What Kind of Math are Fractals?

Date: 10/23/2004 at 20:16:37
From: Lauren
Subject: What branch of mathematics do fractals belong?

Can you tell me what branch of mathematics includes fractals?

Date: 10/25/2004 at 17:45:43
From: Doctor Douglas
Subject: Re: What branch of mathematics do fractals belong?

Hi Lauren.

Thanks for writing to Dr. Math.  Excellent question!  

If I interpret your question literally, then my answer is this: 
fractals are geometric objects in the same way that lines and 
triangles and spheres and ellipses are geometric objects.  Each of 
these geometric figures possesses certain properties that make them 
useful or interesting for one reason or another.  For example, the 
interior angles of a triangle add up to 180 degrees.  Or, the sphere 
gives the maximum volume enclosed for a given surface area.  The 
special property of fractals is that they are each "self-similar" in 
some way:

  What is a Fractal?
    http://mathforum.org/library/drmath/view/54525.html 

However, you may be more likely to encounter fractals from other 
branches of mathematics, just as you may encounter circles in branches 
of mathematics other than geometry (for example, you might use circles 
when you work with imaginary numbers in complex analysis, or if you 
use the unit circle in trigonometry).  Some of the branches of 
mathematics that give rise to fractal objects are the following:  
differential equations and dynamical systems theory (e.g. "strange 
attractors"), measure theory (Cantor set), and complex analysis 
(Mandelbrot set).

You may also encounter fractals in applications of mathematics, such
as image compression, or computer art, or a whole host of scientific
and engineering problems:

  Fractal Image Compression
    http://mathforum.org/library/drmath/view/54530.html 
  
  Fractals in Real Life
    http://mathforum.org/library/drmath/view/54539.html 

I hope that this answers your question.  Feel free to write back if 
you need more help with this.

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