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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:

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

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

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).

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.

Haha, so did I almost for a split second!

Great renders anyway; nice and shiny 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.

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

Hello

If you look in this post: 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.