FOTD -- November 02, 2010 (Rating 7)
Fractal visionaries and enthusiasts:
In my fractal explorations I constantly find interesting and
unexpected slices of the Julibrot. Usually I discard these
almost as soon as I find them, but now I have decided to make
November the month of unexpected slices of the Z^2+C Julibrot,
which is the abstract four-dimensional monster that holds the
Mandelbrot set as well as all the perturbed Mandelbrot sets and
all the Julia sets. Yesterday's image was the first of these
slices.
At first glance, today's image appears to be a Julia set of
Seahorse Valley. But it is not. Seahorse Valley Julia sets
have either XY-axis symmetry or origin symmetry. The mini-thing
in today's scene has X-axis symmetry, something that is
impossible for a Julia set to have. The reason is that it is
not a mini-Julia set, it is an Oblate set corresponding to the
Seahorse Valley of the large minibrot on the main spike of the
Mandelbrot set.
The scene of the image is off to the side of the valley in the
real(Z) direction at a distance of 0.5477. The entire image has
been stretched about 2-3/4 times in the horizontal direction to
bring it into what is considered normal relative dimensions.
The brilliant line through the center of the image is actually
the narrow space between an extension in the real(Z) direction
of the two branches of Seahorse Valley of the large minibrot on
the main spike of the M-set
I put more coloring effort into the image than it deserves, but
the effort raises the worth a full point from 6 to 7. The name
"An Undefined Mini" means that, as far as I know, there is no
generally accepted name for the type of object that appears in
today's image. So maybe I'll invent a name.
The calculation time of 2 minutes is about what the image is
worth. A bargain may be found on the FOTD web site at:
<http://www.Nahee.com/FOTD/>
where the finished image is posted for near-instant viewing, and
if not total, then at least partial satisfaction.
The air remained chilly here at Fractal Central on Monday, with
a high temperature of only 46F +8C, but the sky was clear and
the sun shone strong on the fractal cats' window shelf most all
day, keeping them happy as two bedbugs in a flophouse mattress.
My day was about average.
The next FOTD will be another unexpected slice of the Z^2+C
Julibrot. It will be posted in 24 hours, give or take a few.
Until then, take care, and our world might be a better place
without religion, but what would we replace it with?
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
An_Undefined_Mini { ; time=0:02:12.52-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot4 passes=1 center-mag=-0.\
00030581/0/269/2.763 params=0/0/90/0/-1.7685291540\
2387/0/0.5477/0/2/0 float=y maxiter=120000
inside=255 logmap=12 symmetry=xaxis periodicity=6
colors=0007Wb7Wc7Vc7Vc7Vc7Ud7Ud7Ud7Ud9VeAVeBVfCWfE\
WgFWgGWhHXhIXiKXiLYjMYjNYjOYkQZkRZlSZlT_mU_mW_nX_n\
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4yl4ym4zn4zo3zq3zr3zs3zt3 }
frm:SliceJulibrot4 {; draws all slices of Julibrot
pix=pixel, u=real(pix), v=imag(pix),
a=pi*real(p1*0.0055555555555556),
b=pi*imag(p1*0.0055555555555556),
g=pi*real(p2*0.0055555555555556),
d=pi*imag(p2*0.0055555555555556),
ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
sg=sin(g), cd=cos(d), sd=sin(d),
p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
r=u*sg+v*ca*sb*cg, s=v*sin(a), esc=imag(p5)+9
c=p+flip(q)+p3, z=r+flip(s)+p4:
z=z^(real(p5))+c
|z|< esc }
END PARAMETER FILE=========================================
Behold a fractal devil. Actually that's the name of the formula, the same one used in the Oct. 2 post.
I zoomed in one of the lobes, rotated 90 degrees giving the image a portrait orientation and re-sized.
The rotating of the image takes it from simple anti-aliasing to the level of post-processing.
To everybody this blogs first post-processed fractal! Go to http://maxitersfractalfollies.blogspot.com
for instant gratification.
Halloween Devil { ;fract495.gif
; blank
; calctime 0:07:22.81
; created Oct 31, 2010
; Fractint Version 2004 Patchlevel 10
reset=2004 type=formula formulafile=_d.frm formulaname=devil
passes=1 center-mag=0.738924/3.33067e-016/3.908693 float=y
maxiter=1500 inside=0
colors=000000000000000000000000000000000100100200400500700900B00D00G00I0\
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0720720620620520520520420420420320320220220220120120010
}
frm:devil {; Copyright (c) Paul W. Carlson, 1998
; Based on one of Noel Giffin's formulas
w = pixel
z = iter = wp1 = wp2 = bailout = 0
:
old_w = w
w = w * w - wp2
wp2 = wp1
wp1 = real(old_w) - flip(imag(old_w))
IF (|w| > 4)
bailout = 1
z = 4 * iter + 30
z = z - real(216) * trunc(z / real(216))
ENDIF
iter = iter + 1
z = z - iter
bailout == 0
;SOURCE: 98msg.frm
}
Roger Alexander
A minibrot in Halloween colors. According to Wikipedia "Black and orange are the traditional
Halloween colors and represent the darkness of night and the color of bonfires, autumn leaves,
and jack-o'-lanterns" Gif and re-sampled jpg. At the usual http://maxitersfractalfollies.blogspot.com
Halloween Colors { ; fract493.gif
; blank
; calctime 0:01:56.33
; created Oct 31, 2010
; Fractint Version 2004 Patchlevel 10
reset=2004 type=mandel
center-mag=-0.04428222277847312/-0.98332827545909710/508323.8/1/170.0000\
00000846683/-9.35095410825592666e-010 params=0/0 float=y
maxiter=2300 inside=0 logmap=163
colors=00000000000000000000000000000000000000000000000000000000100100100\
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}
Roger Alexander
FOTD -- November 01, 2010 (No Rating)
Fractal visionaries and enthusiasts:
Today's image was created in the same manner as the image of
three days ago -- the `Offside' image of October 29 -- though
instead of going off from the center of the period-2 bud in the
Rectangular direction, we go off in the other perpendicular
direction, the Oblate, until we reach the edge of the Julibrot.
By some very curious coincidence, in this direction the Julibrot
ends on the real(Z) axis at a distance of plus and minus
1.61803...., which strangely enough is the golden ratio.
Since today's image was created in the same manner as the
earlier image and is a virtual repeat, I declined to give it a
rating. The name "Maximum Credulity" came to mind Sunday after-
noon while I was watching a skeptical show about the existence
(read non-existence) of the supernatural.
Perhaps the best feature of today's image is its fireball-fast
calculation time of 5-1/2 seconds, which guarantees that
virtually no time will have been wasted if the image fails to
inspire awe and wonder for natural numbers.
Viewing the finished image on the FOTD web site at:
<http://www.Nahee.com/FOTD/>
will involve no wishful thinking, and will also guarantee that
no time will be wasted if a let-down happens.
Blustery conditions, accompanied by dark wind-shredded clouds
and a temperature of 54F 12C made Sunday a typical autumn day
here at Fractal Central in central Pennsylvania. The fractal
cats had their thrill when a pigeon landed on the outside of
their lookout window, only inches from Nicholas' hungry jaws.
But as soon as Nicholas made his move, the pigeon flew away,
leaving Cassie to wonder what all the tail thrashing was about.
My day was average, even though the FOTD turned out to be below
average. The next FOTD, an improved one to be sure, will be
posted in 24 hours. Until then, take care, and only the credu-
lous believe in ghosts. Since I do believe in ghosts, I guess
that makes me credulous as well as superstitious.
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
Maximum_Credulity { ; time=0:00:05.32-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot4 center-mag=+0.000039265\
06831894/0/176501/0.0009777 params=0/0/-24.095/0/-\
1/0/1.61803/0/2/0 float=y maxiter=1500 inside=bof61
periodicity=6
colors=0004_q3Rc2IR19DCAv65TdBXR7MD3BJdO9KC00Z00W0\
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EUG7F8uMbTBJAum5TPaDbJ6JLrdARKUwEKc9AK4v3xg2hT1UE0\
FZypNdZBKHx8XU4GLqfE_S7IE }
frm:SliceJulibrot4 {; draws all slices of Julibrot
pix=pixel, u=real(pix), v=imag(pix),
a=pi*real(p1*0.0055555555555556),
b=pi*imag(p1*0.0055555555555556),
g=pi*real(p2*0.0055555555555556),
d=pi*imag(p2*0.0055555555555556),
ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
sg=sin(g), cd=cos(d), sd=sin(d),
p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
r=u*sg+v*ca*sb*cg, s=v*sin(a), esc=imag(p5)+9
c=p+flip(q)+p3, z=r+flip(s)+p4:
z=z^(real(p5))+c
|z|< esc }
END PARAMETER FILE=========================================
FOTD -- October 31, 2010 (Rating 6)
Fractal visionaries and enthusiasts:
Today's rather harmless image lies in a valley of the fractal
that results when various negative parts of Z^(-4.5) and Z(-1.7)
are mixed together and (1/C) is added. Like most fractals of
this double-negative type, this parent fractal cannot be des-
cribed, though it may easily be seen by backing out of today's
image.
I rated the image at a 6. I searched for something to justify a
higher rating, but nothing is there.
The name "Shadow of the Wind" has nothing to do with an old
radio hero who could make himself invisible. It's simply a
phrase that came to mind from apparently nowhere.
The calculation time of 9 minutes means that the time required
to run the parameter file is more than the resulting image is
worth. The problem may be eliminated by viewing the finished
image on the FOTD web site at:
<http://www.Nahee.com/FOTD/>
Too many clouds and a temperature of 54F 12C kept the fractal
cats suppressed here at Fractal Central on Saturday. My day was
relaxing enough, though for some reason the Halloween trick-or-
treaters were out one evening early. The next FOTD will be
posted in 24 hours. Until then, take care, and if a spook
spooks you, spook him back. (They do exist, you know. Whether
they are real is another question.)
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
Shadow_of_the_Wind { ; time=0:09:02.88-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=MandAutoCritInZ function=recip float=y
center-mag=+0.56620315067701/+0.03728036351951/\
8.3e+009/1/2.5/0 params=-0.6/-4.5/-3.6/-1.7/0/0/0/0
maxiter=10000 inside=0 logmap=1140 periodicity=6
colors=000HChCAkF7mE5oE3qD4oD5nC6mC7kB8jB9iAAgABf9\
Ce9Cd9Db8Ea8F`7GZ7GY6IX6JVCHU5LT5LSFHU8JVAIWHGYDGZ\
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ffz`bzVZzPWzK1zB1zC1zD1zE }
frm:MandAutoCritInZ {; 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)+(p4)),
k=real(p3)+1, esc=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < esc }
END PARAMETER FILE=========================================
I let my imagination run wild when I generated today's post and I thought I saw scales on a plant.
The Oct. 28 post (Swirly Wallpaper) formula with different parameters.
Ready made at http://maxitersfractalfollies.blogspot.com
Green Scales { ; fract491.gif
; blank
; calctime 0:00:42.19
; created Oct 29, 2010
; Fractint Version 2004 Patchlevel 10
reset=2004 type=formula formulafile=kerrym.frm
formulaname=LKM-quiltm2 function=cos/cosxx
center-mag=-0.899274/-0.684337/0.3272439/1/180
params=1.75/-0.875/0.375/-0.375 float=y maxiter=1500 inside=0
decomp=256 periodicity=0
colors=0F00F00F00F00F00G00G00G00G00H00H00H00H00H00I00I00I00I00J00J00J00J\
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}
frm:LKM-quiltm2 {; Kerry Mitchell
; quilted backgrounds
; uses 2 iterations of 2 mandelbrot variables with different c's
; c1 and c2 are tiled periodically, with different frequencies
; z1 & z2 are multiplied together to add their angles
; use "decomp=256" coloring
; r=bailout=0 for swirly exteriors and no interiors
; real(p1), imag(p1) = x1, y1 frequencies for c1
; real(p2), imag(p2) = x2, y2 frequencies for c2
; try different fn1 and fn2 for added joy
x=real(pixel), y=imag(pixel)
a1=pi*real(p1), b1=pi*imag(p1)
a2=pi*real(p2), b2=pi*imag(p2)
x1=2*cos(a1*x), y1=2*cos(b1*y)
x2=2*sin(a2*x), y2=2*sin(b2*y)
c1=x1+flip(y1), c2=x2+flip(y2)
z1=c1, z2=c2, r=0:
z1=sqr(z1)+c1, z2=sqr(z2)+c2
z=fn1(z1)*fn2(z2)
|z| < r
;SOURCE: lkm.frm
}
Roger Alexander
FOTD -- October 30, 2010 (Rating 6)
Fractal visionaries and enthusiasts:
Today's image is an afterthought. On Friday I was exploring the
(-Z)^(1.5)+C Mandeloid, with no thought of finding a FOTD there,
but when I stumbled upon today's image, it struck me as so
unusual that I decided to declare it FOTD for October 30.
Actually, what we have here is a scene in the Mandeloid that
results when the formula is iterated with Z initialized at -0.5.
I named the image "Reciprocal World" because of the apparent
negative curvature of the surface at the bottom of the frame.
If this were a real planet, it would be in the shape of a pseudo-
sphere.
The rating of a 6 makes the 4-1/2 minute calculation time barely
worth the effort. I recommend viewing the finished image on the
FOTD web site at:
<http://www.Nahee.com/FOTD/>
Friday turned blustery here at Fractal Central, with alternating
periods of sun and heavy clouds, and even a few light showers.
The temperature of 48F +9C was too chilly for the fractal cats,
who grew tired of watching the blowing leaves and spent most of
the day sleeping near the heat or asking for their treats.
My day was slightly busy, but nothing to get excited about. The
next FOTD is due to be posted in 24 hours. But if an unexpected
antiquing trip comes up on Saturday, it could be dalayed how-
ever. Until whenever, take care, and be filled with wonder.
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
Reciprocal_World { ; time=0:04:26.88-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot5 passes=1 float=y
center-mag=-0.0079014/+0.7369293/90/1/-72.5/0
params=0/0/0/0/0/0/-0.5/0/1.5/0 maxiter=24000
inside=0 logmap=-120
colors=000KAHKAIKAJKAKKALKBMKCNJDOIENHFMGGLFHKEIKD\
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zx`zxZzxYzxXzuYzsYzqYznYz }
frm:SliceJulibrot5 {; draws all slices of Julibrot
pix=pixel, u=real(pix), v=imag(pix),
a=pi*real(p1*0.0055555555555556),
b=pi*imag(p1*0.0055555555555556),
g=pi*real(p2*0.0055555555555556),
d=pi*imag(p2*0.0055555555555556),
ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
sg=sin(g), cd=cos(d), sd=sin(d),
p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
r=u*sg+v*ca*sb*cg, s=v*sin(a), esc=imag(p5)+9
c=p+flip(q)+p3, z=r+flip(s)+p4:
z=(-z)^(real(p5))+c
|z|< esc }
END PARAMETER FILE=========================================
A Fractint generated fractal with nary a minibrot in sight! Fractint - a little something for everybody.
And for those who the whole point of the internet is clicking on links for instant satisfaction
try http://maxitersfractalfollies.blogspot.com
Swirly Wallpaper { ; fract486.gif
; blank
; calctime 0:00:41.91
; created Oct 28, 2010
; Fractint Version 2004 Patchlevel 10
reset=2004 type=formula formulafile=kerrym.frm
formulaname=LKM-quiltm2 function=tan/sinh
center-mag=1.33227e-015/-4.44089e-016/0.4644958
params=0.1115146336252937/0.4293343913083285/1.394390697958312/1.4464552\
75124363 float=y maxiter=1500 inside=0 decomp=256 periodicity=0
colors=05F05F05F16G16G16G16G16G27H27H27H27H27H38I38I38I38I38I49J49J49J49\
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}
frm:LKM-quiltm2 {; Kerry Mitchell
; quilted backgrounds
; uses 2 iterations of 2 mandelbrot variables with different c's
; c1 and c2 are tiled periodically, with different frequencies
; z1 & z2 are multiplied together to add their angles
; use "decomp=256" coloring
; r=bailout=0 for swirly exteriors and no interiors
; real(p1), imag(p1) = x1, y1 frequencies for c1
; real(p2), imag(p2) = x2, y2 frequencies for c2
; try different fn1 and fn2 for added joy
x=real(pixel), y=imag(pixel)
a1=pi*real(p1), b1=pi*imag(p1)
a2=pi*real(p2), b2=pi*imag(p2)
x1=2*cos(a1*x), y1=2*cos(b1*y)
x2=2*sin(a2*x), y2=2*sin(b2*y)
c1=x1+flip(y1), c2=x2+flip(y2)
z1=c1, z2=c2, r=0:
z1=sqr(z1)+c1, z2=sqr(z2)+c2
z=fn1(z1)*fn2(z2)
|z| < r
;SOURCE: lkm.frm
}
Roger Alexander
FOTD -- October 29, 2010 (Rating 6)
Fractal visionaries and enthusiasts:
Today's scene lies off to the side of the large period-2 bud of
the Mandelbrot set at an angle of 90 degrees. But since there
is a complete 360-degree circle of lines in the Julibrot perpen-
dicular to the M-set, we must be more specific. Today's image
lies in the direction of the imag(Z) axis, and is centered at
the point 0.78615 on that axis.
The image is not a perturbed Mandelbrot set. It is rotated at
an angle of -19.37 degrees from the Mandelbrot orientation, the
angle at which it is tangent to the edge of the Julibrot in the
Elliptic orientation. When rendered in the conventional manner,
today's image is a rather boring fractal. The Fractint program
cures this monotony with its various inside and outside fills.
I rendered today's image with the outside set to 'real' and the
inside set to 'bof61'.
I could rate the image no higher than a 6. It has too much
program magic and not enough true fractal stuff for any higher
rating.
I named the scene "Offside" because it lies off to the side of
the large bud on the main stem of the M-set.
The calculation time of under 6 seconds is brief enough to hold
disappointment at bay if the resulting image is seen as lacking.
The easiest way to view the image is to visit the FOTD web site
at:
<http://www.Nahee.com/FOTD/>
where it is posted already calculated and ready to be
appreciated at leisure.
With a temperature of 68F 20C and sunny skies, Thursday was
pleasant enough here at New Fractal Central, though the wind was
a bit too blustery for absolute comfort. The fractal cats spent
most of their time in the window, watching occasional flurries
of leaves that fluttered past on the gusts.
My day was uneventful; FL's day was the same. The next FOTD
will be posted in 24 hours. Until then, take care, and a month
ago my broker told me I could make a pile in the guano market,
but I must have done something wrong because I dropped a load.
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
Offside { ; time=0:00:05.59-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot4 center-mag=0/-0.0000596\
847054883/272487.8/0.0009874/90/0 params=-19.37/0/\
0/90/-1/0/0/0.78615/2/0 symmetry=none inside=bof61
float=y outside=real maxiter=1500 periodicity=6
colors=000WuVTtYQt`NtcKsfHsiEskDGyCleAe_8ZU7SO5LIz\
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FH15OC3G8184hKHUDBF65cKveQrgWnhajbYeYVaSSYNPUHLQCI\
M6FI1CE9MHHVJOdLWmNbvPZuS }
frm:SliceJulibrot4 {; draws all slices of Julibrot
pix=pixel, u=real(pix), v=imag(pix),
a=pi*real(p1*0.0055555555555556),
b=pi*imag(p1*0.0055555555555556),
g=pi*real(p2*0.0055555555555556),
d=pi*imag(p2*0.0055555555555556),
ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
sg=sin(g), cd=cos(d), sd=sin(d),
p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
r=u*sg+v*ca*sb*cg, s=v*sin(a), esc=imag(p5)+9
c=p+flip(q)+p3, z=r+flip(s)+p4:
z=z^(real(p5))+c
|z|< esc }
END PARAMETER FILE=========================================
Today's fractal is from Fractint's built in formula fn(z*z). The design is of spheres
within spheres colored in blue and cyan. Ready made at http://maxitersfractalfollies.blogspot.com
Cyan Spheres { ;fract487.gif
; blank
; calctime 0:00:31.09
; created Oct 28, 2010
; Fractint Version 2004 Patchlevel 10
reset=2004 type=fn(z*z) function=log
center-mag=1.56626/-0.0173035/0.7732086/1.0031/90/3.88578058618804789e-0\
16 maxiter=3000 inside=0 outside=atan cyclerange=0/255
colors=hzzgzzfzzezzdzzczzbzz`zz_zzZzzYzzXzzWzzVzzTzzSzzRzzQzzPzzOzzMzzLz\
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}
Roger Alexander
