Introducing the Kalibrot

[Pauldelbrot]

Introducing the Kalibrot



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=7867

Want to add some texture to those low-iteration Mandelbrots? Kalibrot to the rescue! Kaliset patterns replace boring old iteration bands, seamless and brimming with detail.

[Duncan C]

I like the low-iteration textures, as well as the coloring for the higher iteration bits. Are you going to share the secret of the Kalbrot with us, then?

[msltoe]

I'm going to take a guess. When the first iterator escapes, move on to a second iterator. Clever.

[Pauldelbrot]

Yes, basically, but with a twist. Or rather, an atan2 and some logarithms, and then folded, spindled, and mutilated. Check the comment thread for this image in the Gallery for more information.

[Duncan C]

Yes, basically, but with a twist. Or rather, an atan2 and some logarithms, and then folded, spindled, and mutilated. Check the comment thread for this image in the Gallery for more information.

The effect is cool, but it sounds slow with all those transcendental functions.

My scheme for coloring the thin "tendrils" around a Mandelbrot/Julia set involves applying a non-integer power to the Distance Estimate (DE) value, but only for pixels that are very close to a Mandelbrot/Julia set point. It only has to calculate a non-integer power on a small percentage of the pixels in a plot, and then only for a single distance value of each pixel. I'm not doing multiple transcendentals on each iteration of each pixel like it sounds like you are doing.

How much slower is your Kalibrot algorithm over a straight escape-time algorithm?

My scheme enables me to feather the boundary of the Mandelbrot set and apply mathematical anti-aliasing. I can get results like this:

http://www.pbase.com/duncanc/image/91190108

I was trying to duplicate the lacy look of the B&W images in the seminal books "The Beauty of Fractals" and "The Science of Fractal images". I think I got it:

[Pauldelbrot]

Yes, basically, but with a twist. Or rather, an atan2 and some logarithms, and then folded, spindled, and mutilated. Check the comment thread for this image in the Gallery for more information.

The effect is cool, but it sounds slow with all those transcendental functions.

Not really. It does just normal Mandelbrot iterations until escape, then a more complex version of what the smoothed-iterations colorings do. Basically I get a smoothed iteration fraction value and a smooth decomp value and apply some piecewise-linear maps to those, then feed the result into a fixed number of Kaliset iterations.

My scheme enables me to feather the boundary of the Mandelbrot set and apply mathematical anti-aliasing.  I was trying to duplicate the lacy look of the B&W images in the seminal books "The Beauty of Fractals" and "The Science of Fractal images". I think I got it.

They did that by simply using ridiculously large image resolutions. I have myself applied a logarithmic map to the DE edge pixels to get a smoother image without needing much AA, years ago, and similar techniques allow avoiding the densest areas from blacking or whiting out without costing you the fine filaments attached near minibrots.