Date: Sun, 22 Nov 1998 14:23:59 -0500 (EST)
Message-Id: <1.5.4.16.19981122142337.29b72286@pop.mindspring.com>
To: fractal-art@icd.com
From: Jim Muth <jamth@mindspring.com>
Subject: [fractal-art] Dr. J's headache explained 

At 06:51 PM 11/20/98 -0800, Jay Hill wrote:

          <rest of FOTN snipped>

>PS. Actually, I started playing with Jim's formula, trying first a
>simple MSet and then attempting to figure out what p2 and p3 are
>for. So I put the center coordinates of the image in for these
>parameters and my simple MSet became a giant PAIN in the NECK (see
>Figure 1.)! Since p0 is still 0+0i, I should have the MSet. It looks
>like the MSet with z0 not 0. Jim, how do we use your formula? 

Jay:

The cause of Dr. J's headache in Fig. 1 is exactly what you suspect
 -- he is not critical.  (In fractospace being in critical condition
means being in perfect health.)  What you have drawn is the perturbed
version of this midget with Z initialized to -0.629...+0.462i...

In order to explain how my formula works I must make a startling
statement: there are two Mandelbrot sets.  Well, not actually two
separate sets, but two ways in which the Julibrot may be sliced to
give a complete critical Mandelbrot set.

To see these two sets, bring up the multirot-xy-zw formula and set
function 1 to ident and function 2 to flip.  Leave all parameters at
zero.  Turn off the symmetry and now draw the fractal.  Ahhh ... 
we see our old friend the Mandelbrot set.

But as I have said, there is a second Mandelbrot set lurking nearby. 
To see this second set, set real(p1) to 45, imag(p1) to 45, and leave
p2 and p3 at zero.  What we have done here is to double rotate our
view halfway to the Julia direction.  

The first thing you will notice is that this set is larger -- in fact
its linear dimensions are 1.4142... times those of the classic set,
making its area twice that of the classic set.  I have named this
second Mandelbrot-shaped slice of the Julibrot the Mirror set.

When p1 (angle of rotation) and p2 (initial Z) are kept at zero, my
formula explores the classic set, and the value assigned p3 (initial
C) determines the point of the M-set that will be centered on the
screen.  But when I rotate classic Mandelbrot objects in hyperspace, 
I set both initial C and initial Z to the value of the C-coordinates 
of the object in the classic M-set.  In doing this, I am actually 
shifting to the Mirror set.  (The symmetry must be turned off to 
explore in this way.)

Thus, to rotate the Julibrot around the point at the mouth of the
prominent period-4 NE bud, I would set both p2 and p3 (both initial 
Z and C) to 0.25+0.5i.  If p1 is kept at 0,0 when this is done, a
perturbed set appears on the screen, centered at what is left of the
mouth of the bud.  The set is perturbed because though we are still
in the Mandelbrot orientation, we are off center in the Julibrot.

Now we come to the good part -- the "then a miracle occurs", we might
call it.  Change p1 to 45,45 and brace for a surprise.  We are now in
the mirror set, which is larger than the classic set.  The mouth of
the bud is once again there in its full critical glory, centered on
the screen and ready to be rotated.

Changing the two parts of p1 will now change the angle of view to any
value wished.  The mouth of the bud will always be centered on the
screen, though in many directions it will be totally unrecognizable. 
When p1 is set to 90,90, the Julia set of this point appears.  Change
the two functions from ident,flip to flip,ident for an entirely new
set of rotations between the Mandelbrot and Julia directions.

The other formula -- multirot-xz-yw -- works the same way, though in
this case we explore the rotations that remain simultaneously at 90
degrees from both the Mandelbrot and Julia directions.

When I rotate an object, I use its apparition in the mirror set as a
starting point because the classic set stretches out to an impossible
degree as we rotate toward the four odd planes.

To cure Dr. J's headache, either change p2 to 0,0 and view his
classic apparition, or keep p2 as it is and change p1 to 45,45 to
view his oversized mirror apparition.

There many more such surprises in the Julibrot.  In between FOTD's
and my regular work, I'm trying to get a book about my explorations
together.  (The FOTD for November 23 will appear in about 10 hours.)

Jim Muth
jamth@mindspring.com

_______ ______ _____ ____ ___ __ _
post:  send message to fractal-art@icd.com
unsub: send "unsubscribe" to fractal-art-request@icd.com
admin: send comments to fractal-art-owner@icd.com