Remember all those funky-looking abstarct patterns (that looked like paisley patterns)? These were visualization of fractal )sp.?) mathematical series. One interesting prpoerty of these fractals is the fact that whether at high or low magnifications, the pattern looks the same-supposedly this would be of use in storing information. Another thing-fractal series generate shapes that look like those found in nature. Well, it’s been several years now, and nobody seems to have found any use for these things (except for odd-looking screen savers, and psychidelic art). My questions:
(a) are there reall applications for these things and
(b) can I go out and buy a fractal-based memory chip, that will store a complete set of Encyclopedia brittanica in the head of a pin?
You can’t store the Britannica fractally and get anything useful back, because fractal compression (like JPEG) is a “lossy” process; what you get out is not exactly what you put in. But it’s definitely useful for photgraphs and the like. You can get a fractal-compression plug-in for Adobe Photoshop at Altamira Group. See also Fractal Image Compression and The Data Compression Library - Fractals.
There are applications for fractals in many areas of science and mathematics.
Mostly, it’s not so much that the fractals themselves have practical use, it’s that they represent other things which do have application. For instance, if you have two stationary black holes, and drop objects into them from various locations, which hole the object ends up in can be represented by a fractal pattern. Ok, so that’s maybe still not too useful, but it’s an application.
I think they really weren’t ever as useful as claimed.
Some “examples” from nature were that canyons viewed from high above look like simple gullies viewed from lower down; that ocean waves from high above look like extra-choppy waves in a pond; that from a certain distance trees look like leaves and flower parts resemble trees.
A lot of this just isn’t really true, except in a vague way. It’s pretty easy to tell from a cropped photo whether the picture is of an ocean or a pond. Some characteristics of water waves simply do not expand uniformly to any scale.
One problem with pure math is it’s hard to explain to journalists as worth doing. They keep asking what it’s for, what the applications are. So mathematicians keep pulling applications out of the air.
We saw a similar effect when the Rubik’s cube came out, with people saying it could aid the study of group theory. No one really tried to do that; it was just a random guess.
The same thing happened with “threshold planes” that appeared a few years ago. They were 3-axis plots that looked like twisted paper, viewed on end. They really had no basis in practicality, but when pushed, the mathematicians said well this one could be plotting a stock market collapse and that one a star that captures a twin.
The other mathematicians were left chuckling at what the press would make of it.
That’s one of my problems with the profession. I used to teach college math, abstract algebra and projective geometry, etc. None of the texts had real world examples from engineering or physics. People who write math texts quickly divert their attention to proofs rather than applications. At one time I constructed a set of simple computer-animated “cartoons” to demonstate the connection of sine waves, regression, etc. to the real world, plotting things like point lights stuck to various parts a bicycle as it rolled along, trying to make wave amplitude visual and the secant more physical.
Anyway, to sum up, I think they really weren’t ever as useful as claimed.
I don’t know how useful fractal equations and whatnot have been in the utilitarian practical sense, but philosophically they have caused many of us to rethink and reconsider an old confrontation between determinism and free will.
To be determined (caused) by some initial condition or set of conditions always implied (to most of us) the following parameters:
• lack of freedom
• predictability
• lack of variation (sameness)
• lack of the beauty of spontaneity
Meanwhile, to arise spontaneously (free will) and NOT as the result of any initial causative circumstances always implied (to most of us) the following parameters:
• lack of external control (unaffected)
• ultimately unknowable (nothing intended)
• chaotic (lack of pattern)
• lack of the beauty of discernable meaning
There were those of us who took sides with regards to human nature and the nature of existence overall, not to mention the relationship of the species human to the abstract cause-and-meaning paradigm called “God”. I have always come down solidly on the pro-freedom, anti-determinism side of the conceptual fence, preferring no structure to the potential for the wrong imposed structure and needing to declare space in which for the participant to evaluate and consider the question of voluntary participation or deliberate nonparticipation in the current pattern of things.
Along comes Mandelbrot (well, OK, it takes awhile for the news to circulate) and suddenly we’ve got a deterministic system that creates/defines an unpredictable nonlinear patterned but infinitely variegated result and it is beautiful, chaotic, recognizable, meaningful, free, variable, self-replicating, yet ever-changing. WHOA!!! So suddenly, if not quite embracing causal explanations, I am at least saying that traditional causal explanations are not WRONG so much as they are incomplete, that somehow there are truths I have missed by being on my side of the fence that were accurately grasped by thinkers who believed in determinism, yet there are also and equally truths that I and my colleagues were right about that were always denied by the deterministic camp.
AND, I might add, the effect has been similarly astonishing over there! Radical empiricists, devotees of stepwise regression statistical analyses, neoSkinnerians and even sociologists have found themselves rethinking individual circumstance, uncategorizable situational motivation, immeasurably small but possibly critical important causal factors that cannot be discerned by mass sampling and categorization and statistical regressing, and so on.
I wish I could say it has brought together formerly opposing wings of conceptual theoretical thought, but nothing is ever that simple. What is has done is create a rift between the old fogey establishment (pre-fractal, pre-chaos theory) and newer folks who are probably more willing to read stuff from across the divide.
As a radical social theorist, I was definitely trying to attain a grasp of the thinking within physics and mathematics, finding more interesting and stimulating stuff in the “peritheoretical penumbra” surrounding the heavy math there than I was finding in the dull grey hoary theories popular within sociology.
But it also made me reconsider what I might see if I were to do quantitative statistical analyses which previously I had had no interest in.
And in addition to what AHunter3 says, they’re a lot of fun!
See http://spanky.triumf.ca/www/fractint/fractint.html for what is probably the best fractal program around.