Fractals Revived

[The Potter]

It is the top third of a mini-Mandelbrot.
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[The Potter]

Oh, a new page. Fancy that.



Last fractal from America!
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[The Potter]

Julia Fractals! All of the updates from this year were from the Mandelbrot set Now the closely related Julia set will be looked at.

Also updates may be less frequent for awhile. But that is a good thing if you ask me.



There is a pot of gold at the center of each X.
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[extropalopakettle]

Quality isn't so great, and it has "evaluation copy" all over it, but here it is anyway ... an animation I did of parameter changing on a Julia set, and also zooming in toward the center. The parameter change is very minor, but still has a big effect on structure at a small scale. It's a looping animation - chrome will loop it, IE didn't for me. It's big too (1920 by 1080).

http://youtube.com/v/sbCGMuFxEtA&loop=1&autoplay=1

[The Potter]

Not bad at all...



A seashell of sorts.
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[The Potter]

So could you do a zoom movie that goes to the point 0.3869889274 + 0.5685893599 i? That is the point my series was based on.
Or if you want to go farther, look at 0.3865889274281 +0.56858935988441i instead.

Does mathematical distance use farther or further?


It is a monster!
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[The Potter]

So Julia sets are either "connected" or "disconnected". In images such as this one, the entire trapped region (black) is completely connected. Because it was taken from a spot very close to the border of in and out, their are long trails which almost separate the regions.
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[Jedo the Jedi]

And here I thought Mandelbrot's were my favorites. That is gorgeous.
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[The Potter]

The Julia set has its moments. One aspect I don't enjoy about it is each set is repetitious. Zooming into any area will only create one style of images.
The thin spirals are about my favorite Julia fractals as well.


This one comes from an area that is inside the Mandelbrot Set. Just inside.


This one comes from an area that is just outside the Mandelbrot Set. From a point very close to the other image.
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[The Potter]

Remember when I was talking about branches in the Mandelbrot set? The number of thin trails is related to the number of branches. I believe I could find any number that I desired.
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[Jedo the Jedi]

So awesome! (The one before is too.)
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[The Potter]

Just stare into the right spot.
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[The Potter]

If you meet someone with a troubled mind, just give them the key...
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[The Potter]

snow fractals (from j_s's thread. But I know this thread will last long. They are really cool-- even if I do remember seeing them a few years before as well.)



I am just not sure how to describe this.
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[The Potter]

Off Topic was quiet today!


It was made pretty by limiting the number of iterations to 1000. The black space would be mostly full if I used my normal iteration count of 16000.
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[The Potter]

Stay away from the void.
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[The Potter]

I am not entirely sure what the idea was.
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[The Potter]

I come across so many pretty spirals but rare focus in on them.
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[extro...*]

Another looping animation (with any luck, the url as posted should make it loop). This is a 30,000x zoom on the center of a Julia set ... or sets, really ... as the seed parameter changes just very slightly. What's interesting about these parameter change animations is that you can zoom in (magnify) in both time and space. I could take two adjacent frames from the animation below and do another animation depicting just the transition between them, in slow motion, and zooming in further on the center of that it would have as much detail, and not appear to be in any slower motion.

http://youtube.com/v/20y8XGCQY40&loop=1&autoplay=1

[extro...*]

Same julia set, rendered two different ways (1920 by 1080):

http://glpics.com/extropalopakettle/f2c.jpg

http://glpics.com/extropalopakettle/f2h.jpg

I'm doing an animation of the above - will take about 5 days to render.

[The Potter]

Awesome! One of my gripes about fractal animations has been using boring colour schemes. I usually spend the majority of my time working out the colors.


Lets take a ride to the seashore! Or maybe "Acid Snow"
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[The Potter]

Merry belated Christmas!
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[extro...*]

Another looping julia set animation - no zooming in, just animating based on changing seed parameter. 1920 by 1080 resolution (actually, cropped to 1920 by 1072, as youtube supposedly re-encodes better when dimensions are a multiple of 16)

http://www.youtube.com/v/3tpqACCk0RE&loop=1&autoplay=1

[Jedo the Jedi]

That was your most interesting one yet, extro. I loved watching the lattice stuff contracting and expanding, then contrasting that with the portions which were stable. Very nice color scheme for it too.
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[The Potter]

Better color schemes certainly help!


I think I will call it "Hands reaching out for something"
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[The Potter]

Farewell old '12!


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[Jedo the Jedi]

Very cool.
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[The Potter]

There is something I really like about this guy.
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[The Potter]

Some areas are filled with quiltlike squares. Double spirals always have a good feel to them.
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[The Potter]

Dancing?
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[Jedo the Jedi]

That's a nice change of pace.
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[The Potter]

Working with black space is usually quite challenging. Getting this one to look good took considerable effort but the result was worthwhile.
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[The Potter]

Stitched together.
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[Zag]

Ahhh. I like the ones where you can really see the recursion, as in this one.

I'm curious, though, how it would look if you removed the parts coming off of the sides of the black diamonds -- that is, make it recurse at the corners. As it is, the extra little nipples created along the sides to provide an extra recursion point seem artificial, to me, just to get more 'stuff' in there, because the shape that those make is not "five diamonds off each intersection point" like the master shape is; it's just four diamonds and a nipple on the side of a much bigger diamond. Would that make it too simple-looking to be interesting? (Probably not for me, I'm guessing; I'm a simple guy.) Is there any chance you could make that one?

[extro...*]

Almost like asking a microscopist if he can make the paramecium have less cilia. I'm not sure how one could do that, given that these objects are more found than designed. Nobody made a decision to have those little parts coming off the edge, and there probably isn't one like that out there to find. Also note that these aren't produced by a recursive process at all, but a simple iterative one (and not an iterative process that simulates a recursive one). The process just reveals the structure of a mathematical object, much like a microscope revealing the structure of an organism. We can use dyes and polarized light and take thin slices with a microtome to see it in different ways, but the structure of what we're viewing is just there, not created by the means we use to view it.

[Quailman]

[extro...*]

I was thinking that if there were a number of people familiar with the mandelbrot and julia sets, one could play an interesting contest game wherein each participant renders a part of the mandelbrot or a julia set, and then all participants are challenged to find the same parts and render the same image. I just tried finding the last one above, and I'm stumped. Without the three-way spiral on the left, it was pretty easy.

[Zag]

That's why I asked "is there any chance...?" -- because I didn't know what the process is by which these are created. I can imagine some software in which you make the basic shape, provide the rules for recursion, and then it builds the image. In fact, I could write such a thing, given a couple of months and the proper financial motivation.

If there's no good way to do it, I won't be offended, I just thought I'd ask.

[extropalopakettle]

Sorry, I assumed the process was understood. In any case, it's the process, the simplicity of it, and how unexpected the results are, that makes it so truly fascinating (to me at least). The pretty pictures are nice, but how they arise from such a simple process is what I find amazing. I'm going to start another thread on just that, in a few moments.

[The Potter]

Try a C value of 0.3855183 +0.5695992i

No, it really isn't possible to control individual aspects of the set. By making the edges smoother (choosing a point farther from an edge of the M-set) I will lose the spiraling characteristic of being close to the edge.

The blue/cyan image before is located at 0.3591539+ 0.6159293i. Some aspects are similar. The edge complexity is the main difference.
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