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Inspired by recent threads on FractalForums, I wrote a little gadget to do inflection mapping. This means translation and squaring of complex coordinates before regular Mandelbrot or Julia set iterations. Check the source for full details, the fragment shader GLSL is the place to start. You can download it here:

inflector-gadget-0.1.1.tar.bz2 (sig) | inflector-gadget-0.1-win.zip (sig)

or get the freshest source from git (browse inflector-gadget):

git clone https://code.mathr.co.uk/inflector-gadget.git

It's a bit rough around the edges, press H to print the help to the terminal you started it from (or read the documentation), but in short, each click wraps the pattern twice around the clicked point, allowing you to sculpt patterns out of Julia sets (or the Mandelbrot set if you press M).

Similar sculpting can be achieved by zooming into specfic locations in the Mandelbrot set. The effect is much the same (the outer decorations are missing in the inflector gadget) but it takes a zillion times longer to zoom so deep as required.