(Figure 1) An example of the Mandlebrot set.
Where did the term "fractal" come from?
Mandelbrot coined the word "fractal" to describe his new object and those like it. He argued
that the edge of the set was more than a line (of dimension 1) and less than an area
(of dimension 2). He claimed it had a dimension somewhere between the two. A fractional dimension.
(Figure 1) An example of the Mandlebrot set.
What is a fractal?
From the man himself- Mandelbrot says:
Fractal, in mathematics, is a geometric shape that is complex and detailed in
structure at any level of magnification. Often fractals are self-similar—that is,
they have the property that each small portion of the fractal can be viewed
as a reduced-scale replica of the whole. One example of a fractal is the
snowflake" curve constructed by taking an equilateral triangle and
repeatedly erecting smaller equilateral triangles on the middle third of the
progressively smaller sides. Theoretically, the result would be a figure of finite area but with a perimeter of infinite length, consisting of an infinite number
of vertices. In mathematical terms, such a curve cannot be differentiated. Many such self-repeating figures can be constructed, and since they first
appeared in the 19th century they have been considered as merely bizarre.
A turning point in the study of fractals came with the discovery of fractal geometry by the Polish-born French mathematician Benoit B. Mandelbrot in the 1970s. Mandelbrot adopted a much more abstract "definition of dimension than that used in Euclidean geometry, stating that the dimension of a fractal must be used as an exponent when measuring its size. The result is that a fractal cannot be treated as existing strictly in one, two, or any other whole-number dimensions. Instead, it must be handled mathematically as though it has some fractional dimension. The "snowflake" curve of fractals has a dimension of 1.2618. Fractal geometry is not simply an abstract development. A coastline, if measured down to its least irregularity, would tend toward infinite length just as does the "snowflake" curve. Mandelbrot has suggested that mountains, clouds, aggregates, galaxy clusters, and other natural phenomena are similarly fractal in nature, and fractal geometry's application in the sciences has become a rapidly expanding field. In addition, the beauty of fractals has made them a key element in computer graphics. Fractals have also been used to compress still and video images on computers. In 1987, English-born mathematician Dr. Michael F. Barnsley discovered the Fractal Transform(TM) which automatically detects fractal codes in real-world images (digitized photographs). The discovery spawned fractal image compression, used in a variety of multimedia and other image-based computer applications. [cite]
Why the study of fractals is so new: The calculations involved are repetitive, boring and number in the millions. To produce the Mandelbrot Set on a single screen takes, it is estimated, about 6,000,000 calculations. [cite]
Why does it take so much calculation?
Fractals are made through iterating an equation. That is, you begin with a simple equation
and take the answer you get when you solve it the first time and use it as a variable in
the same equation when you solve it the second time. You keep feeding your answer back
into the equation, so that the result is constantly changing. Since the answer is
constantly changing, the system isn't predictable. But since it is based
on a simple equation, it is orderly. The other part is that since the
equation keeps going, you can "zoom" in on part of the answer and it is also unique.
Similar, but different.
(Figure 2) An example of the Mandlebrot set.
(Figure 3) A zoom of the area in Figure 2.
(Figure 4) A zoom of the area in Figure 3.
What are "strange attractors"?
Chaos theory is riddled with strange patterns which underlie seemingly random and unpredictable behaviour. Although we don't know which weather
pattern will occur ten days down the track, there are weather patterns which are possible, and likely, and others which are not. The likely patterns -
snow in Antarctica, heat in India - are called ATTRACTORS. These patterns ATTRACT the system into their state. In examining the process of the
iteration of many non-linear equations it is found certain patterns occur and often lead to some kind of bounded behaviour. These are the attractors for
the equation.
[cite]
What if I have other questions?
Try the Fractal FAQ (Frequently Asked Questions)
This answers questions like:
Spanky's is probably the most well-known fractal site on the web.
Dick Oliver has written a number of entertaining articles on fractal graphics and chaos science, many of which can be found on the web.
Chaos in the classroom. A way to get math students interested in complex mathematics. Includes instructions and explanations for how to play the chaos game and what it all means. Excellent reference!
Carlson's Fractal Gallery is a nice place to go look at really cool pictures.
Another gallery that has a different look than most fractal sites is 3D Strange Attractors and Similar Objects.
Chaos & Fractals is a good place for more explanation, in fairly easy to understand terms, of chaos theory and fractals. You could also try the Chaos theory, dynamic systems, and fractal geometry site. It has definitions, etc. in easy terms.
Scientific American has a lot of articles on the cusp on intellectualism. Right on that cusp are articles on chaos and associated topics.
Antichaos and Adaptation : Biological evolution may have
been shaped by more than just natural selection. Computer models suggest that certain complex
systems tend toward self-organization.
"Has chaos theory found any useful application in the social sciences?"
From Complexity to Perplexity: Can science achieve a unified theory of complex systems?
Chaos Lab at Georgia Tech. A cool site form my alma mater, if you like geekin'. (And if you don't, there's a pretty cool pem to read.)
Michael Bayne explains how to draw a fractal, in easy to understand algorithms.
Dynamical Systems and Technology Project: This project is a National Science Foundation sponsored project designed to help secondary school and college teachers of mathematics bring contemporary topics in mathematics (chaos, fractals, dynamics) into the classroom, and to show them how to use technology effectively in this process. At this point, there are several interactive papers available. These are designed to help teachers understand the mathematics behind such topics as iterated function systems (the chaos game) and the Mandelbrot and Julia sets.
Clifford Pickover has written many books on math, complexity, etc. His primary interest is "finding new ways to continually expand creativity by melding art, science, mathematics and other seemingly-disparate areas of human endeavor." This site has pictures and a bibliography of his publications.
A lawyer discusses the philosophical implications of chaos theory on the philosophy of science and life in general. Also provides a good history of Mandelbrot, chaos and fractal basics.
Society for Chaos Theory in Psychology and the Life Sciences: The Society is an international forum bringing together researchers, theoreticians, and practitioners interested in applying dynamical systems theory, far-from-equilibrium thermodynamics, self-organization, neural nets, fractals cellular automata, and related forms of chaos, catastrophes, bifurcations, nonlinear dynamics, and complexity theories to psychology and the life sciences. Our members hail from numerous specialties within psychology and the social sciences as well as from biology, physiology, neuroscience, mathematics, philosophy, physics, computer science, economics, education, management, political science, engineering, and the world of art. As of January 1997 we have approximately 300 members worldwide.
COMPLEXITY INTERNATIONAL:An Electronic Journal of Complex Systems Research
Complexity International is a refereed journal for scientific papers dealing with any area of complex systems research. The theme of the journal is the
field of complex systems, the generation of complex behaviour from the interaction of multiple parallel processes. Relevant topics include (but are not
restricted to): artificial life, cellular automata, chaos theory, control theory, evolutionary programming, fractals, genetic algorithms,
information systems, neural networks, non-linear dynamics, and parallel computation. Papers dealing with applications of these topics (for
example, to biology, economics, epidemiology, sociology) are also encouraged.
Complexity On-line: Complexity On-line is a scientific information network about complex systems. You will find access to sites, publications (including the Complexity On-line Journal ) and other sources of interest.
Fractint: A FREE software program for exploring fractals. It's the one that everybody refers to and uses.
Fractal Music Lab: All about software that allows you to make fractal music (ordered, but unpredictable.). Examples, too, if you have the Crescendo plug-in. (More about that on the site.)
Quantum Mechanics, Chaos and the Conscious Brain: An article to appear in Journal of Mind and Behavior. By Chris King - Mathematics Department, University of Auckland.
Chaos, Fractals, and Arcadia. Arcadia is a play by Tom Stoppard about two mathematicians and includes many aspects of chaos theory in the play, both as theme and plot. This is a brief introduction and description of the play.
WEB AND PRINT BIBLIOGRAPHIES
Web AND print
Web
Web
Web
Web: All kinds of math links
Print
Web and print, from the University of Maryland. Actual academic articles, as well as general references.
Print: A search engine for print articles. Just type in "chaos".
Web
Web
Web: Fractals, chaos, complexity, and nonlinearity.
Web: Nonlinear sites.
Print: Many, many general reference books.