ok so here's the my first volumetric rendering of the 3dmandelbulb.  still tweaking the orbital distance algorithm.  i think it kinda looks like a wood cut for some reason...

Hello all.  Very interesting progress so far.

Twinbee...  any chance of you extending your article so it has a page 3 titled "for coders" or something similar?  ie, a pseudo code structure for someone wanting to try these formulas out who is more programming inclined rather than mathematically inclined.

Assuming that the reader has a basic raytracer or raymarcher coded up and have the ability to have a ray starting from the "camera" and know the "direction" the ray is going, then you step a certain amount along the ray and repeatedly try the mandelbulb formula to see if the point is inside or outside the set and plot the point in 3D space accordingly.

The basic code would be along the lines of

for y=1 to bitmapheight do
begin
      for x=1 to bitmapwidth do
      begin
             <calculate vector of ray direction from camera to pixel location in 3d space>
             <now we have the starting point in XYZ coords (this would be the bitmap XY coordinate converted to XYZ space in 3d)>
             <feed the point XYZ into distance estimator>
             <test point returned from distance estimator by iterating it through the iteration loops>
             if ((x^2+y^2+z^2)>4) or (iterations>256) then giveup  else XYZ is a valid point so raytrace the point
     end
end


Questions... 
1. The z triplet and c triplet both have 3 xyz parameters
2. So Z starts as (1,0,0) and C starts as (X,Y,Z) when iterating the formula?
3. Does the distance estimator calc happen while the iteration happens?
4. Am I on the right track?

It would also be nice to have a series of "DistanceEstimatorNylander" "IterateNylander", "DistanceEstimatorMakin" "IterateMakin" snippets of code depending on who first came up with the idea.  Looking back over this thread shows many attempts that may not be the true 3d mandelbrot type, but never the less result in interesting results.  It would be great to have some info along the lines of "the above picture uses the following formula by whoever" and have the relevant snippet/change in code for distance estimation and iteration.

Or even a basic sample code for distance estimation and iteration of the now classic ^8 result.  Enough to get the basic framework going and have a play with these sort of images.

Keep going regardless guys.  There is some great results in this thread so far.

Spy, I think the change of sign was an fix they made to correct the wrong cartesian2polar change of coordinates they were making (same for all those +PI/2 etc you see in the beginning of the thread). I believe most of the latest images in the thread are created with the """correct""" formula (in whatever xyz, xzy system). At leas I'm using the correct version (for the "standard" academic xyz coordinate system). I'm not going to re-post it here, it's just a few pages before this one.

oh man, word of this thing has spread so far... it's really incredible (and to think i have a picture of a quadratic 3d mandelbrot on a mug from april last year!). even my (usually uninteresting) google alerts for "ray tracing" are turning up things related to this, it looks like there are a lot of people doing gpu ray tracing of this thing: http://frictionalgames.blogspot.com/2009/11/fractional-fun.html

i'm interested to hear from the experiences of others on implementing this on the gpu. personally i'm seeing less-than-ideal numerical results from the gpu (need a reference cpu implementation), especially in the finite difference normal estimate, and the polynomial versions are also WAY faster than the ones using trig.

to be more specific about the gpu precision issues, they don't support denorms yet and i'm sure this makes a difference for finite difference normal estimation since the subtracted values should necessarily be very close.

of course, ideal would be to get analytic normals; it would be nice if you could post your jacobian matrix for the z8

<a href="http://www.youtube.com/v/afa7lU_yHy8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/afa7lU_yHy8&rel=1&fs=1&hd=1</a>

I calculated the 3d mandelbrot and exported it as a povray-script. Then added a hollow green box around it with some lights. Texture is glass. The results are not by far as stunning as some others i've seen here though.