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From: wbp@hou2d.UUCP (W.PINEAULT)
Newsgroups: net.math
Subject: Re: fractals
Message-ID: <479@hou2d.UUCP>
Date: Mon, 27-Aug-84 16:10:38 EDT
Article-I.D.: hou2d.479
Posted: Mon Aug 27 16:10:38 1984
Date-Received: Tue, 28-Aug-84 00:22:28 EDT
Organization: AT&T Bell Labs, Holmdel NJ
Lines: 92


	I am surprised that no mathematicians have yet responded to the
question of what fractals are with a simple example or two, but I guess,

	Real Mathematicians don't read net 	::-)

	(I defected to Computer Science)

	Forgive me if this reference was given before, but a very good
introduction to fractals is:
		"The Fractal Geometry of Nature", by Benoit B. Mandelbrot,
	Freeman and CO, San Francisco.

	(The material here is paraphrased from this book.)

	   The concept of fractals is actually a theoretical
	mathematical construction which seems to have implications
	for the real world.  Part of the historical impetus for
	fractals comes from the concept of plane-filling curves
	(Peano curves), where a one dimensional line passes through
	every point on the plane, and so isn't its dimension really
	two?  There are other historical roots, some are from real
	analysis dealing with Cantor sets and measures on them.

	The following is a concrete example:

	1.  Take a line of unit length:    		______

	2.  Replace the middle third by two lines
	    each equal to a third of the unit:		__/\__

	3.  Repeat for each of the 4 new line segments
	    with lines 1/3 of last produced lines.

	4.  Repeat for all new figures indefinitely.

	On the n'th application the length of the figure is (4/3)**n, and
the length of the limit curve is therefor infinite (this takes a bit of
proof, but is elementary calculus.)

	But saying that the length of this bounded curve, which mearly
wiggles a lot, is infinite is not satisfying.  Therefore a new concept was 
devised to somehow measure the unboundedness of this construction.

Intuitive explanation:

	We measure the rate of groth of the series of curves defined
above.  Let L(n) be the length of the curve when using pieces of
length n.  Then we know:

  (1)		L(n/3) = (4/3)L(n).

	Now we define the fractal dimension D of a curve to be that number
which allows the following to be true:

  (2)		L(n) = n**(1-D).

	In this case we substitute (2) into (1) and get:

		(n/3)**(1-D) = (4/3) (n**(1-D))

		(1/3)**(1-D) = 4/3

		(D-1) log(3) = log(4) - log (3)

		D = log(4) / log(3).

Actual:

	The idea of fractal dimensions comes in part from the
concept of Hausdorff measures.  The essece of this idea is to take
minamal coverings of a curve using disks of a fixed radius.  Letting
the radius approach 0, look at what happens to the sum of the diameters
of the covering circles.  Calculations are not hard and is a 
fairly interesting exercise.

	For each curve there is a unique number which makes the above
work (the concept of fractional dimension is well defined), and for
any number between 1 and 2 there exists a fractal with that dimension.

	Mandelbrot is the "father of fractals" and states that he
thinks that they express some fundamental nature of the universe.
His book certainly ties in enough disciplines from relativity to
snowflakes.  What is certain is that the concept of fractals takes care
of many anomalies that have been troubling mathematicians for decades.

	Incidently only crves with a dimension not a unit are considered
fractals.
					Wayne Pineault
					AT&T Consumer Products
					Holmdel, N.J.