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From: billd@fps.com (Bill Davids_on)
Newsgroups: comp.graphics
Subject: Re: Excluding Mandelbrot set
Message-ID: <4233@celit.fps.com>
Date: 23 Nov 89 00:34:51 GMT
References: <3544@quanta.eng.ohio-state.edu> <480003@hpsad.HP.COM> <380@fsu.scri.fsu.edu>
Reply-To: billd@fps.com (Bill Davids_on)
Organization: FPS Computing Inc., San Diego CA
Lines: 38

In article <380@fsu.scri.fsu.edu> prem@geomag.UUCP (Prem Subrahmanyam) writes:
>  Actually, the true definition is that the number does not "blow up" to
>  infinity.  2 is chosen, as there is a theorem that was developed that proves
>  that if the magnitude of a number exceeds 2, then it will "blow up" to 
>  infinity very shortly afterwards, 2 is simply the smallest number at which
>  it is certain that the magnitude will blow up.  FYI, there is also a theorem
>  that was developed that proves that the entire Mandelbrot Set is connected,
>  i.e., that between small "pockets" of Mandelbrot numbers, there also exist
>  tiny "filaments" of Mandelbrot numbers that connect them, so there is no
>  isolated pocket of Mandelbrot numbers within the set.  Some of the larger
>  "filaments" are what create the lightning-like effects surrounding portions
>  of the set.

Gee, THEOREMS!  I assume that means that there are proofs!  If there
are proofs then what is Krantz complaining about?  If someone is
going to the trouble to prove things then it's just as valid as any
other mathematics (much of which is only of interest to mathematicians).

>  Oh, I forgot to mention, a Mandelbrot number is actually one that does not
>  blow up to infinity after an infinite number of recursions.  Of course, we'd
>  be waiting around forever to find one Mandelbrot number.

I seem to recall mention of a theorem that extrememely few numbers
will "blow up" if they haven't already within 1000 iterations.
By experiment, I've found most numbers that diverge, do so well
before hitting 100.  The gain in non-Mandelbrot points from 100
to 256 is fairly small.  The gain from 256 to 1000 is even smaller.
(often there is no difference depending on resolution and sample
space).

Does anyone have references for these theorems (including proofs)?
English is prefered though I suppose I could dig out my old french
book and wade through them in french if I had to.  I'm tired of
reading picture books on fractals.

I'd like to know why 2 is magic.  Why not 1.99?

--Bill Davidson