What the Mandelbulb teach me

[trafassel]

In this thread I will share some results of my private Mandelbulb science.

Working with the Mandelbulb was sometimes very frustrating. The whipped cream - you know. But there are some interesting unanswered questions left.
And it has still some nice places.

[trafassel]

The first frustrating finding was, that most interesting places got distracted at higher iteration.

[trafassel]

I don't know, how the singularities are called. Each singularity has a well defined center. Perhaps this points are related to some roots of the Mandelbulb polygon approximation. Perhaps each singularity is related to an attractor. I don't know.

Two pictures of the same singularity, but with different iterations. Here I used the power 2 formula, as introduced in Reply #10  in the original Mandelbulb thread of Twinbee. http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/

 

[trafassel]

The power 2 variant is nice because it is so far the best locking 3d extension of the 2d Mandelbrot Set. The Hausdorff dimension of the boundary of the Mandelbrot set equals 2, so all surface normals at y=0 (in CAD coordinates) in each true 3d extension of the Mandelbrot Set must be orthogonal to z. No good news for the hunt of the holy grail.

 

[KRAFTWERK]

The power 2 variant is nice because it is so far the best locking 3d extension of the 2d Mandelbrot Set. The Hausdorff dimension of the boundary of the Mandelbrot set equals 2, so all surface normals at y=0 (in CAD coordinates) in each true 3d extension of the Mandelbrot Set must be orthogonal to z. No good news for the hunt of the holy grail.

 

Sad to hear, but WHAT a beautiful cut!
Zoom out slightly and tilt the camera a tiny bit upwards and it is perfect!

[cKleinhuis]

what it teaches at most is that sometimes the simplest ideas come in handy for fantastic results, i mean the simple idea to extend complex multiplication to 3d the way it was done... and this special formula teaches about problems in 3d space in general ( hairy ball theorem ) that become imanent, and become a visible representation, although the whipped cream effect isnt fully understood yet, it seems that it is a result of the problem of extending the complex plane to a 3d space, but this limitations have held us back, and when the spherical method was invented, everyone was astonished that it produced nice results, although it isnt mathematical "correct"

and at least it teached us that it was only the beginning ( mandelbox, kaliset, .... ) and as a result the hybrid forms popped out and became common sense, which opened up an even wider field for fractal exploration ...

[trafassel]

I must admit that I don't thought much about the formula theory. For me it was just a combination of two rotations and multiplying with a power of radius. So i take the formula as given and search for nice places in the Mandelbulb - which becomes difficult.

On unanswered question is, how to color the Mandelbulb surface. My best approach so far is to use the normalized position in the last iteration step as (rgb)-Color values. At low iterations the gives a very sharp coloring. In higher iterations just some sections (i.e. the "bubbles") got a nice color. So i tried a combination of this color at different iterations and publish my results in http://www.fractalforums.com/theory/coloring-the-mandelbulb/
I could shot some nice pictures, but an overall method with works for all sections and all iterations is still unknown.

Here are two new colorized pictures. First  I used the described method, which works well if you choose the used iterations by hand. In the second picture I color the blue part with the minimal radius in all iterations. So the "bubbles" becomes yellow.

 
 

[trafassel]

Sin Minibulb with low and high iterations.

The Sin Mandelbulb is interesting, because the cut at y=0 is equal to the power 8 Mandelbrot. And so, near the surface there could be some interesting structures.

[cKleinhuis]

to your first point, that things get "unineteresting" when increasing iteration depth:

you have to differ between: "what do i want to behave it! " and "what does it have to behave like?" i mean, the mandelbrot in 2d, shows the same behaviour, but we color the outside of the mandelbrot, so, usually we see in 2d the orbit trapped outside of the mandelbrot, in the 3d mode, we just have plain inside/outside, just check a black/white mandelbrot 2d image, you would encounter the same behaviour - dottet dots the stuff mandelbrot saw on his screen late in the 70s when he discovered the mandelbrot set - and this is exactly what happens here ... sure, you would like to see creative explosions.... but you are just increasing the accuracy of your calculation when increasing the iteration depth, we all know that the whole mandelbrot set is connected, but when you watch the spiral views of the 2d mandelbrot, e.g. in seahorse valley, the connections between the single "minibrots" couldnt be visible, just because they can be arbitrary thin, and this behaviour is showed on your first mention (complaint ) ....

[cKleinhuis]

i say: lets talk about how to color the outside of the mandelbulb, any one implemented orbit trapping visible as own element ?!??!
i start a thread in the mandelbulb3d forum ...

[trafassel]

The Sin Mandelbulb with p=32 and p=2.