It’s Fractal Friday! As always you can click the images below to see them full-size.
This week’s fractal images highlight two new features added to the application I’m developing: color cycles, and Nth-degree fractal expressions.
A color cycle is an an array of colors used to render the fractal pixels. The renderer cycles through them for each output value, rather than drawing a gradual shift of color. The result is a striped output that can be pretty dramatic, and sometimes psychedelic!
Here’s the simplest case, using just two values (black and white) and cycling through them to render the Julia set:
Of course any two colors can be substituted. Here’s a bit of Mandlebrot set in bumblebee colors, black and yellow:
A color cycle can use more than two colors; the renderer just uses them all before repeating. Here’s an example with eight colors (plus black, used as the maximum did-not-escape color in the background). Also, this image is using the altered Julia set equation: instead of z^2 + C, this is z^3 + C evaluated recursively.
The app now generally supports z^N (where N is a positive integer) for Julia sets, and the different shapes that emerge are worth exploring. Here is an N=4 set, with a two-color cycle:
And finally an N=5 set centered on the origin, again with a two-color cycle:
All of the julia sets above used C = -0.223 + 0.745i, with breakout value at 1000.0 and max iterations set to 32.
Coming soon: screenshots of the application in development, and other types of mathematical graphics output. And more fractals, every Friday!
Follow Mathaesthetics on Twitter, Instagram, and Facebook for updates, and more mathematical art.