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fractint-digest       Tuesday, December 5 2000       Volume 01 : Number 514




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

Date: Sat, 02 Dec 2000 19:14:46 -0500
From: "Multiple Bogeys" <[e-mail]>
Subject: (fractint) General Cubic Exploration Par

27 pars. Some take a while, others are fast, all look good. Lines limited to
72 characters -- hope this stops Hotmail from mangling them. Use with
formula in previous message -- no other formulas necessary, nor colormap
files.

- --- cubeg2.par ---

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- --- cubeg2.par ---

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

Date: Sat, 2 Dec 2000 20:18:33 -0500 (EST)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 03-12-00 (4D-03 [NA])

FOTD -- December 03, 2000 (Rating NA)

Fractal visionaries and enthusiasts:

Yesterday's image demonstrated that the Northeast period-4 bud
of the Mandelbrot set exists as an interrupted cylinder in the
three-dimensional real(C)-imag(C)-real(Z) slice of the four-
dimensional Julibrot.  In today's image, which is centered on
the same point as yesterday's, we'll rotate the view and observe
what happens to the ghost of the bud.

The orientation of today's image has been rotated 52 degrees
around the Y-axis (imag[C]) toward the Rectangular direction.
The slice of the image now cuts the cylinder at an oblique
angle, causing the brilliant remnant of the bud to appear as an
ellipse.  I have done no stretching or skewing; the features of
the image appear in their natural undistorted shapes.

Though the Mandelbrot bud enlongates in an orderly manner, the
banded Julia stuff stretches at random, with no apparent order
at all.  The same 4-armed spiral convergence still exists within
the outline of the Mandelbrot bud, but the arms have undergone a
complete change of appearance.

In tomorrow's FOTD I'll rotate the image much further, leading
to some totally unexpected results.

Today's image is once again a very fast one from the parameter
file.  If the 30 seconds is still too slow, the GIF file of the
image 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 sunny but quite cold, with a
temperature of 34F (1C) that kept the cats glued to their
sources of heat.

And speaking of being glued, I'm going to spend the rest of the
evening glued to the TV, watching a junky old movie or two.
Until tomorrow, take care, and see you in 24 hours.


Jim Muth
[e-mail]


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

4D-03              { ; time=0:00:37.54 -- SF5 on a P200
  reset=2000 type=formula formulafile=multirot.frm
  formulaname=multirot-XY-ZW function=flip/ident
  passes=t center-mag=-1.11022e-016/8.32667e-017/10.4\
  1667 params=0/52/0.36775/0/0.281/0.531 float=y
  maxiter=3600 inside=0 logmap=yes periodicity=10
  colors=000BAABWfCAACVfDAAEVfEAAFUfFAAGUfGAAHUfHAAITf\
  IAAJTfJAAKSfKAALSfLAAMSfNAANRgOAAORgPAAPQgQAAQQgRAAR\
  QgSAASPgTAATPgUAAUOgV9AWOgW9AXOgX8AYNgY8AZNgZ8A_Mg_7\
  A`Mg`7AaMga6AbLgb6AcLgc6AdMhd8AdNid9AdOjeAAeQjeCAeRk\
  eEFeSlfGKfTmfJPfUnfLUgVngOZgWogRcgYpgShhZqhUjh_rhWlh\
  `siZniasi`pibtibricujdtjev<6>kiykiykjz<11>nqznqznrz<\
  4>nsznsznsz<31>vzzwzzwzz<2>xzzxzzvyztwzruz<3>jmwhkwf\
  iw<2>`cxZbxXbyVbyTby<9>NbzMbzMbz<2>KbzJbzKdz<4>PnzQp\
  zRrz<2>UxzVzzWzz<3>Zzz_zz_zz<5>_zz_zz`zz<35>Nzz
  }

frm:multirot-XY-ZW {; draws 6 planes and many rotations
;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J
;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J
a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994,
z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p2,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p3:
z=sqr(z)+c,
|z| <= 36  }

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


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Date: Sun, 3 Dec 2000 23:28:11 -0500 (EST)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 04-12-00 (4D-04 [NA])

Classic FOTD -- December 04, 2000 (Rating NA)

Fractal visionaries and enthusiasts:

Today brings us to the fourth in the series of 12 images
illustrating a double rotation from the Mandelbrot orientation
to the Julia orientation.  We started with an everyday image of
the period-4 bud on the northeast shore of the main bay of the
Mandelbrot set.  Then we shifted our view 0.36775 along the
real(Z) axis, and saw that the bud was gone, but its ghost
remained, showing that the bud is actually part of a cylinder.
Next we rotated the view 45 degrees around the imag(C) axis,
which caused the ghost of the bud to elongate into an ellipse.
In today's image, which is rotated 80 degrees toward the
Rectangular direction, we continue that rotation.

When the plane of our view is rotated 80 degrees, it cuts the
Mandelbrot bud-cylinder at such a sharp angle that the once
circular bud is elongated almost beyond recognition.  The low-
iteration Julia material in the foreground is also elongated, in
this case to an incredible degree.  And whereas the ghost-bud is
elongated in an orderly, predictable manner, the foreground
material is stretched haphazardly, with no sense of order.  This
random stretching is characteristic of all Julibrot slices
except those oriented in the Julia or Mandelbrot directions, or
in certain equal double-rotations between the two.

The parameter file of today's image once again renders in less
than a minute.  Also, the GIF image has been posted to:

          alt.binaries.pictures.fractals

and soon will be posted to:

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

The fractal weather today was mostly sunny but very cold.  The
temperature of 30F (-1C) kept the cats snug indoors.

I see by the wall clock that it's 11pm -- time to shut down.
But I'll return in less than a day with the surprising fifth in
the series of images.  Until then, take care, and you'll rarely
get dizzy from rotating a fractal.


Jim Muth
[e-mail]


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

4D-04              { ; time=0:00:55.04 -- SF5 on a P200
  reset=2000 type=formula formulafile=multirot.frm
  formulaname=multirot-XY-ZW function=flip/ident
  passes=t center-mag=-1.11022e-016/8.32667e-017/10.4\
  1667 params=0/80/0.36775/0/0.281/0.531 float=y
  maxiter=3600 inside=0 logmap=yes periodicity=10
  colors=000BAABWfCAACVfDAAEVfEAAFUfFAAGUfGAAHUfHAAITf\
  IAAJTfJAAKSfKAALSfLAAMSfNAANRgOAAORgPAAPQgQAAQQgRAAR\
  QgSAASPgTAATPgUAAUOgV9AWOgW9AXOgX8AYNgY8AZNgZ8A_Mg_7\
  A`Mg`7AaMga6AbLgb6AcLgc6AdMhd8AdNid9AdOjeAAeQjeCAeRk\
  eEFeSlfGKfTmfJPfUnfLUgVngOZgWogRcgYpgShhZqhUjh_rhWlh\
  `siZniasi`pibtibricujdtjev<6>kiykiykjz<11>nqznqznrz<\
  4>nsznsznsz<31>vzzwzzwzz<2>xzzxzzvyztwzruz<3>jmwhkwf\
  iw<2>`cxZbxXbyVbyTby<9>NbzMbzMbz<2>KbzJbzKdz<4>PnzQp\
  zRrz<2>UxzVzzWzz<3>Zzz_zz_zz<5>_zz_zz`zz<35>Nzz
  }

frm:multirot-XY-ZW {; draws 6 planes and many rotations
;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J
;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J
a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994,
z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p2,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p3:
z=sqr(z)+c,
|z| <= 36  }

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


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Date: Mon, 04 Dec 2000 13:37:51 -0000
From: "Andrew Coppin" <[e-mail]>
Subject: (fractint) General Cubic Exploration

I have a FRACTINT.EXE which identifies itself as 20.01.03 Float-Only. When I
try to run cubeg2.par, I get

  Oops. I couldn't understand the argument:
    colors=000uh6<16>XE6VC6UB6<2>P56O46O57<37>OaRObSOcS<3>OfU<2>XtZ<63>rvB

  Any key to continue...

Any ideas, anyone?

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Date: Mon, 04 Dec 2000 14:20:10 -0000
From: "Andrew Coppin" <[e-mail]>
Subject: (fractint) General Cubic Exploration

Eh oh! What the heck...

  Oops. I couldn't understand the argument:
    params=-0.5216333017975402/0.2199346348460341/-0.3903440525528733/-0.2

  Any key to continue...

*now* I'm puzzelled. What's there not to understand? Hmmm...

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

Date: Mon, 04 Dec 2000 14:30:06 -0000
From: "Andrew Coppin" <[e-mail]>
Subject: Re: (fractint) General cubic exploration

>--- cubeg2.frm ---
<SNIP>
>--- cubeg2.frm ---
>
>Those iterate z^3 + 3a^2z + b.

Funny. It looks more like z^3 - 3a^2z + b. 8-|]

>The first one slices the M-set in the a
>plane, with b fixed; the second slices the M-set in the b-plane (which
>interects the a-plane in only a point!) with a fixed; the third produces
>straightforwardly the Julia sets. (Want slices of other orientations
>through
>the 4D M-set? Write your own formula. :-)) The M-set images show four kinds
>of colored regions, and show the fate of both critical points. The regions
>are based on whether both, one, the other, or neither critical point
>escapes. Generally you see two Mandelbrot shapes overlapping -- the
>Mandelbrot images for each critical point -- but they influence one
>another,
>as will be shown by some of the images in the par to follow.

G. I'm going to have to analyse these formulas. I really did some of those
pars (8, 15, 20 especially). I love the way the finaments seem to join to
each other. Like I say, I'll have to have a dam good goggle at those
formulas and try to figure how they work.

PS. I bet no-one can make a formula that will show the A or B plane *and*
the Julia sets using the ismand variable! It kinda fudges up for 4D
mandeloids...

PPS. Oh how I wish I could visualise the 4D cubic Mandeloid with POV-Ray...

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

Date: Mon, 04 Dec 2000 16:44:16 -0500
From: "Multiple Bogeys" <[e-mail]>
Subject: Re: (fractint) General cubic exploration

>Funny. It looks more like z^3 - 3a^2z + b. 8-|]

Oops :-) You're right, of course. That's the form that gives +/-a as
critical points.

I am afraid I have no idea why the parameter file won't work with the
float-only version. You'll just have to use a bona-fide fractINT to view
them...
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------------------------------

Date: Mon, 04 Dec 2000 18:21:08 -0600
From: Tim Wegner <[e-mail]>
Subject: Re: (fractint) General cubic exploration

(Tim wakes up ...)

Multiple Bogey's wrote:

> I am afraid I have no idea why the parameter file won't work with the
> float-only version. You'll just have to use a bona-fide fractINT to view
> them...

I am interested in anything that works in regular fractint but not in
the float-only version (with float=yes of course). I'd appreciate your
emailing me the formula and par that shows the problem.

Tim


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

Date: Tue, 05 Dec 2000 02:34:34 +0100
From: [e-mail] (Thomas Beyer)
Subject: (fractint) a fractint-manzpower-bug

Hi everyone,

Try the parameter-set below, or have a look at:

http://www.freenet.de/beyerth/newbug.htm

The star-shaped figure within the bud is obviously an artefact caused by
a bug in the manzpower-integer-routines.

Did anyone notice that before, or has anyone a clue, how that bug
arises?


manzpower-bug      {
  reset=2000 type=manzpower passes=1
  center-mag=-0.583134/0.751948/6.441843 params=0/0/8/0 maxiter=1000
  logmap=yes
colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzzzLzzz000555<3>HHHKKKOO\
O<3>ccchhhmmmssszzz00z<3>z0z<3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zV\
V<3>zzV<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz00S<3>S0\
S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES<3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2>EHSKK\
S<2>QKSSKSSKQSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS00G<3>G0G<3\
>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8GG8EG8CG8AG88<2>GE8GG8EG8CG8AG88G\
8<2>8GE8GG8EG8CG8AGBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>BGFBG\
GBFGBDGBCG000<6>000
  }

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

Date: Mon, 4 Dec 2000 20:50:23 -0500 (EST)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 05-12-00 (4D-05 [NA])

Classic FOTD -- December 05, 2000 (Rating NA)

Fractal visionaries and enthusiasts:

With today's Rectangular Plane image we enter new territory.  In
the previous images we saw that the shape of the Mandelbrot set,
with all its buds, exists as an interrupted cylinder in the 3-D
section of the Julibrot defined by real(C),imag(C),real(Z).  In
fact the cylinder shape exists in all 3-D slices of the Julibrot
that contain the whole C-plane of the classic Mandelbrot set.

In yesterday's 4D-04 image, we saw that by slicing this cylinder
at an increasingly acute angle, the cross section becomes an
increasingly eccentric ellipse.  The limiting case comes in
today's image, where the slice has been rotated a full 90
degrees from the Mandelbrot orientation, and cuts the cylinder
along its length.  In this case, we see the walls of the
cylinder as straight edges.  The lower wall appears as what I
call a bridge.  This bridge is actually a side view of the lower
valley of the original period-4 bud, which has now stretched to
its limit.

Such bridges and straight-edged areas exist in all the odd
planes, (Elliptic, Oblate, Parabolic, Rectangular), of the
Julibrot, in all the rotations between the odd planes, and in
all the simple rotations between the odd planes and the Julia
plane.  It is immediately apparent that Mandelbrot midgets are
impossible in the images that contain these straight-edges and
bridges, for in such images the Mandelbrot shape has been
stretched to infinity, and therefore any midgets will also be
stretched to infinity.

I have attached the parameters for two versions of the same
image.  The second image is identical to the first except that
the center around which the image rotates has been dropped so
that it falls on the straight upper edge of the lower bridge.
It is the center of this second image that we will use as the
center of the rotation to the Julia orientation.

Notice also in today's image that the banded Julia stuff does
not stretch to infinity, but it does twist and distort in a most
curious manner in the Rectangular plane.

For tomorrow's image, the 6th of the series, we shall leave the
comforting familiarity of 3-D objects and launch out into the
fourth dimension.  The surprises we will find will make the trip
worth the effort.

The parameter files of both of today's images render in well
under one minute.  To save bandwidth, I have posted only the
first image to:

          alt.binaries.pictures.fractals

and to:

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

The fractal weather today here at Fractal Central was sunny and
not too cold.  The afternoon temperature of 45F (7C) and warm
sun lured the cats outdoors for a brief romp before that began
worrying for their meal.

That's it for another day, but I'll be back in 24 hours, when
we'll take off on a trip to Julia land.  Until then, take care,
and keep those extra dimensions coming.


Jim Muth
[e-mail]


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

4D-05              { ; time=0:00:40.42 -- SF5 on a P200
  reset=2000 type=formula formulafile=multirot.frm
  formulaname=multirot-XY-ZW function=flip/ident
  passes=t center-mag=-1.11022e-016/8.32667e-017/10.4\
  1667 params=0/90/0.36775/0/0.281/0.531 float=y
  maxiter=3600 inside=0 logmap=yes periodicity=10
  colors=000BAABWfCAACVfDAAEVfEAAFUfFAAGUfGAAHUfHAAITf\
  IAAJTfJAAKSfKAALSfLAAMSfNAANRgOAAORgPAAPQgQAAQQgRAAR\
  QgSAASPgTAATPgUAAUOgV9AWOgW9AXOgX8AYNgY8AZNgZ8A_Mg_7\
  A`Mg`7AaMga6AbLgb6AcLgc6AdMhd8AdNid9AdOjeAAeQjeCAeRk\
  eEFeSlfGKfTmfJPfUnfLUgVngOZgWogRcgYpgShhZqhUjh_rhWlh\
  `siZniasi`pibtibricujdtjev<6>kiykiykjz<11>nqznqznrz<\
  4>nsznsznsz<31>vzzwzzwzz<2>xzzxzzvyztwzruz<3>jmwhkwf\
  iw<2>`cxZbxXbyVbyTby<9>NbzMbzMbz<2>KbzJbzKdz<4>PnzQp\
  zRrz<2>UxzVzzWzz<3>Zzz_zz_zz<5>_zz_zz`zz<35>Nzz
  }

4D-05a             { ; time=0:00:44.46 -- SF5 on a P200
  reset=2000 type=formula formulafile=multirot.frm
  formulaname=multirot-XY-ZW function=flip/ident
  passes=t center-mag=-1.11022e-016/8.32667e-017/10.4\
  1667 params=0/90/0.36775/0/0.281/0.4871 float=y
  maxiter=3600 inside=0 logmap=yes periodicity=10
  colors=000BAABWfCAACVfDAAEVfEAAFUfFAAGUfGAAHUfHAAITf\
  IAAJTfJAAKSfKAALSfLAAMSfNAANRgOAAORgPAAPQgQAAQQgRAAR\
  QgSAASPgTAATPgUAAUOgV9AWOgW9AXOgX8AYNgY8AZNgZ8A_Mg_7\
  A`Mg`7AaMga6AbLgb6AcLgc6AdMhd8AdNid9AdOjeAAeQjeCAeRk\
  eEFeSlfGKfTmfJPfUnfLUgVngOZgWogRcgYpgShhZqhUjh_rhWlh\
  `siZniasi`pibtibricujdtjev<6>kiykiykjz<11>nqznqznrz<\
  4>nsznsznsz<31>vzzwzzwzz<2>xzzxzzvyztwzruz<3>jmwhkwf\
  iw<2>`cxZbxXbyVbyTby<9>NbzMbzMbz<2>KbzJbzKdz<4>PnzQp\
  zRrz<2>UxzVzzWzz<3>Zzz_zz_zz<5>_zz_zz`zz<35>Nzz
  }

frm:multirot-XY-ZW {; draws 6 planes and many rotations
;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J
;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J
a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994,
z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p2,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p3:
z=sqr(z)+c,
|z| <= 36  }

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


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

Date: Mon, 04 Dec 2000 21:15:04 -0600
From: "Paul N. Lee" <[e-mail]>
Subject: (fractint) Wada basins (4 sphere fractals)

Paul Bourke has added a new page to his web site that has "rendering
chaotic scattering":

    http://www.swin.edu.au/astronomy/pbourke/fractals/wada/

Some interesting images with plenty of information.


Sincerely,
P.N.L.
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Date: Tue, 05 Dec 2000 13:56:14 -0000
From: "Andrew Coppin" <[e-mail]>
Subject: Re: (fractint) General cubic exploration

>From: Tim Wegner <[e-mail]>
>Reply-To: [e-mail]
>To: [e-mail]
>Subject: Re: (fractint) General cubic exploration
>Date: Mon, 04 Dec 2000 18:21:08 -0600
>
>(Tim wakes up ...)
>
>Multiple Bogey's wrote:
>
> > I am afraid I have no idea why the parameter file won't work with the
> > float-only version. You'll just have to use a bona-fide fractINT to view
> > them...
>
>I am interested in anything that works in regular fractint but not in
>the float-only version (with float=yes of course). I'd appreciate your
>emailing me the formula and par that shows the problem.
>
>Tim

No, I just downloaded FractInt 20.03.01 - the one *with* integer math - and
I still get

>  Oops. I couldn't understand the argument:
>    >colors=000uh6<16>XE6VC6UB6<2>P56O46O57<37>OaRObSOcS<3>OfU<2>XtZ<63>rvB
>
>  Any key to continue...

This isn't a float-only problem. I see two posibilities. 1) the .par I'm
using is somehow messed up, 2) it's a patch 3 thing.

PS. The .pars all say they were made with 20.01 patchlevel 1. How do I get
my hands on that? I can't seem to find the file on the fraction FTP site...

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

Date: Tue, 5 Dec 2000 20:26:26 -0500
From: Lee Skinner <[e-mail]>
Subject: (fractint) a fractint-manzpower-bug

Hi Thomas, =


An very unusual artifact of integer math.  Set float=3Dyes and the artifa=
cts =

will go away.  And put float=3Dyes in your sstools.ini file to keep such =

artifacts from showing up in the future.

>> manzpower-bug      {
  reset=3D2000 type=3Dmanzpower passes=3D1
  center-mag=3D-0.583134/0.751948/6.441843 params=3D0/0/8/0 maxiter=3D100=
0
  logmap=3Dyes
colors=3D00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzzzLzzz000555<3>HHHKKKO=
O\
O<3>ccchhhmmmssszzz00z<3>z0z<3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zV\=

V<3>zzV<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz00S<3>S0\=

S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES<3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2>EHSKK\=

S<2>QKSSKSSKQSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS00G<3>G0G<3\=

>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8GG8EG8CG8AG88<2>GE8GG8EG8CG8AG88G\=

8<2>8GE8GG8EG8CG8AGBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>BGFBG\=

GBFGBDGBCG000<6>000
  }
<<

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Date: Tue, 5 Dec 2000 21:40:00 -0500 (EST)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 06-12-00 (Minibrot Interlude [6])

Classic FOTD -- December 06, 2000 (Rating 6)

Fractal visionaries and enthusiasts:

Due to an unexpected rush, (these things happen), I was unable
to complete the discussion for the sixth in the series of
illustrations of a four-dimensional rotation.  But all is not
lost, for I substituted a nice, (there's that word again),
minibrot from the M-Mix4 formula.  And the 4-D series will
continue one day behind schedule in tomorrow's FOTD.

The expression that was iterated to produce today's fractal is
2Z^(-1.2)+0.4Z^(-12)+(1/C).  I named the picture "Minibrot
Interlude" because there is a Minibrot at the center and it is
an interlude between the two parts of the 4-D rotation.  I rated
the image a 6, which seems about right.

The parameter file is a slow one, requiring over 16 minutes to
render, and making a download of the GIF image file the better
choice.  The image file may be found on Usenet at:

          <alt.binaries.pictures.fractals>

and in an hour or so at:

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

The fractal weather today was partly sunny, mild in the morning
but turning cold in the afternoon.  The fractal cats, unable to
decide where to spend the day, passed the time indoors in a bad
mood.

That's it for today.  I'll return tomorrow with more time and
further discussion about rotation in four-dimensional space.
Until then, take care, and see you soon.


Jim Muth
[e-mail]


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

Minibrot_Interlude { ; time=0:16:22.22 -- SF5 on a p200
  reset=2000 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix4 function=recip passes=1
  center-mag=+4.95915103552808600/-0.41325256410315570\
  /1.798354e+007/1/-22.499 params=2/-1.2/0.4/-12/0/0
  float=y maxiter=2400 inside=0 logmap=283 periodicity=9
  colors=000gme<3>S`ROYOKVKGSGCPC3OA<3>QH7VF7`D6<2>q84\
  w41<2>tD9<2>rLP<3>nXYnZ_maakdcieegcg<5>WSnUQoSOp<2>M\
  IsKGtKHs<5>ONjPOiQPgQQfRRd<2>SSb<3>iD9<25>FW5DX5CY5<\
  3>8_5<15>hoEjpElqF<2>stGuuHyxI<18>Xb9Va9U`8<3>OX7<3>\
  QT5RS4RR4<2>TO2TO2WQB<2>bVaM9ndWiyre<3>qqaop`mp_kpZw\
  uPjpY_lj<3>ZjSZjOZjJ<2>Zi61om<10>NYpPWpRVq<3>ZPq<3>y\
  oF<3>ujEthEsgErfEqeDpcD<4>kgDjhCiiC<3>emCamNeoC<3>ew\
  OevRezUezXezZZzcTzhJzm<3>WzjZziazidzhkzl<2>_z`czc<6>\
  czc
  }

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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