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fractint-digest        Saturday, April 28 2001        Volume 01 : Number 553




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

Date: Tue, 24 Apr 2001 11:34:08 +0200
From: "Edward Barton" <[e-mail]>
Subject: Re: (fractint) C-FOTD

Jim Muth,
Throughout the short time I have been part of this discussion list, I have
noticed that in your FOTDs, you continually state that a faster rendering
speeds are to be searched for, even if that means getting a new computer.
I don't mean to sound like I am criticising you, but I ask you to remember
what the waiting time was when fractals were discovered, as well as the
quality of the resulting image, and remember that the more interesting
fractal images are those which are hidden deep inside the pattern of a
fractal, and so take significantly longer to render.

Edward Barton

BTW: Despite the tradition of waiting for a week for a fractal, I do so wish
that I didn't have to.  I use my Pentium 133 for other things as well, and
during the rendering, I can do nothing.
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------------------------------

Date: Tue, 24 Apr 2001 09:31:04 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) Re: C-FOTD

At 11:34 AM 4/24/01 +0200, you wrote:

>Jim Muth,
>Throughout the short time I have been part of this discussion list, I have
>noticed that in your FOTDs, you continually state that a faster rendering
>speeds are to be searched for, even if that means getting a new computer.

When one is working with fractals, it is obvious that CPU speed is
important.  I do not need to state the fact.  When I do state it, I am
merely repeating the obvious.  I have no memory of stating that one
should go out and spend money they cannot afford on a new computer, though
I suppose it could be implied in my words.

>I don't mean to sound like I am criticising you,

Your post reads more like a letter of frustration than one of criticism.

>but I ask you to remember
>what the waiting time was when fractals were discovered, as well as the
>quality of the resulting image,

I was there, and I remember well the overnight waits to see a fractal
that we now see in a few minutes.

>and remember that the more interesting
>fractal images are those which are hidden deep inside the pattern of a
>fractal, and so take significantly longer to render.

True!  This is where I search for my FOTD's.


>Edward Barton
>
>BTW: Despite the tradition of waiting for a week for a fractal, I do so wish
>that I didn't have to.  I use my Pentium 133 for other things as well, and
>during the rendering, I can do nothing.

Yes, sometimes the wait does become annoying.

FOTD coming in 15 hours.

Jim Muth


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

Date: Tue, 24 Apr 2001 13:39:04 -0400 (EDT)
From: O Bd <[e-mail]>
Subject: (fractint) Targa Output

Hi! I've been trying to get a Targa output file but the file seems to be
created in the wrong format. Instead of an image I get one that's sort of
looking like a slinky!!!

Is there a way to fix this?

Or am I using the wrong version? Im using 19.2

Any help will be appreciated.

Thanks

O Bd.

The No.1 Utah Jazz fan in England


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

Date: Tue, 24 Apr 2001 14:16:26 -0500
From: "Paul N. Lee" <[e-mail]>
Subject: Re: (fractint) Targa Output

O Bd wrote:
>
> Hi! I've been trying to get a Targa output
> file but the file seems to be created in
> the wrong format.  Instead of an image I get
> one that's sort of looking like a slinky!!!
> Is there a way to fix this?

Are you trying to do 3-D transforms??  And are you doing any Overlays??
Have you set any of the Light Source Parameters??

>
> Or am I using the wrong version?  Im using 19.2

The most current version being developed is 20.1.07 and may be acquired
from the Development Team's web site:

     http://www.fractint.org/

But the latest released version of Fractint is 20.0 and is available
from the Spanky Fractal Database location:

     http://spanky.triumf.ca/www/fractint/fractint.html

>
> Any help will be appreciated.

What program are you using to open the Targa file that was created??


Sincerely,
P.N.L.
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------------------------------

Date: Tue, 24 Apr 2001 20:07:33 -0500
From: "Jonathan Osuch" <[e-mail]>
Subject: Re: (fractint) Targa Output

> But the latest released version of Fractint is 20.0 and is available
> from the Spanky Fractal Database location:

I believe you will find that the Targa output was broken in version 20.0.
It's been fixed in the developer's version.

Jonathan



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

Date: Tue, 24 Apr 2001 23:11:51 -0500
From: "Paul N. Lee" <[e-mail]>
Subject: Re: (fractint) Targa Output

Jonathan Osuch wrote:
>
> I believe you will find that the Targa
> output was broken in version 20.0.
> It's been fixed in the developer's version.
>

Yes, I knew some Patches had been applied, just did not go through all
of the notes to see exactly what version/release/mod-level the repairs
were made at.

Hopefully "O Bd" will choose to upgrade from the 19.2 level that is
being used by him/her, and use the 20.1.07 level.  And maybe do some
further testing for you guys within the area of Targa files.

Sincerely,
P.N.L.
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------------------------------

Date: Wed, 25 Apr 2001 11:51:43 -0000
From: "Andrew Coppin" <[e-mail]>
Subject: Re: (fractint) Re: C-FOTD 18-04-01 (Fractured Fractal [6])

>From: "Paul N. Lee" <[e-mail]>
>Reply-To: [e-mail]
>To: [e-mail]
>Subject: Re: (fractint) Re: C-FOTD 18-04-01 (Fractured Fractal [6])
>Date: Thu, 19 Apr 2001 00:16:05 -0500
>
>Andrew Coppin wrote:
> >
> > Thankyou paul.
>
>Just thought credit should be mentioned where due.    :-)
>
> >
> > It seems only fair to mention that I had
> > help from others on this list thought...
> > (Wish I could remember whom, but I definitly
> > had help!)
>
>You first started discussing the "Branch Cuts" on Fri, 07 Apr 2000.  And
>some of the individuals involved with that topic on the List were as
>follows:
>
>     Barry N. Merenoff (Collin Merenoff)
>     Morgan L. Owens
>     Tim Wegner
>     Jim Muth
>     Kenneth Childress
>     Damien M. Jones
>
>This went on until Tue, 18 Apr 2000.  I am sure that the archives will
>show the details of who did what.

Just spend an hour trolling through the FractInt archives. Gee! I really
must do this more often!!!

The archives show that Morgan provided his usuall deep insights, and Jim
produced a couple of top-notch FotDs with the resulting formula, but the
formula itself basically came from Mr. Merenoff (with a type-error
correction form Kenneth).

Basically, I posted a formula that was partially broken, thus:
>From: "Andrew Coppin" <[e-mail]>
>Subject: (fractint) Branch Cuts
>Date: 10 Apr 2000 10:22:48 GMT
>
>So log(Z) is a cork screw, and you have to take a 360 degree slice of >it.
>Hmmm... This formula (I think) varies the slice taken, but only by >another
>360 degrees. If k is below -3 then you get what the formula parser
> >normally does. If you now raise k, you get a nice branch cut shift...
>until you >reach +3, and thereafter there is no further change. Any ideas
>how to >correct this?
>
>--BEGIN
>comment
>{
>  Andrew Orphi Coppin, 10 May 2000AD
>}
>
>BrotBC = { ; Z^2.5 + C
>           ; real(p1) = phase shift.
>
>  p = real(p1)
>  Z = C = Pixel:
>    Z = log(Z)
>    IF(imag(Z) > p) Z = Z + flip(2*PI) ENDIF
>    Z = exp(2.5*Z) + C
>  |Z| < 4
>}
>--END
Later that day, the following message appeared:
>From: Barry N Merenoff <[e-mail]>
>Subject: (fractint) Branch Cuts
>Date: 10 Apr 2000 08:31:09 -0400
>
>Andrew, try this:
>
>p=3Dreal(p1)+PI
>q=3D2*PI*floor(p/(2*PI))
>r=3Dreal(p1)-q
>Z=3DC=3DPixel:
>Z=3Dlog(Z)
>IF(imag(Z)>p) Z=3DZ+flip(2*PI) ENDIF
>Z=3Dexp(2.5*(Z+flip(q)))+C
>|Z|<4
>}
>
>Note: I don't know if Fractint has a floor function, but it should >have
>something equivalent to it.
>
>Sincerely,
>        Collin Merenoff
As said, the formula was corrected (the "IF(imag(Z)>p)" should actually be
"IF(imag(z)>r)"). And then I posted the message Paul quoted, which is
basically the same but with different variable names. (But still, Jim posted
half a dozen FotDs with my name on 'em. And why not? I *thought* of it!)

Interestingly, the archive also shows that this is the thread that
eventually lead to the addition of extra .FRM parameters in a developer's
patch (can't remember which number tho).

Also during this period there was some cool descussion on why the Mandelbrot
set pops up everywhere. And I asked if fractint's parser could have an arg()
function added. (It never happened!) There was some talk from Tim about
making movable branch-cuts an integral feature of fractint (which also never
happened).

Thanks.
Andrew.

PS. Hey Morgan... What the hell _are_ "sheets in the Riemann surface"?

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

Date: Wed, 25 Apr 2001 09:20:35 EDT
From: [e-mail]
Subject: (fractint) C-FOTD  25-04-01  (OOOF! [3])

Classic FOTD -- April 25, 2001 (Rating 3)

Fractal visionaries and enthusiasts:

Yes, I'm late again today with the FOTD.  It's getting to be a
habit.  My excuse is the same old one which needs no repetition.

Today's fractal image, which looks like a rotten melon with a
face drawn on it, being squashed, fairly cries "OOOF!".  So this
is what I named it.  Normally, I avoid images that resemble
funny faces, but today's picture is so ridiculous that I had no
choice but to declare it FOTD for April 25.  I displayed my
disgust with comedy fractals by rating the image a lowly 3.

The formula behind the image is Z^2+C -- the Mandelbrot formula.
The mouth of the squashed melon-head is actually a Mandel-midget
on the tip of the longest northwest filament shooting out from
the fifth bud along the negative tail of the M-set.  The midget
is so distorted because it has been sliced in a different
direction through the 4-dimensional Julibrot, which is the
assemblage of all Julia sets and all perturbed Mandelbrot sets.

Though the Mandelbrot formula drew the image, the image is not
part of any Mandelbrot set.  Nor is it part of any Julia set.
It is kind of a hybrid, with both Mandelbrot and Julia
characteristics, and a few entirely new characteristics, such as
the stretching effect, thrown in for extra measure.

I seem to have forgotten the Julibrot and the fourth dimension
since becoming involved with the M-Mix4 formula and its
limitless variety of midgets.  But perhaps it's time to get back
to basics as well as the unknown, and nothing is more unknown
than the fourth dimension.

Since the FOTD is so,late, the parameter file is the fastest in
many months, rendering in less than a minute on a 200mhz
Pentium.  It will take longer than that to reach the GIF image,
which is posted to:

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

and to:

          <http://home.swbell.net/sdboyd56/fotd/>

The fractal weather was very warm again today.  The cats,
comfort seekers that they are, took full advantage of the 86F
(30C) temperature, by spending the entire afternoon lounging in
the yard.

That's it for today.  It's time to rev up the fractal shoppe and
get busy with other things.  Until next time, take care, and
life is never so good that it can't get better.


Jim Muth
[e-mail]


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

OOOF               { ; time=0:00:54.58--SF5 on a p200
  reset=2001 type=formula formulafile=multirot.frm
  formulaname=multirot-xz-yw passes=b
  center-mag=-0.00000000002107355/-0.000000000029447\
  31/8.638531e+009/14.687/-180/35.229
  params=45/45/-1.4041038040019/0.00422258114480\
  08/-1.4041038040019/0.0042225811448008 float=y
  maxiter=600 inside=255 logmap=147 periodicity=10
  colors=000Eg`JgbOgdSgfXgh`gjeglignfijckg`ldYnaVpYS\
  qVPsSMtPVrJbpD_rFXsGUtIRvJOwLMxMKpQIiTGaXFV_DNbBGf\
  98i81lYQN`UQbYTdaWgdZihakldmogelgZjgShgLfgEdg7bg0`\
  gFcMTf091V74X67Z5A`4Db3Gd2Jf1MgJaYDaP7aG1a7TF6UO9U\
  WBUdEUlGUtIQjTM`bIRmFIwNLqVNlbQfjSaqUXpSVpRTpPRpOP\
  pNOjLLeJI`HFWFCQD9LB6G93B7195Y9ak8Vj7Pj6Ji6DiCKcHQ\
  YNWTSaNXgIfbVoYgxUskdr_nrcgkfaeiW_lPUoJOrDIu7CkCQa\
  GcbLebQfcVhcZidckdhlemneqofiofaogUogMogFoaLmWQkRVj\
  L_hGdgI`cLY_NUWQQRSNNVJJXGFaJZfLrZLqRLpKLoMQpNVpO_\
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  RAsN9rJ8qF7pB6o85k58h2Ae0CaBPZM`WXlaThgPdmM`sIXxFU\
  ZPM9ZEAaFAcFBeGBgGCfEDeDEeBFdAGd8Hc7Ic6HZ7HV7GR76h\
  l4ilGcdRYYaSQlMJwHCgQL000
  }

frm:multirot-XZ-YW {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p2, c=e*imag(pixel)+p3:
z=sqr(z)+c,
|z| <= 36  }

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

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

Date: Thu, 26 Apr 2001 00:40:06 EDT
From: [e-mail]
Subject: (fractint) C-FOTD  26-04-01  (Baubles [6])

Classic FOTD -- April 26, 2001 (Rating 6)

Fractal visionaries and enthusiasts:

Today's fractal formula takes Z^(-11.5), adds 4 times as much
Z^(-1.15), then adds the reciprocal of C.  Adding 1/C instead of
the usual C turns a fractal inside-out.  The Mandelbrot set,
instead of appearing as a lake in an infinite continent, appears
as an island in an infinite lake.

But fractals created by negative powers of Z normally appear as
islands in infinite lakes, so turning these fractals inside-out
changes them into lakes surrounded by infinite continents.
Combining two different negative powers of Z in an inside-out
fractal and initializing Z to a critical value produces a
fractal with quadratic midgets.  This is the type of fractal
that is the parent of today's scene.

This parent fractal is a huge thing filled with circular
features resembling zippers, with a size exceeding even the full
outzoom capability of the Fractint program.  Today's midget lies
in an area rich with these zipper-circles.

I named the image "Baubles".  The name came to mind after I
spent 1/2 hour with the image on the screen, doing other tasks
and thinking of nothing better.  I rated the image an above-
average 6.

The parameter file renders in under 6 minutes, which is slow
enough to make a download of the GIF image file the better
choice.  That download will ba available within the hour at:

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

and at:

          <http://home.swbell.net/sdboyd56/fotd/>

The fractal weather today here at Fractal central was cloudy and
much cooler than yesterday.  The temperature of 52F (11C) was
just cool enough to keep the fractal cats confined to their
windows, where they sit to watch the outside when they can't
actually be out there.

It's now time to call it a night. . . It's a night! . . . Until
next time, which, within an accuracy of 67 percent, will arrive
in 24 hours, take care, and life as well as cats are better with
fractals.


Jim Muth
[e-mail]


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

Baubles            { ; time=0:05:53.56--SF5 on a P200
  reset=2001 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix4 function=recip passes=1
  center-mag=456.568/-70.8747/1.353355/1/132.499
  params=1/-11.5/4/-1.15/0/300 float=y maxiter=1200
  inside=0 logmap=55 periodicity=0
  colors=00020K30N60S80V90YA0bD0eE0kG0oH0tN6vSCwYHyb\
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  KjzJjzHkzEmzDmzAoz9oz6pz5
  }

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: Thu, 26 Apr 2001 14:31:24 -0500
From: "Paul N. Lee" <[e-mail]>
Subject: (fractint) Web Based Archives

Greetings,

This information is for those that do not know how to use the Usenet
Newsgroups or may have problems using them.  Since some of the other web
based Usenet Newsgroup Archives are either no longer available or not
functioning (such as NailNews), another one has become available:  ETIN
Technologies

Chronologically relevant searching of messages.  Browsing of text and
binary newsgroups.  Posting available.  Free, Public, Complete, and
Anonymous.

Text messages are archived and retained permanently.  Binaries are
retained 10 to 20 days.

Here are the direct URLs for fractal related NGs:

A.B.P.F.
    http://www.etin.com/?folder=alt%2Ebinaries%2Epictures%2Efractals
A.F.P.
    http://www.etin.com/?folder=alt%2Efractals%2Epictures
A.F.
    http://www.etin.com/?folder=alt%2Efractals
S.F.
    http://www.etin.com/?folder=sci%2Efractals
A.B.F-A.
    (this one is not available)


Sincerely,
P.N.L.
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Date: Fri, 27 Apr 2001 07:40:42 -0000
From: "Rupert Millard" <[e-mail]>
Subject: (fractint) Sinusoidal gradient maker

Hello everybody!

It seems that Polynomial gradient maker was a bit boring for you colourful
lot - So I've made Sinusoidal gradient maker.

This program is the same as polynomial gradient maker, except that it puts
sine waves in the red, green and blue channels, instead of polynomials.

You can see the page and donwload the program at:
http://www.geocities.com/kangarupert/sinu_grad_maker.html

From,

Rupert
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Date: Fri, 27 Apr 2001 08:03:58 +0000
From: "Thierry B." <[e-mail]>
Subject: Re: (fractint) Sinusoidal gradient maker

Rupert Millard wrote:
>
>
> This program is the same as polynomial gradient maker, except that it puts
> sine waves in the red, green and blue channels, instead of polynomials.
>
> You can see the page and donwload the program at:
> http://www.geocities.com/kangarupert/sinu_grad_maker.html
>

    Same thing, but in fortran with a web interface
    http://la.buvette.org/fractales/palette.html

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Date: Fri, 27 Apr 2001 08:29:52 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) C-FOTD  27-04-01  (Object Around Midget [5])

Classic FOTD -- April 27, 2001 (Rating 5)

Fractal visionaries and enthusiasts:

As I pondered today's image, I was reminded of nothing in
particular.  The decorations around the midget do resemble
several computer animations I have seen.  And they sometimes
impressed me as the tentacles of some fantastic monster lurking
on the sea floor.  But all these impressions are so vague that I
named the image with the very simple and non-descriptive name,
"Object Around Midget".

The MandelbrotBC formula is responsible for the image, which is
part of the infinite Z^(sqrt2)+C fractal.  This fractal is my
most frequently explored with an exponent of Z between 1 and 2
because in this fractal the midgets are easier to find than in
most fractals lying in this exponent range.

Confining explorations to the fractal created by only one
exponent between 1 and 2 is not limiting, since when drawn by
the MandelbrotBC formula, the fractal is infinite in surface
extent as well as depth and variety.

On my 200mhz Pentium the image takes almost 3/4 of an hour to
render.  It will therefore be an hour or so before it is
available on the Web at:

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

and at:

          <http://home.swbell.net/sdboyd56/fotd/>

The fractal weather today (actually yesterday, April 26) was
just about average for this location at this time of year, with
warm sun and a temperature of 65F (18C).  These conditions had
the fractal cats in the yard most of the afternoon, acting like
kittens.  Such behavior is most undignified for cats of their
venerable age of 9 years.

As for me, it's past 8:15am, and I've got a busy day before me.
So until next time, take care, and enjoy the flowers of Spring
of the leaves of Autumn.


Jim Muth
[e-mail]


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

ObjectAroundMidget { ; time=0:43:53.22--SF5 on a P200
  reset=2001 type=formula formulafile=branchct.frm
  formulaname=MandelbrotBC passes=1 center-mag=-0.15\
  578257934389980/+1.535940899307265/4.223478e+008/1\
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  OuzRzzUzzX7z0Bz4hziczfZza
  }

frm:MandelbrotBC   { ; Inspired by 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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Date: Sat, 28 Apr 2001 17:07:15 +1200
From: "Morgan L. Owens" <[e-mail]>
Subject: Re: (fractint) Re: C-FOTD 18-04-01 (Fractured Fractal [6])

At 23:51 25/04/2001, Andrew Coppin wrote:

>PS. Hey Morgan... What the hell _are_ "sheets in the Riemann surface"?

One of the hassles involved with many functions is that they are many-
valued (and hence, strictly speaking, not functions at all). Even the
humble square root is guilty of this - z^(1/2) usually has two values.
And as for log(z), or arcsin(z)...

What would be nice is if there were some way of turning these many-
valued functions into single-valued functions. Bermhard Riemann came up
with a geometrical stunt that did just this by replacing the complex
plane with a suitable generalisation (mathematicians just loooove to
generalise things).

His idea was to separate each value of the function by placing it on a
separate "sheet", which on its own looks like an entire complex plane.
For example, if the values of z^(1/2) for a given z were a_z and -a_z,
then a_z would go on one sheet and -a_z would go on another. Then you
could take one sheet and say that the values on it are _the_ values of
the function. To drop down into real arithmetic for a second, this is
analogous to saying that _the_ values of the square root function are
the nonnegative ones.

A function may require more than one sheet. The log(z) function, as
immortalised in the MandelbrotBC formula, has an infinite number of
sheets, since log(z) usually produces an infinite number of values for a
given z (if w is one such value, then so are w+2pi, w+4pi, w+6pi...,
among others).

But of course there is more to it than just slapping down extra layers
on the complex plane like tracing paper. For a start, there are often
points where the sheets merge (such as the point z=0 for the function
z^(1/2)). Also significant is the question of where the function moves
from one sheet to another. After all, if this is supposed to be a nice
continuous function, you don't want to introduce a nasty discontinuous
artefact by suddenly jumping from one sheet to another.

So you want to stitch all these sheets together in some way into a
single continuous surface, and it is exactly this that Riemann
described.

Let's take the log function as an example, mainly because that way the
MandelbrotBC formula is available to play with. The log function goes
blecchh at z=0, so we'll stick with nonzero complex numbers. As I said,
if log(z)=w, then log(z)=w+2npi, for any integer n. So w and w+2pi,
w+4pi, w-2pi, w-4pi, w+158pi, ... are all on separate sheets (which we
can index by n, if we are so inclined). The surface is sort of helical.
We start on the ground floor (n=0), and we can walk around the origin
until we've made a complete circle, whereupon we find ourselves on the
first floor (n=1). Another complete circle and we're on the second floor
(n=2). Going the other way from the ground floor we find ourselves on
the first basement level (n=-1), then the second basement (n=-2), and so
on in that direction.

But where precisely do we change floors? What the Riemann surface shows
is that such a question is more a matter of convention than anything to
do with the property of the log() function itself. We could draw a line
out along the positive real axis (on each sheet) and say "Here is where
we change sheets", but we could just as easily draw it along the negative
real axis, or either imaginary axis or anything in between. Wherever you
place it, you can extract a sheet out of the surface that covers the
entire complex plane (except of course for the origin) by cutting along
your chosen axis on any sheet and an adjacent sheet.

It should be pretty obvious that these lines I've been talking about are
the branch cuts. Conventionally, the log() function is cut along the
negative real axis. So is the square root function, for that matter,
while the arctangent function is cut along the negative imaginary axis
starting at 0-i and extending downwards towards -infinity.

To provide a bit of a summary, I'll describe the Riemann surface of the
z^(1/2) function. Since this function is at most two-valued (the only
exception being of course z=0, which has one value), the Riemann surface
is made by stitching two sheets together. Laying them atop each other,
we see that they have to actually meet at z=0 (because they have the
same value there). Now, what do the two values of z^(1/2) have in
common? They're negations of each other, of course, which on the complex
plane is equivalent to rotation about the origin of 180 degrees (or
more appropriately, pi radians). So if we were to start at one value and
walk around the origin and arrive back at our starting point, we'd
find ourselves on the other sheet. Another walk around and we really
will be back where we started. We can manage this by cutting the sheets
as follows. Starting at the origin, we cut a slit out to infinity
(conventionally as I mentioned above), join the lower edge of the lower
sheet to the upper edge of the upper sheet, and the upper edge of the
lower sheet to the lower edge of the upper sheet. Don't worry if this
violates your three-dimensional imagery - it's not a stunt that can be
pulled off in 3D without the optical illusion of the two sheets
apparently interpenetrating. And there's the surface. The (illusory)
crossover line along the axis can be waved and wiggled about all over
the complex plane - as long as one end remains rooted at the origin -
and every such wave and wiggle can be used to define a branch cut for
separating the surface back into sheets again.

Its precisely the positioning of the branch cut for the log() function
that is explored by the MandelbrotBC formula. If you build an animation
where you steadily increase the branch cut parameter, you can almost see
the spiral staircase effect as you climb higher and higher.

See also:http://library.wolfram.com/examples/riemannsurface/

Morgan L. Owens
"I _had_ to ask..."


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

Date: Sat, 28 Apr 2001 08:39:43 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) C-FOTD  28-04-01  (Semi-Chaos [4])

Classic FOTD -- April 28, 2001 (Rating 4)

Fractal visionaries and enthusiasts:

Today's image is of a type that I rather enjoy exploring --
chaotic, yet filled with just enough patches of order and
theme-sections to make it interesting.  I named the image
"Semi-Chaos" mostly as a description.

The midget in today's image is of the order 1.618..., and, for a
change, is not at the dead center of the frame.  Actually, the
midget is an added attraction, the main interest lying in the
fragmentary elements scattered throughout the scene.

Since I find only minor artistic merit in the image, and only a
modicum of math interest, I could rate it no higher than a 4.
But it's still worth a look, if only to see what a Z^1.618+C
Mandeloid midget looks like.

The MandelbrotBC formula, which is rather a committee project,
once again did the work of rendering the image.  The render was
completed in 13-1/2 minutes on a 200mhz Pentium.

The GIF image, already rendered, will soon be available on the
Web at:

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

and at:

          <http://home.swbell.net/sdboyd56/fotd/>

The all-important fractal weather was quite pleasant today.  The
cats approved of the warm sunshine and temperature of 77F (25C),
and showed their approval by sleeping on the porch most of the
afternoon.

While searching this evening for today's fractal, my mind
drifted into heavy philosophical speculation, and I pondered
about gravity and dark matter.  The problem is that the outer
parts of the galaxies are rotating too fast to be caused by the
gravity created by the visible matter of the galaxies.  To
account for this discrepancy, astronomers have hypothesized that
the galaxies are filled with large amounts of undetectable
material they have named dark matter, and this unseen material
is causing the excess gravitational acceleration.  But this
explanation is not satisfactory, since if the matter cannot be
detected, it will be most difficult if not impossible to confirm
its existence.

There may be another possible way of accounting for the excess
rotational velocity, and this is to assume that, like so many
other natural phenomena, the Newtonian law of gravity is valid
only within a limited range, which includes our scale of
experience in everyday life, as well as the solar system.  But
perhaps, beyond this scale, gravity is quantized, and there is a
minimum possible gravitational acceleration.

If this is the case, then the outer parts of the galaxies would
be subject to this minimum acceleration, which, being greater
than the Newtonian laws of gravity would produce, would cause
the excess velocities observed in the outer parts of the
galaxies.  I'll not speculate on how this might affect the
ultimate fate of the universe.

Since this alternate explanation is so obvious, it must have
already been thought of by many astronomers, and probably
discarded because it contains a fatal flaw.  But it did supply
me an hour or so of diversion while working on a dull routine
job.  Perhaps I'll continue this line of speculation on the
philofractal list in the next FOTD.

I now see that it's post time, so until tomorrow, take care, and
could the big bang turn into a big bust?


Jim Muth
[e-mail]


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

Semi-Chaos         { ; time=0:13:37.34--SF5 on a P200
  reset=2001 type=formula formulafile=semikaos.par
  formulaname=MandelbrotBC passes=1
  center-mag=-0.501937/-0.59952/582.5136/1/49.999
  params=1.618034/0/122/0 float=y maxiter=30000
  inside=0 logmap=145 periodicity=10
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  gzzgzzgzzgzzgzzgzzgzzgzzg
  }

frm:MandelbrotBC   { ; Formula by 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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------------------------------

Date: Sat, 28 Apr 2001 08:45:26 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: (fractint) Re: C-FOTD 18-04-01

At 05:07 PM 4/28/01 +1200, Morgan Owens wrote:
>At 23:51 25/04/2001, Andrew Coppin wrote:
>
>>PS. Hey Morgan... What the hell _are_ "sheets in the Riemann surface"?

<long reply from Morgan explaining branch cuts snipped>

Great explanation Morgan!  This is the first time I've actually
understood exactly what the branch-cut discontinuities are.

Jim M.


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End of fractint-digest V1 #553
******************************