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From: sinclair@aero.ARPA (William S. Sinclair)
Newsgroups: net.math
Subject: Re: Is the Mandelbrot set a fiction??
Message-ID: <480@aero.ARPA>
Date: Fri, 27-Sep-85 20:38:58 EDT
Article-I.D.: aero.480
Posted: Fri Sep 27 20:38:58 1985
Date-Received: Wed, 2-Oct-85 08:50:59 EDT
References: <418@aero.ARPA> <646@petsd.UUCP> <221@epicen.UUCP>
Reply-To: sinclair@aero.UUCP (William S. Sinclair)
Organization: The Aerospace Corp., El Segundo, CA
Lines: 34

Dear Joe;

I'm the one that started this Mandelbrot discussion. The reason I wondered
about the reality of it is for certain numbers near the boundary, it is
near impossible on a finite machine to carry thru the iterations without
the error propagation overwhelming the result. Of course, there are isolated
cases where you can prove a number is within the set, and it is easy to
show that certain numbers are outside the set if the number of iterations is
small enough. The really tough ones are those that require more than 20 or so
iterations, becuase by then the error has complely swamped the result.

I did work out a way where you can generate a large number of numbers in the
set when given just a few--but that doesn't help establish whether a number
picked at random is in or out of the set. The way it works is: Suppose a number
c1 is in the set. Then the number c2=c1**2+c1 is also in the set, and likewise
the 2 roots of this equation:

c3**2+c3=c1

are also in the set. The same inference can be made for numbers NOT in the set.
This gives you a way, by induction, to get a very large number of numbers either
in or out of the set, by starting from one or two numbers. Play with this, see if
you agree. This is interesting, but doesn't help with the pixel map a whole 
lot.


                                   Bill Sinclair
 
 
P.S. To get a good idea, take a complex number like (-1/4,1/4) and do the
iterations using *exact* arithmetic. You'll see how quickly you need to
start rounding off the operations.


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