At the end of my exploration of the fractal-universe, I had my tireless Arduino Uno create a couple of ‘classics’. The video shows the growth of the famous Barnsley Fern.
This is a special IFS fractal: each iteration step transforms just one single pixel (instead of the usual geometric shape), using one out of four possible transformations. By changing the parameters and/or the probabilities of the transformations, different kinds of fern patterns will appear.
After I’d finished the sketch (the simplest of all my fractal sketches), I could hardly believe that it could generate such a beautiful image (after 50,000 iterations).
And then, of course, there is the mother of all fractals, named after the (disputed) father of all fractalists: Mandelbrot. The sketch to produce it is much simpler than most backtracking sketches, but it may take up to 256 iterative function calls for every pixel to calculate its color. With 153,600 pixels, it took the Uno a couple of minutes to finish the picture.
The black area is a 2D representation of the actual Mandelbrot Set, but the fascinating stuff is happening at its border. By changing the coordinates and zoom level inside the sketch, you can zoom in at any area. Key factors for getting spectacular results are: selection of the area, maximum number of iterations and color mapping.
This short exploration wouldn’t be complete without mentioning the elegant Julia fractals. Like Mandelbrot fractals, they are created by iteration of the function f(z) = z² + c (in the complex plane). For Julia fractals, c is fixed and we examine the function’s behaviour for all z values within our range of interest. So there’s a Julia set Jc for every c in the complex plane, but the most visually appealing Julia fractals will arise from c values at the border of the Mandelbrot set.
Julia fractal (c = -0.79 -0.15i) produced by an esp8266 on a 480×320 TFT display