Mandelbrot’s Fractals
· If you take a triangle and put another triangle on each of its three sides and then put triangles on the three sides of those triangles and keep on doing that, you get an idea of fractals. Fractals are a curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation. (Are the spirals on the tops of our heads miniature versions of the Milky Way Galaxy?)
· The First Dimension is a line. The second is a square. The third is a cube. Fractals are somewhere in between the square and the cube: a “hidden” dimension found in nature. We can use mathematics—and computers—to write equations for fractals. Nature is mathematical! Infinite complexity explained with simple rules.
· A coastline is a good example fractals. The top of a mountain range is fractal. One tiny portion looks exactly like the whole thing. A tree is fractal. An entire forest looks exactly like the three. Any part of the tree is a tiny version of the entire tree. Branches are like those triangles on the sides of triangles. This is called “self-similarity.”
· The patterns of heartbeats are based on fractal! Blood vessels are fractal! We are NOT machines after all!
· Fractals are like self-replicating networks. Fractals seem to be the underlying pattern of nature. Clouds are fractal. Rivers, too.
· Natural selection seems to be based on fractals. And we can use our knowledge of fractals in practical ways. Cell phones use tiny fractal antennae. The tinier the antenna, the wider the range of frequency. We can detect cancer by looking at “chaotic” blood vessels vs. “orderly” self-similar blood vessels. Fashion designers make “fractal” patterns on clothing using mathematical software. Film editors use fractals to make special effects like lava using mathematical software. Our eyes look at things in a fractal pattern. By mapping that pattern, we can make better safety decisions about the layout of traffic signals and the control panels in our cars.
· Makes you wonder: what other “hidden” laws are at work in the universe?
· Thanks, Beoit Mandelbrot for “discovering” fractals in the early days of computers! Mathematicians called them monsters, but you knew they were beautiful. (There’s a song about it!)
· “I certainly never had the feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science.”
· Check out this website: http://fractalfoundation.org/resources/