Title: Herman Ring Post by: Pauldelbrot on February 02, 2010, 07:10:36 PM (http://u5789.direct.atpic.com/26216/0/1692653/1024.jpg) (http://pic.atpic.com/1692653/1024) 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) |