Complex function iteration: Further results and a puzzle

I used the method of the post “Fractal surprise from complex function iteration” for the function

fzwhere c is a constant. To get a better image I now use dark blue for all numbers z that grow without limit in the iteration. These are the numbers outside of the Julia set and the bright colors correspond to numbers inside the Julia set. For c=0.000002 I get


Result of complex iteration of a polynomial of 6-th order. The numbers outside the Julia set are shown in dark blue.

An magnified image of the tip at the lower right shows the self-similarity:


Magnified image of the tip at the lower right in the image above. Note the self-similarity.

But there is a puzzle. Why has the image approximate 9-fold rotational symmetry ? From the 6-th power of z alone I only get 6-fold rotational symmetry. Similarly, why do I get 6-fold rotational symmetry from a polynomial of 4th degree in the last post ?

Note added later (5th March 2013):

Here too I made a similar mistake as in the last post.

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