Title: Exponent increment Post by: HPDZ on March 21, 2009, 10:02:51 PM These are based on idea by gamma that started in another thread.... The formula is the standard third-order Newton's method, but with a twist -- each pixel's loop begins with the exponent set to 3, then at each iteration of the loop, the exponent increases by a fixed amount. For the first three image, the exponent increment is 0.01, 0.5, and 1.0. The fourth image has an imaginary increment (0,1). Aggressive 9x9 anti-aliasing was used because this gets severely noisy in the negative real half-plane. First image: exponent increment = 0.01. You can see the effect beginning to develop in the portion of the Julia set that lies on the negative real axis. (http://www.fractalforums.com/gallery/0/359_21_03_09_9_39_02_2.jpg) Second image: exponent increment = 0.5. (http://www.fractalforums.com/gallery/0/359_21_03_09_9_39_01_1.jpg) Third image: exponent increment = 1.0. (http://www.fractalforums.com/gallery/0/359_21_03_09_9_39_02_3.jpg) Fourth image: exponent increment is imaginary (0,1). The coloring (unlike the others in this series) is based on the histogram of the counts with a 2X repeat of the underlying gradient. The gradient starts at white and ends at black, so there is a discontinuity from white to black when it wraps around. I felt this thematically matched the presence of the branch cuts induced by the complex exponents. (http://www.fractalforums.com/gallery/0/359_21_03_09_9_39_01_0.jpg) |