From - Mon Nov 18 10:53:41 1996
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From: Andrew Dalton <asdalton@umich.edu>
Newsgroups: sci.fractals
Subject: Fibonacci sequence in the Mandelbrot set
Date: Sat, 16 Nov 1996 23:18:46 -0800
Organization: University of Michigan
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Someone requested this information by e-mail, so here it is.  I am
assuming that those who read this post are already familiar with the
Fibonacci sequence and the basic features of Fractint.

In my parameter file, the "nodes" refer to the structures along the edge
of the set which contain varying numbers of radiating "spokes."  When
the M-set is viewed in Fractint, node 2 is on the left and lies along
the axis of symmetry.  Node 3 corresponds to the structures at the top
and bottom of the screen, each containing 3 spokes.  Node 5 is the
largest node that lies between node 2 and node 3, and it contains 5
spokes.  Node 8 is the largest node that lies between node 3 and node 5,
and it contains 8 spokes.  You get the idea.

To find flower-like structures with the Fibonacci arrangement, first
zoom into one of the nodes (try node 8 or higher).  There will be a
zillion spirals, but the important one is at the end of one of the
spokes, and it is on the _left_ side of the node**.  At this point, you
will want to pick a Julia set that lies near, but not on, this spiral. 
In Winfract, this is done by clicking the right mouse button on the
desired point.  You should get a discontinuous Julia set made of
opposing spirals which correspond to two adjacent numbers on the
Fibonacci sequence.  In the case of node 8, the numbers will be 5 and 8.

[**For node 8, it's on the left side.  For node 13, it's on the right
side.  For node 21, it's on the left side again.  This alternating
pattern is somewhat of a bitch, but hey, I didn't design the Mandelbrot
set or the complex plane.  :)  If you click in the wrong spot, you will
probably get a spiral Julia set with the wrong number of opposing
spirals--such as 8, 3 instead of 8, 5.  I had to figure this out by
experimenting.]

Here are some Fractint parameters that should clear up what I've been
trying to say.  Copy the text below and save it into a text file with a
.PAR extension.  Feel free to respond to this post with questions.

nodes_2_&_3        {
  reset=1821 type=mandel
  corners=-1.914268/0.04818487/-0.2779631/1.194491 inside=0
  }

nodes_3_&_5        {
  reset=1821 type=mandel
  corners=-0.78935752/0.05555374/0.46195099/1.0961634 maxiter=500
  inside=0
  }

nodes_5_&_8        {
  reset=1821 type=mandel
  corners=-0.63073839/-0.32618964/0.50747852/0.73617637 maxiter=1000
  inside=0 logmap=yes colors=@BLUES.MAP
  }

node_8             {
  reset=1821 type=mandel
  corners=-0.420336751/-0.328094801/0.625168255/0.694464784
  maxiter=1000 inside=0 logmap=yes colors=@BLUES.MAP
  }

julia_8            { ; 5,8 phyllotaxis
                     ; compare to cauliflower
  reset=1821 type=julia corners=-2/2/-1.5/1.5
  params=-0.40707697068749998/0.63983602030499998 maxiter=1000
  inside=0 logmap=yes colors=@BLUES.MAP
  }

node_13            {
  reset=1821 type=mandel
  corners=-0.44244417/-0.386793908/0.575152192/0.616959503
  maxiter=1000 inside=0 logmap=yes colors=@BLUES.MAP
  }

julia_13           { ; 8,13 phyllotaxis
  reset=1821 type=julia corners=-2/2/-1.5/1.5
  params=-0.39994128239750004/0.604626346255 maxiter=1000 inside=0
  logmap=yes colors=@BLUES.MAP
  }



-- 
_____________________________________________________________
Andrew Dalton
asdalton@umich.edu

"Faith, n. Belief without evidence in what is told by one who
speaks without knowledge, of things without parallel."
					Ambrose Bierce
_____________________________________________________________