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From: cjh@petsd.UUCP (Chris Henrich)
Newsgroups: net.math
Subject: Re: Is the Mandelbrot set a fiction??
Message-ID: <649@petsd.UUCP>
Date: Fri, 20-Sep-85 15:08:35 EDT
Article-I.D.: petsd.649
Posted: Fri Sep 20 15:08:35 1985
Date-Received: Sat, 21-Sep-85 04:53:26 EDT
References: <418@aero.ARPA> <646@petsd.UUCP> <273@steinmetz.UUCP>
Reply-To: cjh@petsd.UUCP (PUT YOUR NAME HERE)
Organization: Perkin-Elmer DSG, Tinton Falls, N.J.
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[]
In article <273@steinmetz.UUCP> putnam@kbsvax.UUCP (jefu) writes:
>Some questions on the mandelbrot set -- but not necessarily having 
>anything to do with it being a fiction.
>
>The Sci. Am. article mentioned that there is an 'amazing theorem' 
>that the Mandelbrot set is connected.  I don't expect to be able to 
>do this myself, but was wondering if anyone would like to sketch out 
>the basic ideas for>the proof.

I have seen discussions of the proof.  Unfortunately the heavy
work on this circle of problems is mostly published in French.
And, needless to say, my references are not where my terminal
is.  A good starting point (with decent graphics) is an
article in the _Mathematical_Intelligencer_ , sometime in
1984, on "Julia" sets.  There is also a survey article in the
_Bulletin_of_the_American_Mathematical_Society_, also early
1984, on iteration and Julia sets.  Further references can be
tracked down there.

The connectedness theorem of Hubbard and Douady can be chased
down through a survey paper by Douady, in French of course, in
the French periodical _Asterisque_. (Exact citation is found
in the _Math_Intelligencer_ article.)  The proof is sketched
in a note in _Comptes_Rendus_.  The fact that it's in French
is not the only barrier to understanding it.  The math is
*heavy.*  The gist seems to be a frightfully ingenious
construction of an analytic function which maps the disc onto
the complement of the Mandelbrot set.  This implies that the
Mandelbrot set, and its complement, are both connected.
>
>Is the complement of the set connected (im pretty sure that the 
>answer to this one is yes, but again, wouldnt know how to prove it
>without a (or many) hint)?
>
>Is there some sort of minimal nice closed bounding curve for the set?
>(nice meaning continuous and derivatives of all orders existing)
>(minimal meaning minimum area in the curve, but not in the set) ?
Well, imagine you were packaging Mandelbrot sets for sale in
the corner drugstore.  You would probably enclose them in a
piece of plastic, to be mounted on a colorfully printed
cardboard backing.  Now, that plastic can be "shrink-wrapped"
to fit the Mandelbrot set.  The more you shrink the wrapper
to fit the Mandelbrot set, the less air is left in your
package, but the more irregular and wrinkly is the packaging.
How far you go is up to you.
>

Regards,
Chris

--
Full-Name:  Christopher J. Henrich
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