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3.2] What are fractals? (nonlinear science)




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This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

3.2] What are fractals? (nonlinear science)

One way to define "fractal" is as a negation: a fractal is a set that does not
look like a Euclidean object (point, line, plane, etc.) no matter how closely
you look at it. Imagine focusing in on a smooth curve (imagine a piece of
string in space)--if you look at any piece of it closely enough it eventually
looks like a straight line (ignoring the fact that for a real piece of string
it will soon look like a cylinder and eventually you will see the fibers, then
the atoms, etc.). A fractal, like the Koch Snowflake, which is topologically
one dimensional, never looks like a straight line, no matter how closely you
look. There are indentations, like bays in a coastline; look closer and the
bays have inlets, closer still the inlets have subinlets, and so on. Simple
examples of fractals include Cantor sets (see [3.5], Sierpinski curves, the
Mandelbrot set and (almost surely) the Lorenz attractor (see [2.12]).
Fractals also approximately describe many real-world objects, such as clouds
(see http://makeashorterlink.com/?Z50D42C16) mountains, turbulence,
coastlines, roots and branches of trees and veins and lungs of animals.

"Fractal" is a term which has undergone refinement of definition by a lot of
people, but was first coined by B. Mandelbrot,
http://physics.hallym.ac.kr/reference/physicist/Mandelbrot.html, and defined
as a set with fractional (non-integer) dimension (Hausdorff dimension, see
[3.4]). Mandelbrot defines a fractal in the following way:

A geometric figure or natural object is said to be fractal if it
combines the following characteristics: (a) its parts have the same
form or structure as the whole, except that they are at a different
scale and may be slightly deformed; (b) its form is extremely irregular,
or extremely interrupted or fragmented, and remains so, whatever the scale
of examination; (c) it contains "distinct elements" whose scales are very
varied and cover a large range." (Les Objets Fractales 1989, p.154)

See the extensive FAQ from sci.fractals at
<ftp://rtfm.mit.edu/pub/usenet/news.answers/

 

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