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fractint-digest         Sunday, August 6 2000         Volume 01 : Number 488




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

Date: Thu, 3 Aug 2000 01:55:40 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: Re: (fractint) Branch Cuts & Julibrot Slices

At 01:13 PM 8/2/00 GMT, you wrote:

>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...

I have tried animating such rotations, but it's quite tricky.  The
entire M-set is not too difficult, but individual objects such as
midgets are quite difficult to rotate smoothly.  To start, the closer
the view comes to the Julia plane, the faster it changes.  Also, the
Mandelbrot objects grow quite a bit larger as the Julia plane is
approached, requiring several outzooms if the Julia aspect is to fit
on the screen.  So far I've gotten nothing but jerky results, but one
of these days when I've got nothing to do, I'll have a serious go at
it.

Jim M.


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

Date: Fri, 04 Aug 2000 11:13:08 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: Fw: Re: (fractint) Re: 2D slices of Julibrot

 >
 >Hello to all,
 >
 >Here is the mathematical derivation of the formula that I posted:
 >
<Well no, it's not here, 'cos I've snipped it.>


The question is whether SliceJB can provide arbitrary two-dimensional
slices through the four-dimensional Julibrot. Its author, John Goering,
has conjectured in the negative. I agree.

Before I begin, I must say that I'll only be considering rotations
through the _origin_ - that is to say, the origin [0,0,0,0] will remain
fixed throughout all of these transformations. If you don't want the
object to remain at the origin, you can (as in Goering's formula) move
it elsewhere after you've done all your rotating.

Following Goering, I label the mutually orthogonal axes of our four-
dimensional Euclidean space P, Q, R, and S. But I'm not going to refer
to these axes in what follows, since rotations are not so much _about_
axes as _in_ planes. This holds even in one, two and three dimensions,
and is a much more sensible way of thinking about rotations if one is
planning on extending any ideas about it to higher dimensions than
three. For example, in two-dimensional space, there is only one plane in
which rotations can occur, and in one dimension there is no room for
_any_.

In four-dimensional Euclidean space one can have rotations in six
mutually perpendicular planes which I'll call PQ, PR, PS, QR, QS and RS.
(How many ways can you choose two letters from a set of four? How many
mutually perpendicular planes are there in 8-space? Hint: there are 56
mutually perpendicular volumes.)

Now that I've got labels for my planes, I need to be able to specify
rotations in them. Rather than try to visualise four dimensions (which
can be done, I'm assured, though personally I'm still having a hard
enough time with three), I'm going to resort to notation. Specifically,
matrices.

Goering describes the rotation of a point in the XY plane by an angle of t
as:

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

In matrix notation, I'll write this as
[x'] = [x][cos(t) -sin(t)]
[y']   [y][sin(t)  cos(t)]

where [x',y'] is Goering's [xnew,ynew] and [x,y] is his [xold,yold].

Let's go the whole hog and jump straight to four dimensions. To rotate a
point in the PQ plane by an angle t:

[p'] = [p][cos(t) -sin(t) 0 0]
[q']   [q][sin(t)  cos(t) 0 0]
[r']   [r][  0       0    1 0]
[s']   [s][  0       0    0 1]

You'll see the effect of those ones and zeros in a second.

To write out this multiplication in something Fracting can work with:

pnew = pold*cos(t) - qold*sin(t) + rold*0 + sold*0
qnew = pold*sin(t) + qold*cos(t) + rold*0 + sold*0
rnew = pold*  0    + qold*  0    + rold*1 + sold*0
snew = pold*  0    + qold*  0    + rold*0 + sold*1

(You can see how the rotation matrix is reflected in this system of
equations, and maybe suss out much of how matrix multiplication works
from it.) Making obvious simplifications gives us

pnew = pold*cos(t) - qold*sin(t)
qnew = pold*sin(t) + qold*cos(t)
rnew = rold
snew = sold

which is precisely what we expect: if we're rotating a point in the PQ
plane, its R and S (or, for that matter, its T, G, or Wibble) coordinates
shouldn't change.

As I said, there are six orthogonal planes through any given point (and,
in particular, through our origin), and each has its own matrix:

         PQ                   QR                   RP
[ cos(t) sin(t) 0 0] [1    0      0    0] [cos(t) 0 -sin(t) 0]
[-sin(t) cos(t) 0 0] [0  cos(t) sin(t) 0] [  0    1    0    0]
[   0      0    1 0] [0 -sin(t) cos(t) 0] [sin(t) 0  cos(t) 0]
[   0      0    0 1] [0    0      0    1] [  0    0    0    1]

         PS                   QS                   RS
[cos(t) 0 0 -sin(t)] [1   0    0    0   ] [1 0   0       0   ]
[  0    1 0    0   ] [0 cos(t) 0 -sin(t)] [0 1   0       0   ]
[  0    0 1    0   ] [0   0    1    0   ] [0 0 cos(t) -sin(t)]
[sin(t) 0 0  cos(t)] [0 sin(t) 0  cos(t)] [0 0 sin(t)  cos(t)]


(Which sin(t) gets the negative sign pretty much depends on whether
you're left-handed or right-handed - whether you're calling the plane
"PQ" or "QP".)

Now, it so happens that we don't need all six. In fact, if we choose
carefully, we can get by with just three!

Let's say we want to rotate an object in the XZ plane, but we only have
XY and YZ rotations (we'll stick to 3D for this if it helps you
visualise things). What we can do is rotate the object 90 degrees in the
XY plane, rotate it by the desired amount in the YZ plane, then rotate
it _back_ 90 degrees in the XY plane again. The first ninety-degree
rotation effectively swaps X for Y, and the second swaps it again.

Doing this with (three-dimensional) transformation matrices (and I'll
(a) use pi/2 radians instead of 90 degrees, 'cos radians are more
natural, and (b) spell out each step in nauseating detail):

[x'] = [x][cos(pi/2) -sin(pi/2) 0][1   0       0   ][cos(-pi/2) -sin(-pi/2) 0]
[y']   [y][sin(pi/2)  cos(pi/2) 0][0 cos(t) -sin(t)][sin(-pi/2)  cos(-pi/2) 0]
[z']   [z][    0          0     1][0 sin(t)  cos(t)][   0           0       1]

= [x][    0         -1     0][1   0       0   ][   0           1       0]
   [y][    1          0     0][0 cos(t) -sin(t)][  -1           0       0]
   [z][    0          0     1][0 sin(t)  cos(t)][   0           0       1]

= [x*0-y*1+z*0][1   0       0   ][   0           1       0]
   [x*1+y*0+z*0][0 cos(t) -sin(t)][  -1           0       0]
   [x*0+y*0+z*1][0 sin(t)  cos(t)][   0           0       1]

= [-y][1   0       0   ][   0           1       0]
   [ x][0 cos(t) -sin(t)][  -1           0       0]
   [ z][0 sin(t)  cos(t)][   0           0       1]

(Oops, -y instead of y! Never mind: it will all come out in the wash.)

= [-y*1 +x*  0    +z*   0   ][   0           1       0]
   [-y*0 +x*cos(t) +z*-sin(t)][  -1           0       0]
   [-y*0 +x*sin(t) +z* cos(t)][   0           0       1]

= [        -y       ][   0           1       0]
   [x*cos(t)-z*sin(t)][  -1           0       0]
   [x*sin(t)+z*cos(t)][   0           0       1]

= [-y* 0 + (x*cos(t)-z*sin(t))*1 + (x*sin(t)+z*cos(t))*0]
   [-y*-1 + (x*cos(t)-z*sin(t))*0 + (x*sin(t)+z*cos(t))*0]
   [-y* 0 + (x*cos(t)-z*sin(t))*0 + (x*sin(t)+z*cos(t))*1]

= [x*cos(t)-z*sin(t)]
   [        y        ]
   [x*sin(t)+z*cos(t)]

...which should, (and does) equal

= [x][cos(t) 0 -sin(t)]
   [y][  0    1    0   ]
   [z][sin(t) 0  cos(t)]

A rotation in the XZ plane! Huzzah!

So three basis rotations are all that's needed for general four-
dimensional rotations (not coincidentally one less than the number of
dimensions.) Let's say they're PS, PQ and QR. Then we can do a rotation
by t in PR by doing a rotation by pi/2 in PQ, a rotation by t in QR, and
a rotation by -pi/2 in PQ. Achieving a rotation in SR can be done with a
rotation in PS, and then one in PR (which we already know how to do),
and then undoing the rotation in PS.



*HOWEVER*

This is all very well; but SliceJB does not provide us with an arbitrary
sequence of rotations about our basis planes, because Fractint doesn't
provide SliceJB with enough hooks for parameters. It provides us with
the opportunity to make _four_ rotations in a _specific_ order. The
first in the QS plane, the second in the QR plane, the third in the PR
plane, and the fourth in the PQ plane.

I'm sorry, but if all you want is a rotation in the PS or RS planes,
you're out of luck. You could get an RS rotation if you could prerotate
in the QS plane, do your work in the QR plane, and then undo the QS
rotation, but by the time you've done the QR rotation, you can't touch
the QS plane again. You can't even construct such a rotation using the
remaining PR and PQ rotations (even if you could use them as many times
as you like).

So Goering's suspicions are justified: his SliceJB formula cannot
provide every two-dimensional slice of the Julibrot.

Morgan L. Owens
"We need eight real parameters!"


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

Date: Thu, 3 Aug 2000 22:07:27 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) Re: 2D slices of Julibrot

At 11:13 AM 8/4/00 +1200, Morgan Owens wrote:

>The question is whether SliceJB can provide arbitrary two-dimensional
>slices through the four-dimensional Julibrot. Its author, John Goering,
>has conjectured in the negative. I agree.
   <very detailed analysis snipped>

I agree also.  Although I once thought the SliceJB formula could provide
all possible rotations in 4-D space, I now realize that I was mistaken.
I discovered this the hard way -- by trying to do the actual rotations.
There are some orientations that simply cannot be reached with the
formula as it is currently written.  Four more parameter entries are
needed if one wishes to achieve any possible rotation centered at any
arbitrary point in the 4-D Julibrot object -- two entries to define the
remaining rotations, two entries to define initial Z.  Perhaps the
developers will take the hint.

Jim M.


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

Date: Thu, 3 Aug 2000 22:35:15 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 04-08-00, (Something Fishy [8])

FOTD -- August 04, 2000 (Rating 8)

Fractal visionaries and enthusiasts:

For today's fractal image it's back to the MandelbrotMix4 and
the surprises therein.  The iterated formula that drew today's
image results from a very small adjustment I made to the formula
that drew the "Circle" and "Eutectic" FOTD images that appeared
a week or so ago.  Actually, it's the recently posted images by
Mike Traynor that convinced me to return to the formula.  The
parameter adjustment involved changing real(p3), which defines a
multiplication factor, from -1.999 to -1.975.

This particular formula and its close variants is one of the
more interesting ones I have stumbled upon in my FOTD
adventures.  It is so interesting because it draws fractals with
inner features unlike any I have seen before -- and where there
are new features, the unusual midgets can't be far behind.

The trick with this formula lies in finding the midgets, which
lurk hidden in places where they would be least expected.  As an
example, reset the logmap to 0 or 1 and back out of today's
image.  See if you would ever suspect that a midget such as
today's would be buried where I searched.

Today's rather spectacular picture rates a slightly optimistic
much-above-average 8 on my 0-to-10 scale of FOTD worth.  While
studying the image, I suddenly got the impression that I was
staring at twisted fish-tails.  The name "Something Fishy" came
to me within a minute.

The 7-minute render time of the parameter file makes the
download the better choice.  That download is available from:

          <alt.binaries.pictures.fractals>

and from:

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

The fractal weather was variably cloudy today, with light rain
at midday and a gentle thunder-shower at sunset -- all in all
nothing to be concerned about.  The fractal cats however dis-
approved of the light midday rain, and showed their displeasure
by sulking indoors all afternoon.

I once again failed in my philosophical aspirations, but I'll
try again tomorrow, and keep trying until I succeed.  Until next
time, take care, and see you in 24 hours.


Jim Muth
[e-mail]


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

Something_Fishy    { ; time=0:07:02.60 -- SF5 on a P200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix4 function=recip passes=1
  center-mag=+2.3504712563716/-2.170429340948094/1.417\
  785e+007/1/2.499 params=-11/-1.1/-1/-11/-1.975/0
  float=y maxiter=2400 inside=0 logmap=142 periodicity=9
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  mzPlzMjzLjzIizGgzDfzCfz9dz7cz4az3fz0az1<5>CzSCzGAz9
  }

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: Fri, 4 Aug 2000 07:48:18 -0500
From: "Jonathan Osuch" <[e-mail]>
Subject: Re: (fractint) Re: 2D slices of Julibrot

Jim,

> Four more parameter entries are
> needed if one wishes to achieve any possible rotation centered at any
> arbitrary point in the 4-D Julibrot object -- two entries to define the
> remaining rotations, two entries to define initial Z.  Perhaps the
> developers will take the hint.

I can add two more complex parameters easily.  And will do so today or
tomorrow.  Beyond that, restructuring will be necessary.  I need to be
spending time on my job search, so the restructuring won't happen any time
soon.

Jonathan



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

Date: Fri, 4 Aug 2000 19:41:47 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) Re: 2D slices of Julibrot

At 07:48 AM 8/4/00 -0500, Jonathan Osuch wrote:

>I can add two more complex parameters easily.  And will do so today or
>tomorrow.

Great!  Even as little as two more parameter entries will increase
versatility an entire order of magnitude.

>Beyond that, restructuring will be necessary.  I need to be
>spending time on my job search, so the restructuring won't happen any time
>soon.

No rush here.  I'll be busy using the two additional parameter entries for
quite a while.

Jim M.


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

Date: Fri, 4 Aug 2000 21:37:07 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 05-08-00, (Flashy Minibrot [6])

FOTD -- August 05, 2000 (Rating 6)

Fractal visionaries and enthusiasts:

Minibrots, Minibrots, Minibrots . . . will I ever get tired of
Minibrots?  Apparently not.  Yes, I realize that they're all the
same basic shape, and that I'm working myself into a rut with
these Minibrots, and that some may find an endless string of
midgets boring.  But Minibrots are fun to find and the variety
is without limit.

To create today's flashy midget I changed the imag(p1) parameter
of yesterday's formula from -1.1 to -1.15.  The change makes a
world of difference in the parent fractal and also in the shape
of the midgets deep inside.  I'm not going to try to describe
the indescribable parent fractal; those who wish to see it may
do so by resetting the logmap to 1 or 0 and outzooming from
today's image until the entire fractal fits on the screen.

Nor will I try to describe the flashy star-like pattern around
the midget in today's image, which I have named "Flashy
Minibrot" for the obvious reason.  The parameter file takes over
11 minutes to render on a modest Pentium.  On a state-of-the-art
machine it will finish in 3 or 4 minutes.

Patience may be preserved by going in a virtual manner to the
Usenet group:

          <alt.binaries.pictures.fractals>

or to the Web site of Paul Lee at:

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

The fractal weather today was very unsettled as one thunder-
storm after another billowed up nearby.  Luckily none billowed
up overhead, and we escaped with only a few light showers.  The
temperature topped out at 84F (29C), which was comfortable
enough for the fractal cats, but the wet grass kept them
indoors.  Fractal cats don't like to get their paws wet.

The big rush ended today as we delivered the final material to
the organizers of the 2000 International Wushu-Kungfu
Championships Festival, which is being held this weekend here in
Baltimore.  Next week should be slow, giving me a chance to
write some of that great philosophy I've been promising for
some time.  I should also have the time to produce even better
(if possible) fractals.

For this evening however, it will be a night of rest and
relaxation.  Until tomorrow, take care, and a good fractal is
nothing to kick about.


Jim Muth
[e-mail]


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

Flashy_Minibrot    { ; time=0:11:20.44 -- SF5 on a P200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=critical.frm
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  }

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: Sat, 05 Aug 2000 13:53:14 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) Re: 2D slices of Julibrot

At 07:48 04/08/2000 -0500, Johnathan Osuch wrote:
>Jim,
>
> > Four more parameter entries are
> > needed if one wishes to achieve any possible rotation centered at any
> > arbitrary point in the 4-D Julibrot object -- two entries to define the
> > remaining rotations, two entries to define initial Z.  Perhaps the
> > developers will take the hint.
>
>I can add two more complex parameters easily.  And will do so today or
>tomorrow.  Beyond that, restructuring will be necessary.  I need to be
>spending time on my job search, so the restructuring won't happen any time
>soon.

It strikes me that it may prove worthwhile in the longer term (it would no
doubt need significant redesign, though) to allow for an arbitrary number
of parameters (p1, p2, p3, ...), rather than hardcoded restrictions.

Morgan L. Owens
"But if you need 50+ parameters, you're probably trying to do too much at
once."



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Date: Sat, 5 Aug 2000 08:02:15 -0500
From: "Jonathan Osuch" <[e-mail]>
Subject: Re: (fractint) Re: 2D slices of Julibrot

Morgan,

> It strikes me that it may prove worthwhile in the longer term (it would no
> doubt need significant redesign, though) to allow for an arbitrary number
> of parameters (p1, p2, p3, ...), rather than hardcoded restrictions.

The major problems with that are how do you display them on the <Z> screen
and how do you store/retrieve them from GIF files.  Realizing that the GIF
file format is eventually going away (the ability to read them will stay for
a while), there may be a way to store an arbitrary number of variables with
the png file format.  That still leaves the problem with the input screens.

Jonathan




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Date: Sat, 5 Aug 2000 11:35:37 -0400
From: John R Goering <[e-mail]>
Subject: (fractint) Re: 2D slices of Julibrot

Thank you, Morgan Owens, for the explanation of rotations in 4D. I have
just a B.S. degree in Mathematics and never formally studied 4D, so I
really appreciate this information.

When Fractint will allow us to write a formula that can do 6 rotations of
a plane in 4D and then move the origin of that plane to any point in
4-space, what would be the best sequence of 6 rotations? Or is there no
*best* sequence?

For example, in PQRS-space, would it be better for the first two
rotations to be independent of each other (e.g., a rotation in QS and one
in PR), or would it be better for the second rotation to be dependent on
the first rotation (e.g., a rotation in QS and then one in QR)? Or
doesn't it matter?


With lots of questions,
John Goering

View a Mandelbrot set image gallery at
http://homestead.juno.com/johnrhg/files/IntroMandelbrot.html
- -------------------------------------
John Ralph H. Goering
[e-mail]
- -------------------------------------
"Now set your heart and your soul to seek the LORD your God." (I Chron.
22:19a)

________________________________________________________________
YOU'RE PAYING TOO MUCH FOR THE INTERNET!
Juno now offers FREE Internet Access!
Try it today - there's no risk!  For your FREE software, visit:
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Date: Sat, 05 Aug 2000 13:43:13 -0400
From: "Damien M. Jones" <[e-mail]>
Subject: Re: (fractint) Re: 2D slices of Julibrot

John,

 - For example, in PQRS-space, would it be better for the first two
 - rotations to be independent of each other (e.g., a rotation in QS
 - and one in PR), or would it be better for the second rotation to
 - be dependent on the first rotation (e.g., a rotation in QS and
 - then one in QR)? Or doesn't it matter?

I too would like to know. I've recently been exploring my own 4D rotation
formulas, and it is hard to predict what the six rotations I've got are
actually doing. One advantage to doing the first two rotations
independently is that, if your 4D object is the Julibrot, you can go from
Mandelbrot to Julia with just the first two rotations. That's easy enough
to figure.

I'm particularly interested in generalized 4D rotations because the same
technique can be used to explore the Julibrot, quaternion M and J,
hypercomplex M and J, Julia2 space, etc. It's just a generally useful tool
in the fractal formula toolbox.

More parameters in formulas would be good, always. As for interface--might
I suggest displaying the formula text on a separate screen (keeping it
scrollable)? That would leave lots of room for additional parameters while
retaining the ability to view the formula text, which often has hints as to
how to use the formula.

Damien M. Jones   \\
[e-mail]  \\  Fractalus Galleries & Info:
                    \\  http://www.fractalus.com/

Please do not post my e-mail address on a web site or
in a newsgroup.  Thank you.


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Date: Sat, 5 Aug 2000 22:37 0000
From: [e-mail]
Subject: RE: Re: (fractint) Re: 2D slices of Julibrot

Hi Folks,

>The major problems with that are how do you display them on the <Z>
screen
>and how do you store/retrieve them from GIF files.  Realizing that the
GIF

Not to mention that the evolver would need a bit of a rethink to cope
with arbitary numbers of parameters too.... Surely enough parms to
define plane and rotation in a 4D volume would do for now!

Cheers,
          Robin.




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Date: Sat, 5 Aug 2000 19:24:14 -0500
From: "Jonathan Osuch" <[e-mail]>
Subject: (fractint) Patch 13

Patch 13 is on the ftp site.  The executable is there as fradev20.0.13.zip.

This patch adds parameters p4 and p5 to the formula parser and now checks
all the parameters when the symmetry options XAXIS_NOREAL and XAXIS_NOIMAG
are used.

I purposely did not add the two new parameters to the evolver.  Maybe
someday.

Jonathan




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Date: Sat, 5 Aug 2000 22:26:54 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) FOTD 06-08-00, (A Flawed Fractal [2])

FOTD -- August 06, 2000 (Rating 2)

Fractal visionaries and enthusiasts:

Today's modest effort rates only a humble 2 on my scale of
worth.  I had originally rated it a 5, but then I realized that
a better image lies closer in toward the midget.  This
realization caused me to consider today's image flawed, and as a
result I dropped its rating to 2 and named it "A Flawed
Fractal".  But it is flawed only in comparison to what it might
have been, and in fact might turn out to be in a soon-to-appear
FOTD.

The formula is once again a combination of various portions of
Z^(-1.1) and Z^(-11) plus 1/C.  Minor variations of this formula
produce images of the most interesting kind, with midgets of an
even more interesting nature lurking within.

The parameter file takes an exasperatingly long time to render
even on a Pentium machine.  But relief is as near as the
internet, where the GIF file has been posted to Usenet at:

          <alt.binaries.pictures.fractals>

and to the Web at:

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

The fractal weather today was perfect -- the first day in
several weeks that I heard no thunder and saw no rain.  The deep
blue sky, puffy clouds, dry atmosphere, and temperature of 83F
(28C) brought out both the neighborhood lawn mowers and the
fractal cats.

The great conditions also brought out the philosophy in me.
Having just completed the program magazine for the 2000 Kung-Fu
Competition, I pondered the nature of the martial-art discipline
known as Kung-Fu.  After working closely with the organizers for
several weeks, I have come to realize that the art of Kung-Fu
involves as much spiritual discipline as physical prowess.  In
fact, when studying the art, one learns how to direct and focus
a spiritual energy known as Qi, or sometimes Chi.

The more skeptical among us tell that such energy does not
exist, that those who claim to be using Qi energy are actually
skillfully using well-known natural laws of physics.  When I
mentioned this to one of the Kung-Fu masters, he merely laughed.
"Who knows more about Qi energy?" he replied.  "Those who have
worked with it all their lives or those who do not yet know that
it exists?"

Then he gave me a brief demonstration of the energy in action.
The tingly heat I felt could have been the power of suggestion,
or a natural physiological effect, or it could have been Qi
energy.  Since the Kung-Fu master knew more about the topic than
I, and he claimed that the heat was caused by an actual
spiritual energy, I must take the word of the expert.  I find it
a bit incredible that Kung-Fu students would devote so much
effort to working with something that does not exist, and after
hundreds of years have still not discovered that it does not
exist.

As for me, I'm undecided whether it exists, but I have no doubt
that 'Qi' is a good scrabble word.  And of course, I'll be here
again in 24 hours with another fractal, this next one guaranteed
to rate higher than today's.  And I'll also have some more words
of wisdom.  Until then, take care, and fractals are forever.


Jim Muth
[e-mail]


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

A_Flawed_Fractal   { ; time=0:25:31.92 -- SF5 0n a P200
                     ;  Version 2000 Patchlevel 9
  reset=2000 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix4 function=recip passes=1
  center-mag=-0.01733612521571942/+1.594853495298927/1.\
  182371e+010/1/125.003 params=-11/-1.1/-1/-11/-1.975/0
  float=y maxiter=6000 inside=0 logmap=149 periodicity=9
  colors=000R62T62<3>gQQjVWn_a<2>xns<3>kskhtjeuh<2>Xxb\
  UyaTwc<8>TnoTmqTlr<2>TivThwUet<4>UWjVUgVSe<3>VKY<3>c\
  YMe`JhcG<3>pp5<7>md_mccmag<3>lXv<3>jpmjtkkrl<7>lfple\
  qmcq<2>m_smZsk_r<3>ebqdcqbdqaeq`eq<6>XMrXJrWGr<3>V5r\
  k2c<8>eHOeJMdLK<3>bRE<3>HZZB`c6bh<3>FlkHokJqlLsl<3>R\
  PXTITUAPV3M<9>eaTfeUghV<2>jsXkvXjuY<5>dt_ct`bs`<3>_s\
  a<2>huZkuYlna<3>nRqnLtoLt<3>qMtqNtrNtrNt<3>oQqVjXnRp\
  <17>_eaZf`Zf`<3>VjXyNB<3>`fTHxq<13>RncRnbSma<3>VjYGb\
  N
  }

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: Sat, 5 Aug 2000 22:30:40 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) Re: Patch 13

At 07:24 PM 8/5/00 -0500, you wrote:

>Patch 13 is on the ftp site.  The executable is there as fradev20.0.13.zip.

I'm eager to get my hands on the latest patch, but I can't find the
address of the ftp site.  What is the full address?

Jim M.


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Date: Sat, 05 Aug 2000 22:46:38 -0400
From: "Damien M. Jones" <[e-mail]>
Subject: Re: (fractint) Re: Patch 13

Jim,

 - I'm eager to get my hands on the latest patch, but I can't find the
 - address of the ftp site.  What is the full address?

Have a look here:

    http://www.fractint.org/ftp/

Damien M. Jones   \\
[e-mail]  \\  Fractalus Galleries & Info:
                    \\  http://www.fractalus.com/

Please do not post my e-mail address on a web site or
in a newsgroup.  Thank you.


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Date: Sat, 05 Aug 2000 21:42:23 -0500
From: Bob Margolis <[e-mail]>
Subject: Re: (fractint) Re: Patch 13

Jim Muth wrote:
>

>
> I'm eager to get my hands on the latest patch, but I can't find the
> address of the ftp site.  What is the full address?
>
http://www.fractint.org/ftp/

Cheers,

Bob

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Date: Sun, 06 Aug 2000 23:38:10 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) Re: 2D slices of Julibrot

At 11:35 05/08/2000 -0400, John Goering wrote:
>When Fractint will allow us to write a formula that can do 6 rotations of
>a plane in 4D and then move the origin of that plane to any point in
>4-space, what would be the best sequence of 6 rotations? Or is there no
>*best* sequence?
>
>For example, in PQRS-space, would it be better for the first two
>rotations to be independent of each other (e.g., a rotation in QS and one
>in PR), or would it be better for the second rotation to be dependent on
>the first rotation (e.g., a rotation in QS and then one in QR)? Or
>doesn't it matter?

There are at least two ways in which a given sequence of rotations might
be considered the *best* - intuitive ease of use and elegance of coding.
Both approaches suggested by John would appear to be the best by one
or the other of these ways.

The most intuitive (for some definition of "intuitive") is probably to
build up rotations in higher dimensions by successively "tuning" in each
dimension in turn: First do a rotation in 2 dimensions: PQ; then the
rest of the rotations in 3 dimensions: PR, QR; then the rest of the
rotations in 4 dimensions: PS, QS, RS; then (if you're feeling
adventurous) you can follow with rotations in PT, QT, RT, and ST....

While this looks like the most straightforward approach for use, it can
lead to some ungainly and lopsided-looking coding:

xnew=((xold*cos(t1)+yold*sin(t1))*cos(t2)+zold*sin(t2))*cos(t4)-wold*sin(t4)
ynew=((yold*cos(t1)-xold*sin(t1))*cos(t3)+(zold*cos(t2)-(xold*cos(t1)+yold*sin(t1))*sin(t2))*sin(t3))*cos(t5)-(wold*cos(t4)+((xold*cos(t1)+yold*sin(t1))*cos(t2)+zold*sin(t2))*sin(t4))*sin(t5)
znew=((zold*cos(t2)-(xold*cos(t1)+yold*sin(t1))*sin(t2))*cos(t3)-(yold*cos(t1)-xold*sin(t1))*sin(t3))*cos(t6)-((wold*cos(t4)+((xold*cos(t1)+yold*sin(t1))*cos(t2)+zold*sin(t2))*sin(t4))*cos(t5)+((yold*cos(t1)-xold*sin(t1))*cos(t3)+(zold*cos(t2)-(xold*cos(t1)+yold*sin(t1))*sin(t2))*sin(t3))*sin(t5))*sin(t6)
wnew=((wold*cos(t4)+((xold*cos(t1)+yold*sin(t1))*cos(t2)+zold*sin(t2))*sin(t4))*cos(t5)+((yold*cos(t1)-xold*sin(t1))*cos(t3)+(zold*cos(t2)-(xold*cos(t1)+yold*sin(t1))*sin(t2))*sin(t3))*sin(t5))*cos(t6)+((zold*cos(t2)-(xold*cos(t1)+yold*sin(t1))*sin(t2))*cos(t3)-(yold*cos(t1)-xold*sin(t1))*sin(t3))*sin(t6)

though other representations are cleaner-looking:
nx=x*cos(t1)+y*sin(t1), ny=y*cos(t1)-x*sin(t1), x=nx, y=ny
nx=x*cos(t2)+z*sin(t2), nz=z*cos(t2)-x*sin(t2), x=nx, z=nz
ny=y*cos(t3)+z*sin(t3), nz=z*cos(t3)-y*sin(t3), y=ny, z=nz
nx=x*cos(t4)-w*sin(t4), nw=w*cos(t4)+x*sin(t4), x=nx, w=nw
ny=y*cos(t5)-w*sin(t5), nw=w*cos(t5)+y*sin(t5), y=ny, w=nw
nz=z*cos(t6)-w*sin(t6), nw=w*cos(t6)+z*sin(t6), z=nz, w=nw

or

x1=x0*cos(t1)+y0*sin(t1), y1=y0*cos(t1)-x0*sin(t1)
x2=x1*cos(t2)+z0*sin(t2), z1=z0*cos(t2)-x1*sin(t2)
y2=y1*cos(t3)+z1*sin(t3), z2=z1*cos(t3)-y1*sin(t3)
x3=x2*cos(t4)-w0*sin(t4), w1=w0*cos(t4)+x2*sin(t4)
y3=y2*cos(t5)-w1*sin(t5), w2=w1*cos(t5)+y2*sin(t5)
z3=z2*cos(t6)-w2*sin(t6), w3=w2*cos(t6)+z2*sin(t6)



The other alternative suggested is to concentrate on independent pairs
of rotations. A rotation in the PQ plane affects neither the R nor S
coordinates of a point, and a rotation in the RS plane leaves the P and
Q coordinates similarly untouched. So rotating by an angle t1 in the PQ
plane then rotating by an angle of t2 in the RS plane has the same net
effect as a rotation by t2 in the RS plane followed by a rotation by t1
in the PQ plane (assuming that the two rotations are consecutive). There
are four pairs of mutually independent planes: PQ/RS, PR/QS and PS/QR
(which reflects the fact that two planes are independent if they have no
axes in common).

The code that's produced by this sequence of rotations is more
symmetric (at least in a long enough line width):
xnew=((xold*cos(t1)+yold*sin(t1))*cos(t3)-(zold*cos(t2)-wold*sin(t2))*sin(t3))*cos(t5)-((wold*cos(t2)+zold*sin(t2))*cos(t4)+(yold*cos(t1)-xold*sin(t1))*sin(t4))*sin(t5)
ynew=((yold*cos(t1)-xold*sin(t1))*cos(t4)-(wold*cos(t2)+zold*sin(t2))*sin(t4))*cos(t6)+((zold*cos(t2)-wold*sin(t2))*cos(t3)+(xold*cos(t1)+yold*sin(t1))*sin(t3))*sin(t6)
znew=((zold*cos(t2)-wold*sin(t2))*cos(t3)+(xold*cos(t1)+yold*sin(t1))*sin(t3))*cos(t6)-((yold*cos(t1)-xold*sin(t1))*cos(t4)-(wold*cos(t2)+zold*sin(t2))*sin(t4))*sin(t6)
wnew=((wold*cos(t2)+zold*sin(t2))*cos(t4)+(yold*cos(t1)-xold*sin(t1))*sin(t4))*cos(t5)+((xold*cos(t1)+yold*sin(t1))*cos(t3)-(zold*cos(t2)-wold*sin(t2))*sin(t3))*sin(t5)

but in reality there is no significant saving of computation, as is shown by
representations analogous to the other two above:

nx=x*cos(t1)+y*sin(t1), ny=y*cos(t1)-x*sin(t1), x=nx, y=ny
nz=z*cos(t2)-w*sin(t2), nw=w*cos(t2)+z*sin(t2), z=nz, w=nw
nx=x*cos(t3)-z*sin(t3), nz=z*cos(t3)+x*sin(t3), x=nx, z=nz
ny=y*cos(t4)-w*sin(t4), nw=w*cos(t4)+y*sin(t4), y=ny, w=nw
nx=x*cos(t5)-w*sin(t5), nw=w*cos(t5)+x*sin(t5), x=nx, w=nw
ny=y*cos(t6)+z*sin(t6), nz=z*cos(t6)-y*sin(t6), y=ny, z=nz

and

x1=x0*cos(t1)+y0*sin(t1), y1=y0*cos(t1)-x0*sin(t1)
z1=z0*cos(t2)-w0*sin(t2), w1=w0*cos(t2)+z0*sin(t2)
x2=x1*cos(t3)-z1*sin(t3), z2=z1*cos(t3)+x1*sin(t3)
y2=y1*cos(t4)-w1*sin(t4), w2=w1*cos(t4)+y1*sin(t4)
x3=x2*cos(t5)-w2*sin(t5), w3=w2*cos(t5)+x2*sin(t5)
y3=y2*cos(t6)+z2*sin(t6), z3=z2*cos(t6)-y2*sin(t6)

Again, the rotations taken in order are t1..t6 for the sequence of
rotations PQ RS PR QS PS QR.

Whether this is easier for a user to work with, however, is
questionable (at least, I find both equally obscure), and requires
experimentation.


In total, there are thirty (6!/4!) distinct orders in which the
rotations can be made (up to a relabelling of the axes):
PQ PR PS QR QS RS       PQ PR PS QR RS QS       PQ PR PS QS QR RS
PQ PR PS QS RS QR       PQ PR PS RS QR QS       PQ PR PS RS QS QR
PQ PR QR PS QS RS       PQ PR QR PS RS QS       PQ PR QR QS PS RS
PQ PR QR QS RS PS       PQ PR QR RS PS QS       PQ PR QR RS QS PS
PQ PR QS PS QR RS       PQ PR QS PS RS QR       PQ PR QS QR PS RS
PQ PR QS QR RS PS       PQ PR QS RS PS QR       PQ PR QS RS QR PS
PQ PR RS PS QR QS       PQ PR RS PS QS QR       PQ PR RS QR PS QS
PQ PR RS QR QS PS       PQ PR RS QS PS QR       PQ PR RS QS QR PS
PQ RS PR PS QR QS       PQ RS PR PS QS QR       PQ RS PR QR PS QS
PQ RS PR QR QS PS       PQ RS PR QS PS QR       PQ RS PR QS QR PS
This assumes that there is nothing to distinguish one axis from another,
though, and most have nothing to commend them. If different axes are
interpreted differently (as they are in the formula) then there can be
as many as 720 different orders.


However you specify a plane in 4-space, you're looking at at least ten real
parameters - six rotations and a point being one example. Other ways of
defining a plane, such as three points or a point and two vectors, might be
considered, but both have twelve parameters and without some preprocessing
one can end up with distortions (if, for example you make the unwarranted
assumption that the two vectors are at right angles when in fact they need
not be). They may not be so useful, but if its easy to find the coordinates
of three noncolinear points in the plane you're interested in, then a
three-point specification of the plane may well be the way to go.


Morgan L. Owens
"Me? I just stick to the combinatorics..."



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End of fractint-digest V1 #488
******************************