Herman Ring

[Pauldelbrot]

An Antimatter Julia set. This is an actual Herman ring. The red gradient is for diverging points, the pink for points tending to 0, and the black set is the Herman ring basin. Herman rings occur for a irrational and c either real or imaginary and greater in magnitude than about three (and sometimes for other values of c but these depend on a).

Technically, the a value given above isn't irrational, though it is pretty close to the golden mean minus 0.4. On the other hand, the computer converts it from polar to rectangular coordinates using trig that can only be approximated, and the rectangular coordinates finally used are rational while those trig functions are transcendental, so the angle of the a-plane unit circle point actually used surely is irrational, and the image thus is of a Herman ring, or at least as close as is practically possible.

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

Detailed statistics:
Name: Herman Ring
Date: January 6, 2010
Fractal: Antimatter Julia set
Location: a = 0.21803398, c = 3.7.
Depth: Very Shallow
Min Iterations: 1
Max Iterations: 1339
Layers: 2
Anti-aliasing: 3x3, threshold 0.0, depth 2
Preparation time: 10 minutes
Calculation time: a bit more than 1 hour (2.5GHz dual-core E5200)