# 3.2] What are fractals? (nonlinear science)

## Description

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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