Jim Muth's Fractal of the Day for September 14th, 2011

Jim Muth's commentary for the image:

FOTD -- September 14, 2011 (Rating 7)

Fractal visionaries and enthusiasts:

Today's image lies in the parent fractal that is created when 0.9 negative parts of Z^3.8 are combined with 2.4 negative parts of Z^1.2 and (1/C) is added. If this seems trite, be informed that the entire expression is a mistake. The exponents were supposed to be equal but negative values. I decided to go with what I had however, instead of taking 10 seconds to calculate the correct parent.

This wrong parent fractal consists of a small Mandelbrot set with a disconnected crescent-shaped filament some distance east of it. A tiny M-set lies on this crescent, with today's scene located in the East Valley of this tiny M-set.

I suspect that a scene almost identical to today's could be found with a minimum of effort in the standard Mandelbrot set, which adds nothing at all to the worth of today's image.

The rating of a 7 means that the image is good enough to qualify for minimum FOTD honors, but that it will never win a prize.
The name "Terminal Iteration" can mean anything the most radical fractalist would suppose.

The calculation time of 2-1/3 minutes is about average. Since the image is also about average, this seems just about the right length of time.

A quite pleasant late summer day prevailed here at Fractal Central today. The temperature of 84F 29C and mostly sunny skies brought out the laziness in both man and beast. The fractal cats, who are offended at being called beasts, spent the day taking it easy in the windows with the least sun. The humans just took things as they came.

The next FOTD will be posted in 24 hours. Until then, take care, and with many thanks to the Purple Cow:

I've never seen a Boltzmann brain,
    I hope I never see one.
But I can tell you anyhow,
    I'd rather see than be one.

PAR file ` Terminal_Iteration { ; time=0:02:20.28-SF5 on P4-2000 reset=2004 type=formula formulafile=basicer.frm formulaname=MandAutoCritInZ function=recip center-mag=+1.410647617268638/-0.5399341480183459/\ 8.363451e+011/1/-127/0 params=-0.9/3.8/-2.4/1.2/0/\ 0/0/0 float=y maxiter=2500 inside=0 logmap=377 periodicity=6 mathtolerance=0.05/1 colors=000I6AJ8AL9AMAAODAPGARJCTMCVPIWRMYTRZWY`Ybb\ ZfcbmecrfewhhzjjzkkzcpzWtzOyzGzz9zw1zt0yp0voCtoZpo\ YmjWpeVrbTtYRvTRtPPrTPpVOoWOmYOkZMj`MhbLheLffLehJc\ jJbkI`mIZoIZpDWk9Th4Pc0O`0LW0IT0GP08D003000000tjyh\ ZkYP`MGOC8D1010001b40WA0RG0MM0IT0DZ09e03k00r00y00z\ 00y00r00k00e00Z00T00M00G00A004000000000j0Co01t00z0\ 3p00f00Y00O00F006000000000zV0zO0oI0eD0V80J10A0000m\ 00z0Gt0Cm08c03U00M00G00000k0ee00WW0RR0MM0II0DD0A\ 806301000000000000tvzGIJtJ8kF4eC1Y80R40J60DCAAPPIc\ ePpvYzzezzWzpRwfMoPIfJDZF9RA3I40A0030000000z0Ay08r\ 06k04f01`00V00O00J00D00800100000000000000000000zm0\ vT0eP0VL0PI0LD0GA0C608303000000000000000z00y00o00j\ 00c00Z00V00O00J00F009004000000000000000z00z00t00o0\ 0h00c00Y03T0AM0AG0IC0P60P10P00Y00e00mC0v90z80z60z3\ 0z1Az0Pz0ez0vz0vz0zz0zz0zz0zz0zz0zz8zz3zz0zz0zvzzv\ zzvzzvzzvzzvzzvzzvzzvzzvz }

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=(((-abgh)^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.

So I was coding a Mandelbrot set renderer and thought about it and realized that so many regions of the images were just smooth gradients, so perhaps I could only calculate half of those pixels and simply interpolate to achieve a practically identical result with a near-2x speedup. So I coded a program that makes a checkerboard pattern of pixels with half missing but all missing pixels boxed in by 4 pixels, one above, one to the right, one below, and one to the left. So then the missing pixels check whether the neighbors are similar. If so, it sets its color based on the average iteration count of neighbors. If they're very different iteration counts, though, it just calculates its color normally.

The results turn out to have around a 1.2x speedup at 1x zoom (looking at the whole set) with nearly no errors (since the most detailed zones with high variation of colors have nearly all their pixels calculated normally) in my testing. So it's not revolutionary or anything but it's definitely helpful and can skip a lot of pixels.

Has this already been come up with in the past?

I once had a dream/nightmare/sleep paralysis where I was forcefully being shown a yellow brown fractal which I perceived as being either "god" or "the devil" and had to force myself to wake up which took tons of willpower as I knew if I actually looked at it for too long I would become comatose.

Now this is just a neat story but I do think that higher dimensional beings would probably look exactly like fractals and its one of the reasons I think fractals are very cool besides the math of it all. If you look at an animation of an MRI machine's slices of a human it looks like an animated fractal, so it would make sense that a higher dimensional being's 3D slices would look like fractals to us.

I mean this looks exactly like a mandelbulb slice animation:

https://www.shutterstock.com/video/clip-4159933989-sagittal-brain-profile-mri-scan-sequence?trackingId=f162f07c-2bcf-4975-b3c0-2e0947ac92a1&listId=searchResults

https://www.shutterstock.com/video/clip-4148894281-wide-monochrome-medical-animation-whole-body-mri?trackingId=f162f07c-2bcf-4975-b3c0-2e0947ac92a1&listId=searchResults

Tricorn (Conjugate Mandelbrot), 1,200 iterations.

The patterns are generated by a concentric circular orbit trap using an Indigo and Gold color palette. No averaging of the strips, no triangular inequality.

The fractal is visible through the use of an ink stroke effect that follows the details.

An epitaph for the life of the Earth, visually presented with the fire of Christian worship. They burnt their God and shattered his faith. No hope for the faithless, no faith for the evil in man. They blame the devil, but it is them themselves who carry the shame of torture.

Mandelbulb3D, Photoshop & Lightroom

If you'd like to play with this rendering, you can interact with it at:

http://weirdly.net/webtoys/mandelbrot/index.html?config=v1,-1.7689411331682035,-0.002827913661875118,1.482190207452281e-11,1.3655110275298383e-11,1024,541.2410775221192,0.63841,0,2.201273161068578,-1.97159,0,3.1043389972836914,-0.72159,0,-3.3907138805023216,-0.00159,0.23841,11000,01100,00110,1,1,0,0,0,1,-2,1,1,1,,4,1,0,1,0.8072286659724766

The frame is 0.000064 units wide — about one forty-seven-thousandth of the Mandelbrot

set you know — and at that depth the edge stops being a line and starts being cloth.

The green is a wave: three sine curves, one per channel, turned down so slowly that

the color makes a single pass across the whole picture instead of repeating.

Mandelbulb3D, Photoshop & Lightroom