Mike, Thanks for the report of your investigation:
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. . . . I really didn't answer your question but I share your curiosity.
Upon reflection, your observation clearly makes sense:
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.
Numerical analysis might possibly be able to shed some light on the aspects of the equations that lead to this phenomenon. One approach might be to find or construct a very simple equation that shows this behavior -- to ease the difficulty of the analysis. General Appeal ============== If anyone knows of a fractal with a simpler equation than Jim Muth's FOTD for February 6th, 2012: http://www.emarketingiseasy.com/TESTS/FOTD/jim_muths_fotd_2012_02_Feb.html or: http://tinyurl.com/FOTD-for-Feb-2012 that has this type -- F120206A.jpg -- of pattern (one feature "winning" over another when they try to overlap,) I'd like to know about it. Perhaps I can get it to a mathematician who does numerical analysis. But analyzing this is beyond me. The only other conjecture about something similar in fractals that I've had, was concerning areas with sharp discontinuities between them. It strikes me that equations could suddenly start producing significantly different values in an area -- with very small changes in input -- if a discontinuous function like the tangent, ceiling or floor were used. In this case it seems reasonable to me for the pattern in the fractal to be discontinuous. But I've not investigated this. - Hal Lane ######################## # hallane@earthlink.net ######################## ==================================================================
Author: Mike Frazier Date: To: Fractint and General Fractals Discussion Subject: Re: [Fractint] Anyone care to venture a thought on...
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
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