Thursday, November 19, 2009

How to define 3-D analogs of Mandelbrot fractals?

In New Scientists there was an article about 3-D counterparts of Mandelbrot fractals. It is not at all obvious how to define them. Quite impressive analogs of Mandelbrot set have been found using so called hypercomplex numbers (which can have any dimension but do not define number field but only ring) and replacing the canonical map z→ z2 +c with a more general map (see this). c must be restricted to a 3-D hyperplane to obtain 3-D Mandelbrot set.

It occurred to me that there exists an amazingly simple manner to generate analogs of the Mandelbrot sets in 3 dimensions. One still considers maps of the complex plane to itself but assumes that the analytic function depends on one complex parameter c and one real parameter b so that the parameter space spanned by pairs (c,b) is 3-dimensional. Consider two examples:

  1. f: z→bz3 + z2+ c, b real,

    for which b=0 cross section gives Mandelbrot fractal. The first two iteration steps are z=0→c→ bc3 + c2+ c →...

  2. f: z→z3 + bz2+ c, b real, suggested by Thom's catastrophe theory. The first two iteration steps are z=0→c→ c3 +bc2+ c →...

Both options might produce something interesting. One could also construct dynamical 3-D Mandelbrots by allowing b to be complex and interpreting real or imaginary part of b of the modulus of b as time coordinate.

One can deduce some general features of these fractals.

  1. These examples represent 3-D generalization of Multibrot fractals defined by z→ zd (this video demonstrates how Multibrot fractals vary as the integer d varies). Obviously these sets look qualitatively very much like Mandelbrot fractal. Hence each b=constant cross section of these 3-D fractals is expected must have the qualitative look of Mandelbrot fractal.

  2. The boundary of Mandelbrot fractal is now a 2-D surface of 3-D space spanned by points (b,c) and corresponds to the points (b,c) for which the iteration of f applied to z=0 does not lead to infinity. The line c=0 belongs in both cases to the fractal since z=0 is fixed point of iteration in this case. As the value of b grows the cross section must get increasingly thinner for both negative and positive values of b with large enough magnitude rougly as c < b-1/3 for the first case and c<b-1/2 by looking what happens in first iterations. The direction of b is clearly in a special position reflecting the 2-D character of the basic iteration process. This holds true for any analytic functions.

  3. To guess what the boundaries of these fractals could look like, one can try to imagine what one obtains as one builds a small pile of slightly deformed 2-D Multibrot fractals above b =0 Multibrot fractal or any piece of it (see the images here). One expects complex caves inside caves structure. The caves are expected to be high. Also a fractal hierarchy of stalagmite like structures is expected. Their tips would reflect a disappearence or appearance of a new structure in 2-D Multibrot fractal as the height coordinate b varies. The objection is that the boundary curve in the case of Mandelbrot fractal- and maybe also of Multibrots- consists of single component. Stalagmites would correspond to an appearence of disjoint component to the curve would not be possible. In a given resolution however there are always invisible thin hairs connected to thicker regions so that only apparent stalagmites due to finite resolution would be possible. Being inside b> 0 part of the fractal might create same spiritual feelings that one experiences in Gothic Cathedral.

  4. It is good to hear what pessistic has to say. In high enough b-resolution the variation of the Multibrot curve as a function of b could be so slow that the fractal looks like a cylinder having Multibrot as intersection and there would be no fractality in b-direction. Therefore the question is whether the fractality in c-plane implies fractality in b-direction. In other words, is the fractal curve for a fixed value of b critical against variations of b.

3-D graphics skills would be needed to disetangle the huge complexity of the situation. Probably this is far from a trivial challenge. In any case, the visualization as a pile of 2-D Multibrots could be used in the construction of these fractals and would make possible discretization in b-direction and the use of existing 2-D algoriths as such. Maybe some Mandelbrot artist might try look what these fractals look like. In New Scientists there was an article about 3-D counterparts of Mandelbrot fractals. It is not at all obvious how to define them. Quite impressive analogs of Mandelbrot set have been found using so called hypercomplex numbers (which can have any dimension but do not define number field but only ring) and replacing the canonical map z? z2 +c with a more general map (see this). c must be restricted to a 3-D hyperplane to obtain 3-D Mandelbrot set.

It occurred to me that there exists an amazingly simple manner to generate analogs of the Mandelbrot sets in 3 dimensions. One still considers maps of the complex plane to itself but assumes that the analytic function depends on one complex parameter c and one real parameter b so that the parameter space spanned by pairs (c,b) is 3-dimensional. Consider two examples:

  1. f: z?bz3 + z2+ c, b real,

    for which b=0 cross section gives Mandelbrot fractal. The first two iteration steps are z=0?c? bc3 + c2+ c ?...

  2. f: z?z3 + bz2+ c, b real, suggested by Thom's catastrophe theory. The first two iteration steps are z=0?c? c3 +bc2+ c ?...

Both options might produce something interesting. One could also construct dynamical 3-D Mandelbrots by allowing b to be complex and interpreting real or imaginary part of b of the modulus of b as time coordinate.

One can deduce some general features of these fractals.

  1. These examples represent 3-D generalization of Multibrot fractals defined by z&rarr; zd (this video demonstrates how Multibrot fractals vary as the integer d varies). Obviously these sets look qualitatively very much like Mandelbrot fractal. Hence each b=constant cross section of these 3-D fractals is expected must have the qualitative look of Mandelbrot fractal.

  2. The boundary of Mandelbrot fractal is now a 2-D surface of 3-D space spanned by points (b,c) and corresponds to the points (b,c) for which the iteration of f applied to z=0 does not lead to infinity. The line c=0 belongs in both cases to the fractal since z=0 is fixed point of iteration in this case. As the value of b grows the cross section must get increasingly thinner for both negative and positive values of b with large enough magnitude rougly as c < b-1/3 for the first case and c<b-1/2 by looking what happens in first iterations. The direction of b is clearly in a special position reflecting the 2-D character of the basic iteration process. This holds true for any analytic functions.

  3. To guess what the boundaries of these fractals could look like, one can try to imagine what one obtains as one builds a small pile of slightly deformed 2-D Multibrot fractals above b =0 Multibrot fractal or any piece of it (see the images here). One expects complex caves inside caves structure. The caves are expected to be high. Also a fractal hierarchy of stalagmite like structures is expected. Their tips would reflect a disappearence or appearance of a new structure in 2-D Multibrot fractal as the height coordinate b varies. The objection is that the boundary curve in the case of Mandelbrot fractal- and maybe also of Multibrots- consists of single component. Stalagmites would correspond to an appearence of disjoint component to the curve would not be possible. In a given resolution however there are always invisible thin hairs connected to thicker regions so that only apparent stalagmites due to finite resolution would be possible. Being inside b> 0 part of the fractal might create same spiritual feelings that one experiences in Gothic Cathedral.

  4. It is good to hear what pessistic has to say. In high enough b-resolution the variation of the Multibrot curve as a function of b could be so slow that the fractal looks like a cylinder having Multibrot as intersection and there would be no fractality in b-direction. Therefore the question is whether the fractality in c-plane implies fractality in b-direction. In other words, is the fractal curve for a fixed value of b critical against variations of b.

3-D graphics skills would be needed to disetangle the huge complexity of the situation. Probably this is far from a trivial challenge. In any case, the visualization as a pile of 2-D Multibrots could be used in the construction of these fractals and would make possible discretization in b-direction and the use of existing 2-D algoriths as such. Maybe some Mandelbrot artist might try look what these fractals look like.

Addition: Paul Nylander kindly reproduced a picture of az3+bz2+c type 3-D fractal. As you see it is from outside and the local cylinder likeness is obvious from the picture which suggests that there is no genuine fractality in b-direction. This hides the complexity of the boundary of Mandelbrot which can be seen only by going inside.

He also told that also Rudy Rucker has suggested a similar approach.

Also Janne (see the discussion) worked with 2-D sections (see the discussion section and his graph) and thinks that fractality in b-dimension is not true. He sent also a nice animation about quaternionic Julia set which also has a local cylinder like structure. This animation demonstrates the 3-D complexity below the surface hidden by the views from outside by using 2-D cross sections. I would guess that this animation catches much about the (b,c)-fractals. The graphics allowing to see the fractal from the point of view of observer at floor of infinitely high fractal Gothic Cathedral might be a fascinating challenge;-).

10 Comments:

At 11:10 PM, Blogger janne said...

I tried this out quickly on matlab. While the approach obviously works (b*z^3+z^2+c mapping) it's clear you need a software with better graphics capabilities. Best looking results were with merging different b slices into one matrix and forming a isosurface out of that one but even then it looks very crude.

I guess it takes some sort of 3D raycasting method.

 
At 12:24 AM, Blogger Matti Pitkanen said...

The basic question is whether there is fractality in b-direction or not. If not you get just Mandelbrot cylinder in high enough resolution for b and this means that you do not have a genuine 3-D fractal.

The best manner to kill the idea is to look how the 2-D Multribrot varies in interesting manner in various scales as b is varied.

 
At 8:32 AM, Blogger janne said...

Yes you're right. I only tried a very rough overall discretization of the whole mandelbrot set. The resulting isosurface looked more like a 3-D Lyapunov fractal, but that's just because there wasn't any real detail to it.

I'll try out some more detailed b direction slicing tonight.

 
At 11:26 AM, Blogger janne said...

Couple more detailed runs reveal that it's more like a projection similar to Julia sets, there's no fractality involved in the fine structure of the different b-slices.

The most interesting results come from b close to 0, with large |b| the radius of convergence just shrinks down to nothing.

I don't think real value as the third dimension cuts it for 'true' 3-D fractality, since the iteration will still always converge to the '2-D' complex plane.

This makes me think about those quaternion approaches. Isn't this basicly the same idea with a few more dimensions added to b.

 
At 6:45 PM, Blogger janne said...

http://www.youtube.com/watch?v=rSTAjSymJAY

 
At 8:51 PM, Blogger Matti Pitkanen said...

Each slice should look very Mandelbrotian since iteration functions of type z^d+c and more generally polyonomial of z must give rather similar structure. Wikipedia contains animation about this.

My guess is also that things changes slowly in b-direction and you get something like Mandelbrot cylinder.

3-D quaternion approach and hypercomplex approaches are different since z is replaced with quaternion. If one considers only Mandelbrot fractals one can use hypercomplex numbers of any dimension and just take the space of c-parameters to be 3-D.

 
At 9:34 PM, Blogger Matti Pitkanen said...

Thank you for a nice picture. You have f(z)= z^b+c, with b varying. I had in mind f(z)= bz^3+z^3+c or f(z)= z^3+bz^2+c. Your choice is more general than the choices based on polynomials since z^b is discontinuous along x-axis (exp(i*2*pi*b) is not equal to 1 for non-integer b).


Your question:
"This makes me think about those quaternion approaches. Isn't this basicly the same idea with a few more dimensions added to b".

In standard approaches the parameter c corresponds to a shift in complex plane, quaternion plane, or hypercomplex plane (Wikipedia contains a page about hypercomplex numbers). In my own case b does not correspond to a shift. Maybe acting as shift is what is essential for fractality.

 
At 9:38 PM, Blogger Matti Pitkanen said...

One manner to kill the idea that a genuinely 3-D fractal is in question, is to draw some vertical slices of the fractal with various zooms for b. If these are featureless, one can forget it.

Let us for definiteness restrict the consideration to f(z)=bz^3+z^2+c.

*Allow b vary freely.

*Constrain the values of c on ray (c= x*exp(i*phi_0), x varies) or a circle around origin (|c|=constant that is c = rexp(i*phi), r=constant). c-plane would be replaced with (b,x) plane or (b,phi) cylinder in the fractal plot. Cylinder can be replaced with plane for plotting purposes.

*This 2-D slice as would give the intersection of the 3-D fractal with the plane defined by ray or with cylinder |c|=constant.

 
At 11:46 AM, Blogger janne said...

I see what you're trying to say here. I'll give it a go when I have some spare time and energy.

Consider the hypercomplex case:
http://www.youtube.com/watch?v=lwB7KzG9awk
Even there the surface of the set rotated thru the dimensions isn't fractal thru these dimensions. The Multibrot 3-D approach isosurface (z->b*z^3+z^2+c mapping to be precise) I got looked very similar to these, without the extra rotations.

 
At 7:51 PM, Blogger Matti Pitkanen said...

Thank you very much for seeing the trouble. And also for video. I added also some comments on main text.

Matti

 

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