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Hi people! I am new here! 

So, according to this article "The Unraveling of 3D Mandelbub", page 2, http://www.skytopia.com/project/fractal/2mandelbulb.html , one can see that both power 2 and power 3 have trivial symmetry, that is one can rotate the set 180º along the main axis and one get the same picture. Power 8 one can count a 7 fold symmetry, and increasing as the power goes up.

This is a really strange pattern, given that the non trivial cylindrical symmetry spontaneously emerge somewhere after power 3, but it is not clear to me when. I guess this is an aspect of what people call interesting but wasn't clarified.

Could anyone show the intermediate powers from 4 to 7?

That one has 5 fold symmetry. So the non trivial symmetry should appear at power 4. So why 8 is more interesting than 4,5,6,7?