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fractint-digest       Wednesday, August 2 2000       Volume 01 : Number 487




----------------------------------------------------------------------

Date: Sun, 30 Jul 2000 23:44:04 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 31-07-00  (Undersea World [5])

FOTD -- July 31, 2000 (Rating 5)

Fractal visionaries and enthusiasts:

Today's discussion will be short because even though it is a
Sunday, I was busy most of the day finishing a job which must
be at the printer's tomorrow morning.

But despite the rush, I found time to find a fractal.  Don't ask
how I find the time to locate all these fractals -- it's as much
a mystery to me as it is to anyone.  Today's FOTD resembles a
midget under a shallow layer of water.  I named it "Undersea
World".

The formula that I used to draw the image is the MandelbrotMix4,
which calculated the expression Z^(-0.95)+Z^(-2.95)+(1/C).  I
have decided to give this overworked formula a rest for the
month of August, and concentrate on things such as new and
unfamiliar slices through the four-dimensional Julibrot figure,
which is the sum of all Julia sets and all warped Mandelbrot
sets.

The parent fractal is a multi-lobed Mandeloid consisting of the
typical fragmented debris that results when negative exponents
of Z are calculated.  But toward the center top lies a strangely
different area.  It is in this area where I found my midget.
The image rates an average 5 on my scale, since I can't imagine
something found in such haste having above-average value.

The parameter file is another slow one, so the trip to:

          <alt.binaries.pictures.fractals>

or to:

          <http://home.att.net/~Paul.N.Lee/FotD/FotD.html>

is strongly recommended.

The fractal weather today was partly cloudy and oppressively
humid with a heavy thunder-shower in the afternoon that caused
local flash-flooding.  Luckily, the rain fell just short of
again flooding the fractal basement.  The temperature of 84F
(29C) was ideal for cats, but cats don't like wet grass, so the
fractal cats spent the day indoors.

Needless to say, with all the work and the rain that threatened
to flood the basement, I found no time to philosophize today,
and probably will find no time again tomorrow.  But sooner or
later the philosophy will come, and when I philosophize, I
philosophize with no holds barred.

So until tomorrow, take care, and be at peace.


Jim Muth
[e-mail]


START 20.0 PAR-FORMULA FILE================================

Undersea_World     { ; time=0:30:39.40 -- SF5 on a P200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix4 function=recip passes=1
  center-mag=-0.05943680542175839/+1.33301484757948400\
  /2.015923e+008/1/-125 params=1/-0.95/1/-2.95/0/0
  float=y maxiter=3000 inside=0 logmap=656 periodicity=9
  colors=000oidpmfppgdqZUqQWYcJqHKUwmwr<3>RStLKtGCt`Gn\
  uKi<7>bRh`ShZTh<3>QWhXeVbnH<3>RWTORWMNZ<3>A4jwwm80lZ\
  SVyrD<2>_Za<8>WZXWZWVZW<3>UZUJTL9ND<3>HDTJAXL8`N5dO3\
  h<3>kFfyGi<3>XLYQMVDM`JNTOOMUPE`R1<3>TQMSQSQQX<2>LQk\
  <3>0Hd<3>bUFlX9u_3SDCQYBMsA<5>WlGYjHZiI<3>edN<6>afB`\
  fA`f8<3>Zf2<3>jPMmKRpGWsB`v7e<3>lDljFngGpeIrcJs<9>`Q\
  l`Rk`Sj<3>_UhUYdO`aIok<2>IH9abWuwqsGF<2>C7M<3>PPFSUD\
  VYC<3>fo5mv0<3>`jGYgKUdO<2>LW_PfvITc8J6<2>G7p<7>PSXQ\
  VURXS<2>UdKVgIVlB<3>WcSX`XXZ`XXdZXg<8>LHpKGqIEr<2>E9\
  uC7vA5y<3>D7sD7rG6p<2>P6j
  }

frm:MandelbrotMix4 {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j,
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l
}

END 20.0 PAR-FORMULA FILE==================================


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------------------------------

Date: Mon, 31 Jul 2000 22:23:41 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 01-08-00  (Seahorse Valley [5])

FOTD -- August 01, 2000 (Rating 5)

Fractal visionaries and enthusiasts:

Every fractalist is familiar with the classic Mandelbrot set.
The most popular area of the M-set is the valley at -0.75 on the
X-axis, otherwise known as Seahorse Valley.  This valley is
filled with fractal scenery of the most interesting kind, making
it the most rewarding area to explore, especially for the
fractal newcomer.

Seahorse valley also has a Julia aspect, which appears as a
string of roughly circular bays, the shorelines infinitely
divided into identical smaller bays.  because of its self-
similarity, this Seahorse Valley Julia set is less interesting
than the Mandelbrot aspect of the valley.

The Seahorse Valley Julia set is merely the completely
perpendicular plane of the four-dimensional Julibrot that
intersects the M-set at the point -0.75 on the X-axis.  But
since Seahorse Valley is actually a four-dimensional object, it
can be sliced in infinitely many other directions.  Today's FOTD
is a slice through Seahorse Valley in one of these new
directions.

The orientation of today's image is but 0.1 degree removed from
the Julia direction, but look at the difference that 1/10 degree
has made.  The familiar Seahorse Valley Julia figure is there,
rotated to a 35-degree angle, but it is filled with a most
incredible background consisting of parallel bands of color.
This background is actually a grossly enlarged side view of the
wall of Seahorse Valley.  The diagonal pink strip marks the line
where the north and south branches of the valley almost meet.

The maxiter of the image is a whopping 2,000,000, and every
iteration is used, since the part of the valley that we're
viewing from the side lies extremely close to the X-axis of the
M-set, where the iterations are well over 1,000,000.

I named the picture "Seahorse Valley" because even though it's
not immediately apparent, it's the Seahorse Valley part of the
Julibrot that we're viewing.  Since the interest of this image
is more mathematical than artistic, I can rate it only an
average 5.  With such a high maxiter, the parameter file takes
well over an hour to render even on a fast Pentium, but being
merciful, I have posted the GIF image to:

          <alt.binaries.pictures.fractals>

The image is also available on Paul's web site at:

          <http://home.att.net/~Paul.N.Lee/FotD/FotD.html>

The formula that drew the image and will draw many more FOTD's
this month of August was posted to the Fractint list last July
by John Goering.  It is the only formula that draws all possible
oblique angles through the 4-D Julibrot.  Unfortunately the
formula is limited by its inability to enter variable starting
points of Z, but this is a limit of the Fractint program, a
limit which could be easily overcome with 2 more variable
parameter entries for type=formula.

The fractal weather today was mostly cloudy and very muggy, with
only a little sun.  Later in the afternoon a light thunder-
shower passed over, but by that time the fractal cats had
already taken their daily romp in the grass.

In the sultry conditions the philosophy languished, but tomorrow
is but 24 hours off.  Check then for a 25 percent chance of
finding philosophy.  Until then, take care, and don't work
yourself into a fractal sweat.


Jim Muth
[e-mail]


START 20.0 PAR-FORMULA FILE================================

Seahorse_Valley    { ; time=1:03:21.27 -- SF5 0n a P200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=julibrot.frm
  formulaname=SliceJB passes=1
  center-mag=-0.00309701/-0.00597368/0.9861933/1/-35
  params=0.4995/0.5/0.499/0.5/-0.75/0 float=y
  maxiter=2000000 inside=0 logmap=yes periodicity=0
  colors=000KSh<3>KbmKdnKgpKjqKlrKnsKovKptKorKopKnn<12\
  >bdRccPebNfbLhaJi`HkZFmXEoVErTKuRQ<4>pHVoFWmDX<18>SB\
  oRBpQBq<3>MBt<8>IV_IXYIZW<3>GfN<2>G`J6ZU8XSAWR<10>Ti\
  DUjCWkB<2>`o7bp6bp6cq5<3>OY4KT4TP4VT3WT3VU3XV3ZV3YW3\
  ZW3_W3`W1_X3<2>a_7aa9adAafC<3>bkIblJcmLcnMcpO<2>dtSd\
  vUcxWdwV<7>k_NlXMmULnRK<2>oII<4>eNScOU<4>UTrSSwTQrTN\
  m<2>UGZUDUUBRU9R<5>_CK`DJaDI<3>eFEfKJ<3>jK_kKclKd<3>\
  pKhqKirKi<3>vMmwNnwNo<5>wQuwRvwRwwSxwSy<33>whz
  }

frm:SliceJB {; by John R. H. Goering, July 1999
  pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1),
  b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), ca=cos(a),
  cb=cos(b), sb=sin(b), cg=cos(g), sg=sin(g), cd=cos(d),
  sd=sin(d), p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
  q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd), r=u*sg+v*ca*sb*cg,
  s=v*sin(a), c=p+flip(q)+p3, z=r+flip(s):
  z=z*z+c
  |z|<=9
  }

END 20.0 PAR-FORMULA FILE==================================


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------------------------------

Date: Tue, 1 Aug 2000 22:28:27 -0400
From: John R Goering <[e-mail]>
Subject: Re: (fractint) FOTD 01-08-00  (Seahorse Valley [5])

On Mon, 31 Jul 2000 22:23:41 -0400 (EDT) Jim Muth <[e-mail]>
writes:

> The formula that drew the image and will draw many more FOTD's
> this month of August was posted to the Fractint list last July
> by John Goering.  It is the only formula that draws all possible
> oblique angles through the 4-D Julibrot.

John Goering responds:

While this formula is able to slice the 4-D Julibrot at many different
angles (including slices that are parallel to each of the six coordinate
planes), unfortunately, I have not proven that it can slice at all
possible angles.  4-D space is much more "vast" than 3-D space, and it is
difficult (maybe impossible for us mortals?) to visualize complex
rotations in that space.

When I developed the SliceJB formula, I despaired of ever understanding
what goes on when a plane is rotated in 4-D.  So I did all of the math
work using rotations in 2-D.  Since I took this simpler approach, I
suspect that this formula cannot produce all possible orientations in
4-D.

Thank you, Jim Muth, for the fantastic pictures that you have been
producing.

Yours truly,
John Goering

View a Mandelbrot set image gallery at
http://homestead.juno.com/johnrhg/files/IntroMandelbrot.html
- -------------------------------------
John Ralph H. Goering
[e-mail]
- -------------------------------------
Jesus says, "Behold, I am coming soon!" (Revelation 22:12, NIV)

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------------------------------

Date: Wed, 2 Aug 2000 00:38:58 EDT
From: [e-mail]
Subject: (fractint) FOTD  02-08-00,  (The Wrong Way Fractal [5])

FOTD -- August 02, 2000 (Rating 5)

Fractal visionaries and enthusiasts:

Despite the latest flooding of the fractal basement, I managed
to find a Fractal of the Day this afternoon.  The image is a
picture of a fairly typical midget in the southern branch of
Seahorse Valley at the coordinates -0.7751187297-0.1232311722i.
Why then does it not look like a midget?

The reason is simple.  We are viewing it from the wrong
direction.  Today's FOTD image is not a scene in the Mandelbrot
set, nor is it a scene in a Julia set.  It is a hybrid, an
Oblate set determined by real(z) and imag(c), with Mandelbrot
characteristics in the vertical direction and Julia characteris-
tics in the horizontal direction.  The name "Oblate" for the
images sliced in this direction is of my own invention.

The familiar Seahorse Valley features are there, but when seen
from this odd direction, they are distorted almost beyond
recognition.  Also in the image is one of those narrow, straight
features that I call bridges.  It crosses near the top of the
open area.

At six minutes, the parameter file is relatively fast by recent
standards.  The download of the GIF image is even faster
however, and may be found by going to the Usenet binary group:

          <alt.binaries.pictures.fractals>

or to Paul Lee's web site at:

          <http://home.att.net/~Paul.N.Lee/FotD/FotD.html>

The fractal weather was partly cloudy and sultry today, with a
temperature of 86F (30C).  Two thunder-storms passed over in the
afternoon, with rain so heavy that it looked like fog.  The
first started the flooding in the basement, the second finished
the job.

I have had it with water in the basement.  I'm now going to
relax and let the water go back down the same drain where it
comes in, by itself, and do no more mud mopping until the
monsoon season ends.  The problem is a too-small storm drain,
which backs up in excessive rainfalls.  The city already knows
about the situation and will replace the drain within 5 years or
so, as soon as they solve the crime problem.  Until then, they
tell me it's God's fault for sending such rain.  Meanwhile, the
fractal cats are complaining about the rain also.

Sorry philosophy-starved readers, there is no philosophy today
unless you consider a wet basement philosophical.  With even
heavier rain due tomorrow, we'll probably have no philosophy
then either.

Until next time, take care, and whistle while you work.


Jim Muth
[e-mail]


START 20.0 PAR-FORMULA FILE================================

TheWrongWayFractal { ; time=0:06:33.81 -- SF5 on a P200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=slices.frm
  formulaname=Oblate passes=1 periodicity=0
  center-mag=0/0/513184.1/0.2883/0/78.728
  params=-0.7751187297/-0.1232318731/-0.7751185803/-0.\
  1232311722 float=y maxiter=12000 inside=0 logmap=277
  colors=000VG0VG0WH0<3>_L7`M8aNA<13>cZZdZ_d_ad`cdaeda\
  fdaf<2>NSfHOfCMhBLiAJlBKlCKnCKpDKpDLq<11>KNaLN_LOZ<2\
  >NOVNOUMSWLVYLWZLYZfBWcFV<3>TSTQVTNYS<3>CjQAnQ7qP4tP\
  2wP<5>Eh_Gf`Icb<3>PWiRUkTSl<3>`HsbFucDv<3>VDdTD_RDWR\
  BT<8>FQQERQDTQ<2>9XP8ZP5`L<3>AZ`BYdCYhDYlDQZDJLBD8<1\
  6>hC6jC6lC6<3>sC6<16>rTYrUZrV`rWbrXcrYerYerYfqU`nQW<\
  3>cAK`5HY5EV4BT48SBYRIcVMcZPg<3>maC<6>ddPceRaeS<3>Zg\
  `<8>RlMQlLPmJOmI<2>LoDNo8<6>BsW9sZ7ta<2>2uk1vn2vm<4>\
  4yh4yg5zf<2>6zc6zc8zZ<2>EzM
  }

frm:Oblate {; Jim Muth  real(z),imag(c)
z=real(pixel)+p1,
c=flip(imag(pixel))+p2:
z=sqr(z)+c,
|z| <= 16
}

END 20.0 PAR-FORMULA FILE==================================

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------------------------------

Date: Wed, 02 Aug 2000 17:34:52 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) FOTD 01-08-00  (Seahorse Valley [5])

At 22:28 01/08/2000 -0400, John Goering wrote:

>On Mon, 31 Jul 2000 22:23:41 -0400 (EDT) Jim Muth <[e-mail]>
>writes:
>
> > The formula that drew the image and will draw many more FOTD's
> > this month of August was posted to the Fractint list last July
> > by John Goering.  It is the only formula that draws all possible
> > oblique angles through the 4-D Julibrot.
>
>John Goering responds:
>
>While this formula is able to slice the 4-D Julibrot at many different
>angles (including slices that are parallel to each of the six coordinate
>planes), unfortunately, I have not proven that it can slice at all
>possible angles.  4-D space is much more "vast" than 3-D space, and it is
>difficult (maybe impossible for us mortals?) to visualize complex
>rotations in that space.
>
>When I developed the SliceJB formula, I despaired of ever understanding
>what goes on when a plane is rotated in 4-D.  So I did all of the math
>work using rotations in 2-D.  Since I took this simpler approach, I
>suspect that this formula cannot produce all possible orientations in
>4-D.

When you originally posted the SliceJB formula, you offered to post your
derivation of the formula also. Perhaps if you did so some of us could
check your reasoning and see whether your suspicions are justified.

(Alternatively, if you don't reckon it's of general interest, you could
just email me privately.)

Morgan L. Owens
"There are only six real parameters to play with -- enough?"



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------------------------------

Date: Wed, 02 Aug 2000 17:31:33 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) FOTD 01-08-00  (Seahorse Valley [5])

At 22:28 01/08/2000 -0400, you wrote:

>On Mon, 31 Jul 2000 22:23:41 -0400 (EDT) Jim Muth <[e-mail]>
>writes:
>
> > The formula that drew the image and will draw many more FOTD's
> > this month of August was posted to the Fractint list last July
> > by John Goering.  It is the only formula that draws all possible
> > oblique angles through the 4-D Julibrot.
>
>John Goering responds:
>
>While this formula is able to slice the 4-D Julibrot at many different
>angles (including slices that are parallel to each of the six coordinate
>planes), unfortunately, I have not proven that it can slice at all
>possible angles.  4-D space is much more "vast" than 3-D space, and it is
>difficult (maybe impossible for us mortals?) to visualize complex
>rotations in that space.

When you originally posted SliceJB, you offered to post your derivation of
the formula. Perhaps if you did so we could have a look to see how you
pulled it off (and whether you succeeded)?

Morgan L. Owens
"Many mathematicians write as if the answer emerged fully-fledged
Minerva-like from their head."



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------------------------------

Date: Wed, 02 Aug 2000 12:45:01 GMT
From: "Andrew Coppin" <[e-mail]>
Subject: Re: (fractint) FOTD 20-07-00 (A Eutectic Mixture [9])

>From: Jim Muth <[e-mail]>
>Reply-To: [e-mail]
>To: [e-mail]
>CC: [e-mail]
>Subject: (fractint) FOTD 20-07-00  (A Eutectic Mixture [9])
>Date: Wed, 19 Jul 2000 23:55:51 -0400 (EDT)
>
>
>FOTD -- July 20, 2000  (Rating 9)
>
>Fractal visionaries and enthusiasts:
>
>No, there is no mistake.  Today's FOTD actually rates a 9 -- at
>least in my opinion.

I am in compleate agreement.

>I almost gave a
>rating of 9 to yesterday's FOTD, but now I'm glad that I settled
>for the 8, since today's picture is notably better and well
>worth the 9.

Right you are!

>I gave the coloring of today's image the extra effort that I
>failed to give yesterday's picture, and that extra effort made
>the difference, as both underlying images have about the same
>potential.

Some of the best colouring I've seen for a while!

Keep up the good work!

(I'm still wading through several weeks worth of emails here!)

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------------------------------

Date: Wed, 02 Aug 2000 12:54:25 GMT
From: "Andrew Coppin" <[e-mail]>
Subject: Re: (fractint) FOTD 27-07-00 (Lotus [4])

>From: Jim Muth <[e-mail]>
>Reply-To: [e-mail]
>To: [e-mail]
>CC: [e-mail]
>Subject: (fractint) FOTD 27-07-00  (Lotus [4])
>Date: Thu, 27 Jul 2000 00:02:32 -0400 (EDT)
>
>
>FOTD -- July 27, 2000 (Rating 4)


Hmmm... I would have rated it a bit higher... I think it's very pretty!

(*still* catching up with me email...)

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------------------------------

Date: Wed, 02 Aug 2000 13:00:56 GMT
From: "Andrew Coppin" <[e-mail]>
Subject: Re: (fractint) FOTD 30-07-00 (An All-New Midget [6])

>From: Jim Muth <[e-mail]>
>Reply-To: [e-mail]
>To: [e-mail]
>CC: [e-mail]
>Subject: (fractint) FOTD 30-07-00  (An All-New Midget [6])
>Date: Sat, 29 Jul 2000 22:21:08 -0400 (EDT)
>
>
>FOTD -- July 30, 2000 (Rating 6)

Very unusual. I like the combination of purple and green. Very nice!

(Only 200 emails to go now...)

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------------------------------

Date: Wed, 02 Aug 2000 13:13:36 GMT
From: "Andrew Coppin" <[e-mail]>
Subject: (fractint) Branch Cuts & Julibrot Slices

Hmmm... I've been doing some work with a little program called "gifcon".
I've been animating my "branch cuts" formula... Very cool stuff! I'd show
you all... but the files a rather large.

But anyway... I wonder if Jim has tried animating Julibrot slices? What does
it look like to take a Mandelbrot zoom, and animate it rotating into the
Julia plane? Surely that's gotta look pretty increadible, right? Hmm... will
try...

BTW, I've been thinking about trying to take a sequence of branch cut frames
and actually construct a 3D model of the complex exp corkscrew... Take the
frames, layer them, and remove the pixels that stay the same colour... I'll
let you know if it works!

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------------------------------

Date: Wed, 2 Aug 2000 16:00:34 -0400
From: John R Goering <[e-mail]>
Subject: Fw: Re: (fractint) Re: 2D slices of Julibrot

Here is the post that I sent some time ago concerning the derivation of
the SliceJB formula (with some slight changes):

Hello to all,

Here is the mathematical derivation of the formula that I posted:

#1

First, I created the following parameterization of a plane in pqrs-space:
  p=u,  q=v,  r=0, and s=0 (where the u-axis is the horizontal axis on
the screen and the v-axis is the vertical axis).

(In my thinking, I oriented the Julibrot so that  the Mandelbrot set is
in the pq-plane and the simplest Julia set [the disk] is in the rs-plane)

Then I did a sequence of counterclockwise rotations in the following
planes: qs-plane, qr-plane, pr-plane, pq-plane.  I performed these
rotations using the following general formula for performing a rotation
in some xy-plane by some angle t: [where the point (xold, yold) is
rotated about the origin to the point (xnew, ynew)]

#2

xnew=xold*cos(t) - yold*sin(t)
ynew=xold*sin(t) + yold*cos(t)

First, the rotation in the qs-plane (where a is the angle of rotation):

#3

q=v*cos(a) - 0*sin(a)
s=v*sin(a) + 0*cos(a)

(Note that qold=v and sold=0 [see #2] according to the first
parameterization [see #1].  This rotation creates a new parameterization
for q and s.)

So now the complete parameterization of the uv-plane in pqrs-space is:

#4

p=u
q=v*cos(a)
r=0
s=v*sin(a)

Then the rotation in the qr-plane (where b is the angle of rotation and
where I substitute the "old" values for q and r from the above
parameterization (#4) into the rotation equations (#2)):

q=[v*cos(a)]*cos(b) - 0*sin(b)
r=[v*cos(a)]*sin(b) + 0*cos(b)

The new parameterization in pqrs-space then is:

p=u
q=v*cos(a)*cos(b)
r=v*cos(a)*sin(b)
s=v*sin(a)

Then the rotation in the pr-plane (where g is the angle of rotation [if
you wonder where this sequence of letters is from, I originally used the
first 4 letters of the Greek alphabet as the angles of rotation]):

p=u*cos(g) - [v*cos(a)*sin(b)]*sin(g)
r=u*sin(g) + [v*cos(a)*sin(b)]*cos(g)

So the new parameterization in pqrs-space is:

p=u*cos(g) - v*cos(a)*sin(b)*sin(g)
q=v*cos(a)*cos(b)
r=u*sin(g) + v*cos(a)*sin(b)*cos(g)
s=v*sin(a)

Then the rotation in the pq-plane (where d is the angle of rotation):

p=[u*cos(g) - v*cos(a)*sin(b)*sin(g)]*cos(d) - [v*cos(a)*cos(b)]*sin(d)
q=[u*cos(g) - v*cos(a)*sin(b)*sin(g)]*sin(d) + [v*cos(a)*cos(b)]*cos(d)

After some mathematical manipulations, the complete parameterization is:

p=u*cos(g)*cos(d) - v*[cos(a)*sin(b)*sin(g)*cos(d) +
cos(a)*cos(b)*sin(d)]
q=u*cos(g)*sin(d) + v*[cos(a)*cos(b)*cos(d) -
cos(a)*sin(b)*sin(g)*sin(d)]
r=u*sin(g) + v*cos(a)*sin(b)*cos(g)
s=v*sin(a)

Then I performed a simple shift of the origin of the uv-plane to the
point (real(p3), imag(p3), 0 ,0).  This point is in the pq-plane ( the
plane that the M-set is in).  So, theoretically, the resulting
parameterizations of p and q are:

p=u*cos(g)*cos(d) - v*[cos(a)*sin(b)*sin(g)*cos(d) +
cos(a)*cos(b)*sin(d)] + real(p3)
q=u*cos(g)*sin(d) + v*[cos(a)*cos(b)*cos(d) -
cos(a)*sin(b)*sin(g)*sin(d)] + imag(p3)

However, to simplify things, I did not really add real(p3) and imag(p3)
in the formula until I initialized the value of c

So in implementing the parameterization, I initially set z = r + s*i and
c = p + q*i + p3 (where p3 is the third parameter. Note that this adds
the real portion of p3 to p and the imaginary portion of p3 to q.)

I included the formula below so that you can compare the parameterization
with how I wrote it in the actual code.

SliceJB {; by John R. H. Goering, July 1999
;This formula produces 2D slices of the 4D Julibrot set. The numbers for
;p1 and p2 describe the rotation of the plane that slices the set (I
;call that plane the uv-plane -- the u-axis is the horizontal axis on
;the screen and the v-axis is the vertical axis). I call the 4 axes
;in 4-space the p, q, r, & s axes. The M-set is in the pq-plane and the
;J-sets are in the rs-plane and planes "parallel" to it. The rotation
angles
;are to be entered as fractions of pi (e.g., pi/4 would be entered as
0.25).
;At first, the uv-plane is identical to the pq-plane.
;Then the rotations are performed counterclockwise in each plane as
follows:
;First: real(p1)--the rotation in the qs-plane.  Then imag(p1)--qr-plane.
;real(p2)--pr-plane. imag(p2)--pq-plane.
;The origin of the resulting skewed uv-plane is then placed at the point
;(real(p3), imag(p3), 0, 0) in pqrs-space. To create the M-set, leave the
;parameters alone, or you may change p3 to change the position of the
M-set
;on the uv-plane.
;To create a J-set, set real(p1)=real(p2)=0.5, then set p3 equal to the
;constant for the J-set (let imag(p1)=imag(p2)=0).
;The parameters for p1 and p2 that are needed to put the uv-plane
parallel to
;the various coordinate planes are as following (the numbers are given in
the
;order -- real(p1), imag(p1), real(p2), imag(p2) ):
;pq-plane: 0, 0, 0, 0   pr-plane: 0, 0.5, 0, 0   ps-plane: 0.5, 0, 0, 0
;qr: 0, 0.5, 0, 0.5  qs: 0.5, 0, 0, 0.5  rs: 0.5, 0, 0.5, 0

  pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1), b=pi*imag(p1)
  g=pi*real(p2), d=pi*imag(p2), ca=cos(a), cb=cos(b), sb=sin(b),
cg=cos(g)
  sg=sin(g), cd=cos(d), sd=sin(d)
  p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd)
  q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd)
  r=u*sg+v*ca*sb*cg, s=v*sin(a), c=p+flip(q) + p3, z=r+flip(s):
  z=z*z+c
  |z|<=4
}


- -------------------------------------
John Ralph H. Goering
[e-mail]
- -------------------------------------
"Let us fix our eyes on Jesus, the author and perfecter of our faith."
(Heb. 12:2a NIV)

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Date: Wed, 2 Aug 2000 16:41:34 -0400
From: John R Goering <[e-mail]>
Subject: (fractint) Re: Animations in the Julibrot

Greetings,

Here is a formula that simplifies the production of animations that morph
the Mandelbrot set into some Julia set (or vice versa):

2DSlices { ; by John R. H. Goering
       ; This fractal produces 2D slices of the Julibrot set.
       ; p1 is the constant for some Julia set while p2 is a number
       ; from 0 to 1.  p2=0 gives the M-set while p2=1 gives
        ; the J-set specified by p1.  To morph from the M-set to the
        ; J-set, vary p2 from 0 to 1.
  w=pixel, a=pi*p2*0.5, c=p1*p2+w*cos(a), z=w*sin(a):
  z=z*z+c
  |z|<=4
}

The basic concepts behind the development of the formula:
1) Start with a plane that slices the Julibrot through the M-set (when
p2=0).
2) As p2 varies from 0 to 1, rotate and translate the plane until the
plane slices the Julia set given by p1.

When making animations, I have discovered that to make a smooth
animation, I need to use smaller increments as p1 gets close to 1.0.  For
some reason, the transitions seem more abrupt as the orientation of the
"slicing" plane gets closer to orientation of the Julia plane.

After initially designing the formula using real numbers, I was
pleasantly surprised by how simple the formula became with the use of
complex numbers.

John Goering

View a Mandelbrot set image gallery at
http://homestead.juno.com/johnrhg/files/IntroMandelbrot.html
- -------------------------------------
John Ralph H. Goering
[e-mail]
- -------------------------------------
"What is lacking cannot be counted." (Ecclesiastes 1:15b NIV)
So zero is not a counting number  :-)

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------------------------------

Date: Wed, 02 Aug 2000 23:31:05 -0400
From: Mike Traynor <[e-mail]>
Subject: Re: (fractint) FOTD 24-07-00  (Ring Around a Midget [7])

Jim,
>
> FOTD -- July 24, 2000 (Rating 7)
>
> Fractal visionaries and enthusiasts:
>
> Naturally, upon finding something I've never before seen in a
> fractal, my immediate impulse is to dive into it in search of
> midgets.

Me too.  The full image itself might have been an FOTD.  There is a
lot of really neat stuff in this one, from these images that look like
they have cables around them to other areas that have a very barnsley look
to them.

Mike

ring22             { ;   (c) Mike Traynor Aug 02, 2000 t=  0:02:21.93
                     ; [e-mail]
                     ; time on PIII-600 1024x768
                     ;  Version 2000 Patchlevel 6
  reset=2000 type=formula formulafile=_m.frm
  formulaname=mandelbrotmix4 function=recip passes=b
  center-mag=+0.01684641914540264/-0.32168366776083770/8487786/1/-164.999
  params=-11/-1.1/-1/-11/-1.99/0 float=y maxiter=1200 inside=bof60
  logmap=62 periodicity=10
  colors=0002742723873AB4BG4EK5HO<3>7Ke7Ki8Ln8Mr8Mv<8>`bycdygfy<3>smzvozsh\
  v<3>iEgf7cd0`<3>hIkiMnjQpf1VZQLRnC<3>T_NUWPUSSUPU<3>iOblOdbIUTDJJ89<3>UM\
  HXPJZTL<3>jdTlhVokXqnZ<3>cTb`NcYIbRD`NE_PFZSGaTHeXIhXJkYKp<2>ZNwZOyZOzYN\
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  H<2>L0F<3>okTvvWpwV<3>VyVPzVKzV<4>DvKBvIAuF<3>4r72r51q30q1<3>4Qk5Jw9Iu<3\
  >PCkTBiVAj<10>r3q
  }

ring21             { ;   (c) Mike Traynor Aug 02, 2000 t=  0:02:42.36
                     ; [e-mail]
                     ; time on PIII-600 1024x768
                     ;  Version 2000 Patchlevel 6
  reset=2000 type=formula formulafile=_m.frm
  formulaname=mandelbrotmix4 function=recip passes=b
  center-mag=+0.01684645676154260/-0.32168364603497580/424389.3/1/-164.999
  params=-11/-1.1/-1/-11/-1.99/0 float=y maxiter=1200 inside=bof60
  logmap=62 periodicity=10
  colors=0002742723873AB4BG4EK5HO<3>7Ke7Ki8Ln8Mr8Mv<8>`bycdygfy<3>smzvozsh\
  v<3>iEgf7cd0`<3>hIkiMnjQpf1VZQLRnC<3>T_NUWPUSSUPU<3>iOblOdbIUTDJJ89<3>UM\
  HXPJZTL<3>jdTlhVokXqnZ<3>cTb`NcYIbRD`NE_PFZSGaTHeXIhXJkYKp<2>ZNwZOyZOzYN\
  y<3>XNpWNnWMk<3>UMfUMeUMd<3>NDULBSJ9PI7N<3>CCNBEN9FN8GN7HN<4>KGMMGMPFL<3\
  >ZFL`FLcFLeFLgFL<3>s_ivdopbej`We_M<3>faJfaIgaH<3>gbDgbCgcBgcAgcA<5>YUEWT\
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  >PCkTBiVAj<10>r3q
  }

ring23             { ;   (c) Mike Traynor Aug 02, 2000 t=  0:04:30.01
                     ; [e-mail]
                     ; time on PIII-600 1024x768
                     ; Version 2000 Patchlevel 6
  reset=2000 type=formula formulafile=_m.frm
  formulaname=mandelbrotmix4 function=recip passes=b
  center-mag=+0.01684642862575926/-0.32168367984283620/1.595448e+007/1/177\
  .5 params=-11/-1.1/-1/-11/-1.99/0 float=y maxiter=1200 inside=bof60
  logmap=62 periodicity=10
  colors=0002742723873AB4BG4EK5HO<3>7Ke7Ki8Ln8Mr8Mv<8>`bycdygfy<3>smzvozsh\
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  j<10>r3q
  }

ring24             { ;   (c) Mike Traynor Aug 02, 2000 t=  0:05:14.06
                     ; [e-mail]
                     ; time on PIII-600 1024x768
                     ; Version 2000 Patchlevel 6
  reset=2000 type=formula formulafile=_m.frm
  formulaname=mandelbrotmix4 function=recip passes=b
  center-mag=+0.01684642862575926/-0.32168367984283620/7.670425e+007/1/177\
  .501 params=-11/-1.1/-1/-11/-1.99/0 float=y maxiter=1200 inside=2
  logmap=175 periodicity=10
  colors=0002742723873AB4BG4EK5HO<3>7Ke7Ki8Ln8Mr8Mv<8>`bycdygfy<3>smzvozsh\
  v<3>iEgf7cd0`<3>hIkiMnjQpf1VZQLRnC<3>T_NUWPUSSUPU<3>iOblOdbIUTDJJ89<6>aW\
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  j<10>r3q
  }

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------------------------------

Date: Thu, 3 Aug 2000 00:18:51 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD  03-08-00,  (Breaking-up Midget [4])

FOTD -- August 03, 2000 (Rating 4)

Fractal visionaries and enthusiasts:

An extremely busy day means a hasty fractal and short discussion.

Today's curious scene lies in the infinite spiral of the
Z^(sqrt(2))+C Mandeloid, a short distance out from the default
view.  The parent fractal is a distorted bay with an even more
distorted main bud.  Today's scene lies in the left branch of the
valley between the bay and bud.

I named the picture "Breaking-up Midget" when I saw how broken
the surrounding elements are.  I can honestly rate the image no
higher than 4, since I put almost no effort into it.  The
11-minute parameter file makes a download preferable.  The
download may be found at:

          <alt.binaries.pictures.fractals>

and at:

          <http://home.att.net/~Paul.N.Lee/FotD/FotD.html>

The fractal weather today  was partly cloudy and very muggy,
with a temperature of 84F (29C) that was perfect for both
fractal cats and grass trimming.  The rain held off until after
dark, when a heavy thunder-storm skimmed past just to the east.
Luckily the rain was not enough to run into the basement.

I had no time for philosophy today, but I'm accumulating a great
backlog of ideas, which sooner or later will be made public.
Despite the rush, I'll return in 24 hours with a new fractal and
a few more words.  Until then, take care, and the best has not
yet arrived.


Jim Muth
[e-mail]


START 20.0 PAR-FORMULA FILE================================

Breaking-up_Midget { ; time=0:11:06.74 -- SF5 on a p200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=branchct.frm
  formulaname=MandelbrotBC passes=1
  center-mag=-0.09561066091537362/+0.78473810657722410\
  /352024.7/1/35 params=1.414213562373/0/0/0 float=y
  maxiter=60000 inside=0 logmap=136 periodicity=10
  colors=000WYl<3>RSoQRpPQp<21>DehCfhCgh<3>Aig<15>VEUW\
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  Nq_<3>ZuRavPdwNfxKiyI<3>uzAwz8vzA<6>rzIrzJqzK<3>ozOn\
  zOozP<17>xzZ
  }

frm:MandelbrotBC = { ; Z=Z^E+C  Andrew Coppin
  e=p1
  p=real(p2)+PI
  q=2*PI*trunc(p/(2*PI))
  r=real(p2)-q
  Z=C=Pixel:
    Z=log(Z)
    IF(imag(Z)>r)
      Z=Z+flip(2*PI)
    ENDIF
    Z=exp(e*(Z+flip(q)))+C
  |Z|<100
}

END 20.0 PAR-FORMULA FILE==================================


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