In July Subblue released another Pixel Blender plugin, called the Fractal Explorer Plugin – for exploring Julia sets and fractal orbit maps. I didn’t get around to try it out until recently, but it is really a great tool for exploring fractals.
Most people have probably seen examples of Julia and Mandelbrot sets – where the convergence properties of the series generated by repeated application of a complex-valued function is investigated.
The most well-known example is the iteration of the function z ← z2+c. The Mandelbrot set is created by plotting the convergence rate for this function while c varies over the complex plane. Likewise, the Julia set is created for a fixed c while varying the initial z-value over the complex plane.
Glynn1 by Subblue (A Julia-set where an exponent of 1.5 is used).
Where ordinary Julia and Mandelbrot sets only take into account whether the series created by the iterated function tends towards infinity (diverges) or not, fractal orbits instead uses another image as input, and checks whether the complex number series generated by the function hits a (non-transparent) pixel in the source image. This allows for some very fascinating ‘fractalization’ of existing images.
A fractal orbit showing a highly non-linear transformation of a Mondrian picture.
Subblue suggests starting out using Ernst Haeckel beautiful illustrations from the book Artforms of Nature, and he has put up a small gallery with some great examples:
An example of an orbit mapped Ernst Haeckel image.
To try out Subblue’s filter, download the Pixel Blender SDK and load his kernel filter and an input image of choice. It is necessary to uncheck the “Build | Turn On Flash Player Warnings and Errors” menu item in order to start the plugin. On my computer I also often experience that the Pixel Blender SDK is unable to detect and initialize my GPU – it sometimes help to close other programs and restart the application. The filter executes extremely fast on the GPU – often with more than 100 frames per second, making it easy to interactively explore and discover the fractals.
As I final note, I implemented a fractal drawings routine myself in Structure Synth (version 1.0) just for fun. It is implemented as a hidden easter egg, and not documented at all, but the code belows shows an example of how to invoke it:
View: (-0.0,0.2) -> (0.7,0.9)
Arguable, this code is not very optimized (it is possible to add an unlimited number of terms, making the function evaluation somewhat slow), but still it takes seconds to calculate an image making it more than a hundred times slower than the Pixel Blender GPU solution.