mandelbrot algorithm for sphere inversion fractals

[vfxsup]

Hi, new member here, I've been scouring the internets and this seemed like a great place to ask this question.

I've been playing around generating some fractals using python and Maya.  I did some easy ones, like Farey-Ford circles, I played around with circle packing and I've done a simple apollonian using Descartes circle theorem and treating the y axis as imaginary to make the math easier.  That was cool and works especially great for me because, for my purposes, I don't want to have any implicit geometry.  It lets me get an explicit list of all the circles I generated so that I could instance circles or torii or whatever geometry I felt like without needing procedural geometry at rendertime.  So I want to extend my code to generate any circle inversion fractal and I'm trying to implement Mandelbrot's Algorithm for circle inversion fractals.   Mostly I've been looking at these two papers

http://www.josleys.com/articles/Sphere_inversion_Article.pdf
https://www.academia.edu/21612485/The_Fractal_Dimension_of_the_Apollonian_Sphere_Packing

with some help from here

http://users.math.yale.edu/public_html/People/frame/Fractals/CircInvFrac/MandelMethod/MandelMethod5/MandelMethod5.html
http://mathworld.wolfram.com/Inversion.html

What I want is to explicitly generate this

https://upload.wikimedia.org/wikipedia/commons/e/e0/Cicle_inversion.svg

as a list of centers and radii.

So, having said all that, I'm creating my initial circles and my inversion circles and now I'm stuck.  I don't understand which circles I'm supposed to invert or which circles to invert them with respect to. My circle inversion code is straight from Wolfram and appears to work as it matches the images on their page but none of my inversions appear to give my any circles that fall into the correct limit sets.  I've searched like crazy but I can't find anything that explicitly lays out the algorithm. Plainly speaking, how do you get from fig. 1 to fig. 2 in the Jos Leys paper and what's the recursion that gives further levels?  I'm might just missing something obvious but I would greatly appreciate any knowledge this community could share. 


Thanks for any help.  I'm a fractal noob but a render expert so hopefully I also have some knowledge that I can give back!

[JosLeys]

In fig.1, the blue circles are the 'generating circles'.
To get to fig.2, invert all the green circles in all the blue circles.
Do this again, so including the new green circles, to get to the next stage, and keep on doing so...

[Adam Majewski]

https://en.wikibooks.org/wiki/Fractals/Apollonian_fractals

[vfxsup]

In fig.1, the blue circles are the 'generating circles'.
To get to fig.2, invert all the green circles in all the blue circles.
Do this again, so including the new green circles, to get to the next stage, and keep on doing so...

Awesome!  A response from the man himself!  As I was organizing my thoughts so that I could reply with more information as to where I was confused I realized that I was writing some bad attributes onto some of my geometry in Maya which was causing my inversions to be incorrect so, sometimes it's not about the answer, sometimes you just have to ask the question.  .  So I just have to get the recursion working and 2D should be good, then it's on to spheres!  Thanks for the sanity check!

[vfxsup]

So I've got a 3d version of circle inversion working and I can create a sphere fractal, which is cool.  I can make something like this

http://www.hiddendimension.com/fractalmath/Images/sphere.gif

What I'm really after is to be able to make something like this

http://i.imgur.com/hQ7WG.jpg

I thought that it might just be a sphere inversion boolean-ed out of a volume but upon closer examination I don't think that this is the case?

What is that second one called?  If there's a paper that anyone could point me to that would help me understand how to generate that as an explicit list of spheres I would be extraordinarily grateful.  It seems like it is some kind of inversion?  I've googled heaps but all I seem to come up with is the general 2D case.