That's cooooool. I remember the first time I used a Mandelbrot image generator when I was a wee hacker and it blew my mind. After waiting like 10 minutes for each image to come up.
I think I were more spiritually minded I might think there's something to all these fractals, what with them emerging naturally from simple mathematical relationships. Its like the underlying nature of everything is fractal, man... it's all interconnected...
Mathematics is like democracy. It's the worst possible language for describing reality, except compared to all of the alternatives.
All our mathematical descriptions so far have at least one major feature that is perfectly unlike the universe we are trying to describe. In the case of our favourite 2nd order equations it's those embarrassing advanced waves. In the case of fractals it's true, scale-independent, scale invariance.
There may be one or more mathematical descriptions of reality that doesn't contain these huge extraneous features that have to be chopped off by hand based on our non-mathematical knowledge about the world, but there is little motivation for that belief in the actual history of mathematical physics.
Well yeah, ultimately the only complete mathematical representation of the universe IS the universe. We can only ever approximate or make generalisations... at least without building another universe for doing experiments in.
In truth it's not that fractals underlie everything, its that they're just another pattern that emerges from mathematical relationships. They emerge in nature because collections of plant cells, for example, are a really specialised computer for making plants that happens to have some recursive fractal parts to its algorithm.
Mathematics is a language, invented by humans to describe the universe. To say that the universe is a complete mathematical representation of itself makes as much sense as saying the universe is a complete English representation of itself (which would be silly, because everyone knows the universe is written in Latin, or possibly Hebrew...)
Saying cells are "really" computers using recursion is exactly as meaningful as saying cells are "really" a choir singing a round. One metaphor is more useful, yes, but both are equally meaningful.
My favourite example of this kind of thing is Richard Dawkin's claim that viruses are "really" biological chain letters.
I could, and in the past have, argued this point at length (Someone is wrong on the Intenet! Oh noes!) so I'll stop here. Feel free to go back to being wrong :-D
I think you have thought about this a lot harder than me and clearly have a lot more invested in the subject. :)
You're absolutely right, of course. I don't mind admitting I'm a lazy writer in LJ comments!
I'm sure you know what I was getting at--not that mathematics underpins everything, more that if you wanted to specify the universe entirely mathematically you'd been a universe's-worth of mathematics. It's incomplete. Just one useful way of looking at things, as you say.
Actually I didn't get that you were saying "if you wanted to specify the universe entirely mathematically you'd been a universe's-worth of mathematics." I pretty much agree with this, although I've spent a depressing amount of time arguing with mathematicians of a Platonic bent who would vehemently disagree with it. Thus my investment :-P
I wrote my own generator to create this animation, so I did a fair amount of waiting this time around as well. The fastest images generated in a few seconds, but ones towards the end, especially when they had a lot of black pixels, took up to half an hour each.
I'm not quite sure what you're asking, so I'll basically tell you what I did and hope this covers your question.
Every frame of the animation has the same center, approximately -1.75+0i. The center, starting magnification, and ending magnification were all determined experimentally. The height of the first frame is around 180 and the height of the last frame is on the order of 10-12. Unfortunately, I don't have the exact numbers in front of me at the moment. The zoom from each frame to the next was calculated based on the duration of each measure of music. I can't really claim that the calculation I used made sense, but I played around with it until I got results I liked.
For each frame of the animation, each pixel corresponds to a complex number c. For each c, I iterated zi+1=zi2+c. If the magnitude of zi ever exceeds 2, the pixel is colored based on i. The hue is set by i mod 256, giving the rainbow effect with red corresponding to an i that's a multiple of 256. The saturation starts at 100% and decreases to 0% (pure white) at i = 8192.
As an aside, my original intent was to just vary the hue and get rainbow striping. This works when i does not change much between adjacent pixels, but as the magnification increases, i starts changing too fast and the picture looks less like rainbows and more like confetti. I originally intended the fade to white to cover the confetti, but found that the all white background was a convenient stopping point for the animation as well.
Pixels are colored black if |zi| does not exceed 2 for a large value of i. The limit where I colored the pixel black depends on the magnification, and is calculated as π divided by the distance between two adjacent points, but not greater than 2 million. (The point -0.75+0i is on the left edge of the central heart shape of the Mandelbrot set. It is in the Mandelbrot set, but -0.75+ai is not in the Mandelbrot set for any a>0. For the point -0.75+ai, the value of i where |zi|>2 is given by π/a. I decided that was as good a formula as any to determine the limit to color pixels black.)
There are many optimizations that can be applied to this calculation, but the only one I bothered with was checking for cycling. For most (maybe all, I don't know) points in the Mandelbrot set, zi converges on a pattern. This is not necessarily a single value. Often it's a repeating sequence of values. I checked for repeated numbers, and if one was found, I immediately bailed out and colored the pixel black.
With all that in place, I started my program running, and about 48 hours later I had 1200 images which I then converted to this animation.
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Date: 2008-07-07 05:59 pm (UTC)I think I were more spiritually minded I might think there's something to all these fractals, what with them emerging naturally from simple mathematical relationships. Its like the underlying nature of everything is fractal, man... it's all interconnected...
no subject
Date: 2008-07-07 06:31 pm (UTC)All our mathematical descriptions so far have at least one major feature that is perfectly unlike the universe we are trying to describe. In the case of our favourite 2nd order equations it's those embarrassing advanced waves. In the case of fractals it's true, scale-independent, scale invariance.
There may be one or more mathematical descriptions of reality that doesn't contain these huge extraneous features that have to be chopped off by hand based on our non-mathematical knowledge about the world, but there is little motivation for that belief in the actual history of mathematical physics.
no subject
Date: 2008-07-07 06:55 pm (UTC)In truth it's not that fractals underlie everything, its that they're just another pattern that emerges from mathematical relationships. They emerge in nature because collections of plant cells, for example, are a really specialised computer for making plants that happens to have some recursive fractal parts to its algorithm.
no subject
Date: 2008-07-08 02:33 pm (UTC)Mathematics is a language, invented by humans to describe the universe. To say that the universe is a complete mathematical representation of itself makes as much sense as saying the universe is a complete English representation of itself (which would be silly, because everyone knows the universe is written in Latin, or possibly Hebrew...)
Saying cells are "really" computers using recursion is exactly as meaningful as saying cells are "really" a choir singing a round. One metaphor is more useful, yes, but both are equally meaningful.
My favourite example of this kind of thing is Richard Dawkin's claim that viruses are "really" biological chain letters.
I could, and in the past have, argued this point at length (Someone is wrong on the Intenet! Oh noes!) so I'll stop here. Feel free to go back to being wrong :-D
no subject
Date: 2008-07-08 05:18 pm (UTC)You're absolutely right, of course. I don't mind admitting I'm a lazy writer in LJ comments!
I'm sure you know what I was getting at--not that mathematics underpins everything, more that if you wanted to specify the universe entirely mathematically you'd been a universe's-worth of mathematics. It's incomplete. Just one useful way of looking at things, as you say.
no subject
Date: 2008-07-09 03:03 am (UTC)Thanks for the clarification!
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Date: 2008-07-08 01:08 am (UTC)no subject
Date: 2008-07-09 03:19 am (UTC)no subject
Date: 2008-07-09 04:35 pm (UTC)Every frame of the animation has the same center, approximately -1.75+0i. The center, starting magnification, and ending magnification were all determined experimentally. The height of the first frame is around 180 and the height of the last frame is on the order of 10-12. Unfortunately, I don't have the exact numbers in front of me at the moment. The zoom from each frame to the next was calculated based on the duration of each measure of music. I can't really claim that the calculation I used made sense, but I played around with it until I got results I liked.
For each frame of the animation, each pixel corresponds to a complex number c. For each c, I iterated zi+1=zi2+c. If the magnitude of zi ever exceeds 2, the pixel is colored based on i. The hue is set by i mod 256, giving the rainbow effect with red corresponding to an i that's a multiple of 256. The saturation starts at 100% and decreases to 0% (pure white) at i = 8192.
As an aside, my original intent was to just vary the hue and get rainbow striping. This works when i does not change much between adjacent pixels, but as the magnification increases, i starts changing too fast and the picture looks less like rainbows and more like confetti. I originally intended the fade to white to cover the confetti, but found that the all white background was a convenient stopping point for the animation as well.
Pixels are colored black if |zi| does not exceed 2 for a large value of i. The limit where I colored the pixel black depends on the magnification, and is calculated as π divided by the distance between two adjacent points, but not greater than 2 million. (The point -0.75+0i is on the left edge of the central heart shape of the Mandelbrot set. It is in the Mandelbrot set, but -0.75+ai is not in the Mandelbrot set for any a>0. For the point -0.75+ai, the value of i where |zi|>2 is given by π/a. I decided that was as good a formula as any to determine the limit to color pixels black.)
There are many optimizations that can be applied to this calculation, but the only one I bothered with was checking for cycling. For most (maybe all, I don't know) points in the Mandelbrot set, zi converges on a pattern. This is not necessarily a single value. Often it's a repeating sequence of values. I checked for repeated numbers, and if one was found, I immediately bailed out and colored the pixel black.
With all that in place, I started my program running, and about 48 hours later I had 1200 images which I then converted to this animation.
no subject
Date: 2008-07-11 04:17 pm (UTC)