Title: Cross Axis fractals Post by: gandreas on December 07, 2006, 05:25:23 PM So if you consider a 4D space {Zr,Zi,Cr,Ci} and the set of points such that iteration Z = Z * Z + C remains less than a bailout value (using complex math), you've got a classic fractal. If you take a two dimensional slice by setting the images X & Y coordinates to two of those four axes, and leave the other two set to 0,0 you can get six different images:
Zr,Zi (standard Mandelbrot) (http://projects.gandreas.com/quadrium/experiments/ZrZi.png) Cr,Ci (standard Julia Set, and since the other two values are 0,0 we end up with a circle) (http://projects.gandreas.com/quadrium/experiments/CrCi.png) Cr,Zi (http://projects.gandreas.com/quadrium/experiments/CrZi.png) Ci,Zi (http://projects.gandreas.com/quadrium/experiments/CiZi.png) Cr,Zr (http://projects.gandreas.com/quadrium/experiments/CrZr.png) Ci,Zr (http://projects.gandreas.com/quadrium/experiments/CiZr.png) All of these were rendered with the same "view point" (centered at the origin), with the inside using a "min" coloring, and nice smooth continuos outside coloring. Title: Re: Cross Axis fractals Post by: gandreas on December 07, 2006, 05:31:26 PM And here's the same thing with Z = Z * Z * Z + C Zr,Zi (standard Mandelbrot3) (http://projects.gandreas.com/quadrium/experiments/ZrZi3.png) Cr,Ci (standard Julia Set, and since the other two values are 0,0 we end up with a circle again) (http://projects.gandreas.com/quadrium/experiments/CrCi3.png) Cr,Zi (http://projects.gandreas.com/quadrium/experiments/CrZi3.png) Ci,Zi (http://projects.gandreas.com/quadrium/experiments/CiZi3.png) Cr,Zr (http://projects.gandreas.com/quadrium/experiments/CrZr3.png) Ci,Zr (http://projects.gandreas.com/quadrium/experiments/CiZr3.png) |