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नमस्ते! (i bow to you!)
please feel free to check out the मण्डलबेथ (maṇḍalabeth), new 3D fractal with various symmetry groups.
http://www-personal.umich.edu/~bethchen/mandalabeth/

actually, it's a 3D analog of the mandelbrot set!
the standard 2D generating function (over the standard unit circle) gives you the usual 2D mandelbrot set.
the 3D generating function (over a bouquet of rings) is the sum of 2D generating functions (over each ring),
where the function is conjugated by the isometry (between the ring & standard unit circle) & its inverse.
if the power–1 is a multiple of the natural symmetry of the bouquet of rings,
then the 3D fractal will have the same symmetry group as the bouquet of rings.

i invite you to explore these new worlds, and have fun drawing lots of pretty pictures!
bethchen@umich.edu