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From: root@lucifer.acc.virginia.edu (Operator)
Newsgroups: comp.graphics
Subject: Re: Fractals
Summary: test for convergence
Message-ID: <1191@hudson.acc.virginia.edu>
Date: 26 Feb 89 14:40:56 GMT
References: <160.2404E090@muadib.FIDONET.ORG> <11150@s.ms.uky.edu>
Sender: news@hudson.acc.virginia.edu
Reply-To: saj@lucifer.psyc.virginia.edu (Steve Jacquot)
Organization: University of Virginia, Charlottesville
Lines: 25

In article <11150@s.ms.uky.edu> jgary@ms.uky.edu (James E. Gary) writes:
>
>Sorry, no source for you, but there was a 'fast way to skip black' algorithm
>described in a recent Scientific American. The basic idea is to generate
>all the points on the perimeter of a rectangle (initially the size of
>the screen), if all these points are in the set, just flood fill the
>rectangle and quit.  Otherwise divide the rectangle into two halves and
>recursively continue. This is possible because the Mandelbrot set is
>connected.

But what if one of the little spidery filaments that connects it to itself
slips between two of the points you test?  Perhaps this isn't a problem
in practice, but it worried me enough that I didn't want to implement
this algorithm.

What I've been doing instead is testing if the iteration converges, in
parallel with the usual test for divergence.  A fairly inexpensive way of
doing this is described in Dewdney's first article on Mandelbrot, in
Scientific American, Aug '85, p. 24.  The best thing about it is that
different starting points converge to cycles of different lengths, and
if you color the points inside the set according to the length of the
cycle you get really neat patterns.  No more "black areas".  It's also 
faster.  Check it out.

Steve Jacquot				saj@lucifer.psyc.virginia.edu