While calculating zooms into the parent fractal of Jim Muth's
FOTD for February 6th, 2012, I ran across an especially
clear occurrence of something that happens quite often in
fractals that I've always found interesting:

When two clearly different features or patterns that dominate
in their own areas of a fractal meet, often one or the other
will "win," and will be the only pattern in evidence in the
overlap area, as opposed to creating a mixture of features.

I've often wondered what aspect of the mathematics involved
makes one feature "win" during these "confrontations."


I have often wondered the same thing. I even did some experiments to try to figure this out on several occasions. The feature that wins is the one that causes it to bail out at the lowest iteration. Tracing that back to which part of the equation that is dominant is another matter. These formulas that mix two powers can create some interesting effects.

I experimented with the non-rotated version of this formula that Jim posted a few days ago and it has some interesting features. When you substitute the parameter numbers, it looks like:

z = (z^2) * (1/(z^n + 400000))) + c

When you look for places to zoom into, it has a lot of two way symmetry just like an ordinary order 2 mandelbrot. The difference is that when you zoom in on a two way symmetry, the minbrots aren't order 2 they are a higher order that is determined by the 1/z^n part.

I really didn't answer your question but I share your curiosity.
  
--
Mike Frazier
www.fracton.org