Fractal Dimension of a Coastline

Date: 01/16/97 at 13:54:17
From: Becky Brotzman
Subject: Fractal geometry

How do you measure the fractal dimension of a coastline?  My group 
tried to locate information on how to calculate this, but we could not 
find any specific information/equations.

Date: 01/16/97 at 15:03:00
From: Doctor Sarah
Subject: Re: Fractal geometry

Hi Becky -

Take a look at David G. Green's paper:

  http://parallel.hpc.unsw.edu.au/complex/tutorials/tutorial3.html   

FRACTALS AND SCALE

David G. Green, 
Environmental and Information Sciences, Charles Sturt University 

Introduction:

How long is the coast-line of Great Britain? At first sight this 
question may seem trivial. Given a map one can sit down with a ruler 
and soon come up with a value for the length. The problem is that 
repeating the operation with a larger scale map yields a greater 
estimate of the length. If we actually went to the coast and measured 
the coastline directly, then still greater estimates would result. It 
turns out that as the scale of measurement decreases, the estimated 
length increases without limit. Thus, if the scale of the 
(hypothetical) measurements were to be infinitely small, then the 
estimated length would become infinitely large! Lewis Fry Richardson 
(quoted in Mandelbrot, 1983) noted this dependence of measured length 
to the measuring scale used.... 
______________

This paper continues and you should read the rest of it on the Web.

Our Dr. Math archive at http://mathforum.org/dr.math/    also 
yields an entry if you search for the word coast:

  http://mathforum.org/dr.math/problems/fractals.html   

The answer also talks about measuring the coast of Britain:

How long is the coast of Britain?  The answer is that there's no right 
answer.  See, you could never measure all the little nooks and 
crannies on the coast, every atoll and bay, and every point, so you 
have to decrease your resolution when you're trying to measure it.  If 
you truly did measure EVERY little crannie and nook, you'd come up 
with the answer that the coast of Britain has an infinite(!) length.  
We say that it is a fractal.  It has dimension between 1 and 2.
_________

There's a lot more about explaining fractals to students in this Dr. 
Math answer, so you'll want to check it out too.  If you still have 
questions after reading it, please don't hesitate to write back to Dr. 
Math.

-Doctor Sarah,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/