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To: fractint-digest@lists.xmission.com
Subject: fractint-digest V1 #569
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fractint-digest Thursday, June 7 2001 Volume 01 : Number 569
----------------------------------------------------------------------
Date: Sat, 2 Jun 2001 23:03:29 -0400 (EDT)
From: Jim Muth
Subject: (fractint) C-FOTD 03-06-01 (Julia mandelbrot [6])
Classic FOTD -- June 03, 2001 (Rating 6)
Fractal visionaries and enthusiasts:
Three FOTD images in a row, all rating an 8, is too much to hope
for. So I had no hopes of producing another 8-rated image when
I sat down for my daily search for fractals this evening. Then,
when the day's effort yielded only a 6-rated image, I was not
disappointed.
But a rating of 6 is still above average, and well worth the 7
minutes required to render the scene from the parameter file.
The formula that created the image is totally whimsical --
1.3Z^(1.3)+0.13Z^(-13)+(1/C). I found the parameters by turning
off my thinking (quite an easy task) and letting my fingers roam
where they wished.
The name "Julia Mandelbrot" came to me as I studied the picture.
The pattern around the central midget is that of julia sets
located in the East Valley area of Mandelbrot midgets, yet the
midget itself is pure Mandel stuff. The combination inspired
the name. The rating of 6 is honest, since the image has too
many imperfections for a higher rating.
Other than running the parameter file, an alternate way of
viewing the image is to download the GIF file from the Web at:
or at:
The fractal weather today was variably cloudy but so far dry,
with a temperature of 77F (25C). The fractal cats must have
approved, for they spent much time in the yard, trying to stay
out of trouble.
Now it's my turn to try to stay out of trouble. It's hard to
imagine how I could get into trouble watching a junky old sci-fi
movie, so that's what I'll do. Until next time, take care, and
take two fractals at bedtime for a good night's sleep.
Jim Muth
jamth@mindspring.com
START 20.0 PAR-FORMULA FILE================================
Julia_Mandelbrot { ; time=0:06:47.82--SF5 on a p200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=-2.485951411296012/+0.00097323098757938\
/3.036155e+007/1/102.5 params=1.3/1.3/0.13/-13/0/0
float=y maxiter=850 inside=0
logmap=-137 periodicity=10 sound=off
colors=000F00J00L00S80WB0_F0cJ0gM1kQ5oU8sZCudHwjLy\
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0jr0jr0ir0ir0ir0ir0hr0hr0
}
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: Sun, 03 Jun 2001 09:01:10 -0000
From: "Andrew Coppin"
Subject: Re: (fractint) C-FOTD 31-05-01 (Fractal Fractles [7])
>From: Jim Muth
>Reply-To: fractint@lists.xmission.com
>To: fractint@lists.xmission.com
>CC: philofractal@lists.fractalus.com
>Subject: (fractint) C-FOTD 31-05-01 (Fractal Fractles [7])
>Date: Thu, 31 May 2001 10:59:51 -0400 (EDT)
>
>
>Classic FOTD -- May 31, 2001 (Rating 7)
Erm... Wow! That is one *trippy* image! I really really dIg those colours!
Very nice work, Jim!
Thanks.
Andrew.
PS. One day (when I have web space) I think I may start doing a fractal of
the month...
_________________________________________________________________________
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------------------------------
Date: Sun, 03 Jun 2001 09:06:06 -0000
From: "Andrew Coppin"
Subject: Re: (fractint) C-FOTD 03-06-01 (Julia mandelbrot [6])
>From: Jim Muth
>Reply-To: fractint@lists.xmission.com
>To: fractint@lists.xmission.com
>CC: philofractal@lists.fractalus.com
>Subject: (fractint) C-FOTD 03-06-01 (Julia mandelbrot [6])
>Date: Sat, 2 Jun 2001 23:03:29 -0400 (EDT)
>
>
>Classic FOTD -- June 03, 2001 (Rating 6)
I'd rate this one higher than that, perhaps 7, on account of the elegant
simplicity of the image and the very nice, harmonious colours.
Thanks.
Andrew.
_________________________________________________________________________
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------------------------------
Date: Sun, 3 Jun 2001 09:55:03 -0400 (EDT)
From: Jim Muth
Subject: (fractint) Re: Wet basement
At 12:00 PM 6/3/01 +1200, Morgan Owens wrote:
>Eeh, you're really going to have to do something about that
>basement. Next thing, you'll have your house rotting out from
>under you.
As long as it's not brain rot, I can live with it. The brick
house is in pretty good shape -- foundation sitting on solid rock
- -- and the water comes in only with very heavy rain, like the 10cm
in 8 hours downpour we had Friday. To fix the problem, it would be
necessary to dig up my yard and the garden of the elderly widow
next door, which would upset not only the widow, but the cats as
well. I have installed a sump pump, and that keeps the wetness
down to a wet floor, so I guess I can live with the dampness a
while -- just as long as some alien fractal fungus doesn't develop
and start growing down there.
Jim M.
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------------------------------
Date: Mon, 4 Jun 2001 01:24:32 -0400 (EDT)
From: Jim Muth
Subject: (fractint) C-FOTD 04-06-01 (Cometary Impression [8])
Classic FOTD -- June 04, 2001 (Rating 8)
Fractal visionaries and enthusiasts:
The hot streak continues unabated. Not only is my fractal
philosophy once again active, but my fractal images are active
also. And all this action piled up on a day with so many
mundane chores needing to be done.
With today's image we're back up to a superior rating of 8. If
the present trend continues, we may reach one of the very rare
9's, or possibly even an unheard-of 10 before long.
To create today's fractal I entered the parameters for
- -2Z^(-1.1)-0.0002Z^(-11)+(1/C) into the MandelbrotMix4 formula,
and let her rip. Today's scene lies at the end of a curving
filament extending from a bud, very near the point where many
filaments converge into a starlike object.
I named the image "Cometary Impression" when I saw the eight
elements around the midgets and imagined eight circling comets.
The parameter file render time of 13 minutes is slow. The
download is fast, and will be available shortly on the Web at:
and at:
The fractal weather today was variably cloudy and breezy. The
wind and temperature of 75F (24C) kept the fractal cats indoors
most of the day, frequently checking the door to see whether it
was still breezy outside. Later in the afternoon the wind died
down and the cats went out, scolding me for not turning down the
wind earlier.
The time is now after 1am -- most certainly time to shut down
the fractal shoppe and throw the big switch. Until next time,
take care, and have faith in your fractals.
Jim Muth
jamth@mindspring.com
START 20.0 PAR-FORMULA FILE================================
CometaryImpression { ; time=0:13:25.10--SF5 on a p200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=-6.312863840603067/-8.071658728774926/1\
80586.5/1/147.5 params=-2/-1.1/-0.0002/-11/0/300
float=y maxiter=1500 inside=0
logmap=261 periodicity=10
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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: Mon, 4 Jun 2001 22:54:11 -0400 (EDT)
From: Jim Muth
Subject: (fractint) C-FOTD 05-06-01 (A Raspy Old Midget [9])
Classic FOTD -- June 05, 2001 (Rating 9)
Fractal visionaries and enthusiasts:
Things have been going well here at fractal central lately. I'm
in one of those periods where I seem to automatically pick the
right places to look for those hard-to-find midgets, and equally
important, good color palettes seem to appear almost by magic.
I named today's image "A Raspy Old Midget". The eight rasp-like
elements surrounding the midget inspired the name. Undecided as
to what rating I might bestow upon the picture, I decided on a
rather liberal 9. An 8-1/2 might be more accurate, but we need
at least a few 9's in the archives.
The formula -5(Z^(-1.15))-0.02(Z^(-11.5))+(1/C) drew the parent
fractal, which is a rather interesting but oversized figure by
itself, with a prominent fan-like element extending eastward
from the origin. Today's midget lies on the south border of
this fractal fan.
The 11-minute parameter file is a bit slow. I advise visiting
Paul's web site or Scott's site, and downloading the GIF image
file from there. Paul's FOTD site can be found at:
Scott's is at:
The fractal weather today was virtually perfect, with sunny
skies, a temperature of 77F (25C), and happy cats.
And having found such a fine fractal, I'm happy too. Until next
time, take care, and I wonder whether fractal seeds will grow.
Jim Muth
jamth@mindspring.com
START 20.0 PAR-FORMULA FILE================================
A_Raspy_Old_Midget { ; time=0:11:41.29--SF5 on a P200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=+1.77096187026955700/-2.487403802992336\
00/3.864825e+009/1/92.499/0.003
params=-5/-1.15/-0.02/-11.5/0/0 float=y
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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: Wed, 6 Jun 2001 08:38:24 -0400 (EDT)
From: Jim Muth
Subject: (fractint) C-FOTD 06-06-01 (Golden Chariot [8])
Classic FOTD -- June 06, 2001 (Rating 8)
Fractal visionaries and enthusiasts:
The formula -0.6(Z^(-1.3))-0.006(Z^(-3))+(1/C), when calculated
with an escape radius of 900, draws an hourglass-shaped figure
composed of almost total chaos. Two out-zooms are needed to see
the entire figure, which is a Mandeloid sitting on its nose,
with the tiny hourglass lying in the center of the main bay.
The figure is unusual in that the filaments extending from the
buds end in rings instead of simply petering out in ever-smaller
details. Today's midget lies deep within the end-ring of the
main southwest filament of the main bud, which in today's
fractal lies on the south shore of the main bay.
I named the picture "Golden Chariot" because when I saw it, I
had an immediate impression of a chariot wheel. I briefly
considered a name such as "Chariot of the Gods", but a similar
name has already been used in a well-known book.
The image rates an 8. The rating might be a bit liberal, but
it's a strange scene in a fractal with even stranger scenes.
I'll present some of these other scenes in the next few FOTD's,
the first of which will be tomorrow's.
A good feature of today's image is that it renders in only 3-1/2
minutes. In an hour or so, the image may also be seen by
visiting the web sites of Paul and Scott. The URL's of those
sites are:
and:
The fractal weather, which everyone is waiting for with bated
breath, was average today. The partly cloudy skies were partly
sunny during the daylight hours and partly starry after dark.
The temperature of 80F (26.5C) was average, and the fractal cats
had an average day on the porch and in the yard.
And it's the start of another average day for me. I've got
about the average amount of work to finish before I can relax
and turn to tomorrow's FOTD, which is all goes well, will be far
above average.
Jim Muth
jamth@mindspring.com
START 20.0 PAR-FORMULA FILE================================
Golden_Chariot { ; time=0:03:30.04--SF5 on a p200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=+7.63095735774279/-41.9285774906314/3.1\
96408e+008/1/162.5 params=-0.6/-1.3/-0.006/-3/0/800
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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: Wed, 6 Jun 2001 11:49:39 -0300
From: "Ricardo M. Forno"
Subject: (fractint) Algorithmic Composer
There are some algorithmic composers over the Web. I think the one I wrote
compares favorably to most of them. You can download it from:
http://www.geocities.com/rmforno/index.html
It does not use a fractal algorithm, but anyway I think this mail is on
topic.
Hope you like the music.
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------------------------------
Date: Thu, 7 Jun 2001 10:23:24 -0400 (EDT)
From: Jim Muth
Subject: (fractint) C-FOTD 07-06-01 (Charged Midget [8])
Classic FOTD -- June 07, 2001 (Rating 8)
Fractal visionaries and enthusiasts:
When a package or roll of photographic film is opened carelessly
in a darkroom in dry weather, black tree-like figures sometimes
appear on the film when it is developed. The figures are caused
by sparks of static electricity that are created as the layers
are peeled apart too roughly. These figures have a name, which
eludes me at the moment, though I think it begins with the
letter 'l'.
If I could remember that word, I would have the perfect title
for today's image, which reminds me of the sheets of ruined film
that I have thrown away over the years. But, being unable to
remember, I have named the image "Charged Midget".
The scene of today's picture lies in the same fractal as
yesterday's, though in a totally different part. Whereas
yesterday's scene was located at the end of a filament, today's
is located deep in an almost featureless valley on the WNW shore
of the main bay.
And speaking of fractal bays, this one has that strange
hourglass-shaped chunk of total chaos sitting in the middle of
it. Usually, such totally chaotic areas hold nothing of
interest, but this one is worth at least a cursory
investigation. If I find something, it will be tomorrow's FOTD;
if not, I'll turn somewhere else.
Today's image rates an 8, keeping the run of exceptional images
unbroken. The total lack of reds is intentional. Sometimes
fewer colors can have greater impact than more.
The image was rendered with the 'passes equal b' option, which I
rarely use, but which sometimes is the fastest of all. With
today's 3-3/4 minute parameter file, the difference of a few
seconds is inconsequential however.
As always, the GIF image will soon be available on the web for
those who would rather not run the parameter file. The URL's
are:
and:
The fractal weather today featured a temperature of 81F (27C),
changeable skies, with nearly every cloud type imaginable, and a
light sprinkle at noon to wet those going to lunch. It also wet
the fractal cats, who scowled as they retreated indoors until
the rain stopped.
And it's now time for me to retreat to the task of accomplishing
the day's work. Until next time, take care, and will we ever
find out who is right?
Jim Muth
jamth@mindspring.com
START 20.0 PAR-FORMULA FILE================================
Charged_Midget { ; time=0:03:42.60--SF5 on a p200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=recip passes=b
center-mag=-14.16793212758799/-11.36345363439138/2\
6147.93/1/19.999 params=-0.6/-1.3/-0.006/-3/0/800
float=y maxiter=1500 inside=255
logmap=43 periodicity=0
colors=0000Mp0Ho0Co09m09l69jD9iH9gM9jPElSJmXOo`Trc\
Xsgaujgvmkyrqzuvzxyzrxzmsxinudhr`coWZlRVjMRgHNdDKa\
8GZ3CW09U09S0AR0AP0BO0AM0BL0BJ0AH0CG0CE0FD0HB0IA0L\
80M60N50P30P10Q00Q00U00V00X00_00`00b00d00e00e00e00\
e00e03e06e0Ae0De0Ge0Je0Me0Pe0Se0We0Ze0ae0de0ge0je0\
le0cj0Wm0Op0Gs08v10y10z30z30z60zA0yB0xE0vG0sJ0rM0p\
O0oR0mS0lW0iZ0g`0ec1dd3bg3`i6_j8_jA_lD_lE_mG_mHYoL\
YoMYpOYpPYrSYrUXsWXsXXu`XuaXvcXvdXvgYuiYsjYslYrmYp\
oYppYorYosYmu_lv_lx_jy_jz_iz_gz_gz_fz_fziaxrZsrWos\
SlsPdizXXzR9zSCzSEzSFzSHzSJzSKzSMzSNzSPzSQzSSzSUzS\
VzUXzUYzUYzU_zW`zW`zWbzWbzXdzXezXezXgzZizZizZjzZjz\
cizgizlizpizuizygzzgzzgzzgzzgzzezzdzzdzzbzzbzz`zz_\
zz_zzYzzYzzXzzVzzVzzUzzUzzSzzSzzQzzPzzNzzMzzKzzJzz\
HzzFzzEzzCzzAzz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9z\
z9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9\
zzAzzAzzAzzKzzSzz_zziz000
}
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, 7 Jun 2001 22:16:27 -0400
From: "Multiple Bogeys"
Subject: (fractint) Some intriguing stuff.
- ------=_NextPart_001_0000_01C0EF9F.7F321100
Content-Type: text/plain; charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable
These formulas make it possible to investigate the escape behavior of the=
Henon map:
x -> a - by + x^2
y -> x
both for the case where x and y are real, and the case where they may be =
complex.
The formulas whose names end in 'J' are Julia-like. Henon_J1 will vary y =
over the screen while fixing x. The parameters fix a,
b, and z. Henon-J2 will vary real x and real y over the screen; setting t=
heir imaginary parts to
zero is accomplished by setting p3 to zero. If p1 and p2 are real (e.g. t=
he classic 1.4 and -0.3) the basin of attraction of the familiar real Hen=
on map appears.
The formulas whose names end in 'M' are Mandelbrot-like. Henon-M3 is most=
Mandelbrot-like; b varies over the screen while a and initial x and y (a=
ll complex) are set with parameters. No known choice of initial x and y a=
cts like the "critical" value 0
for the classic Mandelbrot set, so the result always seems to look pertur=
bed. (I looked long and hard with the evolver/explorer to find a good app=
roximation to a "critical" value; nothing seems to cut it, perhaps becaus=
e there is no
"critical" value for such multi-recurrence maps as this. The Jacobian has=
nonzero determinant if b is not zero -- if b is zero,
both x and y do the usual Mandelbrot set with x =3D 0, y =3D anything "cr=
itical". However, the imaginary parts of initial x and y should be zero t=
o get "Mandelbrot-like" it seems.) Henon-M2 varies real a and real b over=
the screen. The result resembles Lyaounov space! Henon-M4 is like Henon-=
M2, but it automatically composites the maps obtained for nine choices of=
initial real x and y. These iterate the real Henon map -- imaginary x, y=
, a, and b are identically zero. Henon-M5 is complex instead of real, lik=
e Henon-M3, but is a composite like Henon-M4 using nine values of initial=
real x and y. Henon-M2, M3, and M5 are best viewed with inside =3D per, =
outside =3D numb, outside =3D 0, maxiter > 255, and a color map that make=
s colors 0 and 255 different and varies a lot in the early colors (try th=
e default VGA palette modified to invert the last 32 colors or so). A zoo=
m down M5's seahorse valley is interesting. The "halo" of color 255 about=
the Set in M5 and in the lower right quadrant of M4 represents the regim=
es where strange attractors can occur in the system. The set of points fo=
r which actual strange attractors occur is buried among "islands" of orde=
r and pockets of escape -- the former look like Lyapunov swallows in M4 a=
nd presumably would appear to be mini Mandelbrots in M5 if "critical" val=
ues for initial real x and y could be found.
WARNING: The below probably has the dreaded "3D" disease, no thanks to MS=
N Exploder. If it does, IIRC someone published a utility to strip the enc=
rufting^H^H^H^H^H^H^Hoding on here a year or so ago.
Henon_M2 { ; p1 is a point in the Henon map. Screen coordinates are param=
eters 'a' and 'b'.
; Escape pixels escaped radius 1000. Try non-standard inside o=
ptions with
; periodicity checking enabled.
a =3D real(pixel), b =3D imag(pixel), z =3D p1:
z =3D a - b*imag(z) - sqr(real(z)) + (0,1)*real(z),
|z| < 1000000
}
Henon_M3 { ; Henon map on complex numbers.
; p1 and p2 are a point in the Henon map. Screen coordinates a=
re parameter 'b'.
; Escape pixels escaped radius 1000. Try non-standard inside o=
ptions with
; periodicity checking DISabled.
a =3D 1.4, b =3D pixel, z =3D p1, w =3D p2:
z1 =3D a - b*w - sqr(z)
w =3D z
z =3D z1,
lastsqr+|w| < 1000000
}
Henon_M4 { ; Screen coordinates are parameters 'a' and 'b'.
; Escape pixels escaped radius 1000. Try non-standard inside o=
ptions with
; periodicity checking enabled. Composite view.
a =3D real(pixel), b =3D imag(pixel), z =3D 0, r =3D 0, done =3D 0:
z =3D a - b*imag(z) - sqr(real(z)) + (0,1)*real(z),
IF (|z| > 1000000)
IF (r =3D=3D 0)
z =3D 1
r =3D 1
ELSEIF (r =3D=3D 1)
z =3D (0,1)
r =3D 2
ELSEIF (r =3D=3D 2)
z =3D -1
r =3D 3
ELSEIF (r =3D=3D 3)
z =3D -(0,1)
r =3D 4
ELSEIF (r =3D=3D 4)
z =3D 1+(0,1)
r =3D 5
ELSEIF (r =3D=3D 5)
z =3D 1-(0,1)
r =3D 6
ELSEIF (r =3D=3D 6)
z =3D -1+(0,1)
r =3D 7
ELSEIF (r =3D=3D 7)
z =3D -1-(0,1)
r =3D 8
ELSE
done =3D 1
ENDIF
ENDIF
done =3D=3D 0
}
Henon_M5 { ; Henon map on complex numbers. Screen coordinates are paramet=
er 'b'.
; Escape pixels escaped radius 1000. Try non-standard inside o=
ptions with
; periodicity checking DISabled. Composite view.
a =3D 1.4, b =3D pixel, z =3D 0, w =3D 0, r =3D 0, done =3D 0:
z1 =3D a - b*w - sqr(z)
w =3D z
z =3D z1,
IF (lastsqr+|w| > 1000000)
IF (r =3D=3D 0)
z =3D 1
w =3D 0
r =3D 1
ELSEIF (r =3D=3D 1)
z =3D 0
w =3D 1
r =3D 2
ELSEIF (r =3D=3D 2)
z =3D -1
w =3D 0
r =3D 3
ELSEIF (r =3D=3D 3)
z =3D 0
w =3D -1
r =3D 4
ELSEIF (r =3D=3D 4)
z =3D 1
w =3D 1
r =3D 5
ELSEIF (r =3D=3D 5)
z =3D 1
w =3D -1
r =3D 6
ELSEIF (r =3D=3D 6)
z =3D -1
w =3D 1
r =3D 7
ELSEIF (r =3D=3D 7)
z =3D -1
w =3D -1
r =3D 8
ELSE
done =3D 1
ENDIF
ENDIF
done =3D=3D 0
}
Henon_J1 { ; Henon map on complex numbers.
; p1 and p2 are a and b. Screen coordinates are y. p3 is x.
; Escape pixels escaped radius 1000. Try non-standard inside o=
ptions with
; periodicity checking DISabled.
a =3D p1, b =3D p2, z =3D p3, w =3D pixel:
z1 =3D a - b*w - sqr(z)
w =3D z
z =3D z1,
lastsqr+|w| < 1000000
}
Henon_J2 { ; Henon map on complex numbers.
; p1 and p2 are a and b. Screen coordinates are real(x), real(=
y). p3 is imag(x), imag(y).
; Escape pixels escaped radius 1000. Try non-standard inside o=
ptions with
; periodicity checking DISabled.
a =3D p1, b =3D p2, z =3D real(pixel) + (0,1)*real(p3), w =3D imag(pixe=
l) + (0,1)*imag(p3):
z1 =3D a - b*w - sqr(z)
w =3D z
z =3D z1,
lastsqr+|w| < 1000000
}
Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com.
- ------=_NextPart_001_0000_01C0EF9F.7F321100
Content-Type: text/html; charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable
These =
formulas make it possible to investigate the escape behavior of the Henon=
map:
x -> a - by + x^2
y ->=
; x
both for the case where x and y are real=
, and the case where they may be complex.
Th=
e formulas whose names end in 'J' are Julia-like. Henon_J1 will vary =
;y over the screen while fixing x. The parameters fix a,
b, an=
d z. Henon-J2 will vary real x and real y over the screen; sett=
ing their imaginary parts to
zero is accomplished by setting p=
3 to zero. If p1 and p2 are real (e.g. the classic 1.4 and -0.3) the basi=
n of attraction of the familiar real Henon map appears.
=
The formulas whose names end in 'M' are Mandelbrot-like. Heno=
n-M3 is most Mandelbrot-like; b varies over the screen while a and initia=
l x and y (all complex) are set with parameters. No known choice of initi=
al x and y acts like the "critical" value 0
for the classic Ma=
ndelbrot set, so the result always seems to look perturbed. (I looked lon=
g and hard with the evolver/explorer to find a good approximation to a "c=
ritical" value; nothing seems to cut it, perhaps because there is no
"critical" value for such multi-recurrence maps as this. The =
Jacobian has nonzero determinant if b is not zero -- if b is zero,
=
both x and y do the usual Mandelbrot set with x =3D 0, y =3D anythin=
g "critical". However, the imaginary parts of initial x and y should=
be zero to get "Mandelbrot-like" it seems.) Henon-M2 varies real a and r=
eal b over the screen. The result resembles Lyaounov space! Henon-M4 is l=
ike Henon-M2, but it automatically composites the maps obtained for nine =
choices of initial real x and y. These iterate the real Henon map -- imag=
inary x, y, a, and b are identically zero. Henon-M5 is complex instead of=
real, like Henon-M3, but is a composite like Henon-M4 using nine values =
of initial real x and y. Henon-M2, M3, and M5 are best viewed with inside=
=3D per, outside =3D numb, outside =3D 0, maxiter > 255, and a color =
map that makes colors 0 and 255 different and varies a lot in the ea=
rly colors (try the default VGA palette modified to invert the last 32 co=
lors or so). A zoom down M5's seahorse valley is interesting. The "halo" =
of color 255 about the Set in M5 and in the lower right quadrant of M4 re=
presents the regimes where strange attractors can occur in the system. Th=
e set of points for which actual strange attractors occur is buried among=
"islands" of order and pockets of escape -- the former look like Lyapuno=
v swallows in M4 and presumably would appear to be mini Mandelbrots in M5=
if "critical" values for initial real x and y could be found.
WARNING: The below probably has the dreaded "3D" disea=
se, no thanks to MSN Exploder. If it does, IIRC someone published a utili=
ty to strip the encrufting^H^H^H^H^H^H^Hoding on here a year or so ago.=
DIV>
Henon_M2 { ; p1 is a point in the Henon map. =
Screen coordinates are parameters 'a' and 'b'.
 =
; ; Escape pixels escaped radius 1000=
. Try non-standard inside options with
&=
nbsp; ; periodicity checking enabled.
a=
=3D real(pixel), b =3D imag(pixel), z =3D p1:
z =3D a - b*imag=
(z) - sqr(real(z)) + (0,1)*real(z),
|z| < 1000000
}
=
Henon_M3 { ; Henon map on complex numbers.
&nb=
sp; ; p1 and p2 are=
a point in the Henon map. Screen coordinates are parameter 'b'.
 =
; ; Escape pixels e=
scaped radius 1000. Try non-standard inside options with
&=
nbsp; ; periodicity checking DI=
Sabled.
a =3D 1.4, b =3D pixel, z =3D p1, w =3D p2:
z=
1 =3D a - b*w - sqr(z)
w =3D z
z =3D z1,
la=
stsqr+|w| < 1000000
}
Henon_M4 { ; Scr=
een coordinates are parameters 'a' and 'b'.
&n=
bsp; ; Escape pixels escaped radius 1000. T=
ry non-standard inside options with
&nbs=
p; ; periodicity checking enabled. Composite view=
.
a =3D real(pixel), b =3D imag(pixel), z =3D 0, r =3D 0, done =
=3D 0:
z =3D a - b*imag(z) - sqr(real(z)) + (0,1)*real(z),
&=
nbsp; IF (|z| > 1000000)
IF (r =3D=3D 0)
&nbs=
p; z =3D 1
r =3D=
1
ELSEIF (r =3D=3D 1)
&=
nbsp; z =3D (0,1)
r =3D 2
&nbs=
p; ELSEIF (r =3D=3D 2)
z =3D -1 r =3D 3
ELSEIF (r =
=3D=3D 3)
z =3D -(0,1)
&=
nbsp; r =3D 4
ELSEIF (r =3D=3D 4)
&n=
bsp; z =3D 1+(0,1)
&nb=
sp; r =3D 5
ELSEIF (r =3D=3D 5)
&nbs=
p; z =3D 1-(0,1)
r =3D 6
ELSEIF (r =3D=3D 6)
=
z =3D -1+(0,1)
r =3D 7
=
ELSEIF (r =3D=3D 7)
z =3D -1-(0,=
1)
r =3D 8
ELSE done =3D 1
ENDIF ENDIF
done =3D=3D 0
}
=
Henon_M5 { ; Henon map on complex numbers. Screen coordinates are paramet=
er 'b'.
;=
Escape pixels escaped radius 1000. Try non-standard inside options with<=
BR> ; periodi=
city checking DISabled. Composite view.
a =3D 1.4, b =3D pixel,=
z =3D 0, w =3D 0, r =3D 0, done =3D 0:
z1 =3D a - b*w - sqr(z)=
w =3D z
z =3D z1,
IF (lastsqr+|w| > 100=
0000)
IF (r =3D=3D 0)
&n=
bsp; z =3D 1
w =3D 0
&nb=
sp; r =3D 1
ELSEIF (r =3D=3D 1)
&nbs=
p; z =3D 0
w =3D=
1
r =3D 2
ELSEIF=
(r =3D=3D 2)
z =3D -1
&=
nbsp; w =3D 0
r =3D 3
&n=
bsp; ELSEIF (r =3D=3D 3)
z =
=3D 0
w =3D -1
&nb=
sp; r =3D 4
ELSEIF (r =3D=3D 4)
&nbs=
p; z =3D 1
w =3D 1
r =3D 5
ELSEIF (r =3D=
=3D 5)
z =3D 1
&nb=
sp; w =3D -1
r =3D 6
&nb=
sp; ELSEIF (r =3D=3D 6)
z =3D -1<=
BR> w =3D 1
&nbs=
p; r =3D 7
ELSEIF (r =3D=3D 7)
 =
; z =3D -1
w =3D -1
&nbs=
p; r =3D 8
ELSE
&n=
bsp; done =3D 1
ENDIF
E=
NDIF
done =3D=3D 0
}
Henon_J1 {=
; Henon map on complex numbers.
&=
nbsp; ; p1 and p2 are a and b. Screen coordinates are y=
. p3 is x.
 =
; ; Escape pixels escaped radius 1000. Try non-standard inside options wi=
th
; peri=
odicity checking DISabled.
a =3D p1, b =3D p2, z =3D p3, w =3D =
pixel:
z1 =3D a - b*w - sqr(z)
w =3D z
z =3D=
z1,
lastsqr+|w| < 1000000
}
Henon_J2 { ; Henon map on complex numbers.
&n=
bsp; ; p1 and p2 are a and b. Screen coordi=
nates are real(x), real(y). p3 is imag(x), imag(y).
=
; Escape pixels escaped radius=
1000. Try non-standard inside options with
&n=
bsp; ; periodicity checking DISabled.
&n=
bsp; a =3D p1, b =3D p2, z =3D real(pixel) + (0,1)*real(p3), w =3D imag(p=
ixel) + (0,1)*imag(p3):
z1 =3D a - b*w - sqr(z)
w =3D=
z
z =3D z1,
lastsqr+|w| < 1000000
}
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- --------------------------------------------------------------
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End of fractint-digest V1 #569
******************************