Re: Mandelbrot and its applications.

On Oct 31, 1:02 am, mike3 <mike4...@xxxxxxxxx> wrote:

Which of course inspires all sorts of questions: Why Does The
Mandelbrot Set Look The Way It Does??? WHY, for example, do
spirals and swirls appear at all? What is the nature of the
ornamentation (stuff that attaches to the bulbs)? What does it
represent? Why not just bulbs on bulbs, ad infinitum? Why does
the period of the bulbs seem to correlate with motifs found in the
ornamentation? Why, around midgets, do the ornaments go in
powers of two (it seems related to the "^2" in "z^2 + c", but HOW?)?
See, it's those type of questions that looking at this thing inspires.
I'm quite curious about those questions myself, and I'd like to
know if anyone has any *rational* theories about them and their
answers. (Not crazy, crackpot stuff like the "Integrity Paradigm"
someone dug up googling for a similar question here.)

There are rational explanations for the shape of the Mandelbrot set
based on an analytical treatment of what is going on. Particularly,
if you ask the question, "What shape must a region have if the points
inside have periodic orbits with a given period?", then the answers
give you the cardioids and the disks/bulbs. Stable points give the
interior and neutrally stable points give the boundaries. Unstable
points give the Misiurewicz points. And realizing that, the larger
the period, the more regions there are, gives rise to the fact that
the midgets and bulbs must get small really quickly to cram an
infinity of them into a finite space. So, there are reasons for the
features, and for me, that enhances its wonder, kinda like knowing how
a rainbow works.

Kerry Mitchell

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