Fractal Dimension of a CoastlineDate: 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/ |
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