FOTD -- October 28, 2010 (No Rating)
Fractal visionaries and enthusiasts:
Today's image gets no rating. Why? Because it's a virtual
repeat of yesterday's image. The difference is that, while
yesterday's image was sliced in the Oblate direction, today's
is sliced in the Rectangular direction, which consists of the
imag(Z) and imag(C) axes of the Julibrot.
IMO, what made yesterday's scene so curious is the fact that the
fractal terminates on the left and right at values of plus and
minus 1.618...., (the golden ratio), while the large valleys
meet at plus and minus 0.618...., the reciprocal of the golden
ratio).
This is curious enough, but when I checked the horizontal
terminal points of today's image, I found a value of plus and
minus 0.78615.... What an incredible coincidence it is that
this is the reciprocal of the square root of the golden ratio.
(The two large valleys in today's image meet at plus and minus
0.521555...., though I have yet to find anything significant in
this value.)
Mathematical interest aside, today's image is simply a variation
on yesterday's image, with the same color palette. Those who
enjoyed yesterday's will enjoy today's almost as much. Those
who thought yesterday's image was boring will be no more
impressed by today's.
The name "Curious Coincidence" refers to the mathematical aspect
of the image.
The calculation time of only 3 seconds is mercifully brief. The
best way to view the image is to enjoy it already calculated on
the FOTD web site at:
<http://www.Nahee.com/FOTD/>
Wednesday began with clouds here at Fractal Central. But the
clouds broke up by midday, leading to a sunny afternoon with a
temperature of 72F 22C. These conditions were perfect for the
fractal cats, who spent several hours in the sun that streamed
in the southwest window. My day was average; FL had a similar
day. The next FOTD will be posted in 24 hours, and is likely to
be another mathematical curiositiy, though one with far more
artistic impact. Until then, take care, and why does the number
epsilon equal minus one when it is raised to the power of
imag(PI)?
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
CuriousCoincidence { ; time=0:00:03.35-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot4 center-mag=0/0/1.398/1/\
90/0 params=90/0/0/90/-1/0/0/0/2/0 float=y inside=0
maxiter=300 inside=0 outside=real periodicity=6
colors=000KZzJXwJVsJTpJRmJQjJPgJNeJMbJL`IKYHJVGISF\
HPEGMDFJCDFBBCA8A958GQKJYKMeKPmKUmKZmKcmKhmKmmIrmF\
vmCzjHzhNzeSzcXzbTzaPz`Lz_HzZDzZ9zSCzLFzEHz7Kz1Mz6\
OzAPzFRzJSzNTzSSzXSzaRzfRzhRzjRzlRzmRzoRzqRzrRzzOz\
zLzzIzzFzzKzzOzzTzzXzzVzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzz }
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 a remix of my Oct. 24 post. Same formula and palette, different parameters.
Like the Oct.24 post ready made at http://maxitersfractalfollies.blogspot.com
Twilight Remix { ; fract484.gif
; blank
; calctime 0:17:07.88
; created Oct 26, 2010
; Fractint Version 2004 Patchlevel 10
reset=2004 type=formula formulafile=kerrym.frm
formulaname=hermanm_man-polar
center-mag=-1.89610111352240000/+0.00222840388810970/9616.574/1/22.49999\
99999827232/-2.86882184674652763e-011 params=4/2/1/60/3/0 float=y
maxiter=1500 inside=0 proximity=1 outside=fmod decomp=256
colors=000302403503504604705805806907A07B08B08C09D1AE1AF1BF1BG1CH1CI1DI1\
EJ1EK1FL1FL1GM1GN1HO2IN2KM2NK2QJ2TH2VG2YE2`D2cB1fA1i81l71n51q41t21w00z03\
z07z0Bz0Fz0Jz0Nz0Rz0Vz0Zz0bz0fz0jz0nz0rz0vz0zz0xz0vz0tz0rz0pz0nz0lz0jz0h\
z0fz0dz0bz0`z0Zz0Xz0Wz0Uz0Sz0Qz0Oz0Mz0Kz0Iz0Gz0Ez0Cz0Az08z06z04z02z00z00\
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0zH0zE0zB0z80z60z30z00x01v02s03q04n06l07i08g19e1Ab1B`1CY1EW1FT1GR1HO2JN2\
IM2HK2GJ2FH2DG2CE2BD2AB19A18817715514413212000000101201
}
frm:hermanm_man-polar { ; Kerry Mitchell 16feb98
;
; real(p1) = z exponent (use integer >= 2; m=n-1)
; imag(p1) = g exponent (integers)
; real(p2) = alpha magnitude (try 1)
; imag(p2) = alpha polar angle (try integers)
; real(p3) = critical point selector (>0 for positive root)
; imag(p3) = unused (<0 for negative root)
; use decomp=256
; zero and infinity bailouts hardcoded to 1e-6, 1e6
; coloring speed hardcoded to 4
;
c=pixel, iter=1, n=real(p1), m=imag(p1), nfac=2*n-1
maxr=1e6, minr=1/maxr, speed=4*pi/128
r=real(p2), t=imag(p2), alpha=r*(cos(t)+flip(sin(t)))
oln=1/log(n), fac=log(0.5*log(maxr))
c2=sqr(c), hypnum=sqr(n)+sqr(m), pn=1
hypden=sqr(n-m), hypfac=hypnum/hypden
if (real(p3)<0)
pn=-1
end if
if (real(c2)>hypfac)
pn=-pn
end if
if (imag(c)<0)
pn=-pn
end if
afac=c*n, bfac=c2*(n-m)+(n+m), cfac=c*n
d=sqrt(bfac*bfac-4*afac*cfac)
z=(bfac+pn*d)/(2*afac)
:
g=(z-c)/(1-c*z), z=alpha*z^n*g^m
iter=iter+1, r=|z|
;
; orbit trap around 0
; renormalize iteration count via decomp angle
; set "iteration done" flag (iter=-1)
;
if (r<minr)
angle=(iter+oln*(fac-log(log(cabs(z)))))*speed
z=cos(angle)+flip(sin(angle))
iter=-1
end if
;
; orbit trap around infinity
; renormalize iteration count via decomp angle
; add pi to angle to separate from 0 orbit trap
; set "iteration done" flag (iter=-1)
;
if (r>maxr)
angle=(iter+oln*(fac-log(log(cabs(z)))))*speed
angle=angle+pi
z=cos(angle)+flip(sin(angle))
iter=-1
end if
iter>0
}
Roger Alexander
FOTD -- October 27, 2010 (Rating 7)
Fractal visionaries and enthusiasts:
The formula behind today's image is Z^2+C, the same formula that
creates the Mandelbrot set and all its associated Julia sets.
But today's image is obviously not the M-set, nor is it any
recognizable Julia set. It is something I call an Oblate set,
a hybrid set half-Mandelbrot and half-Julia.
The X-axis of the image is the real(Z) axis of the Julibrot,
while the Y-axis is the imag(C) axis. The image has a Mandel-
brot nature in its vertical direction and a Julia nature in its
horizontal direction. The center of the image is the center of
the large period-2 bud of the M-set. No shape manipulation such
as stretching or skewing was done in the creation of the image,
though I did add a bit of life by rendering it with the outside
set to 'real'.
A most curious relation exists in the location of the elements
along the X-axis of the image. The fractal terminates on the
left and right at the points plus and minus 1.61803398875....
which of course is the Golden Ratio, while the two major valleys
closest to the center meet at plus and minus 0.61803398875....
which is the reciprocal of the Golden Ratio. There must be a
logical reason why this is so, though I have not been able to
find it.
I rated the image at a 7, a good part of which is due to the
mathematical interest. Artistically, the image rates a 5 or 6.
I named the image "A Golden Slice" because of the appearance of
the Golden Ratio in it.
Despite all the golden glitter, the image is blazingly fast,
finishing on my machine in under 3 seconds, almost as fast as
the display can switch to super-VGA mode. The trip to see the
finished image on the FOTD web site at:
<http://www.Nahee.com/FOTD/>
may not be quite as fast, but it will involve much less set-up
work.
Rain fell most all Monday night here at Fractal Central. The
remaining clouds blocked the sun on Tuesday, but the temperature
of 70F 21C was mild enough that the lack of sun was never
noticed. The fractal cats spent an hour chasing a mechanical
toy mouse we picked up Monday evening. Nicholas finally killed
it when its battery went flat. The rest of the day was occupied
with routine work.
Remembering yesterday, I dared not ask FL what was happening on
the Spanish channel. The next FOTD will be posted in 24 hours.
It should be no surprise if another odd slice of the Z^2+C
Julibrot shows up. Until then, take care, and let your negative
side shine through.
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
A_Golden_Slice { ; time=0:00:02.85-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot4 center-mag=0/0/0.7544715
params=0/0/90/0/-1/0/0/0/2/0 float=y maxiter=300
inside=0 outside=real periodicity=6
colors=000KZzJXwJVsJTpJRmJQjJPgJNeJMbJL`IKYHJVGISF\
HPEGMDFJCDFBBCA8A958GQKJYKMeKPmKUmKZmKcmKhmKmmIrmF\
vmCzjHzhNzeSzcXzbTzaPz`Lz_HzZDzZ9zSCzLFzEHz7Kz1Mz6\
OzAPzFRzJSzNTzSSzXSzaRzfRzhRzjRzlRzmRzoRzqRzrRzzOz\
zLzzIzzFzzKzzOzzTzzXzzVzzzzzzzzzzzzzzzzzzzzzzzzzzz\
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zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzz }
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 26, 2010 (Rating 5)
Fractal visionaries and enthusiasts:
Today's image is a view of the western part of the Mandelbrot
set rotated so that west is facing up. It is sliced at an angle
rotated 66 degrees toward the Oblate orientation. The image is
actually a scene in the four-dimensional Z^2+C complex known as
the Julibrot, which holds all the perturbed Mandelbrot sets as
well as all the Julia sets.
Over the years I have stumbled upon countless other Julibrot
scenes sliced in odd directions, which look nothing at all like
either Julia or Mandelbrot images. Perhaps I'll use some of
these odd images as FOTD's before too long. They certainly are
interesting enough even though they might not always excel in
artistic worth.
The spurious buds in today's image are neither Julia nor
Mandelbrot things. They exist only at the orientation of
today's image. Changing the P1 or P2 parameters will show how
critical the orientation is.
The name "Blazing Slices" has no connection to an old western
comedy movie. It refers to the fiery colors I intentionally
gave the image.
The calculation time of 10 seconds makes even a 5-rated image
such as today's a bargain. The image may also be viewed, this
time in finished form, on the FOTD web site at:
<http://www.Nahee.com/FOTD/>
I made a big mistake Monday afternoon when I caught FL watching
one of those mushy latin soap operas and asked her what the show
was about. When she described the plot, it went something like
this: `that woman loves that man but he doesn't love her and the
woman he does love doesn't know that he is her brother. This
other woman wants a divorce from her no-good husband, but he has
just won a huge lottery and now she doesn't know what to do
about the man she really loves. This other man loves the first
woman but he has a crippled wife who depends on him.' On and on
the convoluted plot description went until, after a few minutes
of trying to fathom the intricacies, I dozed off. When I jerked
to my senses, I asked FL where does the action fit in. The
resulting laughter made my day.
A fairly pleasant day passed almost un-noticed here at Fractal
Central on Monday. The fractal cats did notice the intermittent
periods of sun and the unseasonably warm temperature of 72F 22C,
but the humans were involved in other things.
The next FOTD will be posted in 24 hours. Until then, take
care, and the golden rule is supposed to work both ways.
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
Blazing_Slices { ; time=0:00:10.10-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot4 center-mag=0/1.10231/\
1.5/2.0709 params=0/66/0/90/-0.75/0/0/0/2/0 float=y
maxiter=450 inside=0 logmap=yes passes=1
colors=00091ZB2_E3`H4aK5bM6cP7cS8cV9cYAc`BccC_fDWj\
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zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzz }
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 25, 2010 (Rating 6)
Fractal visionaries and enthusiasts:
Today's image is a "Seahorse Abomination", which is what I named
it.
Seahorse Valley never deserved to have such things done to it.
The valley is a nice little place, a four-dimensional complex,
so what have we done to the innocent valley to cause the gross
disruption apparent in today's image?
To begin, the image is not a Julia set of the familiar Seahorse
Valley of the Mandelbrot set. It is a Julia set of Seahorse
Valley of the Quaternion Mandelbrot set, which is actually an
eight-dimensional Julibrot complex, with a 4-D Mandelbrot aspect
and a 4-D Julia aspect.
I'll not go into details about the wonders of the eighth
dimension; I'm still working on understanding the fourth. But
one thing certain is that in 8-D space, 28 two-dimensional
planes pass through every point, with every plane perpendicular
to all the others.
(There is something puzzling about the two quaternion formulas.
Today's image was created by the type=quatjul formula, which
plots Julia sets. This formula has six variable parameters, all
of which are necessary if we are to move off from the screen in
the six perpendicular directions. Yet the type=quat formula,
which plots Mandelbrot sets, has only two changeable parameters
[plus two unused parameters] both of which have the same effect,
giving the impression that the quaternion Mandelbrot set is a
simple three-dimensional thing that would result from rotating
the classic M-set around the X-axis.)
Be that as it may, today's image, with its minor mathematical
interest and only slightly better artistic worth, rates a 6.
The fireball-fast calculation time of under 3 seconds will leave
little time to wonder about higher dimensions before the image
bursts onto the screen in a blaze of mediocrity.
Those who prefer to remain burst-free may see the finished image
on the FOTD web site at:
<http://www.Nahee.com/FOTD/>
Sunday here at Fractal Central featured a mixture of sun and
clouds, and a very pleasant temperature of 70F 21C. The fractal
cats complained that there was not enough sun then went to sleep
on the living room carpet.
After my trip back to the nostalgia-infested land of Old Fractal
Central on Saturday, Sunday seemed relaxing. (Though I still
wish the Phillies had won the NL pennant.) FL occupied herself
wondering why I end my FOTD discussions with so many 'silly and
unanswerable' comments.
The next FOTD will be posted in the generally accepted 24 hours.
Until then, take care, and if the belief in God and an afterlife
is a defense mechanism evolved to enable us to cope with the
awareness of our ultimate mortality, how would society react if
it were ever demonstrated beyond all doubt that these things do
not exist?
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
SeahorseAbomnation { ; time=0:00:02.70-SF5 on P4-2000
reset=2004 type=quatjul center-mag=-0.0300864/0.01\
38653/1.05286/1/32.5/0 params=-0.75/-0.055/-0.113/\
-0.013/-0.048/-0.134 float=y inside=0 periodicity=0
colors=000KIcLJcMKcNLcOMcPNcQOcTQcWScYUc`WcbYceZch\
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mfMleNjdOhdPgcPecQcbRbaS`aSZ`TY_UW_VUZbfWadX`cX_aX\
Z`YZ_YYYYXXZWVZVUZ8p9OmYPkZQiZQgZRfZRdZSbZT`ZT_ZUY\
ZUWZVUZ_LiZNfYPdXQbWS`f0V }
END PARAMETER FILE=========================================
FOTD -- October 24, 2010 (Rating 5)
Fractal visionaries and enthusiasts:
Elephants are everywhere in Seahorse Valley of the Mandelbrot
set. They come in all sizes, shapes and colors. They also
enjoy doing tricks. The elephants in today's image have formed
themselves into a wreath, a bit too early for the season to be
sure, but decorative just the same. The name "Elephant Wreath"
describes the scene well enough.
The image is stretched more than 1000 times (1083 to be exact)
in the horizontal direction. Such stretching is necessary to
correct the natural distortion of the scene, which lies in an
orientation rotated 10 degrees from the Rectangular toward the
Oblate.
The rating of a 5 might be a bit too conservative, but how much
can a holiday wreath decorated with elephants be worth?
The calculation time of 11-1/2 seconds is fast by any standards.
Viewing the finished image on the FOTD web site at:
<http://www.Nahee.com/FOTD/>
is equally fast, and could be a lot more fun.
A frosty morning gave way to warm sunshine and an afternoon
temperature of 64F 18C here at Fractal Central on Saturday. (It
reached 73F 23C at Old Fractal Central 150 miles 240km south of
here.) The fractal cats slept through most of it. My day,
which was occupied with a trip back to Old Fractal Central, was
otherwise uneventful. The next FOTD will likely be posted
within 12 hours, which will bring the FOTD up to date if it
happens. Until next time, take care, and keep alert for that
quantum stuff. It's lurking everywhere.
Jim Muth
jamth(a)mindspring.com
START PARAMETER FILE=======================================
Elephant_Wreath { ; time=0:00:11.52-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot4 center-mag=-0.003041525\
45290287/0/16.30635/1083 params=100/90/0/90/0.265/\
0/0/0/2/0 float=y maxiter=1500 inside=0 logmap=48
symmetry=xaxis periodicity=6 mathtolerance=0.05/1
colors=000BaMCbODcOEdQFeSGfUHgWIhYJiZKjYMkYOlXQmXS\
nWUoWZrVcuVhxUmzUruVvrVwnVzkWwhWmeWcbWUZXKWWJVVHTV\
EQUCNTAKS9IS8HS7GS6FR5DS4CQ3AO25M10K00K40K70KA0KD0\
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IrcKpcLmdNkeOheQffRcgT`gUZhWWiXUiZRj_PkaMkbKhaMe`O\
c_Q`_RYZTWYVTXXRXYOW_LVaJUcGUdDTfBSh8Rj6Rk8ShATfBU\
cDUaFV_GWXIXVKUULVUNWUOXUNYUNYUMYUMXVLXWLXXLWYKWZK\
W_JV`JVaJVbIScIQdHNeHLfGMgGNhGNiGOjGPkGPlV0PX1OY1N\
Z1M_1L`2La2Kb2Jc2Ie3Hf3Hg3Gh3Fi4Ej4Dk4Dl4Cn5Bo5Ap5\
9q59r68s67t66u66r9BoCGlFLiHQfKVcN_pIZaPcP_mOWhNTcN\
Q_MNVMJRLGMLDIKADK79M9ANBBODCPEDRGESIFTJGULGVNHXOI\
YQJZSK_ULaVMbXNcZNd_OeaPgcQhdRifSjhTjlVkiTkfSkdQka\
Pk_NlXMlUKlSJlPHlNGlKEmHDmFBmCAmA8m77m56l66k76k86j\
96jA6iB6iC6hD6hE6gF6gG6fG6eH6eI6dJ6dK6cL6cM6bN6bO6\
aP6aQ6`R6`R6_TD_UK_VQZWXZXcZYibVeeSaiPZlMVoJRsGOvD\
KyBHo9DqjdsleunfvT9vpghQY }
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 post is a minibrot in twilight colors. At the usual http://maxitersfractalfollies.blogspot.com
Twilight Minibrot { ; fract483.gif
; blank
; calctime 0:03:23.94
; created Oct 24, 2010
; Fractint Version 2004 Patchlevel 10
reset=2004 type=formula formulafile=kerrym.frm
formulaname=hermanm_man-polar
center-mag=-2.00319/-0.00306882/27.01637/1/90/3.88578058618804789e-016
params=3/2/1/45/1/0 float=y maxiter=1500 inside=0 outside=atan
decomp=256
colors=000302403503504604705805806907A07B08B08C09D1AE1AF1BF1BG1CH1CI1DI1\
EJ1EK1FL1FL1GM1GN1HO2IN2KM2NK2QJ2TH2VG2YE2`D2cB1fA1i81l71n51q41t21w00z03\
z07z0Bz0Fz0Jz0Nz0Rz0Vz0Zz0bz0fz0jz0nz0rz0vz0zz0xz0vz0tz0rz0pz0nz0lz0jz0h\
z0fz0dz0bz0`z0Zz0Xz0Wz0Uz0Sz0Qz0Oz0Mz0Kz0Iz0Gz0Ez0Cz0Az08z06z04z02z00z00\
y10x20v20u30s40r50p50o61n71l81k91i91hA1gB1eC1dC1bD1aE1_F1ZF1YG1WH1VI1TJ2\
SJ2RK2PL2OM2MM2LN2JO2IM2GJ2EH2CE1AC18916614312000101302503605806A07C19D1\
9E1AF1AF1BG1BH1CI1DI1DJ1EK1EL1FL1FM1GN1HO2IO2JQ2IS2HV2GX2F_2Da2Cd2Bf2Ah1\
9k18m17p15r14u13w12z00z20z50z70zA0zC0zF0zH0zK0zM0zP0zR0zU0zX0zZ0za0zc0zf\
0zh0zk0zm0zp0zr0zu0zw0zz0zw0zu0zr0zo0zl0zj0zg0zd0za0z_0zX0zU0zS0zP0zM0zJ\
0zH0zE0zB0z80z60z30z00x01v02s03q04n06l07i08g19e1Ab1B`1CY1EW1FT1GR1HO2JN2\
IM2HK2GJ2FH2DG2CE2BD2AB19A18817715514413212000000101201
}
frm:hermanm_man-polar { ; Kerry Mitchell 16feb98
;
; real(p1) = z exponent (use integer >= 2; m=n-1)
; imag(p1) = g exponent (integers)
; real(p2) = alpha magnitude (try 1)
; imag(p2) = alpha polar angle (try integers)
; real(p3) = critical point selector (>0 for positive root)
; imag(p3) = unused (<0 for negative root)
; use decomp=256
; zero and infinity bailouts hardcoded to 1e-6, 1e6
; coloring speed hardcoded to 4
;
c=pixel, iter=1, n=real(p1), m=imag(p1), nfac=2*n-1
maxr=1e6, minr=1/maxr, speed=4*pi/128
r=real(p2), t=imag(p2), alpha=r*(cos(t)+flip(sin(t)))
oln=1/log(n), fac=log(0.5*log(maxr))
c2=sqr(c), hypnum=sqr(n)+sqr(m), pn=1
hypden=sqr(n-m), hypfac=hypnum/hypden
if (real(p3)<0)
pn=-1
end if
if (real(c2)>hypfac)
pn=-pn
end if
if (imag(c)<0)
pn=-pn
end if
afac=c*n, bfac=c2*(n-m)+(n+m), cfac=c*n
d=sqrt(bfac*bfac-4*afac*cfac)
z=(bfac+pn*d)/(2*afac)
:
g=(z-c)/(1-c*z), z=alpha*z^n*g^m
iter=iter+1, r=|z|
;
; orbit trap around 0
; renormalize iteration count via decomp angle
; set "iteration done" flag (iter=-1)
;
if (r<minr)
angle=(iter+oln*(fac-log(log(cabs(z)))))*speed
z=cos(angle)+flip(sin(angle))
iter=-1
end if
;
; orbit trap around infinity
; renormalize iteration count via decomp angle
; add pi to angle to separate from 0 orbit trap
; set "iteration done" flag (iter=-1)
;
if (r>maxr)
angle=(iter+oln*(fac-log(log(cabs(z)))))*speed
angle=angle+pi
z=cos(angle)+flip(sin(angle))
iter=-1
end if
iter>0
}
Roger Alexander
I usually don't give my posts names but if I did I would call this one "Algae" because it reminds me of algae.
Am I consistent or oh so predictable? Pre-calculated at http://maxitersfractalfollies.blogspot.com
Algae { ; fract482.gif
; blank
; calctime 0:05:29.50
; created Oct 22, 2010
; Fractint Version 2004 Patchlevel 10
reset=2004 type=formula formulafile=kerrym.frm
formulaname=conic-near_man
center-mag=+0.00133886277847312/+1.00493995512520900/10743.32
params=-0.5/-0.4000061037018952/0.03488265633106479/0.125/0.557969908749\
6567/-0.06808679464094974 float=y maxiter=1500 inside=0 decomp=256
periodicity=0
colors=0100200300400500700800900A00B00D00E00F00G00H00J00K00L00M00N00P00Q\
00R00S00T00V00W00X00Y00Z00`00a00b00c00d00f00g00h00i00k00k01k02k03l04l05l\
05m06m07m08n09n0An0Bo0Bo0Co0Dp0Ep0Fp0Gp0Hq0Hq0Iq0Jr0Kr0Lr0Ms0Ns0Ns0Ot0Pt\
0Qt0Ru0Su0Tu0Tu0Uv0Vv0Wv0Xw0Yw0Zw0Zx0_x0`x0ay0by0cy0dz1dz1ez1ez1fz2fz2gz\
2gz2hz3hz3iz3iz3jz4jz4kz4kz4lz5mz5mz5nz5nz6oz6oz6pz6pz7qz7qz7rz7rz8sz8sz\
8tz8uz9uz9uz9tz9tz9tz9sz9sz8sz8rz8rz8rz8qz8qz7qz7pz7pz7oz7oz7oz6nz6nz6nz\
6mz6mz6mz5lz5lz5lz5kz5kz5kz4jz4jz4iz4iz4iz4hz3hz3hz3gz3gz3gz3fz2fz2fz2ez\
2ez2dz1dz1cz1bz1ay1`y1_y1_x1Zx1Yx1Xw1Ww1Vw1Uv1Uv1Tv1Su1Ru1Qu1Pt1Ot1Ot1Nt\
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06g06f05e05d05c05b05a05`05_05Z04Y04X04W04V04U04T04S04R03P03O03N03M03L03K\
03J03I02H02G02F02E02D02C02B02A0190180170160150140130010
}
frm:conic-near_man { ; Kerry Mitchell 11may98
zc=0, cc=pixel, maxr=1e12, minr=maxr, iter=1
a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
e=real(p3), f=imag(p3):
iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
conic=|x*(a*x+b)+y*(c*y+d)+e*x*y+f|
if (conic<minr)
minr=conic
end if
if ((|zc|>maxr)||(iter==maxit))
iter=-1
t=log(minr)
z=cos(t)+flip(sin(t))
end if
iter>0
}
Roger Alexander
