Jim Muth's Fractal of the Day

TheParentUnleashed

Tue, 21 May 2013 07:35:55 -0000

2013.05 The Parent Unleashed [M-4,A-6]

FOTD -- May 17, 2013 (Rating M-4,A-6)

Fractal visionaries and enthusiasts:

Today's image is not a scene in a parent fractal; it is the parent itself. This parent came to be when I checked to see what would happen when the formula Z^(1.237)+C is iterated 14 levels up the hyperladder.

What happened was hardly overwhelming, but when rendered with the inside set to bof60 and the outside set to tdis, it was interesting enough to rate a 4 for math value and a 6 for artistic value.

The name "The Parent Unleashed" is intentional hyperbole, but it might get a few reluctant fractalists to check the implied wild-animal ferocity in the image.

The running time of the parameter file is 1-3/4 minutes, which is about average, but setting up and running a parameter file is far more complicated than checking the FOTD web sites, where the image is already rendered.

Check the finished image at:

and the high-definition renderings at:

Back images may be accessed at:

Today brought ample sun along with a fine display of cirrus clouds and a temperature of 68F 20C to Fractal Central. The fractal cat, now nearing senior cat status, slept through most of it, while the humans busied themselves with human things.

There will be no FOTD for May 18; the next FOTD will be posted most likely on the 19th. Until whenever, take care, and a lot of people are being injured by . . . so let's pass a law to ban . . . (fill in the blanks)

Jim Muth jimmuth@earthlink.net

Patterns in a Void

Tue, 21 May 2013 00:10:04 -0000

2013.05 Patterns in a Void [M-8,A-6]

FOTD -- May 19, 2013 (Rating M-8,A-6)

Fractal visionaries and enthusiasts:

Today's image highlights one of the mystery buds of the 4-D HyperMandelbrot set. But the mystery bud is not the reason I posted the image. The most striking part of the scene is the fine display of moire patterns filling the bud. And this pattern will change with every change, however small, of overall image size, resolution or magnitude.

0+0i is the critical starting point of the 2-D Mandelbrot set, so I assume 0+0i+0j+0k would be the critical starting point of the 4-D HyperMandelbrot set. At the same time, I have no idea what the perturbed Hyper-sets might look like. Today's scene might be critical, it might be perturbed. One thing for sure is that much more exploration needs to be done.

The name "Patterns in a Void" refers to the Moire patterns filling the open area of the bud.

The math interest rates a very respectable 8, the artistic value however is worth no more than a humdrum 6. I leave it to the individual fractalist to decide whether the image is worth the quarter hour required to calculate it. If the decision is negative, the web sites are there to bring relief.

The finished image is posted on the official FOTD web site at:

High-definition variations are online at:

Back images are available at:

Heavy clouds, (stratocumulus for the weather-wise), prevailed all day yesterday and today here at Fractal Central, though no rain fell. Because of the high humidity, the temperature of 66F 19C felt just about right. The fractal cat had an active day, often dashing up and down the hallway as though chasing or being chased by an invisible cat. The humans had a slow day, which is boring to read about but was relaxing to experience.

The next FOTD will be posted soon. Until soon arrives, take care, and in June 1976 I spent two days in Seattle. On both days the weather was sunny and quite warm, proving that despite its reputation of being a cloudy and rainy city, Seattle is a sunny and warm city.

Jim Muth jimmuth@earthlink.net

17-year Minibrot

Fri, 17 May 2013 00:21:53 -0000

2013.05 17-Year Minibrot [M-7,A-7]

FOTD -- May 16, 2013 (Rating Math-7,Art-7)

Fractal visionaries and enthusiasts:

Today's image lies on the main spike of an oversized Mandelbrot set that is just starting to morph into a Z^17+C Mandeloid. The first signs of transition are already visible from the surface, in the finer detail. Today's scene lies just inside the point where the main spike is cut off by the escape radius. A small outzoom will show how close it lies.

The name "17-Year Minibrot" refers to the generating formula, and also to the 17-year cicadas, (magiciada septemdecem), which are due to emerge in the Fractal Central vicinity in a few weeks.

I rendered the image with the outside set to 'tdis'. This gave the image enough detail to make it interesting.

Both the mathematical and artistic ratings are 7, which means that both are somewhat above average. The calculation time of 15 seconds will probably be cut by at least half on a SOTA (state of the art) machine, but the minor annoyance of setting up and running this letter as a parameter file will remain. This minor annoyance can be eliminated at the web sites.

Thanks to the wonders of modern technology, the image may be viewed without the need of calculating it by surfing over to the official FOTD web site at:

High-definition variations are posted at:

and scads of back images are online at:

Really pleasant weather was appreciated as best as possible by those here at Fractal Central today. The fractal cat questioned the partly cloudy skies, but approved of the temperature of 82F 28C. The fractal humans alternated between commercial work and weather enjoyment.

The next FOTD will be posted in the near future. Until the near future arrives, take care, and terrorists are said to be enemies of freedom, which I guess gives us the right to over-run their land and drop bombs on them. But I fail to see how anyone could be an enemy of freedom.

Would the terrorists rather be slaves than free people, or could they simply be fighting back as best they can against what they see as our aggressive intrusion into their sovereign territor- ies? After all, some people say we must keep them suppressed if we are to maintain access to the raw materials that lie in their territories but that we need to maintain our comfortable but extravagant life style.

Jim Muth jimmuth@earthlink.net

Crystal Cities

Fri, 17 May 2013 00:10:05 -0000

2013.05 Crystal Cities [M-4,A-6]

FOTD -- May 15, 2013 (Rating M-4,A-6)

Fractal visionaries and enthusiasts:

In today's image we take a brief break from hyperland and check the parent fractal that results when the formula Z^(1.37)+C is iterated 1.37 levels up the hyperladder with no function applied. This parent appears as a shapeless main bay with an equally large and shapeless period-2 bud attached to the northwest side. Today's image is located in a twisting filament just inland of the eastern shore line of the parent's main bay.

I named the image "Crystal Cities" for no apparent reason.

The math interest rates a rather boring 4, the artistic worth, a not much better 6.

The calculation time of a mere 45 seconds will remove some of the disappointment if the image lays an egg.

The finished image should be posted on the official web site at:

but the host site has been having problems and has been hard to access lately. It might be down. If it is down, check the hi-def version of the image at Hal's site at:

Hopefully, his site is up to date:

Regardless, the back images are online at:

A rather pleasant day passed virtually un-noticed today here at Fractal Central, with ample sun and a temperature of 75F 24C. The fractal cat enjoyed the few beams of sunlight that managed to sneak into F.C. (We get little inside sunlight during the high-sun-angle time of year.) The humans made the best of things.

The next FOTD will be posted in a matter of hours. How many hours is yet to be determined. Until whenever, take care, and get hot under the collar about climate-change-global-warming.

Jim Muth jimmuth@earthlink.net

Surprise Test

Wed, 15 May 2013 00:10:04 -0000

2013.05 Surprise Test [M-7,A-7]

FOTD -- May 13, 2013 (Rating M-7,A-7)

Fractal visionaries and enthusiasts:

The discussion is brief because the day was very busy.

Today's image shows a tiny minibrot in the East Valley area of a larger but still small minibrot on the main stem of the Mandelbrot set. The minibrot has been almost totally filled in by hyper-stuff, with the only remaining open area a black dot at the center of the main bay.

The math value is down 2 points from yesterday's, since the effect is not as surprising as the title would imply. The artistic worth is up a point however, due to the extra effort I hurriedly put into the coloring.

The name "Surprise Test" is an exaggeration. I was really not very surprised when I saw the image

The calculation time of a mere 30 seconds will make all fractal troubles vanish like a black cat on a dark night.

See the light! Enjoy the finished image posted at:

Thrill in the hi-def version at:

And if today's image leaves you hungry for more, check the thousands of FOTD back FOTD fractals and hair-raising discussions online at:

A definite chill was in the air here at Fractal Central today, with a mix of sun and clouds and a temperature of 52F 11C. The low tonight is forecast to be 27, which could be the lowest ever recorded in May. No wonder the local farmers are worried. The fractal cat is not worried however. An extra treat of tuna kept him happy most all day. The fractal people made it through a very busy day with a minimum of distractions.

The next FOTD will be posted, tomorrow if all goes well, the next day if it does not. Until whenever take care, and where is the outrage about the bombing in Afghanistan and the 19 people shot at the parade in New Orleans? I guess people get outraged only when the news media jumps on a story, thereby telling the public, (Are we a flock of such easily manipulated sheep?) that it's time to be angry again.

Jim Muth jimmuth@earthlink.net

Oh I See You

Tue, 14 May 2013 03:10:03 -0000

2013.05 Oh_I_See_You [M-9,A-6]

FOTD -- May 12, 2013 (Rating M-9,A-6)

Fractal visionaries and enthusiasts:

Today's rating is in two parts, a math part and an art part. The mathematical interest is quite high, the artistic interest, just so-so.

The scene lies well inside the main bay of the Mandelbrot set, where nothing but a blank screen should be expected, but today's image is anything but blank. The 'ghost bud' peeking out from behind the dark foreground stuff on the left is the first sign I have found of minibrots in the hyper part of the HyperMandelbrot set. True, the scene is not critical, but it is a sign that interesting discoveries might lie ahead. So far however I have been unable to bring the partially hidden bud out into the open.

The hole at the center of the parent fractal closely resembles the mandel-object found by Mike Frazier at the bottom of what I had previously assumed was an endlessly shrinking series of concentric fractal starbursts. Today's scene lies inside a second-order bud at the edge of the circluar hole at the center of the main bay of the M-set.

The superior math rating of a 9 is held down by the merely average artistic-worth rating of a 6. The name "Oh I See You" describes how I felt when I first caught sight of the unexpected bud.

The calculation time of 7 minutes is slow, let's face it. The way to avoid time frustration is to find relief on the web sites.

Enjoy the finished image posted at:

Thrill in the hi-def version at:

And if today's image leaves you hungry for more, check the thousands of FOTD back fractals online at:

A mix of sun and ragged clouds, a temperature of 59F 15C and a brisk northwest wind made today feel unseasonably chilly here at Fractal Central. The fractal cat managed to find a stray shaft of sunlight in the afternoon, and curled up in comfort, but a cloud covered the sun after only 10 minutes and he retired to claim my favorite chair. With no mothers to visit, the fractal humans had a pleasant day at a local barn that has been conver- ted into an antique shop.

The next FOTD will be posted in the proper time. Until that wonderful moment arrives, take care, and if the right is right, how can the left be right?

Jim Muth jimmuth@earthlink.net

Hyperscene

Sun, 12 May 2013 06:10:10 -0000

2013.05 Hyperscene [7.5]

FOTD -- May 11, 2013 (Rating 7.5)

Fractal visionaries and enthusiasts:

Today's image begins an investigation of the 'hyper' parts of the HyperMandelbrot set. The investigation will continue in the days to come, when hopefully, we will stumble upon some far better hypervistas.

Today's hyper vista lies in the East Valley area of the minibrot on the main stem of the parent M-set.

The name "Hyperscene" is a description. The rating of 7-1/2 is a come-down, but still a bit above average. The calculation time of 2-1/4 minutes is fair enough, but the wait can be cut somewhat by visiting the web sites.

The calculated image is at:

A hi-def rendering is at:

FOTD back images are at:

An all-day rain watered down the planned outside activities here at Fractal Central today. The temperature of 66F 19C was fine, that is for the ducks. The next FOTD will be posted when it is ready. Until then, take care, and don't look here for a clever closing remark.

Jim Muth jimmuth@earthlink.net

Variation

Sat, 11 May 2013 00:10:03 -0000

2013.05 Variation [8.5]

FOTD -- May 10, 2013 (Rating 8.5)

Fractal visionaries and enthusiasts:

Today's image lies in the same area of the same parent fractal that gave us yesterday's image, though it is located deeper in the parent's East Valley than yesterday's. It would have rated a 9 if it had not been a virtual repeat, but the actual rating of 8-1/2 is still high enough to do justice to its dark somber mood.

Once again, there is no minibrot lurking at the center of the image, which apparently continues down forever, with the number of arms increasing in the same irregular manner. In fact, I have yet to find a single minibrot in any of the strange higher- dimensional parts of the HyperMandelbrot set. This is really not surprising, since HyperMinibrots would be four-dimensional and unable to be squeezed onto a flat 2-D screen without huge distortion.

The name "Variation" points out that the image is a variation of yesterday's theme, rather than something new.

The good news is the fireball calculation time of 40 seconds, which will be cut substantially if the parameter file is run on one of the present-day multi-processor hot-rod units that the Fractint program cannot yet (if ever) take advantage of. The not-so-secret path to total freedom from parameter-running chores lies on the FOTD web sites, where fractal enlightenment abounds.

The official FOTD web site, where the basic finished image is posted, may be accessed at:

with studies of the image in high-definition at:

The FOTD back fractals from April 1997 are filed at:

The day began with dense fog here at Fractal Central. The fog lifted later in the morning, but the sun had a hard time break- ing through higher clouds. We did have enough sun however to raise the temperature to 73F 23C, which made the fractal cat happy even from his lookout post inside FC. The humans had one more in the apparently endless string of normal days. But this does not imply boredom. After all, who needs sensational happen- ings in their daily lives when there is quite enough sensation coming from the mass media.

The next FOTD will be posted in a nominal amount of time. No temporal anomaly is implied however. Until next time, take care, and mythological stories are symbols that, when properly understood, describe the spiritual world; just as mathematical expressions are symbols that, when properly understood, describe the physical world.

Jim Muth jimmuth@earthlink.net

Out in Left Field

Fri, 10 May 2013 00:10:04 -0000

2013.05 Out in Left Field [9]

FOTD -- May 09, 2013 (Rating 9)

Fractal visionaries and enthusiasts:

I have no idea why I named today's image "Out in Left Field", but that's how it goes sometimes. The lack of a logical reason for the name is no problem however. With a rating of a 9, the image needs no fancy, logical name.

The image is a scene in the East Valley area of a Hyper- Mandelbrot set that is filling in around the edges of the main bay and inside the buds. The brilliant white hole at the center is not an under-resolved minibrot. It is all that remains of the period-21 bud on the south side of East Valley of the M-set. The rest of the bud is filled in with patterns similar to those in today's image, which, like the stack of world turtles, goes down forever.

One of these days, someone like Mr. Mandelbrot (may he rest in fractal land) will determine exactly what is going on in images like today's. Unfortunately, I am not that person.

The rating of a 9 is no mistake. The unusually brilliant colors, for which I *am* responsible, earn the rating.

The calculation time of 50 seconds is a joy to experience. It's so fast that the calculation is dangerously close to finishing before it starts, thereby creating a time paradox. Checking the image on the web sites will eliminate this threat.

Be safe! Check the finished image at:

and the augmented version at:

And don't forget the back images at:

After yesterday's deluge, today, even with its heavy clouds and chilly temperature of 68F 19C, was an improvement. It was also the fractal cat's day for a check-up at the doctor. He endured the discomfort bravely, but showed a bit of resentment when he returned to FC.

The next FOTD will be posted when it is ready. Until this readiness arrives, take care, and the paranormal does indeed exist. It is nothing more than natural things we do not yet understand.

Jim Muth jimmuth@earthlink.net

Sinuous Discharge

Thu, 09 May 2013 03:41:01 -0000

2013.05 Sinuous Discharge [8]

FOTD -- May 08, 2013 (Rating 8)

Fractal visionaries and enthusiasts:

Today's image with its lightning-like filaments reflects the weather today here at Fractal Central. The formula causing all the bad weather on the screen takes 1.5 parts of Z^(-1.1) and combines it with 0.3 part of Z^(-7.4) before adding (1/C) on each iteration.

The resulting parent fractal is a hugely oversized thing resembling an upside-down figure eight, with today's scene lying on the filament extending east from the east branch of the valley separating the two large lobes of the parent.

The name "Sinuous Discharge" refers to the image's sinuous outer curves as well as the electrical effect of the brilliant inner curves. The rating of an 8 is getting to be a habit, a habit that I hope continues.

The calculation time of 1-1/3 minutes is appropriate for the image, neither too fast nor too slow. But a chore remains -- the chore of setting up this e-mail as a parameter file and running it. Thankfully, this is when the web sites are there for the rescue.

Check the finished image at:

and the hi-def variations at:

The back images are posted at:

The weather took a turn for the worse here at Fractal Central today, as a series of severe thunder-storms moved in during the morning and dropped 3.15 inches, 8cm, of rain by noon. During the storm an apartment building was struck by lightning and set afire. The temperature hung around 61F 16C most all day. Luckily, the fractal sump pump did its job and we had nothing worse than a couple puddles on the basement floor. The fractal cat, who doesn't like thunder, became uneasy until the noise stopped.

The next FOTD will be posted in a reasonable time. Until reason prevails, take care, and presidential assassins are buried in the U.S. with no fuss; the Oklahoma City bomber, who killed well over one-hundred, was buried in the U.S. with no fuss; the Newtown school shooter, who killed 20 children was buried in the U.S. without a fuss. Why then is there so much silly fuss finding a place to bury the Boston bomber, who killed only four people? If we claim it is because he used a weapon of 'mass destruction', then shouldn't the automatic gun used by the Newtown shooter and the bomb used by the Oklahoma City bomber also be considered weapons of mass destruction?

IMO, everyone, even those we consider the most evil, deserves a proper burial! God will deal with the evil-doer.

Jim Muth jimmuth@earthlink.net

Busy-Body

Thu, 09 May 2013 00:10:04 -0000

2013.05 Busy-Body [8]

FOTD -- May 07, 2013 (Rating 8)

Fractal visionaries and enthusiasts:

Today's image has some very unlikely parameters, which is a sign that I was led to the parent fractal by the Fractint evolver feature. But regardless of where the image came from, it still rates an 8, which has been the standard for the past several days.

The parent combines 2.475 negative parts of Z^(-1.05) with 0.275 negative parts of Z^(-5.5) then adds (1/C) on each iteration. This parent is a huge Mandeloid with a circular main bay edged by several large buds and a series of rings. Today's scene lies in a filament leading to a degenerate ring on the eastern edge of the large circular bay.

The rating of an 8 is perhaps a bit more than the image deserves, but then again, perhaps I'm a bit too humble. The name "Busy Body" refers to the clutter of detail filling the frame.

The calculation time of 12 minutes is slow. It's at times like this that the web sites come in handy however.

The finished image is posted at:

with hi-def variations at:

The back images are online at:

The weather was near perfect all day today here at Fractal Central. The fractal cat enjoyed it, as did the fractal humans. The next FOTD will be posted well before the world disintegrates into a de-sitter soup. Until whenever, take care, and maybe I'm a Boltzmann brain inside a larger Boltzmann brain, and it's Boltzmann brains all the way up. BTW, when will these fascinat- ing hypothetical creatures be mentioned on one of the pop-science TV shows?

Jim Muth jimmuth@earthlink.net

In Deep Territory

Wed, 08 May 2013 00:10:05 -0000

2013.05 In Deep Territory [8]

FOTD -- May 06, 2013 (Rating 8)

Fractal visionaries and enthusiasts:

Today's image lies very deep in the parent fractal that comes about when 0.1 part of Z^(4.4) is combined with 2 parts of Z^(0.4) and straight C is added on each iteration. This parent resembles a Mandelbrot set facing northeast with a large triangular area sprouting behind it from its East Valley area. Today's scene is located in the northwest valley forming the entrance to this triangular area.

The name "In Deep Territory" refers to the unusual magnitude of the image, one of the greatest of all time.

The rating of an 8 reflects my opinion of the image at the time I rated it. I may soon shift to an more informative rating system that gives two ratings, one for the artistic value, the other for the mathematical interest.

A good part of the artistic effect was achieved by rendering the image with the inside set to 'fmod'. The magnitude of over 2.5*(10^13) is at the verge of resolution breakdown. In fact, the breakdown is already becoming visible along some of the sharp edges.

Luckily, the image calculates in 3-1/2 minutes, surprisingly fast for a scene that uses the 'fmod' feature. And the web sites are always there for instant enjoyment.

Check the finished image at:

and the high-definition renderings at:

The back images are at:

Another absolutely perfect day prevailed here at Fractal Central today. The few angel-hair cirrus clouds that appeared in the morning, and the few cotton-puff cumulus clouds that popped up over the ridges in the afternoon did nothing but add to the perfection. The temperature of 73F 23C and gentle southeast wind were also perfect. The fractal cat admired the outdoor conditions . . . from the inside. The fractal humans also admired the outdoor conditions.

The next FOTD will be posted in a number of hours. Until that number passes, take care, and I am still waiting for someone to solve the very interesting and quite puzzling 'Boltzmann brain' paradox, which currently has physicists pondering and rational skeptics a bit concerned also. I'll forget the convoluted line of thought about entropy that led to the problem, since it is on the internet. At this time, I'll merely state that either the existence of the universe involves more than pure blind chance, or the universe is filled with disembodied 'Boltzmann brains' which somehow can exist and be conscious while floating in empty space.

The apparently 'logical' deduction is that, if the universe is merely a random accident, then it is many googols of times more likely that, instead of existing in a real physical universe, I am a pathetic deluded Boltzmann brain, floating in empty space and imagining everything I have ever experienced, including the physical world and everything else that appears to be so real. I doubt that this is true, but, despite convoluted claims to the contrary, no one has yet been able to show beyond doubt that it is untrue without invoking an intelligence of some kind to elimi- nate the random element. And we know what rational scientific skeptics think of the idea of 'God'.

Jim Muth jimmuth@earthlink.net

Lopsided Basin

Tue, 07 May 2013 00:10:03 -0000

2013.05 Lopsided Basin [8]

FOTD -- May 05, 2013 (Rating 8)

Fractal visionaries and enthusiasts:

Today's image lies in the parent fractal that results when the simple formula Z^(1.75)+C is iterated 9 levels up the multi- dimensional complex-logarithmic hyperspiral. This fractal consists of a large main bay split along the X-axis, with a prominent disconnected smaller Mandeloid lying some distance to the southeast. Today's scene lies in a valley in this smaller Mandeloid.

The name "Lopsided Basin" refers to the unbalanced nature of the basin surrounding the central minibrot, where the iteration count and therefore the appearance of the fractal elements on the left and right sides are vastly different.

The rating of an 8 includes a bonus for the coloring effort, though I'm not saying how much of the rating includes the bonus.

The calculation time of 1-2/3 minutes will pass fast enough to keep boredom in the realm of remote possibilities. Boredom may be done away with altogether by viewing the image on one of the FOTD web sites.

Accept the obvious. Fractals are great! For rational proof, check the official FOTD web site at:

and if further proof is needed, check the high-definition renderings at:

The ultimate in proof may be found at:

where thousands of similar fractals are posted.

Cloudless skies prevailed all day on the 4th and 5th here at Fractal Central, with a temperature of 72F 22C both days and gentle breezes. The fractal cat saw the sun outside and wondered why it failed to reach his shelf. The humans spent the 4th working and the 5th relaxing.

The next FOTD will be posted soon enough to prevent withdrawal and with enough delay to prevent fractal overload. Until whenever, take care, and to me it seems as though we're so obsessed with political correctness that we have no way to even recognize the causes of our problems much less to accept the causes and rationally discuss possible solutions.

Jim Muth jimmuth@earthlink.net

Well Shielded

Sat, 04 May 2013 03:10:02 -0000

2013.05 Well Shielded [8]

FOTD -- May 03, 2013 (Rating 8)

Fractal visionaries and enthusiasts:

Today's image resembles a shield decorated by fractal heraldry. I appropriately named it "Well Shielded". The fractal, which is rotated 90 degrees, lies on the main stem of its parent, an oversized Mandelbrot set that is on its way to becoming a cubic Mandeloid. The exact location of the scene is in a blunt valley jutting eastward into the second bud out the main spike, which is already splitting into the cubic-bud shape, with two lobes.

The rating of an 8 is based on the unusual cubic shield-like pattern and also on my semi-expert coloring, which some say is great and others say is amateurish. All I can say is that great art is not created in the matter of the minutes that I spent on today's image.

I went back to the 'DivideBrot5' formula for today's image because this formula does some types of mixed-power images better than the 'FinDivBrot-2' formula that I usually use.

The calculation time of 2 minutes might be just slow enough to give restlessness a chance to appear. The wiser way to see the finished image is to visit the FOTD web sites.

Find your happiness at:

where the image is posted already calculated.

And clarify your vision at:

where the high-definition versions are posted.

The thousands of back images are online at:

bringing joy beyond comprehension to those who venture there.

Perfect weather still prevailed here at Fractal Central today, with a cloudless sky, gentle breeze and temperature of 70F 21C. The fractal cat took it in stride, being more concerned with pigeons roosting in the area. The humans had an average day, which is pleasant to experience but boring to write about.

The next FOTD will be posted quite soon, and that is not boring. Until soon arrives, take care, and we should not be angry at big corporations such as oil and phone companies for making such obscene profits at our expense. We should be angry at ourselves for having allowed ourselves to become addicted to their flashy products, thereby assuring these powerful conglomerates that their obscene profits will continue to grow.

Jim Muth jimmuth@earthlink.net

Aliens Among Us

Fri, 03 May 2013 00:40:40 -0000

2013.05 Aliens Among Us [7.5]

FOTD -- May 02, 2013 (Rating 7.5)

Fractal visionaries and enthusiasts:

Today's image bears footprints, which is not surprising, since it lies in yesterday's parent fractal, very close to yesterday's scene. But these footprints were not made by bigfoots, they were made by aliens from outer space, which explains why they resemble the pea-pods of the body snatchers.

The actual location is East Valley of the large minibrot on the main stem of the parent, where a tiny minibrot is located. Today's scene lies at the tip of the main spike of this minibrot.

The name "Aliens Among Us" merely tells it like it is. For all anyone knows, the space aliens may very well be living among us, unrecognized and disguised as the man and woman next door.

The rating of 7-1/2 is quite average. The extra half point is a reward for the coloring, which was more difficult than first appears.

The calculation time of one minute will leave just enough time to go into the next room and pour a drink. Viewing the image on the web sites will avoid the drink.

Be cheerful anyway! View the finished image at:

Be joyful! Check the high-definition variations at:

Be in total bliss! Check the thoudands of FOTD images at:

Today brought another round of cloudless skies, gentle winds and dry air to Fractal Central. The temperature of 73F 23C made it the second consecutive perfect day. Nicholas, the cat of the fractal variety made it through the day uneventfully, sleeping all afternoon in my favorite chair. The fractal humans made it through the day by working.

The next in the inexhaustible series of FOTD fractals will be posted soon. Until whenever, take care, and everyone wants peace, but the evil other side will not let it happen, and this is true regardless of which side a particular person is on.

Jim Muth jimmuth@earthlink.net

Sasquatch Bigfoots

Thu, 02 May 2013 06:10:10 -0000

2013.05 Sasquatch Bigfoots [7.5]

FOTD -- May 01, 2013 (Rating 7.5)

Fractal visionaries and enthusiasts:

Today's image reminds me of bigfoot footprints. Bigfoots (not 'bigfeet') are also known as sasquatches, so I named today's image "Sasquatch Bigfoots". These enigmatic creatures cannot possibly exist. If they did, we would long since have found hard evidence of their existence such as their bodies, excrement and dwelling places. But despite the impossibility, many people still claim to have seen bigfoots, and unless one is a hard core devoted CSICOP-type skeptic, it is hard to believe that every one of these people are mistaken or lying.

Today's image gives proof that bigfoots are lurking around FC. I mixed 1/10 part of Z^2 with one part of Z^(-1000) and found the proof I was not really searching for. This unlikely combination of parameters produced an oversized parent fractal that appears to be a Mandelbrot set with flat disks filling the smaller buds and floating in the center of the larger buds as well as the main bay.

The situation looked rather hopeless at first, but knowing how tricky fractals can be, I checked Seahorse Valley of the large minibrot on the main stem of the parent, where two disks are tangent, and found an interesting area of chaos. This area is where the bigfoots thrive, and it took little additional time to find the evidence.

If I had found an actual bigfoot, I might have rated the image at a 10. As it is, with only the mysterious footprints to show, I could give a rating no higher than 7-1/2.

Perhaps the best feature of the image is its starship speed of 20 seconds. But be warned, when calculated at faster-than-light warp speed, the image finishes before it is started. So be sure to avoid the time paradox that would result if you failed to start the calculation after the image had appeared on the screen. Or better still, forget such time-travel paradoxes by checking the image on one of the web sites.

Be happy in your fractal work by viewing the finished image at:

and check the glories of high-definition at:

More FOTD back images than one can shake a stick at are posted at:

After viewing the back images, forget about shaking the stick. You'll be too exhausted to lift it.

The weather was absolutely perfect here at Fractal Central today. Perfection is described by a cloudless deep blue sky, pleasantly warm sun, gentle breezes, low humidity and a temperature of 75F 24C. The fractal cat took it in stride. FL spent the afternoon puttering in her garden, while I took a pleasant walk to the Susquehanna River, which is 1/2 mile from FC, 1/2 mile wide, and home to a variety of common eastern water fowl.

The next FOTD will appear, though not by magic, in the proper universal time. Until whenever, take care, and be aware that, according to relativity, there is no universal time.

Jim Muth jimmuth@earthlink.net

Deep Green

Fri, 03 May 2013 00:10:04 -0000

2013.05 Deep Green [8.5]

FOTD -- April 30, 2012 (Rating 8.5)

Fractal visionaries and enthusiasts:

Today's 8.5 rated image lies in the parent fractal that results when 0.2 part of Z^2 is combined with 15 parts of Z^(-1.5). This parent appears as an oversized but skeletal Mandelbrot set with a well-defined much smaller M-set in the area of its East Valley and two minibrots symmetrical to the X-axis in the area of its main spike. Today's scene lies in the intricate debris between these two minibrots.

I named the image "Deep Green" because of the vivid green ring surrounding the central minibrot. The rating of 8-1/2 includes a sizable and perhaps too liberal bonus for my coloring. The calculation time of 1-3/4 minutes borders on slowness, making the web sites a good calculation alternative.

Find this calculation relief on the official FOTD web site at:

and classy variations at:

Hordes of back images are posted at:

Lots of clouds hung overhead here at Fractal Central today, and a few flurries of drizzle drifted down now and then, but the sun also broke through several times, and the temperature of 66F 19C was so close to average that it left no grounds for complaint. The fractal cat took it all in stride, satisfying himself when he evicted me from my favorite chair and took it over with a satisfied smirk on his face. The humans meanwhile checked the latest news, seeing nothing new, of course.

The next FOTD will be posted when I get a round tuit. Until I can find a tuit shop, take care, and there is a question I want a clear answer to. It is, 'why are we here on earth?' (Answers such as 'to get to heaven' 'to propagate the species' 'it's a lucky accident' or 'it's up to each of us to decide' are rational enough but not really satisfactory.)

Jim Muth jimmuth@earthlink.net

Five

Tue, 30 Apr 2013 03:10:02 -0000

2013.04 Five [7]

FOTD -- April 29, 2013 (Rating 7)

Fractal visionaries and enthusiasts:

Today's image is a nice soft, soothing, low-contrast scene showing a quintic minibrot against a greyish-blue background. I named it "Five" because of the Z^5 energies infesting it. (There is also an almost-forgotten movie named "Five".)

The parent fractal is a Mandelbrot set caught halfway through the process of morphing into a quintic Mandeloid, at the moment when it is at the peak of shapelessness. Today's scene is located near the X-axis of the parent, at the western extremity of its main body.

The circular colors ringing the minibrot came about when I rendered the scene with the inside set to 'fmod'. (Having the background a flat black also works quite well.)

The rating of a 7 is getting quite commonplace, but with it's unspectacular effect, that's what I think the image is worth.

The calculation time of 5-3/4 minutes is slow. Luckily, the web sites are always there to make life easier.

Enjoy life to the max. See the image on the FOTD web site at:

with variations at:

and slews of older images at:

Heavy clouds kept a blanket over Fractal Central today. The rain was very light, yet just enough to keep things wet. The fractal cat spent the day watching the local pigeons and asking for food, while the fractal humans caught up on the news.

The next FOTD will be posted, that much is sure. The big question is exactly when. Until whenever, take care, and if it takes two to tango, how many does it take to boogaloo?

Jim Muth jimmuth@earthlink.net

A Round Pentagram

Mon, 29 Apr 2013 03:10:02 -0000

2013.04 A Round Pentagram [7]

FOTD -- April 28, 2013 (Rating 7)

Fractal visionaries and enthusiasts:

Today's scene shows a quintic minibrot, which unlike most minibrots in the Z^5+C Mandeloid is actually interesting. This is because it is combined with the Z^2+C Mandelbrot set in such a way that the excessively large and numerous minibrots found in the Z^5+C Mandeloid are separated much more widely and the in-between space filled in.

The parent fractal is an oversized Mandelbrot set already starting to look a little weird in places. Today's scene is located in the basin of a larger quintic minibrot that lies in a small island of debris inside the bud located just beyond the large bud on the X-axis of the parent.

I named the image "A Round Pentagram" after the rounded pentagram immediately surrounding the central minibrot. A pentagram is the familiar 5-pointed star, which is a religious symbol when oriented with a point facing upward, and a symbol of occultism when rotated 36 degrees so that it resembles a goat's head, with a point facing downward.

The rating of a 7 is in the ball park of average. If I had put extra effort into the coloring, the image might have rated an 8.

The calculation time of a speed-of-light 45 seconds will pass in a flash. The flashing, which might be confused with bomb testing, thereby causing a sudden unwelcome arrival of security forces, may be avoided by checking one of the web sites.

The finished image is posted online on the official FOTD web site at:

High-quality renderings are at:

and the back images are at:

A near perfect day prevailed here at Fractal Central today, with lots of sun, gentle breezes and a temperature of 70F 21C. Clouds arrived later in the afternoon, but by then all the outdoor activities were nearly finished. The fractal cat seems to be resigned that no more sun will be falling on his window shelf until autumn, and has claimed my favorite chair as his bed. The humans are resigned that autumn will eventually come, followed by winter.

The next FOTD will be posted in the divinely ordained time. Until whenever that is, take care, and believe it or not, some people actually are puzzled by the way I feel at peace when I am out alone without a cell phone. The truth is that it is quite easy to be at peace when out of contact with others. After getting hooked on computers, I simply refused to become addicted to any additional absolutely necessary conveniences of modern technological life, without which survival would be in doubt.

Jim Muth jimmuth@earthlink.net

Fairies on theLawn

Sat, 27 Apr 2013 06:10:08 -0000

2013.04 Fairies on the Lawn [6.5]

FOTD -- April 26, 2013 (Rating 6.5)

Fractal visionaries and enthusiasts:

Today's image is named "Fairies on the Lawn".

I have heard it said that even though we never see fairies dancing on the lawn, it is no sign they are not there. Well, in today's image the fairies are dancing on the screen and they are quite visible. True, they do not resemble tiny people, and why should they? They are 'arupa' fairies, basically bright little bundles of energy with no definite shape.

I came upon the fairies as I was exploring the fractal that results when 0.1 part of Z^2 is combined with 2 parts of Z^(-3) and the whole thing is iterated with the bailout radius set to 900. This fractal is an oversized, well-decorated Mandelbrot set overflowing with extra goodies such as islands inside the main bay and superfluous minibrots everywhere. Today's scene lies just off the main stem near the spot where the large minibrot should be.

The rating of 6.5 includes the half-point bonus for my coloring efforts. I'm not sure that the bonus was fully earned, but it gives me a feeling of satisfaction to pat myself on the back occasionally.

The calculation time of a fireball-fast 30 seconds will pass in about 8 breaths. Checking the finished image on the FOTD web sites will save your breath.

The fairies dance online in finished form at:

where the image has been calculated and is posted.

The high-definition variations are at:

The back images are at:

With photoshop blue skies, a few puffy white clouds, gentle breezes and a temperature of 61F 16C, today's weather was near perfect here at Fractal Central, and the fractal cat made the most of it. The fractal humans made the most of it also, while we were busy on more important things however.

The next FOTD will be posted in the near future. Until the future arrives, take care, and most everyone, even the wealthiest, has occasional feelings that despite modern technology something is missing in his or her life. But it's not so simple to pin down exactly what is missing or where it might be found.

Jim Muth jimmuth@earthlink.net

Butterfly Cookies

Fri, 26 Apr 2013 00:12:39 -0000

2013.04 Butterfly Cookies [7]

FOTD -- April 25, 2013 (Rating 7)

Fractal visionaries and enthusiasts:

Today's image is named "Butterfly Cookies". The obvious reason is that it reminds me of butterfly cookies. In fact, it reminds me so much of the cookies that, when I saw the image, I ran to the local store to pick up a box.

The recipe behind the cookies combines a measly 0.1 part of Z^2 with a full part of Z^(-3) then adds C on each iteration. The resulting parent fractal is an oversized thing closely resem- bling some of those created by the parallel-resistance formula posted on the list a week or so ago. Its Mandelbrot shape is apparent, but the normally empty inside parts are filled with all kinds of superfluous decorations. Today's scene lies on the X-axis just beyond a large minibrot on the skeletal negative spike.

The rating of a 7 is an improvement over yesterday's, but still leaves lots of room for improvement. Hopefully, tomorrow's image will rate an 8, but there is no guarantee that it will not check in at a rating of an embarrassing 6.

One thing not in dispute is the speed of today's image. On my far from SOTA machine it finishes in 20 seconds. Some of the current units will finish it in far less time, that is if they can run the old Fractint DOS program at all. Regardless, the image is available on the FOTD web sites.

The official FOTD web site with the finished image may be found at:

Augmented variations of the image are online at:

and the FOTD back images are at:

The weather was near perfect here at Fractal Central today, with photoshop (old-kodachrome) blue skies and white cotton-puff cumu- lus clouds. The only thing not perfect was the temperature, which at a high of 57F 14C was notably chilly for late April. The fractal cat would have preferred more sun on his shelf. But now that the sun is at a much higher angle it strikes only the edge of the shelf closest to the window. The humans, not very concerned with sun angles, had an average day.

The next FOTD will be posted in a number of hours. Until that mysterious number of hours passes, take care, and thanks to science we have life better today than at any time in the past. Why then are so many people dis-satisfied with their lives?

Jim Muth jimmuth@earthlink.net

A Spiked Fractal

Fri, 26 Apr 2013 00:10:04 -0000

2013.04 A Spiked Fractal [6]

FOTD -- April 24, 2013 (Rating 6)

Fractal visionaries and enthusiasts:

Today's minor fractal diversion is a scene on the X-axis of the fractal that results when 0.1 part of Z^(-4) is combined with Z^2, and straight Z is added on each iteration.

The resulting parent fractal is an oversized but skeletal Mandelbrot set, with today's scene located in a blank area on the phantom X-axis.

With too little time to really work up a good coloring, (that is if any colors could have helped the image), I could rate the scene no higher than a 6. The name "A Spiked Fractal" refers to the sharply pointed dark areas surrounding the rather large minibrot.

Without doubt, the best feature of the image is its unbelievably fast calculation time of under 5 seconds. But of course, this does not include the time required to set up and run the parameter file. The web sites will eliminate the need to run the parameter file however.

The completely calculated image is posted online at:

with high-definition variations at:

The FOTD back imaged may be accessed at:

Sunny weather and a temperature of 72F 22C made today near ideal here at Fractal Central. Clouds and a rain shower moved in at 6:00pm, but by then the fine day was in the bag. The fractal cat and the fractal humans both enjoyed the fine weather while it lasted.

The next FOTD will be posted before too long. Until whenever that happens, take care, and I hear that some teen-aged girls around town are swooning over the surviving Boston bomber, and even planning to have his picture tattooed on their bodies. I guess they think he's cute or something, but to me it seems strange to be honoring a 19-year-old man who is almost certainly a mass murdering terrorist. At the same time, it is also a relief for any worried men, since it appears that such frivolous women will be totally unable to replace men in any significant numbers in the prestigious but formidable high-paying fields of science and technology.

Jim Muth jimmuth@earthlink.net

A Long Journey

Thu, 25 Apr 2013 00:10:04 -0000

2013.04 A Long Journey [5]

FOTD -- April 23, 2013 (Rating 5)

Fractal visionaries and enthusiasts:

Today's quick fractal diversion shows a minibrot in the East Valley area of what is left of the large minibrot on the skeletal main stem of the fractal that results when 3 parts of Z^(-2) are combined with 0.333 parts of Z^2 and straight C is added on each iteration.

The rating of a meager 5 shows that the image is little more than a filler to help bring the lagging FOTD up to date.

The name "A Long Journey" came to mind when I had the impression of a wheel with 16 spokes surrounding the minibrot, with a shoe attached to each spoke.

There is little else to be said about the image except that it is fireball fast even on non-SOTA units. And the web sites are always ready to simplify things by eliminating the need to run the parameter file.

The official web site may be accessed at:

The augmented versions of the image are at:

and the thousands of FOTD back images are at:

The morning was dreary here at Fractal Central today, with low clouds and a temperature of 43F +6C. But at noon the clouds evaporated, leading to a sunny afternoon with a temperature of 61F 16C. The fractal cat tried to enjoy the sun on his shelf, but the sun angle is so high in spring and summer that it mostly misses the shelf. The humans spent the day at work with no play.

The next FOTD will be posted before long. In all likelihood, it will have a higher rating than today's. Until whenever, take care, and why don't we all try to stop fighting about religion? We could begin by agreeing that the deep truths of all religions are the same even as the surface details differ widely.

Jim Muth jimmuth@earthlink.net

BlackGold Minibrot

Tue, 23 Apr 2013 00:10:04 -0000

2013.04 Black-Gold Minibrot [No Rating]

FOTD -- April 21, 2013 (No Rating)

Fractal visionaries and enthusiasts:

On April 17 Mike Frazier posted an interesting image named "Black Gold". Today's FOTD is named "Black Gold Minibrot". The name indicates that I checked the center of Mike's image and not surprisingly found a minibrot. That minibrot appears at the center of today's FOTD image.

I tweaked the colors to add a little variety, but made no changes to the parallel resistance formula itself.

Since the image is a borrowed one I could give it no rating, but it's still a rather spectacular one to look at.

The calculation time of one brief minute will cause no pain even for those with the busiest schedules. And the web sites are always ready to shorten even that brief minute.

The finished image is posted at:

Variations are at:

The FOTD back images are at:

Nothing but sun fell on Fractal Central today, though the temperature of only 50F 10C was hardly spring-like. The fractal cat passed most of the day on his window shelf, watching for passing stray cats. When he finally saw one, his growl of defiance could be heard throughout the room. FL spent several hours working in her garden. As for myself, I simply spent the day.

The next FOTD will be posted within a reasonable time. Until a reasonable time passes, take care, embrace the good and eschew the evil.

Jim Muth jimmuth@earthlink.net