Ivgenny," by
FROM THE WEBSITE "": "A fractal is a geometric figure with
two special properties. First, it is irregular, fractured, fragmented,
or loosely connected in appearance. Second, it is self-similar; that
is, the figure looks much the same no matter how far away or how close
up it is viewed. The term fractal was invented by Polish French mathematician
Benoit Mandelbrot (1924– ) in 1975. He took the word from the Latin
word fractus, which means "broken." The idea behind
fractals is fairly simple and obvious when explained. But the mathematics
used to develop those ideas is not so simple. Most objects in nature
do not have simple geometric shapes. Clouds, trees, and mountains, for
example, usually do not look like circles, triangles, or pyramids. Instead,
they can best be described as fractals. Fractals are used by geologists
to model the meandering paths of rivers and the rock formations of mountains;
by botanists to model the branching patterns of trees and shrubs; by
astronomers to model the distribution of mass in the universe; by physiologists
to model the human circulatory system; by physicists and engineers to
model turbulence in fluids; by economists to model the stock market
and world economics." [CHP editor's
note: And by movie special effects and CGI specialists to create
natural-looking artificial landscapes, and artists like Janet Parke
(see image above) to produce gorgeous abstract images.]
Most of the links below point to scientific discussions, but many point to fractal art, as opposed to fractals as used in "practical" applications--astrophysics, materials engineering, etc. But they are all enlightening and, in many cases, very beautiful.
Michael Frame, Benoit Mandelbrot, and Nial Neger, Fractal Geometry (Yale University, December 3, 2004)
Truly amazing. This is an on-line collection of pages meant to support a first course in fractal geometry for students without especially strong mathematical preparation, or any particular interest in science. In includes a vast array of practical explanations and other resources, working Java applications, etc., covering a vast array of applications of fractal theory.
NOVA: "Hunting the Hidden Dimension." One-hour video divided into five chapters. This documentary highlights a host of filmmakers, fashion designers, physicians, and other researchers who are using fractal geometry to innovate and inspire. Also lots of textual material. Original PBS Broadcast Date: October 28, 2008.
MAIN LINK:
1. "Fractal Basics."
2. "The Mandelbrot Set."
3. "On the Defense."
4. "Fractals in the Body."
5. "Nature's Fractal Nature."
Internet Resources for Fractals (an extensive web bibliography with links)
What is a fractal?
The Fractal Geometry of the Mandelbrot Set
A Mandelbrot set animation
from http://members.rogers.com/ender.othc/images/mandelzoom.mpeg
And the created mathematically via a Java applet (pretty neat!)
http://www.thorsen.priv.no/services/mandelbrot/
Fractal bacterial growth
Fourth International Symposium: Fractals in Biology and Medicine, Ascona, March 10-14, 2004
Fractals in Human Physiology
About fractals
How fractal art images are made
Fractals, in Layman's Terms
Fractal Art Images
More fractal images
Fractal Easter-egg designs (these images are created entirely by a set of mathematical parameters, with no artistic post-processing).
How artists actually do this stuff--part of a tutorial on a program called UltraFractal).
Return to "Clausewitz and Complexity"
Return to The Clausewitz Homepage
The Mandelbrot Set