NationStates Jolt Archive

Consider this:

http://upload.wikimedia.org/math/d/d/5/dd546c6e374152cd846701ecd67f9719.png

A simple mathematical function, no? Deceptively simple...

Deceptively simple because it invokes the complex number system, derived from the imaginary number (http://en.wikipedia.org/wiki/Imaginary_number) i. When graphed, this basic-looking equation produces a figure of infinite complexity and limitless beauty... the Mandelbrot set (http://en.wikipedia.org/wiki/Mandelbrot_set).

http://upload.wikimedia.org/wikipedia/commons/thumb/5/56/Mandelset_hires.png/322px-Mandelset_hires.png

Now at this point it looks pretty uninteresting. It is a simple black-white dichotomy. Either a point is in the Mandelbrot set, or it's not. The set can be made more interesting with the addition of color:

http://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Mandel_zoom_00_mandelbrot_set.jpg/322px-Mandel_zoom_00_mandelbrot_set.jpg

The graph can have color because the Mandelbrot set is a fractal (http://en.wikipedia.org/wiki/Fractal). Fractals are formed by taking a shape, changing it in some fixed way, and then performing that same change over and over again, ad infinitum. For instance, once could take a triangle, draw an upside-down triangle inside of it, draw right-side-up triangles in each of the resulting triangular sections, etc., etc., etc. (That particular fractal is called a Sierpinski triangle (http://en.wikipedia.org/wiki/Sierpinski_triangle).)

Since the graph is a fractal, the answer to the equation is inexact. The first calculation, or "iteration", produces a sphere; the second, a blobby shape. Continued iterations refine this shape, sharpening and deepening it, making its edges more and more complex. The colors mark the boundaries between different iterations.

Because the equation is irrational, it produces an irrational answer. Just like the digits of pi repeat forever no matter how far you calculate it, the graph of the Mandelbrot set can always be refined, made more accurate and more complex. But, unlike the boring numbers of pi, this infinity can be seen. Seen and explored.

http://www.complexlab.com/Members/nicolaantonucci/immagini/mandelbrot-zoom5.gif

When I said that the Mandelbrot set contained infinite beauty, I was not exaggerating. No matter how deeply you zoom (http://en.wikipedia.org/wiki/Mandelbrot_set#Image_gallery_of_a_zoom_sequence) into its whorls, there will always be more detail, more complexity. To give you an idea of what that means, watch this video (http://www.youtube.com/watch?v=ATWrMlIKRBk).

The first frame of that video is 1x10^89 times bigger than the final frame. To put that into perspective, the ratio between a proton and the known universe is "only" 1x10^26. That fractal is many orders of magnitude deeper than the universe itself. And it goes even deeper than that, forever.

It took a lot to produce that video. Its creator had this to say:

The entire sequence took a full 8 months, using 2, and for the last half 3, systems, all running essentially 24/7 except for a few brief interludes when I rendered other fractal animations. The last stretch, 300 frames, going from E+79 -> E+89 alone took exactly 3 months. As zoom depth increases, so does rendering time! The final frame took 18 hours on my fastest system.

Also of note is the presence of fractals in nature. Since many biological structures are based partly on fractals, many fractal shapes can appear to be natural. Watch this video (http://www.youtube.com/watch?v=G_GBwuYuOOs&mode=related&search=) of a different Mandelbrot zoom to spot forms that look like waves, leaves, brain matter, galaxies, and even sea anemones.

If you'd like to explore Mandelbrot and other fractals as far as your computer can allow, I suggest you download the free program XaoS (http://wmi.math.u-szeged.hu/xaos/doku.php?id=downloads:main). It allows continuous fractal zooming via left-click, and the resolution for any figure can be increased as high as your CPU can handle via Calculation>Iterations in the Menu Bar. It also has a great tutorial program that can explain fractals visually, and much better than I have here. I've had it for a few days now, and I can say that it's quite satisfying to probe deep into the mysterious art hidden in these numbers.

http://www.portablefreeware.com/graphics/xaos/screenshot.gif

Enjoy...

My god... it's full of stars!

My Dad used to be obsessed with fractals about 10 or so years ago, he really did find some strange patterns.

That's not deceptively simple. As an experienced mathematical simpleton I can say with authority that that looked absurdly difficult from the beginning.

http://www.clicksmilies.com/s1106/mittelgrosse/medium-smiley-014.gif

http://i3.photobucket.com/albums/y79/Goomg/other/blahhh/thingyggfhdf.jpg

http://i3.photobucket.com/albums/y79/Goomg/other/blahhh/gfhjkjhgfdfghj-1.jpg

http://i3.photobucket.com/albums/y79/Goomg/other/blahhh/dertyhujikl.jpg

Stop it, you're going to make my brain 'asplode.

Stop it, you're going to make my brain 'asplode.

http://i3.photobucket.com/albums/y79/Goomg/other/blahhh/dfghjklplkjhgf.jpg

I don't get what's the big deal. It's a bunch of mathematically rendered curvy computer graphics. Yawn, we've all seen this a hundred times, there's nothing philosophical or deep about it!


:p

http://www.cringehumor.net/forums/images/smilies/emot-suicide.gif

It's making me think about the universe.. :(

I don't get what's the big deal. It's a bunch of mathematically rendered curvy computer graphics. Yawn, we've all seen this a hundred times, there's nothing philosophical or deep about it!


:p
There's something about the idea of infinite detail, an incredibly complex and beautiful pattern literally wrapped up in itself, never-ending, that's really cool and mildly creepy. And the idea that it's all just numbers, self-evident math -- nothing man-made about it... it's really interesting. :cool:

It's making me think about the universe.. :(

Exactly, watching that video made me realize how incredibly big the universe is. Hence the brain explosion/suicide.

http://www.cringehumor.net/forums/images/smilies/emot-suicide.gif

http://img337.imageshack.us/img337/2535/spiraloy6.png

Yeah, not just the universe, but dude. Like. Everything!

Like I always figure, there's some point - beyond atoms, electrons, quarks - where it stops. Where there's no more smaller particle, like when you look really closely at a wall and there's a point where you just can't see any more. The highest resolution, the maximum zoom. And that level is just the base matter or stuff that the universe is made of. But this made me realize this might not be true. It NEVER STOPS getting smaller! There is an infinite amount of smaller and smaller ... things ... in even the smallest subatomic particle we know of. It just keeps on going. All kinds of weird ... blobby, fucked-up quantum realities beneath each other in an endless series that no matter how well you could possibly see, you could never see them all. Each and every one of us is made up, literally, of an infinity of infinities, and each of those infinities, made up of an infinity of infinities. We - you, me, that bum scratching his ass on the bus - we're all infinite.

And yet, because it works both ways, we're all infinitely small too. We're all no more significant than some quark on an electron on an atom of a molecule of semen. No, much less significant! The universe gapes and gapes and each and every one of us is - nothing - in comparison to the infinite stretches of space, and time, which swallows everything in an endless loop, like an Ouroborous.

I dunno, at first I felt really big and infinite, and then I felt really small and meaningless, and I guess in the end it balances out and I'm just the right size. Like Goldilockses porridge.

Now I just need an infinite woman to eat me.

http://www.nvnews.net/vbulletin/images/smilies/zombie3.gif
I'm already dead. Your fractals don't effect me!

Consider this:

http://upload.wikimedia.org/math/d/d/5/dd546c6e374152cd846701ecd67f9719.png

A simple mathematical function, no? Deceptively simple...

Deceptively simple because it invokes the complex number system, derived from the imaginary number (http://en.wikipedia.org/wiki/Imaginary_number) i. When graphed, this basic-looking equation produces a figure of infinite complexity and limitless beauty... the Mandelbrot set (http://en.wikipedia.org/wiki/Mandelbrot_set).

http://upload.wikimedia.org/wikipedia/commons/thumb/5/56/Mandelset_hires.png/322px-Mandelset_hires.png

Now at this point it looks pretty uninteresting. It is a simple black-white dichotomy. Either a point is in the Mandelbrot set, or it's not. The set can be made more interesting with the addition of color:

http://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Mandel_zoom_00_mandelbrot_set.jpg/322px-Mandel_zoom_00_mandelbrot_set.jpg

The graph can have color because the Mandelbrot set is a fractal (http://en.wikipedia.org/wiki/Fractal). Fractals are formed by taking a shape, changing it in some fixed way, and then performing that same change over and over again, ad infinitum. For instance, once could take a triangle, draw an upside-down triangle inside of it, draw right-side-up triangles in each of the resulting triangular sections, etc., etc., etc. (That particular fractal is called a Sierpinski triangle (http://en.wikipedia.org/wiki/Sierpinski_triangle).)

Since the graph is a fractal, the answer to the equation is inexact. The first calculation, or "iteration", produces a sphere; the second, a blobby shape. Continued iterations refine this shape, sharpening and deepening it, making its edges more and more complex. The colors mark the boundaries between different iterations.

Because the equation is irrational, it produces an irrational answer. Just like the digits of pi repeat forever no matter how far you calculate it, the graph of the Mandelbrot set can always be refined, made more accurate and more complex. But, unlike the boring numbers of pi, this infinity can be seen. Seen and explored.

http://www.complexlab.com/Members/nicolaantonucci/immagini/mandelbrot-zoom5.gif

When I said that the Mandelbrot set contained infinite beauty, I was not exaggerating. No matter how deeply you zoom (http://en.wikipedia.org/wiki/Mandelbrot_set#Image_gallery_of_a_zoom_sequence) into its whorls, there will always be more detail, more complexity. To give you an idea of what that means, watch this video (http://www.youtube.com/watch?v=ATWrMlIKRBk).

The first frame of that video is 1x10^89 times bigger than the final frame. To put that into perspective, the ratio between a proton and the known universe is "only" 1x10^26. That fractal is many orders of magnitude deeper than the universe itself. And it goes even deeper than that, forever.

It took a lot to produce that video. Its creator had this to say:



Also of note is the presence of fractals in nature. Since many biological structures are based partly on fractals, many fractal shapes can appear to be natural. Watch this video (http://www.youtube.com/watch?v=G_GBwuYuOOs&mode=related&search=) of a different Mandelbrot zoom to spot forms that look like waves, leaves, brain matter, galaxies, and even sea anemones.

If you'd like to explore Mandelbrot and other fractals as far as your computer can allow, I suggest you download the free program XaoS (http://wmi.math.u-szeged.hu/xaos/doku.php?id=downloads:main). It allows continuous fractal zooming via left-click, and the resolution for any figure can be increased as high as your CPU can handle via Calculation>Iterations in the Menu Bar. It also has a great tutorial program that can explain fractals visually, and much better than I have here. I've had it for a few days now, and I can say that it's quite satisfying to probe deep into the mysterious art hidden in these numbers.

http://www.portablefreeware.com/graphics/xaos/screenshot.gif

Enjoy...
...

...

You lost me after "Consider this:".

That's not deceptively simple. As an experienced mathematical simpleton I can say with authority that that looked absurdly difficult from the beginning.Finally someone who talks some sense.


The pictures sure are pretty, though. :p


Oh, also: Mandelbrot is German for almond bread. See? Now I totally contributed to the on-topic discussion, too. :)

http://i3.photobucket.com/albums/y79/Goomg/other/blahhh/dfghjkllkjhgf.jpg

http://i3.photobucket.com/albums/y79/Goomg/other/blahhh/dtyuioiuytr.jpg

http://i3.photobucket.com/albums/y79/Goomg/other/blahhh/dfghjklkjhgfd.jpg

I'm having too much fun..
*exits program*

How do you get the colour out of the equation again?

I'm having too much fun..
*exits program*
Ah, but there's more to it than that. Did you see the other fractal types? The Julia sets? The pseudo-3D mode?

There's infinite fun to be had...

This reminds me of the time when I tried to prove the Riemann Zeta hypothesis while cataclysmically stoned...

How do you get the colour out of the equation again?
A fractal isn't really a shape -- it's a process. Like the triangle example I gave. You can keep adding more and smaller traingles, but you'll never be finished. It keeps getting more detail forever.

Same with the Mandelbrot, except more complicated and not symmetrical. You can only have an approximation, not the real thing. For every "iteration" of the graph, it gets more detailed and you can zoom in further. The different colors mark how many iterations it took to get to that "resolution".

http://upload.wikimedia.org/wikipedia/en/9/95/Mandelbrot_bulbs.jpg

That isn't the best picture, but it'll do. The first iteration is a circle, then a blob, then a sharper blob. With each iteration the form of the shape gets closer and closer to the true shape. But the true shape, like the last digit of pi, can never be found.

Old hat. It's mildly interesting, but I'm not moved to thoughts of a cosmic nature.

A fractal isn't really a shape -- it's a process. Like the triangle example I gave. You can keep adding more and smaller traingles, but you'll never be finished. It keeps getting more detail forever.

Same with the Mandelbrot, except more complicated and not symmetrical. You can only have an approximation, not the real thing. For every "iteration" of the graph, it gets more detailed and you can zoom in further. The different colors mark how many iterations it took to get to that "resolution".

http://upload.wikimedia.org/wikipedia/en/9/95/Mandelbrot_bulbs.jpg

That isn't the best picture, but it'll do. The first iteration is a circle, then a blob, then a sharper blob. With each iteration the form of the shape gets closer and closer to the true shape. But the true shape, like the last digit of pi, can never be found.
Ah, all right. So the colours are arbitrarily assigned to determine what stage you're at, then?

This is mad cool. I'm actually looking forward to doing complex numbers now.

Ahhh, Mandelbrot sets. When we first got a computer (a 386, just to prove how old I'm getting...), my dad went out and got this program that demonstrated chaos theory, things like the pool balls moving and stuff. The pool thing was kinda neat, actually, you could adjust the coefficient of friction and put little magnetic thingies all over the place to affect the movement of the balls. But the thing I remember most was being dazzled as only a twelve year old can be by the fact that no matter how much you zoomed in (or out) in the set, the patterns were always beautiful. Trippy, but beautiful.

Staring into the maw of Infinity



Oh yeah? Well your Maw is so infinite, that when she lays on the beach, people shout "Free Willy!"

The Julia sets?:eek:
What are those?

Oh yeah? Well your Maw is so infinite, that when she lays on the beach, people shout "Free Willy!"

/me applauds

Oh yeah? Well your Maw is so infinite, that when she lays on the beach, people shout "Free Willy!"

pwnd!

...
Oh, also: Mandelbrot is German for almond bread. See? Now I totally contributed to the on-topic discussion, too. :)

Or it could be a name. Of a guy called Mandelbrot. Which would still mean almond bread.

Ahhh, Mandelbrot sets. When we first got a computer (a 386, just to prove how old I'm getting...), my dad went out and got this program that demonstrated chaos theory, things like the pool balls moving and stuff.

Our first computer wasn't so modern.

The Mandelbrot set...I believe Piers Anthony made one of his various Modes in the Mode series a real-life universe that was essentially a Mandelbrot Set...it was freaky.

I don't get what's the big deal. It's a bunch of mathematically rendered curvy computer graphics. Yawn, we've all seen this a hundred times, there's nothing philosophical or deep about it!


:p
But it is rather psychotropic.

Or it could be a name. Of a guy called Mandelbrot. Which would still mean almond bread.Er... I kinda assumed these things are named after a guy named Mandelbrot, yes. Which would still mean almond bread indeed. ;p

:eek:
What are those?
I'm not quite sure how those are derived, myself. From how the XaoS tutorial explained it, each fractal is like a "map" of an infinite number of Julia sets.

Practically, it works like this: press "j" to go into "Fast Julia mode", which will place a little window in the corner. Click-and-drag across the fractal, and the mini-Julia will shift and change like a kaleidoscope. Each Julia is based on the point you select -- it will look superficially like the base area, or "seed", but quickly diverges out into a totally different fractal. It's pretty cool. :cool:

Our first computer wasn't so modern.

my parents resisted the siren call as long as they could. I remember sitting in my dad's office at work, dictating my fourth grade geography report. I'm pretty sure he was typing in the DOS text editor...

My husband, on the other hand, had a TI994A. I presume that means something to someone...

I'm not quite sure how those are derived, myself. From how the XaoS tutorial explained it, each fractal is like a "map" of an infinite number of Julia sets.

Practically, it works like this: press "j" to go into "Fast Julia mode", which will place a little window in the corner. Click-and-drag across the fractal, and the mini-Julia will shift and change like a kaleidoscope. Each Julia is based on the point you select -- it will look superficially like the base area, or "seed", but quickly diverges out into a totally different fractal. It's pretty cool. :cool: :eek:

I have a mini me!

And you have no idea how much I wish pressing "j" would have that effect in real life, too. :rolleyes:

:eek:

I have a mini me!
Lawl. I didn't even think of that...

When one stares in to infinity, infinity also stares in to you.

Hey guys, have you ever realized that like, we're all like, molecules? And that wall is to.

Have you ever looked at your hands, man? I mean, REAAAAALYYY looked at your hands.

Here is a good little one for this: http://youtube.com/watch?v=vtmBzoPCTWA

Hey guys, have you ever realized that like, we're all like, molecules? And that wall is to.

Have you ever looked at your hands, man? I mean, REAAAAALYYY looked at your hands.

But have you ever looked at your hands, on weed?!?!?

There are like finger prints that are like valleys man, and tiny little spires of hair that jut out of holes in your skin man.

Rhaomi, thank you for finally providing something that confirms a thought I've had for some years now. It's about time I have something that shows my consideration to be more sane than sanity itself.

And to think that the video only shows the following of one little path. . . to think what one could see if one deviated in every single unending manner.

Which, or course, is here shown to be an impossible feat to "fully" appreciate. Right. Still splendidly delightful!

"The substructure of the universe regresses infinitely towards
smaller and smaller components. Behind atoms we find electrons,
and behind electrons quarks. Each layer unraveled reveals new
secrets, but also new mysteries."


I was playing Alpha Centauri (as the University:) ) earlier and this quote seemed appropriate to this thread.

Non-mathematician that I am, I thought this was a mandelbrot set.

http://kosherfood.about.com/od/passoverdesserts/r/mandel_pesach.htm