Mandelbrot kaleidoscope/IFS

[Duncan C]

This image could be said to be a meta-fractal.

It starts with a square section of 3D rendering of a Mandelbrot set fractal, turned into a kaleidoscope by taking 1/2 of the image a long a diagonal, flipping it along the diagonal, and then flipping the resulting square first along a vertical edge, and then along the horizontal edge. This gives a kaleidoscope like this:




I then applied the same transformation twice more, and got this image:



I tried applying the transformation a couple more times. The results are good, but the details get too small to see except at very large image sizes (too large to post to the internet.)

This would seem to meet the definition of an IFS, but using a Mandelbrot fractal as the base image. Thus, a meta-fractal.


Duncan

[David Makin]

Nice images !

I don't think it can be called an IFS because (if I understand correctly) all you're doing is transforming the image of a Mandelbrot i.e. it's just a transformed Mandy - in fact you could produce the same result by applying a kaleidoscope transform to "pixel" before plugging it into the Mandelbrot iterations.
Even applying the transforms on every iteration of the Mandelbrot wouldn't really by an IFS.
It would be if you were creating a tree of transforms so the result is the mixture of all variations of the transforms that you are applying.
e.g.

              ----------------------- flip along diagonal----------------------
            |                                            |                                           |
  --flip along diagonal-                 ----flip along vertical-----           ---flip along horizontal----
 |             |            |                |             |                  |            |            |                   |
diagonal vertical horzontal      diagonal      vertical       horizontal   diagonal   vertical       horizontal

etc.

Of course actually doing the above would just produce a mess, ideally the transforms should be contractive (or at least mostly contractive).

An IFS basically requires at least two separate functions to be mixed together i.e. in all possible combinations.
In the examples of Mandelbrot IFS that I posted (http://www.fractalforums.com/ifs-iterated-function-systems/julia-ifs-example-%27kissing-circles%27/msg7147/#msg7147) there are truly 4 separate functions because each of the 4 (although all basically Mandelbrots) has different affine transforms applied, so each is a separate function (actually it's possible to produce the same thing using 4 quadratic functions f(z) = a*z^2 + b*z + c + pixel).