u/LegalizeAdulthood

Jim Muth's Fractal of the Day for August 19th, 2008

Jim Muth's commentary for the image: > FOTD -- August 19, 2008 (Rating 8) > > Fractal visionaries and enthusiasts: > > The formula for today's image adds a very small bit of Z^(-20) > to Z^2 to produce a Mandelbrot set with disks of Z^(-20) stuff > at the centers of all the bays and buds. The image is located > on the X-axis, in the detail at the east edge of a disk almost > filling the large period-2 bud of the M-set. More than anything > else it resembles a surrealist's spider-web. > > I named the image "Stars of Iteration", which sounds a little > poetic and in a way describes not only today's fractal, but all > fractals. And then I rated it at an 8. I considered adding a > half point for my coloring, but decided that I did not do quite > enough color tweaking to merit the extra half-point. > > The calculation time of only 42 seconds is fast enough to cause > no impatience. > > With a temperature of 88F 31C here at Fractal Central on Monday, > it was just a bit too warm to be perfect, but all else was the > picture of perfection. The dynamic fractal cats, who never stop > reiterating, agreed. Just after lunch they spotted an unfamil- > iar manx cat in the yard next door, and with a volley of hisses > and growls, sent it on its way. I did not tell them that the > cat was ready to move on anyway, and their empty threats had > nothing to do with its departure. > > Otherwise the day was uneventful. The next FOTD will be posted > in 24 hours. Until then, take care, and to become famous, solve > the measurement problem.

PAR file

Stars_of_Iteration { ; time=0:00:42.84-SF5 on P4-2000
  reset=2004 type=formula formulafile=allinone.frm
  formulaname=MandAutoCritInZ function=ident passes=1
  center-mag=-0.9160783767764306/0/8.62e+008
  params=1/2/1e-018/-20/0/0/0/0 float=y maxiter=1450
  inside=0 logmap=153 symmetry=xaxis periodicity=10
  colors=0000EC0FF0GI0HL0IO0JQ0LT0NW0PZ0Ra0Td0Vg1XiD\
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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, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 18th, 2011

Jim Muth's commentary for the image: > FOTD -- August 18, 2011 (Rating 7) > > Fractal visionaries and enthusiasts: > > I believe in honesty in advertising, and when a fractal falls > flat, I admit it. I realize that no one gets rich by being > totally honest about their product, but no one gets rich from > finding fractals either, so I've got nothing to lose by telling > it like it is. > > With a rating of an everyday 7, today's deep scene in the > Mandelbrot set barely deserves the word 'deep'. At the modest > magnitude of only 7*(10^15) it is barely beyond conventional > math range, but at least IMO, it has a certain interesting > quality about it that raises it to FOTD standards. > > The scene lies in the East Valley of a larger minibrot on a side > filament of the main filament extending from the bud on the > north shore line of the period-2 bud of the M-set. > > I named the image "Elephant Trek". I see a few hints of > elephant parts in the scene, though not quite enough trunks and > heads to fully justify the name. > > The parameter file takes 25 minutes to finish. > > The day began mostly sunny here at Fractal Central, and > continued this way through a good part of the afternoon. At > post time, showers were in the area, though none had yet reached > FC. The temperature of 86F 30C brought out the best in the > fractal cats. They spent most of the time sleeping. > > While the cats were taking it easy, the humans were doing what > needed to be done. When the necessities were finished, we took > it easy also. It is hardly necessary to state that the next > FOTD image will be posted in 24 hours. Until then, take care > and when I was a youngster, I knew that science and technology > would be our salvation. But now I'm not so sure. What it is > doing is making us ever more dependent on the high priests of > the 'Church of Science and Technology'. > > I recently overheard a young boy (probably an internet addict), > ask how people ever got along when they didn't have cell phones > and the internet. Well, I was around before cell phones and the > internet, getting along quite well, the same as everyone else, > and I can state that we managed life the same as we are doing it > now without the next big technological advance, which will not > be developed for another 50 years.

PAR file

Elephant_Trek { ; time=0:25:10.06-SF5 on P4-2000
  reset=2004 type=mandel passes=1 center-mag=-1.0188\
  8639656655182417/0.30447073015893949152/7.411e+015\
  /1/-10/0 params=0/0 float=y maxiter=7500 bailout=16
  inside=0 logmap=1110 periodicity=6
  colors=000F0AF0AF0AE0AE0AE0AE0AE0AE1AE2AE3AE4AE5AD\
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  `gziazfWzdRzbLz`GzZPzcTzO }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 17th, 2012

Jim Muth's commentary for the image: > FOTD -- August 17, 2012 (Rating 6) > > Fractal visionaries and enthusiasts: > > Today we check the 'Zexpi' fractal, or at least one of the > infinity of Zexpi fractals. The one we are checking looks > nothing at all like the familiar one, which closely resembles > the cubic Mandelbrot set. Instead of sitting upright, today's > parent Zexpi has been turned on its side, with two main buds on > the western lobe of the main bay and only half of a bud on the > eastern lobe. > > I created this parent by calculating the formula Z^(PI)+C at a > level 15.5 rungs up the logarithmic hyperladded with the 'ceil' > function in effect. I think this is the first time I have used > this function, which is one of the rounding functions. Today's > scene is located on the west side of the main bay, in the debris > near the X-axis. > > The rating of a 6 is average and indicates that I see nothing in > the image to get excited about. The name "Primarily PI" shows > that this is the first in a series of images based on the Zexpi > fractal. The calculation time of a mere 20 seconds is about > what the image is worth. > > Increasing clouds marred the early part of the day here at > Fractal Central, and rain spoiled the latter part. The fractal > cats approved of the temperature of 86F 30C however. > > The humans made it through the day with little of the excitement > that makes interesting reading. The next FOTD will be posted > before too long. Until whenever, take care, and why are the > 99-percent so offended by the wealth of the 1-percent? Could it > be envy?

PAR file

Primarily_PI { ; time=0:00:20.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=MandelbrotBC3 function=ceil logmap=37
  center-mag=-0.9369868294821131/+0.0378608092734622\
  /1.568021e+011/1/132.5/0 params=3.14159265358979/0\
  /15.5/0 float=y maxiter=2400 inside=0 periodicity=6
  colors=0000FI0EL0DN0CP0BR0AT08V56aA4iG3tM8wTBwUEwV\
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  `HzaHzaHzaHzdHzgHziIzhJzh }

frm:MandelbrotBC3   { ; by several Fractint users
  e=p1, a=imag(p2)+100
  p=real(p2)+PI
  q=2*PI*fn1(p/(2*PI))
  r=real(p2)+PI-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|<a }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 16th, 2004

Jim Muth's commentary for the image: > FOEFD -- August 16, 2004 (Rating 7) > > Fractal visionaries and enthusiasts: > > As has become the usual state of affairs lately, the work is > still heavy here at Fractal Central, and the discussion will be > brief. > > Today's fractal image features spirals -- brilliant, sparkling > spirals everywhere, a good reason to name it "Spiral Spangled".
> Those sparkling small spirals, and the large main spiral > surrounding the central midget, raise the rating of the image to > a respectful 7. The conveniently short render time of 4 minutes > gives the image an even more respectful overall value of 162. > > The image lies near the end of a filament extending from a bud > on the north shoreline of the main bay of its parent fractal.
> This parent was created by combining 4 parts of Z^(0.5) with 2 > parts of Z^(-1.5) and adding (1/C). > > The parameter file renders in just over 4 minutes on my now-antique > fractal-dedicated machine. It will go faster on a newer > unit. > > Storm Charley passed by Saturday with 0.08 inch 2mm of rain and > a wind gusting to 5mph 8kph. The fractal cats watched the non-excitement > from the dryness of the porch. Sunday stayed dry all > day, giving the dynamic duo much time in the fractal yard.
> Today is starting sunny. It looks like another good day for the > fractal duo. > > For me it looks like work, and there is no better time to get > started than now. Until the next FOEFD appears in a few days, > take care, and dig deeply.

PAR file

Spiral_Spangled { ; time=0:04:19.91--SF5 on a P200
  reset=2003 type=formula formulafile=allinone.frm
  formulaname=MandelbrotMix4 function=recip passes=b
  center-mag=-0.21650624140427070/+0.019943127229515\
  34/4.234284e+009/1/37.5/-0.000450000450211679448
  params=2/0.5/1/-1.5/1/-70 float=y maxiter=1000
  inside=255 outside=imag logmap=113 periodicity=10
  colors=000G0oD0jB0e80a50Z10W03R09O0QN0UN0YN0aN0cN0\
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  9Dz6Gz3Jz0Nz0Oz0Lz3IzD000 }

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 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 15th, 2008

Jim Muth's commentary for the image: > FOTD -- August 15, 2008 (Rating 8) > > Fractal visionaries and enthusiasts: > > In today's fractal I divided Z^2 by (Z^(0.000001)+5), or something > like that, and then added C. At least, that's what I > think happened when I entered the parameters. But regardless of > what actually happened, the resulting parent fractal resembles > an oversized Mandelbrot set, though deep inside it is different. > > I found today's scene on the edge of the infinitely divided main > spike, in the area just east of the large minibrot. This is > quite a rich area, filled with surprises, an area which until > now I have largely ignored. > > I named the image "Little Critters" when I noticed the dark > insect-like things sitting on the webs surrounding the central > minibrot. Then I rated the image at an 8, which includes a > small reward for my coloring. > > The brief calculation time of 1-1/3 minutes will make everyone > but the most impatient fractalist happy. And even those > impatient few will be only slightly discombobulated. > > Too many clouds and a threat of rain kept Thursday from being > perfect here at Fractal Central. But the temperature of 82F 28C > was quite pleasant and the rain held off until nightfall, so the > fractal cats had nothing to complain about. > > I had nothing to complain about either. So I'm not complaining. > The next Fractal of the Day will be posted in 24 hours. Until > then, take care, float like a duck and sting like a flea.

PAR file

Little_Critters    { ; time=0:01:20.40-SF5 on P4-2000
  reset=2004 type=formula formulafile=allinone.frm
  formulaname=DivideJulibrot center-mag=-10.49111003\
  76656/+0.00000459624450347/5.459246e+011/1/177.5/0
  params=0/0/0/0/0/0/0/0/2.000001/5 float=y inside=0
  maxiter=1800 logmap=200 periodicity=10
  mathtolerance=0.05/1
  colors=000K8_K9bK9dKAgKAiKEkKHmKKoKNqKQsKUsKXvJ`vG\
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  MgzOlzRqzTuzVezRQzNlzVkzY }

frm:DivideJulibrot   {; draws 4-D slices of DivideBrot Julibrots
  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),
  aa=-(real(p5)-2), bb=(imag(p5)+0.00000000000000000000001),
  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),
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=sqr(z)/(z^(aa)+bb)+c
  |z|< 1000000 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 14th, 2004

Jim Muth's commentary for the image: > FOEFD -- August 14, 2004 (Rating 6) > > Fractal visionaries and enthusiasts: > > For today's image we return to the overworked MandelbrotMix4 > formula. The very fast image takes us into the fractal created > by combining various negative portions of Z^(-1.3) and Z^(-3.9), > then adding (1/C). > > This parent fractal is a grossly bloated thing far too immense > to fit onto the screen even at the minimum possible magnitude.
> Today's image lies near the end of a filament extending from a > distorted bay of the largest Mandel-like feature. > > I named the image "Electric Discharge" because of the many > zig-zag electric-arc-blue filaments surrounding the central > midget. I rated it at a 6 because I consider it a bit above > average. > > I rendered the scene with the passes set to 'b'. Not only is > this the fastest method with this particular image, it is also > the most fun to watch appear on the screen. > > The render time of 1-1/2 minutes combines with the rating of 6 > to give an overall value of an impressive 378. The fastest way > to see the image is to run the included parameter file. > > The unsettled weather continued Thursday and Friday here at > Fractal Central, with enough rain to keep the fractal cats > confined to the indoors. They were understandably sulky. Today > is starting cloudy, and the remains of storm Charley are due in > the vicinity later today with deluges of rain. We'll see how > the cats take this weather. > > I still have enough backed-up work to keep me hopping. The > worst (or best) part is that the new work comes in almost as > fast as the old work is finished. The fractals are temporarily > slowed down; the philosophy is all but dead. But the catch-up > point is visible on the distant horizon. Look for the next > FOEFD around Monday the 16th. Until then, take care, and maybe > nothing is real.

PAR file

Electric_Discharge { ; time=0:01:35.35--SF5 on a P200
  reset=2003 type=formula formulafile=allinone.frm
  formulaname=MandelbrotMix4 function=recip passes=b
  center-mag=-2.02795783865152300/-0.531289001289163\
  20/7.840581e+010/1/-122.5/0.129435658686167326
  params=-1/-1.3/-0.3/-3.9/0/-20 float=y maxiter=900
  inside=255 logmap=45 periodicity=10
  colors=000zaJzeBzj2zn2zr2zl2zh2te2lZ2cW2WS2OM2HI47\
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  zzzzzzyzzuzzqzzmzzizzczz_zzXzzTzzPzzLzzIzzPzzVzz_z\
  zezzkzzqzzwzzzzzzzzzzzzzzzzzzzzzzzzzzyzzyzzwzzwzzu\
  zzszzszzqzzqzzmzzkzzizzgzzezzczz_zzZzzXzzVzzTzzRzz\
  PzzPzzPzzPzzPzzPzzgzz_000 }

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 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 11th, 2003

Jim Muth's commentary for the image: > FOTD -- August 11, 2003 (Rating 5) > > Fractal visionaries and enthusiasts: > > I spent too much effort being lazy all day Sunday, so the > fractal I eventually found rates only a 5. But after all, > even an average fractal is worth seeing. > > Some fractals, such as the well-known Mandelbrot set, are > connected. This means that every apparently disconnected midget > is actually connected to the main bay by an infinitely thin > filament. Other fractals, such as the parent of today's image, > are not connected, and contain areas totally cut off from the > main body of the fractal. > > Today's midget lies in one of these areas that is cut off from > the main body of the parent fractal, which appears as a slightly > distorted M-set. This is why I named it "Disconnected Midget".
> I rendered the image with the outside set to 'tdis' to emphasize > the disconnected nature of the scene. The 'tdis' outside also > makes a better picture. > > The short render time of a nice neat 0:02:02.02 is pleasant > enough. > > Sunday was an unusually pleasant day here at Fractal Central.
> The rain held off and the temperature reached 84F 29C, which > made the cats happy, and I got through the day without finding > work that needed to be done, which made me happy. Hopefully, > today will be a repeat, but there is still a bit of work to be > finished first. > > Until next time, take care, and stay connected.

PAR file

DisconnectedMidget { ; time=0:02:02.02--SF5 on a P200
  reset=2002 type=formula formulafile=allinone.frm
  formulaname=MandelbrotMix4 function=ident passes=1
  center-mag=-1.83854548477051500/+0.410334133501795\
  70/4.195225e+007/1/85/-3.35216519486980502e-007
  params=1/3/2/-0.5/0/0 float=y maxiter=500 inside=0
  outside=tdis periodicity=10
  colors=000vPnyOuqSrjWpc_nXblQfiJjgCne5qc4sj3uq3wwA\
  nxGeyNXyTOzZFz`JvbNrdQnfUkhYgj`cld_mgXkXWiMWgCWbEV\
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  JhIIhEGgAFg7IiDKjJMkPOlVQm`SnfUolWprUomTnhRncQmZPm\
  VYUMXOOWIPgYFjWMmUTpT`sRg }

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 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 10th, 2005

Jim Muth's commentary for the image: > FOTD -- August 10, 2005 (Rating 8) > > Fractal visionaries and enthusiasts: > > I see on the news that someone has just cloned a dog. It's not > much of a surprise, since they have already cloned mice, sheep, > horses, cats, etc. But it does raise a theological question.
> Is the fact that men have created living animals evidence that > God is unnecessary for the emergence of life, or is it evidence > that life can arise only through the working of a higher intelligence? > This question is one that I will discuss in detail when > (if ever) the next outburst of philosophy happens. > > For now, we have a fractal to discuss, and a rather good one at > that. The parent fractal came about when I combined a negative > portion of Z^(-1.1) with 5 negative portions of Z^(-11), then > added (1/C) to create one of those rambling, indescribable > fractals, boring on the surface but filled with deep mysteries. > > Today's parent fractal is dominated by one huge open ocean. The > scene of the image is located on the northern shoreline of this > ocean, deep in a rather strange inverted 'V' shaped valley. I > rendered it with the outside set to 'tdis', resulting in the > venetian blind effect that inspired the name "Venetian Blinds". > > I rated the image at an 8. It is quite unlike a typical image > of a typical midget, and this uniqueness gives it its value.
> The render time of 6 minutes is slightly slow, but this time is > on my 10-year-old machine. On most modern machines the render > time will be cut to around 2 minutes. > > The clouds hung heavy on Tuesday here at Fractal Central, but no > rain fell from them and the temperature stopped rising at 77F > 25C. These conditions proved unexpectedly satisfactory for the > cats, who spent over five hours in the holly thicket. Only a > little tuna was needed to end their day happily. Today is starting > partly cloudy with rain again forecast, but something tells > me that the experts are wrong with their rain forecast, and the > cats will be happy. I'll be happy when I get the work caught up, > a condition that is not yet showing signs of coming about.
> Unless something goes wrong, the next FOTD will appear in 24 > hours. Until then, take care, and does ultimate truth always > need to make sense?

PAR file

Venetian_Blinds { ; time=0:06:17.73--SF5 on a P200
  reset=2004 type=formula formulafile=allinone.frm
  formulaname=MandelbrotMix4 function=recip passes=1
  center-mag=-0.36941160733038520/+1.098637553115404\
  00/2.600405e+008/1/-122.5/4.31742300780260657e-005
  params=-1/-1.1/-5/-11/0/0 float=y maxiter=2000
  inside=0 outside=tdis periodicity=10
  colors=000sVgrXhqXipYjoYkn_lm_ll`mk`njaoiaohbofbpe\
  dqddrbesaet_ftYftXguVgvUhvShvUgvVfuVftXetXdtYds_br\
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  a0aY0ZV0VR0SL0MG0H506_ugYueXscVrbUpaSn`RlZRjWPhVNf\
  TLbSJ`QGYOGVKDSH9NE5JA0D6 }

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 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 8th, 2008

Jim Muth's commentary for the image: > FOTD -- August 08, 2008 (Rating 8) > > Fractal visionaries and enthusiasts: > > There can be no debate about the correct format of today's date. > Any way one arranges the numbers, it's correct. And in honor of > the occasion, as if it needed honoring, we have a fractal. > > Today's fractal combines elements of Z^2+C and Z^2.5+C, and ends > up looking quite like an oversized version of a Julia set of the > Z^2.003 Mandeloid. In fact, today's scene is a Julia set of the > East Valley of the large minibrot on the main spike of its corresponding > Mandeloid. It closely resembles one of the Julia sets > of the same area of the Z^2.003 Mandeloid, the Julia set that > holds the curious rectangle where I occasionally return in > search of a FOTD. I have not yet found any rectangles in > today's Julia set however, but then I have hardly started to > search. > > I named today's image "Railyard Symphony" because the tangle of > filaments reminds me of the well-known (at least to rail buffs) > Enola Railyards, about 65 miles down the river from F.C. > > I rated the image at an 8, leaving it to the individual > fractalist to decide whether it's worth this rating. > > One thing that can not be exceeded is the brevity of the calculation > time, which at a faster-than-light 19 seconds, actually > threatens to go backward in time. > > A near perfect day here at Fractal Central was compromised by a > brief but heavy thunder-shower in the afternoon, which dropped > the temperature from an already pleasant 81F 27C to an almost > chilly 68F 20C. The fractal cats ignored the weather and spent > most of the day playing hide-and-seek in the nooks and crannies > of the bookshelves in the long hallway of FC. > > My day was pleasant enough, and with any luck at all, tomorrow > will be the same. The next FOTD is due to be posted in 24 > hours. Condidering the regularity of the FOTD postings, I feel > it will appear on schedule. Until then, take care, walk lightly > and carry a big fractal.

PAR file

Railyard_Symphony  { ; time=0:00:19.01-SF5 on P4-2000
  reset=2004 type=formula formulafile=allinone.frm
  formulaname=DivideJulibrot
  center-mag=0/0/0.3133196/1/-90/
  params=90/0/90/0/-43.762/0/0/0/2.5/25 float=y
  maxiter=1500 inside=0 periodicity=10
  colors=000U71U81U94UA7UBAVCDWDHXEKYFNUGQWHTYIX_J_a\
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  gJu`HuWmuVmuOhuHduA`u4Xu5Yu6Yu7Yu8Zu9ZuAZuAZuDXuFW\
  uHUuJTuLSuSUuYWucXuiZu_7u }

frm:DivideJulibrot   {; draws 4-D slices of DivideBrot Julibrots
  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),
  aa=real(p5)-2, bb=imag(p5)+0.00000000000000000001,
  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),
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=sqr(z)/(z^(-aa)+bb)+c
  |z|< 1000000 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 6th, 2005

Jim Muth's commentary for the image: > FOTD -- August 06, 2005 (Rating 6) > > Fractal visionaries and enthusiasts: > > The brevity of today's discussion is a result of the > continuing rush here at F.C. > > Today's fractal subtracts Z^1.01 from Z^2. The result shifts > the Mandelbrot set to the east and splits the negative real > axis. Today's image is located in the East Valley area of the > large midget on the negative axis. > > I have named the image "Streamliner". This type of image always > reminds me of a railroad. The glory days of the railroad are > long past, but the word 'streamliner' still evokes feelings of > the rails. > > We have seen this type of image many times before, which limits > the rating to a 6. The render time of a little over 3 minutes > is quite reasonable. > > The Friday high temperature of 95F 35C limited the cats' outdoor > time to under one hour. Around 7pm it began thundering, but no > storm arrived and the cats ended their day peacefully. Today is > starting cloudy and a bit cooler. If all else goes well, it > will be a better day for the cats. For me the work is moderate, > threatening to get heavy. We'll see how it goes. The next FOTD > will appear in 24 hours. Until then, take care, and things are > never so good that they cannot get better.

PAR file

Streamliner { ; time=0:03:22.02--SF5 on a P200
  reset=2004 type=formula formulafile=allinone.frm
  formulaname=MandelbrotMix4 function=ident passes=1
  center-mag=-0.99882094606212310/+0.001072594239131\
  53/892730.3 params=-1/1.01/1/2/0/0 float=y
  maxiter=5000 inside=0 logmap=62 periodicity=10
  colors=00006n9FsIOwJWwL`xGdyBhy6lz2oz1jr0dh0_`8WIF\
  SMNOQUKU`GYhCao8ev5it9msCprFtqbwzjtqwqwopweozWouTn\
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  EzMHzSKzYNzcQziTznWzgZz`azU_zZZzbYzgXzkWzpVztZzqbz\
  oezlizjmzgpzeZzcIza1z`0zg }

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 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 4th, 2009

Jim Muth's commentary for the image: > FOTD -- August 04, 2009 (Rating 8) > > Fractal visionaries and enthusiasts: > > Today's skeletal scene returns us to yesterday's image, but have > no fear. It is far from a boring repeat. The image is actually > a modest blow-up of a scene in the 'tree' at the eastern edge of > yesterday's image. > > I named the image "What Numbers Do". While studying it, I began > wondering how mere abstract numbers could possibly create such a > scene. But that's what makes a fractal a fractal. > > The image rates an eight. I gave it this rating because I think > it deserves such a rating. > > A near perfect day passed un-noticed here at Fractal Central on > Monday. The temperature of 82F 28C was made even more enjoyable > by the lower humidity, while the electronically-intensified blue > skies and puffy cumulus clouds lent a perfect backdrop to any > outdoor activity. But my day was spent indoors, and the fractal > cats seemed more interested in the worn-out catnip rat than the > sunny window. > > The next FOTD will be posted in 24 hours. Until then, take > care, and is it an oxymoron to have faith in science?

PAR file

What_Numbers_Do { ; time=0:04:16.01-SF5 on P4-2000
  reset=2004 type=formula formulafile=basic.frm
  formulaname=DivideJulibrot passes=1
  center-mag=+5.51760082/0/345/1/-90/0
  params=89.98/0/89.98/0/-0.77/0/0/0/1.05/1.5 float=y
  maxiter=750 inside=0 logmap=370 periodicity=10
  colors=000zzzmvzUpznjzUqzihzUdzrazUZzm_zI6IJ8JKAKL\
  CLMEMNGNOIOPKPQMQRORSQSTSTzUUzZZzcczhhzmmzzzzzzzzz\
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  PezKyzNtzQozTkzWfzZbzaYzYYz`WzbVzdUztRzoSzjTzfTzaU\
  zXVzTVzOWz70zCBzGMzKWzOfz }

frm:DivideJulibrot   {; draws 4-D slices of DivideBrot Julibrots
  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), aa=-(real(p5)-2),
  bb=(imag(p5)+0.00000000000000000000001),
  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),
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=sqr(z)/(z^(aa)+bb)+c
  |z|< 1000000 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for August 3rd, 2009

Jim Muth's commentary for the image: > FOTD -- August 03, 2009 (Rating 6) > > Fractal visionaries and enthusiasts: > > Don't be shocked if today's image seems strangely familiar.
> It's merely a near-repeat of yesterday's image, with only a > little tweaking of the same color palette and an 0.02-degree > double rotation from the Julia orientation. > > The name "The Worm Turns" refers to the tiny departure from > the true Julia direction. The rating of a 6 is generous for a > near-repeat. > > The result is a bit more interesting than yesterday's image, > though still far below top-FOTD quality. > > But why all this obsession with quality, whatever fractal > quality might be? Fractals are for enjoyment. And today's > image can be enjoyed to the fullest only 56 seconds after the > included parameter file starts running. > > Sunday started with light rain here at Fractal Central, but by > mid-afternoon the sun appeared and the temperature reached 82F > 28C, making the rest of the day quite pleasant. The fractal > cats forgot their fractal roots when they got into a spat over > the one remaining catnip rat. > > The next FOTD, not another repeat, will be posted in 24 hours.
> Until then, take care, and the asteroid that will eventually > strike earth is already on the way.

PAR file

The_Worm_Turns { ; time=0:00:56.37-SF5 on P4-2000
  reset=2004 type=formula formulafile=basic.frm
  formulaname=DivideJulibrot center-mag=4.94067/0/1.8
  params=89.98/0/89.98/0/-0.77/0/0/0/1.05/1.5 float=y
  maxiter=600 inside=0 logmap=25 periodicity=10
  colors=000I6IJ8JKAKLCLMEMNGNOIOPKPQMQRORSQSTSTUUUZ\
  ZZccchhhmmmzzzzzzzzzmmmmmmcccUUUPRoMDUBIBJHARH9ZG8\
  c32eA5fG7hNAmR0kS6iTCkQ9mNCmQFmTImWLmZOmaRrdUre_rg\
  bqddpb`q`YrYUtWQwUNzRLzOJzLHzIFzFDzDBzHPzLazOnzHZz\
  AKzF5zF8zKAzQDzWFzaHzgKzmMzmOzmZzmZzmdzmazmZzmXzhL\
  zcKzZKzeKzbJz`IzZHzaOzcVze`zhgzjnzltzfqzanzWkzRhzL\
  ezGbzB`zNkzYvzWizVYzMOzDFvPIr_KmVRZQYcLdZOgUQjPSlK\
  UoMWlNRfON`PIVQEQK9UO5YS1bb0fl0zeDz_PzT`zNlzrzzjwz\
  eszUqzPmzQkzN`zKRzHGuF6pJ9kNCkQEpUHuXJz2oz9izFczFp\
  zFjzFdzFZzFTzFOzJSzMVzPYzT`zWczZfzWezUdzSczQbzObzL\
  dzJez5szBlzHfzN_zTUzZNzcHz`JzYLzVNzSPzhazaYzWVzQRz\
  JOzDKz1Bz7HzDMzJRzPWzVaz`fzfkzf8zkpzgozcnz_mzWlzSk\
  zOjzHlzKjzMhzPfzRdzTbzW`zYZzScz_XzXazVezTjzRnzOMzO\
  PzORzOUzJ1zLCzMMzOWzPezKyzNtzQozTkzWfzZbzaYzYYz`Wz\
  bVzdUztRzoSzjTzfTzaUzXVzTVzOWz70zCBzGMzKWzOfzsBzUv\
  zSpzQjzhqzWhzPdzIazBZz0_z }

frm:DivideJulibrot   {; draws 4-D slices of DivideBrot Julibrots
  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),
  aa=-(real(p5)-2), bb=(imag(p5)+0.00000000000000000000001),
  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),
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=sqr(z)/(z^(aa)+bb)+c
  |z|< 1000000 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for July 31st, 1997

PAR file

Nested_Cogwheels   { ; time=0:00:08.07-SF5 on P4-2000
  reset=1960 type=formula formulafile=jul-97.par
  formulaname=Mytest10 passes=1 center-mag=-0.472468\
  /0/0.9363296 params=2/0/3e-008/0/0/7 float=y
  maxiter=300 inside=253 logmap=yes symmetry=xaxis
  periodicity=10
  colors=000000J9GKGPLLTNSaOWeP_iRbmSfqTjuOhbIgKDe1L\
  i9SmH_qQfuYnyehw_cuVYsPTpKNnEIl9Cj3Dh3Eg3Fe3Gd3Hb3\
  Ia3J_2KZ2LX2MW2NU2OT2PR2hBEfDHeFKcHNaJQ`LUZNXXP_VR\
  bUTeSVhRWhPWhOXhMYiLYiJZiIZiG_iF`iD`iCaiAbj9bj7cj6\
  cj4dj6cg8cdBbaDb_FaXH`UK`RM_OOZLQZJTYGVYDXXAZX8bSF\
  eNLiISlDYp8ds3jp3kl3ki3lf3lc3m_3mX3nU3nR3oN3oK3pM7\
  qPArRErUHsWLtZOu`SucVveZwgawjcsmdpoemrfjugfwhczi`y\
  h`xh`wg`vfaufateaseaqdapcaocanbbmablabk`bj`bfceceg\
  `giYikUknRmpOorLqtpcVnbVlaWi`Wg_WeZWcYX`XXZWXXVXTd\
  fPmoOjkOghNddMa`LZXLXUKUQJRMIOIILFHIBQcTZxjXxhVwfT\
  wdRvbPv`NuZLuXJuVGtTEtRCsPAsN8rL6rJ4qH2qF5mH8iJAdK\
  D`MGXOJTQLOROKTRGVQJXPNZOQ`NUbNXdM_fLchKfjJjlImnUU\
  mdAmUEaIIQloXagSS`NHTIDD3AI90b23X65RAcr43EGsrHVaFk\
  sg8Px8Ol7N`7MPqxcUdRhm7xngjf_CABB8BB6BL6DV6Fd7Gn7I\
  x7Kv9JsAIqCHnDGlFFjGDgICeJBbLA`M9jDXt4tAQ09P48O77N\
  BVNqNMcFMR_`mD9JzzGE5DG9I
}

frm:Mytest10  {; Jim Muth
  z=c=pixel:
  z=z^p1+p2*((z^p3)^(-p3))+c,
  |z|<=100
}

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for July 30th, 2002

Jim Muth's commentary for the image: > Fractal visionaries and enthusiasts: > > Today's fractal reminds me of another real-world object. In > this case, I see the graceful swaying of seaweed, a few meters > under the surface near some tropical shore. The scene is also > filled with schools of small, colorful tropical fish, though > these are more difficult to pick out. > > With all this underwater activity in mind, I named the image > "Aquatic Life". Then I stepped back to evaluate the result.
> After much mental wrestling, I rated the image a much-above- > average 8. > > The formula of the aquatic scene is Z^(-1.1)+Z^(-11)+(1/C), > a simple expression that does more than it should. But as I > stated recently, it is not the formula that works the magic; it > is the 280 iterations. > > The parent fractal of today's scene appears as a large open bay > with many buds and valleys around its perimeter. Today's scene > is located in the center of a blunt valley on the south shore of > the main bay. The aquatic effect was achieved by rendering the > scene with the outside set to the <tdis> option, followed by > careful coloring. > > The render time of just over 1 minute makes running the attached > parameter file the quickest way of seeing the seaweed. > > The weather Monday here at the (IMO) center of the fractal world > was hot, hot, hot. A breeze and slightly lower humidity made it > feel less oppressive than it otherwise might have been, but 100F > 38C is still hot -- there's no escaping it. The heat was so > severe that it buckled the rails of a local rail line, causing > a train to derail. > > The dynamic fractal cats avoided the excessive heat by passing > the day in the air-cooled indoors, though they did check several > times just to be sure it really was that hot outside. > > Today is Tuesday, and it's another scorcher -- a perfect excuse > for me to stay indoors and get a little ahead on the upcoming > convention program that needs to be finished within a week.
> Until next time, which will come in 3 days on August 2, take > care, and why are tropical shores with all their humidity and > heat considered paradise on earth?.

PAR file

Aquatic_Life { ; time=0:01:27.77--SF5 on a P200
  reset=2002 type=formula formulafile=allinone.frm
  formulaname=MandelbrotMix4 function=recip passes=1
  center-mag=+0.856206964880088/-1.054799381804275/2\
  124528/1/-160/-3.37734701108538538e-008
  params=1/-1.1/1/-11/0/525 float=y maxiter=280
  inside=0 outside=tdis periodicity=10
  colors=000z01z03y04u04o06j06f07`07X09S09M0AI0AD0C7\
  0C30D00D13I47M7CQAGTDLYGPaJTfLYiOamQfrTjvXoy_szaxz\
  czzUzzKzz7zz6vz4lz3`z1Qv0Gs6FuCDvIDxOCyTAzYAz`CzcF\
  zfGziIzlLzoMzrOzuQzxSzyTzxVzxXvxYrvYmv_iv`dv``uaXu\
  cSudOsdJsfFsgAsg6mc9g`CcYFYVGTSJOOMJLPDIQ9FT3CX09Y\
  6C_DD_LF_SG_TI_TPcPXgLaiGikCmaJqSPu0Mz6LuAJoGGjLFd\
  QD_VAT`9Pd7Jj4Do37u13y00z00z00z10z60zC0xG0sM1oQ4iX\
  7d`A_fDVjGQpJLuMGzPAzUUzc1zmYzuVzzSzzQzzOzyMzxJzvI\
  zuFzsDzrAzp9zo6zm4yl1yj0xl3vl6vl9umCumFsmGroJroMpo\
  PppSopVopXmrUlsUluUjuUjvUitUimUmcUqUPvCdz0sv4osCjo\
  JflQaiYYddValQ_sMVzISzDPzATzMXsX`jfc`pgQzjIzgMzdPz\
  cSx`Vs__oXajVdfSgaQjYYXcdIgl4lgAicGg_MfVSdQYaMc`Ii\
  _DoY9uV4zT0zS0zQ0zV0z_0za0yf0vj0sm0pr0mv0jy0gz0dz0\
  az0_z0Yz0Vz0Tz0Qz0Pz0Mz0Ly0JxIiTJfOJdILcDLa7M`1M_0\
  OY0OX0ld3sXJzQYy6ls0yo0xl0xi0xf0va0v_3vX6uT9uPCuMF\
  sJIsGLsCi09i07i0`_7zPTz9F
  }

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
}

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for July 29th, 2005

Jim Muth's commentary for the image: > FOTD -- July 29, 2005 (Rating 7) > > Fractal visionaries and enthusiasts: > > The work is still backed up here at good old Fractal Central, so > the discussion will need to be short and to the point. > > Today's image is a scene in the east branch of the second valley > of the northern period-3 bud-complex (radical) of the Mandelbrot > set. > > Most of the effect of swirling comets is due to the unusual > coloring, which I worked out before things fell behind. Many > registers of the color palette are dead black, as can be seen by > striking the 'e' key with the image on the screen. I could > probably have gotten the same effect more easily by setting the > 'logmap' to a much higher number, but the image turned out > pretty well the way it is, so I let it be. > > I decided on the name "Magic Mandelbrot" when nothing better > came to mind. The rating of a 7 is satisfactory if not > exceptional. The render time of a fast 5 minutes makes running > the included parameter file a breeze. > > Wednesday was extremely hot and humid here at Fractal Central, > with the reported temperature in the city reaching 104F 40C.
> The heat confined the fractal cats to the coolest indoor spot > they could find. At 6pm, when a heavy rain squall blew up, the > electricity went out and the cool air went off. The cats went > out onto the porch when the rain stopped, where it was now > cooler than inside. Thursday was notably cooler and drier. The > duo enjoyed several hours in the yard. Today is starting cool, > but it is cloudy and threatening rain. I hope the cats are > ready for an inside day. For me, the work is still very heavy.
> If nothing goes wrong, the next FOTD will appear in 24 hours.
> Until then, take care, and be at peace with whatever you are at > peace with.

PAR file

Magic_Mandelbrot { ; time=0:05:20.60--SF5 on a P200
  reset=2004 type=mandel passes=1
  center-mag=-0.11315234783132390/+0.837744818390413\
  40/1.362938e+010/1/155/-6.62844733679418541e-006
  params=0/0 float=y maxiter=75000 inside=0
  logmap=3366 periodicity=10
  colors=0000000000000000000000000000000000000000000\
  00000000000000000000000000000000000000000000000000\
  00000000000000000000000000000000000000000000000000\
  00000000000000000000000000000000000000000000000000\
  00000000000000000000000000000000000000000000000000\
  00000000000000000000000000000000000000000000000000\
  00000000000000000000000000000000000000000000000000\
  00000000000000000000000000000000000000000000000000\
  00000000000000000000000000000000000000000000000000\
  000000000000000000000000000200400600800A00C00E00G0\
  0I00K00M00002Q04S06U0800AY0C_0Ea0G00IZ0KU0MP0OK0PF\
  2RC4TD6UE8WFAYGCZHE`IGbJIdKKeIMgHOiGQjESlDUnBWoAYq\
  8_s7av4cw6ex8gyAizBkzDmzFozGqzHtzHwzKzzMzzOzzQzzSz\
  zUzzWzzYzz_zzazzczzezzgzzizzkzzmzzozzqzzszzuzzvzzw\
  zzxzzyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
  zzzzzzzzzzzzzzzzzzzzzzzzz }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for July 28th, 2002

Jim Muth's commentary for the image: > Fractal visionaries and enthusiasts: > > I named today's 7-rated image "The Helm" because of the > resemblance it bears to a ship's wheel. > > Almost all my FOTD fractals are named after a resemblance they > bear to a real-world object. This tendency to find the familiar > in the unfamiliar is a universal human trait, perhaps best > illustrated by the "faces in the clouds" illusion. > > No ship's wheel exists in today's image, which is merely a > mapping to the screen of the number of iterations required > before the iterated points of the image pass beyond the bailout > radius. The coloring does not exist in the formula, but was > consciously added by myself to enhance the resemblance to a > wheel. > > Today's image is in fact nothing but one of the many ways in > which the basic underlying mathematical abstraction could have > been made manifest. The same formula could just as easily have > created a piece of fractal 'music'. (Though I have yet to hear > a piece of fractal 'music' that was worth being listened to more > than once. [Or even once, for that matter.]) > > Today's generating formula -- 1.5(Z^1.1)+1.5(Z^11)+(1/C) -- is > unusually simple. This is not surprising. When working with > fractals, long convoluted formulae are unnecessary, and in fact > often not worth the effort of writing them. Complex formulae > are not needed, because the iterations work the magic. Great > images come from skillful searching, not necessarily from 200- > line-long formulae. > > The above statements are my personal opinion of course, > especially the statement about fractal 'music' being worthless.
> Since beauty exists in the mind of the beholder, it may be that > my mind is not tuned to the mental frequency of the appreciation > of fractal 'music' and all the beauty in the music passes right > through me like light through the invisible man. > > Forgetting the worth or non-worth of fractal music, the render > time of today's image is a fast 3 minutes on my brave little > P200 fractal machine. > > The weather Saturday here at Fractal Central was not perfect, > but it could have been a lot worse. The temperature was held to > a muggy 83F 28C by heavy clouds, which threatened all day but > produced only a few drops of rain. The cats, who dislike high > humidity, briefly checked the outdoor conditions, then decided > to spend the day indoors. > > And I'm about ready to spend today relaxing. Until next time, > take care, and where is that "Theory of Everything for Dummies" > book?

PAR file

The_Helm { ; time=0:03:11.41--SF5 on a P200
  reset=2002 type=formula formulafile=allinone.frm
  formulaname=MandelbrotMix4 function=recip passes=1
  center-mag=-6.14601973153377700/-1.981123197536113\
  00/1.483931e+010/1/-28.746056810576011/-0.02765394\
  21627924401 params=1.5/1.1/1.5/11/0/0 float=y
  maxiter=900 inside=0 logmap=63 periodicity=10
  colors=0006AI7AJ8AK9APAJUBMXCNcvl4kc8UNCzu8faBTODm\
  A3`B7KBBk_ocTbWNVOHNb`dVSWNJNzfvbR`labcVXWOROHLu9x\
  gAhUBUOZ9KNCDwBEcDFPErCkeB`TBQg_t`TiUN_NHP80PB4LE8\
  I2Y_z8cj9X`ARQBLfl1FwzGfjGRVvrt1GQ6EMBCI6Y8AQBDIDq\
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  KJavGUfFNRFNboA5NC7KE9HdwuYjjSZ_MNP6JxBFa9nFDVFONZ\
  Vy4Uba_h4TX8MMCP4MM6KK8IIAG5wK8jIBZHENGjwlvfomiddl\
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  qQ4szAub1wKzyjnzbbzVRzNqz5gz8ZzAPzDzzPbzKQz83zK7zI\
  AzHDzGYzBSzDMzEDzNszH_zG2zm7zaCzQxzflz_azTRzMOzELz\
  FIzFQz2Nz6Lz9IzC6ztBz_OzW }

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 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for July 27th, 2007

Jim Muth's commentary for the image: > FOTD -- July 27, 2007 (Rating 7) > > Fractal visionaries and enthusiasts: > > Today's image takes us to our old friend the Mandelbrot set, > where we examine a remote valley in the basin of the large bud > on the northwest shore of the main bay. The name I gave the > image, "In A High Order Valley", is a description of the > location of the scene. > > I rated the image at a 7, mostly to reward myself for the extra > work I put into the coloring. I still don't quite have the > colors right, but the color palette is good enough to make > today's FOTD a 7-rater. > > The calculation time of 3-1/3 minutes on the fast P2000 machine > would be extended to about one hour on the slow P200. For some > reason, the fast machine is almost 20 times as fast as the slow > machine with type=mandel fractals, while it is only 5 times as > fast with type=formula fractals. > > A warm but rather muggy day with a temperature of 82F 28C here > at Fractal Central on Thursday was interrupted by a light > thunder shower in the afternoon. When the rain ended, the day > continued cooler and even more pleasantly. The fractal cats > enjoyed the reappearing sun. > > My day was busy enough to be satisfying. The next FOTD fractal > will appear in 24 hours. Until then, take care, and search for > other worlds.

PAR file

InAHighOrderValley { ; time=0:03:22.57-SF5 on P4-2000
  reset=2004 type=mandel passes=1
  center-mag=-0.50566260915093210/+0.604501129421405\
  50/5.622984e+009/1/-140/-4.46438176447683244e-005
  params=0/0 float=y maxiter=1200000 inside=0
  logmap=42541 periodicity=10
  colors=000YT_YT_YT_YU_YU_YV_ZV_VWZSWYPXXMXWJXVGYVD\
  UVARW7NY4J_1Ga0Cc09d0Hc0Ob0Vb1ba3ia4p`5w`6w_7w_7w_\
  8w_9w_9w_Aw_Aw_BwZCwZCwZDwZEwZEwZFwZFwZIwXLuVOsTQq\
  RToPWnNZlL`jJchHffFidDkcBYgJKkR6oZ9sUBvQEsSHqUKoWM\
  mYPk_SiaVfcXde_bgb`ieZkgXmf_nfanednefneiodkodnocpo\
  cspcupbwpbwpbwpZtkVmfRhaNhXJhS0h00h00h00h0VhHafGgd\
  GnbFt`FrZIpXLnVOlTRjRUhQXfO_dNbbMeaOe`Pe_QeZReYSeX\
  TeWUeVWeUXeTYeSZeR_eQ`ePaeHab9a_2aY4ZZ5X_6V`7Ta8Qb\
  9OcAMdBKeCIfFHeIGeLGdNFdQFcTEcWDbYDb`CacCafB`hB`eI\
  bbOc_VeX`fUfhRmiOskLzlMz`MzQMzFKzJJzNIzQHzUGrYFm`E\
  gdDahCWkFRfHMbKHZMIVPLRRONVQJYTFaWBdd7mi3tq0th0qlS\
  lgQgcOb`NYZLTSKONIJIHEDF9DD4DC0DA0I90N70N69NDcQEiP\
  EoPEu7Ez8Ez9EzBEzDEzFEzHEzJEzLEzNEzPEzREzTEzVEzXEz\
  ZEz`EzbEzdEzeEzfEzgEzhEziEzjEzlGzmIznKzoMzpKzqUzrY\
  zsYzuYzvYzwYzyYzyYzyYzyYzyYzzYzzYzzYzzYzzYzzYzzYzz\
  YzzYzzYzzYzzYzzYzzYzzYzzY }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for July 26th, 2003

Jim Muth's commentary for the image: > FOTD -- July 26, 2003 (Rating 5) > > Fractal visionaries and enthusiasts: > > In today's image we take another look at the rectangle that > caught fire in yesterday's image. That rectangle, which is > located in the Z^2.003+C Julia set with a C-value of -1.7435, at > the Z-coordinates of 0.00019,0.07388 and at a magnitude of 100, > is one of the most curious things I have yet discovered in the > world of fractals. > > When viewed as a Julia set, it is as perfect a rectangle as one > will find in any random fractal. But the slightest deviation > from the Julia orientation has a devastating effect on it. A > rotational deviation of as much as 1 degree totally destroys the > perfection of the rectangular shape and produces a curvilinear > quadrilateral such as we have in today's image. Changing the > imaginary part of the C-value even the slightest distance from > zero has an even more devastating effect. The entire rectangle > rapidly falls apart into fractal debris and vanishes into the > fractal ether. > > Since today's image is another sequel, I have named it "Blazing > Rectangle-2". As an FOTD, it rates an average 5, but hey, the > average FOTD is nothing to be sneezed at. And the render time > of 5 minutes is fast enough to make the attached parameter file > a near-joy to run. > > Friday, with its brilliant sun and temperature of 86F 30C, > turned out to be one of those rare perfect summer days here at > Fractal Central. The fractal cats were in one or their rare > near-perfect moods as they passed the afternoon in the yard. My > mood was a bit less than perfect as the work piled up faster > than I could accomplish it. > > Today's weather is promising to be less than perfect, but still > quite pleasant. If I can finish the modicum of work by noon, > I'll have a pleasant enough day. And I'm certain to find an > interesting fractal before the day ends. To see that interes- > ting fractal, check back here in 24 hours. Until then, take > care, and is truth a fractal?

PAR file

BlazingRectangle-2 { ; time=0:04:59.97--SF5 on a P200
  reset=2002 type=formula formulafile=julibrot.frm
  formulaname=SliceJB-new passes=1 center-mag=+0.004\
  99268009490636/-0.00325779513771971/76.89922
  params=0.5076/0.5037/0.49256/0.49876/2.003/0/-1.74\
  35/0/0.00019/0.07388 float=y maxiter=1000
  inside=bof61 logmap=51 periodicity=10
  colors=000GADIAFJAHKAJMALNANPAPQARSBTTBVUBXWBZXC`Z\
  Cb_Cd`CfbChcDjeDlfDngDpiErjEtlEvmExnEzoGzpIzpKzqMz\
  qOzrQzrSzsUzsWztYzt_wubtueqvhnvklwnjwqhytgywhzwhzw\
  hzmczjfzgizdczaYzZTzWNzTHzQCyN6xK1wJ5uI9sHDqGHoFLm\
  FOkESiDWgC_eBccBfaAj_9nY8rW7vU7yS9wQAuOBsMCqKDoKEm\
  MFkOGjQHhSIfTJdVKbXL`ZMZ`NYbOWcPUeQSgRQiSOkTMlULhS\
  MdQM`OMXMMTKMPIMLGMHEMDCM9AM59MF8MO7MX6Me6Mc7La8L`\
  9KZAKXBKWCJUDJTEJRFIPGIOIHMKHKMHJOGHQGGSGEUFCWFBYE\
  9_E7aE6cD4eD3gD6iE8kFBmGDnHGnIImJLlKNkLPjMSiNUhOXg\
  PZfQaeRcdSecTd`SdYRdWQdTPcQOcONcLMcILcGKbHNaIP`JSZ\
  KVYLXXM_WMaUNdTOgSPiRQlPRoOSqNTtLSwMTvMTvMTvMTvNTv\
  NTvNTvNTvNUvOUvOUvOUvOUvPUvPUvPUvPVuRVuQVuUVuQVuXV\
  uRVu_VuRWubWuRWueWuSWuhWuSWukYvSXunWuTVtqVtUUttTsU\
  SswSsVRrzQrWPrzPqXOqzOqYQpzSpZUozWo_Yoz_n_anzcn`em\
  zgmaimzklbmlzopaqnzslcukzwiezhzzfgzdzzcizazz`kzZzz\
  YlzWzzUnzTzzRpzQzzODzfzza }

frm:SliceJB-new {; by John R. H. Goering, July 1999
  pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1),
  b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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), c=p+flip(q)+(p4), z=r+flip(s)+(p5):
  z=z^(p3)+c
  |z|<=9 }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Jim Muth's Fractal of the Day for July 25th, 2003

Jim Muth's commentary for the image: > FOTD -- July 25, 2003 (Rating 6) > > Fractal visionaries and enthusiasts: > > After discovering that the fourth spatial dimension was in my > back pocket, I went searching. Maybe I searched in the wrong > pair of pants, but I suspect that the extra dimension slipped > off into hyperspace while I was not looking, and has escaped for > good. What a disappointment -- having a visualization of the > fourth dimension so close and then losing it. I guess I'll > never see a hypersphere in the condition of double rotation, or > a pseudosphere in its infinite entirety. > > Today's image appears to be rotating. It is a slightly enlarged > and distorted view of a small part of yesterday's image, on > which I have used a more flexible Fractint generating formula so > that I would have more angles to examine. > > The angle of today's slice is once again very close to, but not > exactly, the Julia direction. That slight rotation explains the > "Blazing Rectangle" name. The feature in the center of today's > image has a roughly rectangular shape, but it is not rectangular > enough to justify the word 'rectangle'. Where then is the > rectangle? > > The rectangle is actually very close, but it is hiding. To see > the near-perfect rectangle, change both parts of the p1 and the > p2 parameters to exactly 0.5 and re-render the image. It takes > only a minute or two. Now the rectangle is visible, revealed in > all its near-perfect glory at the center of the frame. > > Actually, I easily might have used the pure Julia version of > today's scene as the FOTD of the day, but the distorted version > has more action, and the pure Julia version doesn't appear to be > blazing. > > I rated the image at a 6, which equals a bit above average.
> After searching carefully, I could find little that has not been > seen before. (Originality in doing old things rates high with > me.) The render time of 2 minutes is unusually short, due > mostly to the low magnitude of only 44. > > A perfect day on Thursday, with a temperature of 81F 27C, > produced perfect cats. Tippy was a bit too perfect when he > viciously attacked my ankle like he used to do long ago when he > was still a kitten and wanted to be played with. To see if he > was feeling playful, I took the feather on a string from its > shelf, where it had rested untouched for several years, and to > my surprise, Tippy actually leaped around chasing it. Thomas > watched the activity with interest, but weighing over 17 pounds, > declined to participate. At the end of the day, both Tippy and > Thomas were exhausted, and needed no additional treat before > settling down for the night. > > Today looks like it will be a repeat. I'll make a full report > in tomorrow's FOTD, which will appear in 24 or so hours. Until > then, take care, and what do you call someone who wastes his > time talking about stuff he does not believe in?

PAR file

Blazing_Rectangle { ; time=0:02:19.51--SF5 on a P200
  reset=2002 type=formula formulafile=julibrot.frm
  formulaname=SliceJB-new center-mag=-0.0021281/-0.1\
  47743/44.02148/1/140/5.12714870559705105e-014
  params=0.5013/0.5007/0.5016/0.5012/2.003/0/-1.7435\
  /0/0.00019/0.07388 float=y maxiter=2000 passes=1
  inside=bof61 logmap=34 periodicity=10
  colors=000zFzzFzzFzyFzxFzvFzuFzsFzrFzpFzoFzmFzlFzj\
  FziFzgFzdFzaFz_FyXFxTFvQFuOFsLFrIFpFFoCFm9Fl6Fi3Ff\
  0Fc0F`0FY0FV0FS0DP0CM4AJC9IJDCQJMXOGcSCjY6ra1yf0zj\
  0zl0zm0zm0zo0zp0yp0xr0xs0vs0vu0uv0uv0sx0ry0ry0pz0p\
  z1oz1mz3mz3lz3lz4jz4jz4gz9dxDasG_oLXjOTfSQaVOY_MTa\
  JOfGJjDFmAAr76u41y10z00z00z10z40z70zA0xD0uG0rJ0oM0\
  lP0iS1fV3cY3``4Yc6Vf7Si9PlAMoALpGPrMSsSVuY_vcaxidy\
  ogzulzzozzrzzuuzrjzoazlSziIzf9xc0v`0u_0xa9ycJzdSzf\
  `zgjziszjzzlyzjvziuygrvgosfmpdjmdgjcfgacd````_Y_XV\
  YTSYSPXPMVMJTLGTIDSFAQD7QTJLgVGxfCzp7zsCzuFyvJuxMp\
  yQmzTizXdz``zcXzgSzjPzmOzoMzpLzrJxsIuuGsvFpxDmyCjz\
  Agz9dz7cz6`z4Yz3Vz1Sz0Qz3Oz4Mz6Jz9IvAGsCDpDClG9iI7\
  fJ6aL3_O1XP0SQ0PS0MQ0OP0OM0PGSOCPJ9MG6JD1G90D60A30\
  7004001003334CA6MJ7VS9d_AmgCxoCzmAzl9uj7oi6ig4cf4_\
  d3Tc1Oa0I`0C`07_1F_3M_4S_7_xoMzmPzjQziTzgXzf_zd`za\
  _z_YxXXuTXpQVmPTiMTfJScGQ }

frm:SliceJB-new {; by John R. H. Goering, July 1999
  pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1),
  b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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), c=p+flip(q)+(p4), z=r+flip(s)+(p5):
  z=z^(p3)+c
  |z|<=9  }

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.