Fractal images

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olesma
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Fractal images

Post by olesma »

Okay - here is a fun little program:Fractal Explorer 2.0. Go download it and use some of the tutorials that are linked to in the side bar of the main page (the link above is to the download page) to learn how to use it (this one is a particularly good list of them - but really, the operation is incredibly simple - nuance is complex, but the basics are really easy).

This is the coolest program. I've been fooling with it since it was just v. 1.24 - but the new 2.0 has good fixes for some of the annoying problems in v. 1.24.

The coolest images are found by using a Mandelbrot set and zooming in on different areas. I don't totally understand the math completely - but here is the best shot I can take at it:

We all worked with geometry in high school. In geometry you take an equasion and find a set of points that satisfy the equasion (for example: x2 + y2= 1). When you instead iterate an equasion instead of solving it, the equasion becomes a process instead of a description - dynamic instead of static. When a number goes into the equasion, a new number comes out; the new number goes in and on and on - points hop from place to place. A point is plotted not when it satisfies the equasion but when it produces a certain kind of behavior. One behavior might be a steady state, another might be a convergence to a periodic repetition of states, another might be an out-of-control race to infinity.

In the Mandelbrot set you simply take a number, multiply it by itself, and add the original number z = z2 + c

Pretty wild. A 2 element equasion with only 4 characters results in what is considered the most complex mathematecal object known today.

(the equasion for this image in the program looks like this: Z=Fn1(Z); Z=Z*Z + Fn2(C) - In the program it looks longer because you have to label the function of Z (Fn1) and function C (Fn2).)

This is what a Mandelbrot set looks like:
Image

Anyway - here are some of the images I've created (click on the thumbnail for the full version) - all of these images except one come from zooming in on a Mandelbrot graph (by as much as 1,000,000x magnification):
ImageImage
ImageImage
Image

These last images were created by altering the colors of the original set:
ImageImage

Or by using a different equasion - in this case a Newton attractor:
Image

Now - go forth and create funky images.

The best images are found along the edges. If you focus on large blank areas (either black or with only one or two colors) - you get nothing.

Don't let the math fool you - creating these images involved no manipulation of the math - it is a graphics program first and foremost. If you understand the math you can do really cool stuff, but it isn't necessary.

Myself - I save the images I like into 1024 x 768 and use them as desktop graphics (all the large images that can be linked to are that size). If you're really into this type of stuff, you can get the 3-D generator as well and create fractal landscapes. That is pretty cool stuff - but I don't have quite the patience for it.
'Weird is a relative, not an absolute.' - A. Einstein
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pfredricks
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Post by pfredricks »

Great to see you Olesma-
I thought you may have permanantely jumped ship. I have to poke at you a bit though.
It believe it is fundamental that in any discussion of fractals you include a requisite discussion of Chaos theory.
Chaos is fascinating/complex/yet simple to understand and the foundation of fractals.

Read here:

http://www.duke.edu/~mjd/chaos/chaosp.html

and become enlightened. It's a pretty nice summary

-Pete
"I'd feel better if we had some crampons. Oh, what the hell, let's go for it..." — Common climbing last words.
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olesma
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Post by olesma »

In response to pfredricks' reply: "Summary" is a good explanation of that. But you're right - that site is a very good summary - I haven't seen one better.

I recently got done reading a truly excellent book on the subject:

Chaos: Making a New Science by James Gleick

Gleick is a very good author and makes the story of the development of the science of chaos understandable and interesting. I also recommend another book by him titled Genius which is a story of the life of Richard Feynman - a dearly missed member of the scientific community and a truly unique character in history (who is quoted at the head of the URL from PFredricks).

Chaos is pretty cool and bizare stuff. I believe the quote goes something like:
"Underneath all the order is an erie sense of chaos, and under the caos is an even more erie sense of order."

A Mandelbrot set is one of the best visualizations of what chaos is all about. Apparent randomness that yields to mathmatical precision and repeating patterns the farther down you go. Or apparent order that yields to apparent randomness that actually turns out to be mathematically based and not random at all.

Mind warping stuff - but very cool.
'Weird is a relative, not an absolute.' - A. Einstein
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GeorgAZ
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Fractals

Post by GeorgAZ »

Just love those things,especially the Mandelbot. Check out http://www.fractalarts.com/ASF/index.html.
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olesma
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Post by olesma »

Link is busted - or they used all their bandwidth.

Also - all but one of the images are from the Mandelbrot.

It is said that you could never see the entire construct of a Mandelbrot in one lifetime - and it is doubtful you could see the whole thing in several lifetimes. It is an infinite construct with no hint thus far of true repetition of form at any level - shades of repetition, but not 100%.
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GeorgAZ
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Fractals

Post by GeorgAZ »

I was just on that site.Maybe it was too busy. How cool that you can do those. It's always been fascinating to me and I am NOT a math person.Good job!
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olesma
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Post by olesma »

Hey, like I said - you don't have to be. Download the program and give it a shot. Trust me - you'll love it once you do a couple.

I've gotten everyone in my office wasting all kinds of production time making fractal images. None of them are math whizzes - or even computer whizzes for that matter. But they do just fine.
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