Note from your fractalist: Sorry, folks, this one is a day late. I discovered early yesterday that my old website had been hacked. It has been fixed, now, although I plan to eventually remove everything from there, and repost it here somewhere. Just getting the bad scripts out has kept me plenty busy. Never fear, I did finish the Friday Fractal. Other (current) posts are forthcoming. -K
I’m not the only one around here who gets into fractals. I’ve noticed a few other science bloggers occasionally blog on the topic. Mark, over at Good Math, Bad Math, has been working on a series describing the various types of fractals. He’s been quite comprehensive, so far:
- An Introduction to Fractals
- Fractal Dimension
- The Mandelbrot Set
- Julia Set Fractals
- The Sierpinski Gadget by Affine
- Iterated Function Systems and Attractors
- Fractal Pathology: Peano’s Space Filling Curve
- Fractal Borders
- Fractal Dust and Noise
- Fractal Mountains
- Fractal Curves and Coastlines
You’ll notice he discusses many of the types of fractals you’ve seen here on Fridays. With Mark going over the math, I can talk a bit more about the aesthetics of fractals. Or, at the very least, I can share a bit of the strange visual convergence that happens when I mix fractal types.
Let’s begin with something simple, like the Mandelbrot set. With various other fractal sets lying within its borders, it always makes a good beginning for complex beauty:
Next, let’s zoom in on the upper limb. (I’ll try to point out a more precise location, later.)
Now, let’s play. Add in a bit of noise (fBm noise, in paticular) which will seem to make the pattern more chaotic, perhaps even less aesthetic:
Finally, let’s add one more twist, with a transforming formula called "TIA glass". Apparently, it must be a complex one, for the author, Damien M. Jones, said "I think I must be insane" when he tried to describe it. The formula, which he suggests has little redeeming mathematical value, but much artistic value, iterates the Mandelbrot set, then "uses triangle inequality average coloring as a turbulence value." It is turbulent, I’ll agree. This transformation distorts the entire fractal radially from a point in the center, rippling outwards. (You can see this effect by clicking the image below for a full-size image.) "Outwards" we find patterns which are familiar, yet alien. It almost appears to be the surface a cratered planet, swept by winds or water:
Painted Craters: An Abstract Fractal Landscape
Strange, isn’t it? This is simply a few complex patterns, each nearly chaotic in form, interacting with one another. If we zoom back out, we can see we haven’t really left the Mandelbrot set. (The black arrow marks the spot we were just visiting.)
It takes a change in perspective, a leap in scale, to see that this really is just a fractal.
As I hinted at the beginning of this post, I’m not the only one around here to enjoy discovering fractals. Chris over at Mixing Memory did a piece a while back, discussing the way Jackson Pollock created fractal forms in his abstract art. Chris even suggests why we may be so fond of such forms: "It might be that a general principle of neuroaesthetics is that people prefer natural (though perhaps exaggerated) patterns to unnatural ones, and chaotic patterns just look more natural."
Sandra of Omnibrain, another fan of fractal art, sent me a link to a delightful animation, called Orthologia Twist, by Mark Dow. (Sandra says, "Mark Dow works on animation software for brain imaging at U of Oregon. Cool stuff.") If you didn’t get enough zoom with the fractal above, be sure to check this one out. I’ve been using it as a bit of inspiration for the story I’m working on... if only there were a way to put an animated image on the wall....
Fractals created by the author using ChaosPro.
Putting an animated image on a wall is simple; just use a LCD connected to your computer. Maybe a tad expensive but would work.
The image you named painted craters looks like frosted glass (after a very hard frost) to me, I need a better imagination.
I also speculate that there is a neuroaesthetic basis for our enjoyment of fractals. Visual perception is tuned for interpreting similarity (scale and rotational invariance) of known objects. For example we need to recognize the actual size and orientation of objects no matter their size or orientation on the retina. Fractals have this self-similarity in spades.
I'm waiting for a dedicated animated T-shirt, using LED's. VGA resolution would do the trick. Ultimate bling.
Chris, your imagination is fine and healthy; it's a nice comparison. Actually, I think "Frosted Craters" might have made an excellent title.
Mark, it's sort of a chicken-and-egg situation, isn't it? Which came first, fractal patterns in nature or the perception of the fractals? I've often thought the recognition of scaling, self-similar patterns in nature would offer a useful advantage. Perhaps, in addition to adapting to the spectrum of light coming from our yellow sun, eyes are also adapting to fractals.