END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

This is one of the interior spirals from <i>Julia's Curls II</i>. The gradients and orbit traps have been tweaked slightly.

The image uses five layers: one gives an iteration gradient; one for each of the two attracting basins tints that basin and adds "ribbons" via orbit traps, in both cases an off-centered ring around the attractor; and one for each basin modifies the luminance in high and low iteration areas.

The ribbon features would not render well without aggressive antialiasing, which, combined with the number of layers, is the cause of the fairly long render time of 45 minutes.

This Julia set does not contain any actual Herman rings. The only attractors are the two obligatory ones, zero and infinity, for this family of mappings. I should probably come up with a better name than "the Herman ring formula" for this family of rational maps. Something snappy and suggesting its nature. How about ... Antimatter? There's a "negative" and a "positive" version of everything, due to the mirror relationship between A(0) and A(infinity). (Indeed, applying f(z,a,c) to (1/z,a,c) and taking the reciprocal yields f(z,-a,c). So the one basin has the same structures for a that the other has for -a, and those are simply mirror images, for example counter-clockwise spirals replaced by clockwise ones.) Twisted mirror, mind you, since the imaginary part of a induces a rotation of the one relative to the other.

Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.

Detailed stats:
Name: Ribbons I
Date: January 30, 2009
Fractal: Herman Ring Julia set
Location: angle parameter = 0.21803398, c = 0.2291666667 + 3.291666667i; zoomed slightly on a prominent spiral
Depth: Very Shallow
Min Iterations: 1
Max Iterations: 128
Layers: 5
Anti-aliasing: 3x3, threshold 0, depth 2
Preparation time: 1 minute
Calculation time: 45 minutes (2GHz dual-core Athlon XP)