Definition of Fractal

Date: 01/17/2005 at 08:15:14
From: Boris
Subject: Fractals

What is the definition of a fractal?  I'm not sure what the definitive
definition is from reading over the posts regarding fractals to date.

I thought that maybe a circle would stretch the limits of the
definition of a fractal?

Date: 01/19/2005 at 12:07:50
From: Doctor Douglas
Subject: Re:  Fractals

Hi Boris.

The key idea in the definition of fractal is "self-similarity".  What
this means is that the object looks the same no matter what power
magnifying glass you use to view it.

In this sense a straight line is a fractal:  if you enlarge it by a
factor of 2X, or a factor of 10X, or a factor of 1000X, or 10^9X, it
still looks like a straight line (of zero width).  In fact it IS a
straight line.  So while it satifies the definition of a fractal in
that it is self-similar (looking like a straight line no matter what
magnification is used), it is not a particularly interesting "fractal".

So to distinguish such familiar shapes as a (one-dimensional) line or
a (two-dimensional) circular disk, often there is an another
defining property to the concept of a fractal, namely, that fractals
possess a non-integer dimension.  Understanding this can involve
some pretty sophisticated mathematics.  But there is an introductory
level (middle-school to high-school) explanation at Cynthia Lanius's
website:

  Cynthia Lanius' Lessons:  A Fractals Lesson
    http://math.rice.edu/~lanius/frac/index.html 

At a higher mathematical level, you can consult

  The Chaos Hypertextbook
    http://hypertextbook.com/chaos/ 

And for word definitions of the term "fractal", you can use the following:

  Fractal
    http://mathworld.wolfram.com/Fractal.html 

  Fractal FAQ
    http://www.faqs.org/faqs/fractal-faq/ 

I hope that this helps.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/