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Hi

I've got the impression that the quest for the "true" 3D version of the Mandelbrot Set was inspired mainly by the intricate and mostly spiral-containing nature of the colorful renderings obtained using this fractal formula, and that there was (and to some extent still is) a certain disappointment that the byzantine structural richness of those images could not be reproduced - in a similar fashion - in 3 dimensions.

Exploration of the power 8 Mandelbulb (from the outside as well as the inside) has shown unlimited detail, but not yielded the variaty of different shapes that a lot of people had probably expected or at least hoped for. But as far as I understand it, the Mandelbulb just presents the (approximated) volume defined by the points (voxels?) that actually belong to the M.bulb set (the 'inside'). The patterns (e.g. spirals) that we associate with the M-set, however, are actually (mainly) represented by points that are 'outside' the set and colored according to their escaping/orbit characteristics.

As it would not make sense to encase the Mandelbulb set in a solid mass of outside points (even colored differently, spirals would not be visible from the outside), I am wondering if the use of orbit traps could reveal spiral details outside the M.bulb set without the view being obscured.

I am aware of some attempts in this direction (mostly using a 3D version of a pickover stalk trap shape) by Paul Nylander, Iñigo Quilez & David Makin

http://www.fractalforums.com/3d-fractal-generation/3d-pickover-stalks/
http://www.bugman123.com/Hypercomplex/PickoverStalks-Tricorn-large.jpg
http://iquilezles.org/www/articles/orbittraps3d/orbittraps3d.htm

but I have the impression that these were not designed to zoom-in close to the surface of the 3D fractal where spiraling might become visible (like you have to zoom into the Seahorse or Elephant valleys to observe spiral structures in the M-set).

So - could it be that the use of e.g. spherical orbit traps (appropriately sized and placed) would make it possible to trace / visualize convoluted three-dimensional patterns close to the M.bulb set, e.g. in the form of beads-on-a-string spirals?

Hi David


So (as I understand it), the DE calculation for the M.-bulb is done in quite a special way which is not based on a combiniation / integration of the raymarching algorithm with the 'standard' iteration process, where the fractal's boundary is found when raymarching hits a solid spot (an 'inside' point that belongs to the M.-bulb set, e.g. for which the max. iteration condition is fullfilled while the bailout value has not been reached)?

Since there is actually not much difference in calculation effort required when switching from iteration coloring to orbit trap coloring with the 2-dimensional M-set, I had assumed that a similar "switch" between these two options for 3D Mandelbulb calculation could be implemented fairly easily ...