View Full Version : Fractals are groovy :)

Archernar7

2006-Feb-16, 02:06 PM

Check it out (http://images.google.com/images?q=fractals&hl=en)

They are awesomely beautiful. :)

http://i47.photobucket.com/albums/f192/CMarek/juliasetfractal.png

It's called the Julia set. No explanation why that name. It's gorgeous.

And this gem:

Image (http://fusionanomaly.net/kilroywasherefractal.jpg)

Mind bender. I love it! Have never seen anything quite like that -- and the colors, wow. :D

tofu

2006-Feb-16, 02:31 PM

It's called the Julia set. No explanation why that name.

They are named after the mathematician who discovered them, Gaston Julia.

Argos

2006-Feb-16, 02:34 PM

It's called the Julia set. No explanation why that name. It's gorgeous.They were described by Gaston Julia. See maths (http://mathworld.wolfram.com/JuliaSet.html) of the Julia sets.

Disinfo Agent

2006-Feb-16, 03:16 PM

A fractal link. (http://math.rice.edu/~lanius/frac/)

You could also check out Mandelbrot's book, The Fractal Geometry of Nature, or Mandelbrot's World of Fractals, available on DVD.

Peptron

2006-Feb-16, 04:03 PM

When I've left college and was still looking for work, I've been playing with fractals to make sure that my programmation skills wouldn't rust.

I even noticed that the Web site I've done during that period still exists! It's completely out of date and I've not even been on it for like 4 years, I'm surprised it still exists :). Besides, I've abandonned it the moment I've got a job, so it's not even half complete.

I experimented with VB by making a program to generate fractals, and I've even put it on that site. That program is awfully out of date, but it's still a working fractal generator.

http://peptron.fateback.com/fractalesprog.html

Some fractals made with that program:

http://peptron.fateback.com/fractalesVB1.html

I've also made other fractals using the program Tierazon (you can google it):

Using Tierazon led me to try to make my own fractal generator, since I've felt that fractals made with Tierazon weren't really fractals I've made myself.

http://peptron.fateback.com/fractales1.html

http://peptron.fateback.com/fractales2.html

http://peptron.fateback.com/fractales3.html

http://peptron.fateback.com/fractales4.html

http://peptron.fateback.com/fractales5.html

http://peptron.fateback.com/fractales6.html

http://peptron.fateback.com/fractales7.html

http://peptron.fateback.com/fractales8.html

http://peptron.fateback.com/fractales9.html

http://peptron.fateback.com/fractales10.html

http://peptron.fateback.com/fractales11.html

http://peptron.fateback.com/fractales12.html

http://peptron.fateback.com/fractales13.html

I'm fascinated about how complex fractals can get when you know how simple the calculation of it is. You can get infinitely recursing twists and spirals in like 5 lines of code.

One of my friend once asked me what fractals are useful for... I know they are useful in some field of science, but all I know for sure is that they make interesting desktop backgrounds :). They make you look like you have an artistic sense while in fact you really just played with numbers, switched variables in a calculation, and other unartistic things like that :).

Archernar7

2006-Feb-16, 06:08 PM

When I've left college and was still looking for work, I've been playing with fractals to make sure that my programmation skills wouldn't rust.

I even noticed that the Web site I've done during that period still exists! It's completely out of date and I've not even been on it for like 4 years, I'm surprised it still exists :). Besides, I've abandonned it the moment I've got a job, so it's not even half complete.

I experimented with VB by making a program to generate fractals, and I've even put it on that site. That program is awfully out of date, but it's still a working fractal generator.

http://peptron.fateback.com/fractalesprog.html

Some fractals made with that program:

http://peptron.fateback.com/fractalesVB1.html

I've also made other fractals using the program Tierazon (you can google it):

Using Tierazon led me to try to make my own fractal generator, since I've felt that fractals made with Tierazon weren't really fractals I've made myself.

http://peptron.fateback.com/fractales1.html

http://peptron.fateback.com/fractales2.html

http://peptron.fateback.com/fractales3.html

http://peptron.fateback.com/fractales4.html

http://peptron.fateback.com/fractales5.html

http://peptron.fateback.com/fractales6.html

http://peptron.fateback.com/fractales7.html

http://peptron.fateback.com/fractales8.html

http://peptron.fateback.com/fractales9.html

http://peptron.fateback.com/fractales10.html

http://peptron.fateback.com/fractales11.html

http://peptron.fateback.com/fractales12.html

http://peptron.fateback.com/fractales13.html

I'm fascinated about how complex fractals can get when you know how simple the calculation of it is. You can get infinitely recursing twists and spirals in like 5 lines of code.

One of my friend once asked me what fractals are useful for... I know they are useful in some field of science, but all I know for sure is that they make interesting desktop backgrounds :). They make you look like you have an artistic sense while in fact you really just played with numbers, switched variables in a calculation, and other unartistic things like that :).

WOW. :D I'm checking out link #1 of fractals you've made.

Thanks to everyone who responded.

I really want to try this for myself, use that fractal generator. My math skills are not all that hot, unfortunately, so this will be an experiment indeed. :p

Wolverine

2006-Feb-16, 07:15 PM

They are awesomely beautiful. :)

Yes they are -- but please don't hotlink images from other websites in accordance with the forum rules (http://www.bautforum.com/showthread.php?p=564845#post564845). Not only does it add to the site's bandwidth (which isn't nice), but they may also be copyrighted.

snarkophilus

2006-Feb-16, 09:45 PM

They are also (apparently) useful in the real world!

http://www.nature.com/nature/journal/v439/n7077/full/439648a.html

The Supreme Canuck

2006-Feb-16, 10:02 PM

Fractals can kill (http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v402/n6761/full/402465a0_fs.html).

5. How does a basilisk operate?

The short answer is: we mustn't say. Detailed information is classified beyond Top Secret.

The longer answer is based on a popular-science article by Berryman (New Scientist, 2001), which outlines his thinking. He imagined the human mind as a formal, deterministic computational system — a system that, as predicted by a variant of Gödel's Theorem in mathematics, can be crashed by thoughts that the mind is physically or logically incapable of thinking. The Logical Imaging Technique presents such a thought in purely visual form as a basilisk image which our optic nerves can't help but accept. The result is disastrous, like a software stealth-virus smuggled into the brain.

8. Are there basilisks in the Mandelbrot set fractal?

Yes. There are two known families, at symmetrical positions, visible under extreme magnification. No, we're not telling you where.

Great series of sci-fi stories by David Langford. Look 'em up!

Melusine

2006-Feb-17, 08:01 AM

There is a great fractal site I found some time ago, ArtByMath.com, but I just looked at it and it appears to have been attacked by some porn deal, so don't go there now. This safe link (http://www.zazzle.com/contributors/home/default.asp?cid=238336099936940168) shows some of what's on there, but not the puzzles, etc.

Archernar7, cool stuff there.

A fractal link. (http://math.rice.edu/~lanius/frac/)

You could also check out Mandelbrot's book, The Fractal Geometry of Nature, or Mandelbrot's World of Fractals, available on DVD.

Thanks for that link, Disinfo. I'm familiar with fractal art, but I'm so math-challenged that the link is about my speed. :( I didn't quite get this from Snark's Nature article:

Human motion is known to display fractal properties when people restore their balance, says Taylor, and films of Pollock seem to show him painting in a state of 'controlled off-balance'. Second, the dripping and pouring itself could be a chaotic process.

Can someone explain that in English, as in English for an English Major? Thanks for the article, Snarkophilus; I am very familiar with Jackson Pollock, but was not aware that his paintings are being analyzed in this way. :)

01101001

2006-Feb-17, 08:25 AM

Can someone explain that in English, as in English for an English Major? [...] I am very familiar with Jackson Pollock, but was not aware that his paintings are being analyzed in this way. :)

If you zoom in on a Pollock, say look at 1/16th of it, and blow that small part up 16 times, it looks extremely similar to the painting it came from.

His works have self-similarity.

The same goes for the muscle tremors that are present in the act of balancing. Look at the amplitude of the movement over, say 10 seconds, perhaps presented as the wiggle on a chart. If you take 1 second of that wiggle, and blow it up ten times, it will have the same character as the original.

Spot the fractal nature of my signature. It is self-similar.

Look at what is the pattern of, say, the appearances of 01 and 10 within it or at what is the pattern of 0110 and 1001. Look what pattern you get if you replace every 0 by 01 and simultaneously replace every 1 with 10.

Melusine

2006-Feb-17, 09:26 AM

Thank you Binary Man, that explanation works for me. And here it is 3:00 a.m. and I'm playing with your signature, but that's what it takes for me, at times, for something to really sink in. Wonderful. :)

Disinfo Agent

2006-Feb-17, 02:23 PM

And here it is 3:00 a.m. and I'm playing with your signature, but that's what it takes for me, at times, for something to really sink in.It's what it takes for most of us.

01101001

2006-Feb-17, 05:49 PM

And here it is 3:00 a.m. and I'm playing with your signature, but that's what it takes for me, at times, for something to really sink in. Wonderful. :)

I forgot one interesting way to zoom in on my signature -- which incidentally is known as the Thue-Morse Sequence (http://mathworld.wolfram.com/Thue-MorseSequence.html).

0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...

Look at every other member:

0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...

Or, look at every 4th member:

0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...

Now, none of these exercises is very interesting for a repeating string, say 0 0 0 0 0 0 0 ... They are self-similar, too, but trivially so. Thue-Morse is cool because it is not repeating. If you stick a decimal in front of it, you get a transcendental (http://mathworld.wolfram.com/TranscendentalNumber.html) number.

It is so non-repeating that you'll never find anywhere within its infinite length any sequence repeating even three times in a row: no 000, no 010101, no 101101101. Yet, when you look within it, using a variety of methods, zooming out and zooming in, you keep finding copies of itself.

Amazingly enough, the sequence is closely related to the good ol' integers. Look what you get if you express each integer in binary and then count the number of 1s that appear, noting whether that count is odd or even, and marking 0 for even and 1 for odd:

Integer Binary 1s-Count Parity Mark

0 0000 0 even 0

1 0001 1 odd 1

2 0010 1 odd 1

3 0011 2 even 0

4 0100 1 odd 1

5 0101 2 even 0

6 0110 2 even 0

7 0111 3 odd 1

8 1000 1 odd 1

9 1001 2 even 0

10 1010 2 even 0

11 1011 3 odd 1

12 1100 2 even 0

13 1101 3 odd 1

14 1110 3 odd 1

15 1111 4 even 0

So although the Thue-Morse sequence has a chaotic nature, it is easy to compute the value of an arbitrary nth member.

That's probably way more than an English major wants to know, but it fascinates me.

01101001

2006-Feb-17, 06:15 PM

Oh, one more thing. Just to tie Thue-Morse back into art, even if it is sort of ugly, dated op-art, I once did a graphic representation of the first 4096 members arranged in a square, using a yellow dot to represent 1 and nothing to represent 0:

1972

(Click to enlarge.)

There are patterns there. There are only two kinds of rows throughout, and only two kinds of columns. It is symmetric about the horizontal, and the vertical, and both diagonals, and rotationally symmetric. Yet, there is no real regularity, just variations on a theme.

Melusine

2006-Feb-17, 09:38 PM

So although the Thue-Morse sequence has a chaotic nature, it is easy to compute the value of an arbitrary nth member.

I trust you on this point. ;)

That's probably way more than an English major wants to know, but it fascinates me.

A long-ago English major, btw, but not all English majors are as innumerate as I am. I was checking out your math link earlier and it was like a vampire seeing the light of day. :eek: I understood your examples up to the box, which I haven't had time to mull over, but thanks. I completely understand your fascination; if you can communicate concepts so that others get it, too, great. (Isn't there some aphorism that says you don't realize how little you know until you're asked to explain something?)

Analogies help, tying it all back to art and nature is even better (for me).

I'll save that link.

snarkophilus

2006-Feb-18, 12:31 AM

So although the Thue-Morse sequence has a chaotic nature, it is easy to compute the value of an arbitrary nth member.

Oh man, that's cool stuff... I can think of another number like that: pi. Easy enough to get the value of an arbitrary decimal place, but totally chaotic. However, I don't know if it has fractal characteristics. And here I thought you had picked that number because in hex it is 69. :)

The Basilisk thing reminds me of the Monty Python sketch where they have the funniest joke in the world, one that causes people to laugh themselves to death... it gets translated into German for use against them in the war, but each translator can only look at one word at a time, or else.

Found a link! http://www.jumpstation.ca/recroom/comedy/python/joke.html

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