**"Ivgenny," by Janet Parke**

**FROM THE WEBSITE "SCIENCE CLARIFIED":** "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– 2010) 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.]

**NOVA Video: "Hunting the Hidden Dimension." Below. This is the best introduction to fractals we've seen. **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. Click on the picture below to watch the video on YouTube. The links below that go the the video's website at PBS.

**MAIN LINK: **http://www.pbs.org/wgbh/nova/fractals/program.html

1. **"Fractal Basics."** http://www.pbs.org/wgbh/nova/programs/ht/tm/3514.html?site=34&pl=wmp&rate=hi&ch=1

2. **"The Mandelbrot Set."** http://www.pbs.org/wgbh/nova/programs/ht/tm/3514.html?site=34&pl=wmp&rate=hi&ch=2

3. **"On the Defense."** http://www.pbs.org/wgbh/nova/programs/ht/tm/3514.html?site=34&pl=wmp&rate=hi&ch=3

4. **"Fractals in the Body."** http://www.pbs.org/wgbh/nova/programs/ht/tm/3514.html?site=34&pl=wmp&rate=hi&ch=4

5. **"Nature's Fractal Nature."** http://www.pbs.org/wgbh/nova/programs/ht/tm/3514.html?site=34&pl=wmp&rate=hi&ch=

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. The background images for most of the *ClausewitzStudies.org* website are fractal; the current page shows the famous "Mandelbrot set."

**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.

*http://classes.yale.edu/fractals/*

Internet Resources for Fractals (an extensive web bibliography with links)

*http://math.fullerton.edu/mathews/c2003/FractalBib/Links/FractalBib_lnk_1.html*

What is a fractal?

*http://www.laubender.de/fractaline/what_is_a_fractal.htm*

The Fractal Geometry of the Mandelbrot Set

*http://math.bu.edu/DYSYS/FRACGEOM/FRACGEOM.html*

A Mandelbrot set animation from *http://members.rogers.com/ender.othc/images/mandelzoom.mpeg*

And the same set created mathematically via a *Java* applet (pretty neat!)

*http://www.thorsen.priv.no/services/mandelbrot/*

Fractal bacterial growth

*http://classes.yale.edu/Fractals/Panorama/Biology/Bacteria/Bacteria2.html*

Fourth International Symposium: Fractals in Biology and Medicine, Ascona, March 10-14, 2004

*http://www.fractals.issi.cerfim.ch/*

Fractals in Human Physiology

*http://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.html*

About fractals

*http://www.infinite-art.com/aboutfractals.html*

How fractal art images are made

*http://www.infinite-art.com/aboutfractals.html*

Fractals, in Layman's Terms

*http://www.fractalus.com/info/layman.htm*

Fractal Art Images

*http://perso.wanadoo.fr/imagitheque/*

More fractal images

*http://gnofract4d.sourceforge.net/gallery1.html*

Fractal Easter-egg designs (these images are created entirely by a set of mathematical parameters, with no artistic post-processing).

*http://www.parkenet.org/jp/ufeggs.html*

How artists actually do this stuff—part of a tutorial on a program called UltraFractal).

*http://www.parkenet.org/jp/ufscratch.html*

**The Mandelbrot Set: See video above, image below.**

**The link in the line above goes to the Wikipedia article on Fractals.**

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