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GaulerTheGoat

Is it fractals?

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Hey all, I accidentally made these images in PSP7 and they look a lot like fractals, but I'm not sure what type they are or even if they are. Sorry about the clickies, too, but yahoo is the only web space I have (maybe someone can upload them to a better server). Images The images were created by starting with a white 200x200 canvas. Go to Effects->Noise->Add Noise=50 Random Go to Effects->Plug-in Filters->Eye Candy 3.1->Jiggle Twisty and repeat the twisty over and over again while the image slowly forms on the edges of the canvas. BTW, the jpegs are fuzzier than than the originals due to compression. I don't know much about fractals or what algorithm "jiggle" uses. Any comments? edit=oops. got the index up. [edited by - GaulerTheGoat on October 20, 2003 7:43:30 PM]

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Looks like a fractal. It looks like the ''jiggle'' effect is a midpoint displacement with rotation, which in theory could be continued recursively forever on smaller and smaller divisions.

It''s more or less the same idea as fractal-generated heightmaps for terrain.

Tom

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Guest Anonymous Poster
For it to be a fractal it would be possible to zoom into the picture indefinitely and see more and more details of the "jiggles".

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ParadigmShift:
I tried googling for ''midpoint displacement rotation'' and found this, except they call it plasma fractals. When I made the images, all the development is on the edges, slowly growing inwards. The center of the image changes very little. The "midpoint" would seem to be OK on the edges but isn''t there also supposed to be a 5th midpoint in the center of the other four? Maybe ''jiggle'' is a variant that doesn''t use the middle point? I''m going to try ''jiggling'' some simple shapes to figure out what''s going on. Thanks for the starting point.

AP:
Yeah, that''s why I don''t know if it''s really a fractal or not. It looks like one but of course, the details are being lost because we can''t see any deeper than one pixel.

Mastaba:
Sounds cool, but could you explain ''folding'' please? Googling for ''image folding'' got me, like nowhere

Thanks for replies

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For a simple folding map try googling for
folding map chaos

or

horseshoe map

They are quite complicated to study ( you''ll find them in post-graduate maths books )

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A fractal is a self-similare mathematical object that involves a repeated process over the course of, typically, an infinit amount of iterations and that retains a constant level of detail irrelevant of the amount of "zooming in". Of course, infinity isn''t something quite feasible on a computer

Here''s an example, an L-system (Lindenmayer system). Take a triangle. Now draw an inverted triangle in the middle of that triangle such as that the three points are the middle of the three sides of the initial triangle (think Triforce from Legend of Zelda). Color that black. For the remaining three triangles, do the same. You should have 9 triangles now. Keep doing that over and over. You''ll end up with a fractal object (the Sierspinsky triangle).

An other fractal, more well-known, is the mandelbroth fractal. Applying the equation z -> z² + c over a complex plane from -2-2i to 2+2i and coloring pixels which never escape a certain threshold black and the others white will result in a rather strange shape somewhat like a twisted sideways 8. Successive zooms into the image will reveal astounding details and a very self-similare structure (well, in fact, you''ll find the basic starting image all over the place).

So no, that''s not a fractal. It''s kinda nice though.

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quote:
They are quite complicated to study ( you''ll find them in post-graduate maths books )


Post Graduate! They were part of my undergraduate degree! Modern Analysis, Dynamical Systems, and Ergodic Theory all covered one or other of them.

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quote:
Original post by Keem
They are quite complicated to study ( you''ll find them in post-graduate maths books )


I wouldn''t really call ''em complex... The basics of it are quite simple. The actual philosophy behind them is rather... "heavy", as it is.

There''s a great book on the subject (well, chaos theory in general) that I consider my personal bible: "CHAOS - The Making of a New Science" by James Gleick. Very, very interesting read. Worth a look if you have the time.

It offers a really nice explanation of fractals, though.

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quote:
Original post by RuneLancer
Here's an example, an L-system (Lindenmayer system). Take a triangle. Now draw an inverted triangle in the middle of that triangle such as that the three points are the middle of the three sides of the initial triangle (think Triforce from Legend of Zelda). Color that black. For the remaining three triangles, do the same. You should have 9 triangles now. Keep doing that over and over. You'll end up with a fractal object (the Sierspinsky triangle).

Yeah, I've seen that before. And I like your Zelda reference. More mathematicians should play video games
Seen the Mandelbrot[h?] set, too.
quote:
Original post by RuneLancer
A fractal is a self-similare mathematical object that involves a repeated process over the course of, typically, an infinit[e] amount of iterations and that retains a constant level of detail irrelevant of the amount of "zooming in".
[....]
So no, that's not a fractal. It's kinda nice though.

I didn't know that in addition to having "infinite detail" (what's the mathematical definition of this, anyway? The boundry has infinite arc-length, or something?), it also has to have similarlty to itself (like at any neighborhood? Because the Mandelbrot looks different at different places.)

Why does this mean that the 'jiggle' images couldn't be fractals, though. They seem to be developing infinite detail (although, there's no way to tell without knowing the 'jiggle' algorithm). Also, the Mandelbrot doesn't look the same everywhere (it's lopsided, to begin with) so I'm not sure what you mean by self-similar (like, it maps to itself linearly, I think is the word?) edit="affine" map was the word I was looking for

When done on simpler objects, the 'jiggle' just looks a distorted expand and contract. It does it at all points, even the center (so I'm not sure why the center of the above images is unchanging).

When done on a straight line, eg, it looks like a really sloppy sine wave. But, if you do it on two perpindicular lines, they both look like perpindicular sine waves. Also, with some other experiments, it looks like the 'jiggle' is always perpindicular to the line (more or less).

Isn't there a fractal that starts out as a triangle, then the middle one-third of each straight edge is replaced with another triangle, and so on. Could the images above be formed by something like that, since it seems to have a similar kind of "straight line deformed by perpindicular motion" thing going on. This is just speculation, though

Thanks for good replies. You guys obviously know quite a bit about this stuff

[edited by - GaulerTheGoat on October 21, 2003 6:07:27 PM]

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By "infinit details", I mean that regardless of how far you scale the image, you end up having a constant level of detail. Take, for instance, the mandelbroth fractal. You can keep on zooming and even when you''ll reach the limits of your computer''s processing power, you''ll still have a constant level of detail. Well, that''s assuming an infinit amount of iterations...

The reason why your image isn''t a fractal is because it doesn''t involve an actual methematical process. Although in a way it is, as you are applying a "simple" process to an astute mapping repetitively. It''s like er... forgot his name. Some guy''s horse-shoe folding. In any event, the main problem is that you can''t zoom in infinitely. You create noise and apply a process to it an infinit amount of time. It''s like creating a random noise mapping and applying a smoothening filter to it. Although the process is fractal, when you zoom in your noise "particles" will become more and more distant and applying the process won''t give you a constant level of detail anymore.

The line->triangle thing you described is, indeed, a well-known fractal called the Koch curve. There''s a variation called the Koch snowflake that involves applying that to a triangle. Pretty dang nice.

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quote:
Original post by GaulerTheGoat
I didn''t know that in addition to having "infinite detail" (what''s the mathematical definition of this, anyway? The boundry has infinite arc-length, or something?), it also has to have similarlty to itself (like at any neighborhood? Because the Mandelbrot looks different at different places.)
If you imagine the Mandelbrot fractal. Good. Then understand that in every "point" in it, the whole image is repeated.
quote:
Why does this mean that the ''jiggle'' images couldn''t be fractals, though.
Because an image isn''t a fractal in itself, it is only a snapshot of a fractal. The fractal is the mathematical algorithm that produces the image. Your image may look similar to an image of a fractal, but that doesn''t quite cut it.

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I didn''t mean that the jpeg image itself was a fractal. I wanted to know if the algorithm used to create the image might be based on a mathematical formula that would generate a fractal. The image was just my basis for suspicion since it looked a bit like one.

Anyways, thanks for the input, especially RuneLancer. I think I should work on it myself a little. I have The Fractal Geometry of Nature by Mandelbrot himself. It''s been at the bottom of a stack of books covered in dust in my room for a couple years now. It looks like a good place to start. Thanks again

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The fact that an image is not a true fractal is meaningless. We also can''t write all the digits of pi but that doesn''t stop us from understanding what pi is all about. Chaotic images may still contain more than enough information to speak volumes about a complex system''s dynamical nature for instance.

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quote:
Original post by RuneLancer
quote:
Original post by GaulerTheGoat
I have The Fractal Geometry of Nature by Mandelbrot himself.


You lucky, lucky man...

How come? Is it hard to find or something? I got it at a used book store for $12.50

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