conformal mappings in 3D are very limited.  However, I think nobody really understood why the Mandelbulb looks like it does, and why does it look strange for n less than 4 or 5.  See also http://www.fractalforums.com/theory/3d-mandelbrot-formula-based-on-the-hopf-map/

[Tglad]

"But why squaring?"
Because I'm talking about direct extensions of the mandelbrot. I don't doubt that there are beautiful variations in 3d.
Take the real valued fractal R^2+C, the mandelbrot is the direct extension of it to 2d. You can't get the equivalent extension to 3d simply because that formula cannot be made in 3d without breaking lots of properties, like being conformal (and universal).

I hope and imagine there are lots of amazing and beautiful formulas in 3d, but I can't see how you could legitimately call one the 'real 3d mandelbrot'... just like there are many polyhedrons but you can't call any of them the 'real 3d octagon'.
However, you could perhaps say 'this is the closest to a 3d version' and I would say that is the marvelous mandelbulb! (or perhaps the imaginary quat mandelbrot... or perhaps a slice of the tetrabrot).

[M Benesi]

   There are no true 3d Mandelbrot, but here is a z^3:


and a z^2 or 3:




or a z^4 straight on: