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"The Mandelbrot Set Meets Indra's Pearls"

As you may know, an alternate way to visualize the Mandelbrot set is by its orbits, which are shaped like a bunch of circles and cardioids. The Mandelbrot set is defined by iterating f(z) = z² + c. For example:
1st level: F(z) = f(z) = (z² + c)² + c
2nd level: F(z) = f(f(z)) = (z² + c)² + c
3rd level: F(z) = f(f(f(z))) = ((z² + c)² + c)² + c

Without going into detail, an orbit is defined as the set of c such that there exists an attractive fixed point z0 where F(z0) = z0 and |F'(z0)| < 1.

I thought it might look nice to render these orbits as 3D "pearls" instead of 2D circles and cardioids. The first image uses a sphere with a dimple ("tomato"-shape) for the 3D cardioids (actually a lathed 2D cardoid), and the second image uses "heart"-shaped 3D cardioids.

In order to further enhance the image, I also added some Mandelbrot polynomial roots to the image as discussed in my previous post here:
http://www.fractalforums.com/theory/the-mandelbrot-polynomial-roots-challenge/

Then I added some thin rings around the pearls (like silver settings) and a hint of the escape time field to fill in some gaps. Here is the result. I think it looks like jewelry:

Now I know what you're thinking... if only there were some way to spread these pearls around in 3D? I suppose a simple rotation around the x-axis looks pretty, but it's certainly no way to create the "true 3D Mandelbrot" that we've been searching for. Perhaps a better approach would be to try to find some fancy algorithm for rotating and branching the pearls in 3D around their parent pearls. But even this is too simple an approach for the true 3D Mandelbrot, because the sizes of the branching pearls might vary as well.

Hi Paul, here's an idea also based loosely on Indra's Pearls - a non-affine IFS of translated quaternionic Julias for z^2+0 i.e. translate z to z-Pn, calculate z^2, translate to z+Pn for each transform n.
I suggested it a while ago in the original thread and was going to try it since one could quite easily use a distance estimation method for rendering the basic idea but I obviously got sidetracked by the Mandelbulb - like everybody else ;)
Here's my original post with a render of a 2D complex version:

http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg7090/#msg7090 (http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg7090/#msg7090)

Just out of interest does anyone know if there's an iterated formula similar to z^2+0 that will produce a cardioid rather than a sphere ?

Paul,

 It looks you are inching ever closer to the holy grail even it's by logical construction.
 The z^n+c multibrots, with y and z coordinates, where n = 3,4,5, seem like the optimal reference for how the spheres should get smaller and smaller and even move inwards as one rotates along the x-axis starting from the top.

 The problem I've had is matching the z^2+c mandelbrot's circles (pearls) exactly with the bulbs of the n=3-5.
 For example in ChaosPro notation:

 a = real(pixel)^(1/1.5);
 b = imag(pixel)^1.5*sqrt(2);
 z = z^3 + a + flip(b);

 This will vertically line up the bulbs of the z^3 multibrot with the z^2.

-mike

Yes, I remember that one, David. But I don't understand it. Can you render a ray-traced version of it?

I can (I think) when I get some time - I really want to get my class-based 3D renderer for UF sorted first :)

As to understanding it, it's really just simple IFS but using non-linear transforms.
The 2D example above was something like this:

IFS transform 1:

Process: p = p - 1 then p = p^2 then p = p + 1     i.e. p = (p-1)^2 + 1

IFS transform 2:

Process: p = p + 1 then p = p^2 then p = p - 1     i.e. p = (p+1)^2 - 1

IFS transform 3:

Process: p = p - i then p = p^2 then p = p + i     i.e. p = (p-i)^2 + i

IFS transform 4:

Process: p = p + i tthen p = p^2 then p = p - i     i.e. p = (p+i)^2 - i


The IFS was rendered using the escape-time method basically using the above transforms (but I think I included some scaling and the offsets may not have been units).
You could probably adapt your MIIM code to do the same using the reverse method ?

Edit: Obviously for the 3D cut-down quaternionic version you'd probably use 6 transforms (and 8 for the full 4D version).

That is beautiful!

Actually I think I can work that out - it just needs the same method applying to z^2 i.e. find a transform that transforms the main cardioid to the unit circle/sphere and apply that first, then apply z^2 and then apply a transform that transforms the unit circle/sphere to the main cardioid.

Congratulations bugman! Building a fake 3D Mandelbrot this way is something I have suggested in several posts. I'm glad you found a way to "emulate" that. To be honest, when I saw the last image, I thought that you had found the Grail. Anyway, these images are beautiful and I'm sure it could be a starting point to get closer to the real thing. Up to the mad coders! ;)

Haha, so did I almost for a split second!

Great renders anyway; nice and shiny :D We should make a collection of all the more 'artistic' attempts to create a 3D Mandelbrot. I know of only around three or four so far.

Cool! :) How did you find the centers and radii of the circles/spheres?

Here are some links on finding Mandelbrot orbits:
Mathematica notebook by Mark McClure: http://facstaff.unca.edu/mcmcclur/papers/CriticalBifurcation.nb
Mandelbrot Components: http://commons.wikimedia.org/wiki/File:Mandelbrot_Components.svg
Period of Hyperbolic Components: http://en.wikibooks.org/wiki/File:Mandelbrot_Set_%E2%80%93_Periodicities_coloured.png
Mandelbrot Orbital Boundaries by Donald Cross: http://cosinekitty.com/mandel_orbits_analysis.html
Introduction to the Mandelbrot Set by Iñigo Quilez: http://www.iquilezles.org/www/articles/arquimedes/arquimedes.htm

Here is the main 3D cardioid for the quadratic Mandelbulb set (based on the Daniel White's original negative z-component formula). The mathematical derivation is shown below.

The image also shows 3 roots for finding the second cycle 3D orbits, although I wasn't able to solve for the shape of those orbits. I also tried to solve the first cycle 3D orbits for higher power Mandelbulb sets, but no luck.

Notice the strange distortions at the poles. I think this might shed some light on what is the problem with the Mandelbulb formula. Suppose instead that we choose the shape of the main 3D cardioid we wish to have and then work backwards to find what squaring function creates it? Unfortunately, I do not think we can simply use a sphere with a dimple for our main 3D cardioid ("tomato" shape) because that will merely take us back to the lathed Mandelbrot (typical of the quaternion squarring function, see below).

Hello

If you look in this post: http://www.fractalforums.com/meet-and-greet/mandelbulb-bitten!!/ (http://www.fractalforums.com/meet-and-greet/mandelbulb-bitten!!/)

You see that a slight modification to the mandelbulb formulae yields the mandelbrot with smooth cardoids..
Dunno if this helps you anny though.

Here is an alternate solution to the same equations. However, I don't think this solution is correct.

Here is my first attempt to render the second period 3D orbits using numerical methods (shown in green). Unfortunately, convergence wasn't very good. You can barely see the circlular outline of the largest second period bulb, but the rest of it looks like a mess.

somehow noone with a nice shading algorithm tried those variations yet, it seems.

And that although some of them looked really interesting...

But actually that caridod also  looks nice and might be what happens if you search for a true spherical equivalent for it...

I wonder how different kinds of cycloids would look like in 3D :)