Title: Embossed Spirals Post by: Pauldelbrot on September 09, 2011, 03:05:59 AM Embossed Spirals (http://nocache-nocookies.digitalgott.com/gallery/8/511_09_09_11_3_05_59.jpeg) http://www.fractalforums.com/index.php?action=gallery;sa=view;id=8444 A familiar Seahorse Valley Mandelbrot with an unusual coloring: the "grad" of the smoothed iterations. Black = the smoothed iteration value increases in the positive real direction, violet = positive imaginary, white = negative real, pinkish = negative imaginary. This coloring method, when used with a cyclic gradient with strong luminance contrast, produces an embossed, three-dimensional look. A second layer with normal smoothed iterations darkens the seahorse cores and turns the edge of the minibrot yellow-brown. Mathematically, the calculations resemble those for distance estimator. The derivative dzn/dc is maintained, starting as 0 (since z0 = 0 is a constant independent of c) and being updated by new-der = 2 times old-der, times zn, plus 1. So far, so identical to distance estimator. The difference is in what's done at the end of the iteration. For distance estimator, the modulus of the escaped z is multiplied by its own logarithm and by half, and divided by the modulus of the derivative. For this, I multiply the escaped z (not its modulus) by half and divide by the derivative (not its modulus), leaving the logarithmic term of the product out entirely and yielding a complex number. That's the grad. It's the argument of this complex number that is used for the coloring: the direction of the gradient. |