What is a Fractal?

Date: 04/01/97 at 17:16:47
From: brandin
Subject: Fractals

What are fractals?  Who first used the term, when, why, and give at 
least two specific examples.

Date: 04/01/97 at 18:29:40
From: Doctor Sarah
Subject: Re: Fractals

Hi Brandin -

There's a lot on the Web about fractals.  Much of it is hard to 
understand, but there are sites that will give you good explanations 
and illustrations.  Here is some information compiled from what the 
Web has to offer, and you will want to read more yourself at the sites 
listed:

On his Web page about fractals, 

  http://www.glyphs.com/art/fractals/what_is.html   

Alan Beck says, "Basically, a fractal is any pattern that reveals 
greater complexity as it is enlarged. Thus, fractals graphically 
portray the notion of 'worlds within worlds' which has obsessed 
Western culture from its tenth-century beginnings."

He further explains that when we look very closely at patterns that 
are Euclidean, the shapes look more and more like straight lines, but 
that when you look at a fractal up close you see more and more 
details.

He illustrates with several graphics.  Here is a particularly nice one 
(it is also 96K, so be patient and wait for it):

  http://www.glyphs.com/art/fractals/images/serpente.jpg   

"Whether generated by computers or natural process, all fractals are 
spun from what scientists call a 'positive feedback loop'. Something - 
data or matter - goes in one 'end', undergoes a given, often very 
slight, modification, and comes out the other. Fractals are produced 
when theoutput is fed back into the system as input again and again."

Students often study Sierpinski's Triangle as an example of a fractal. 
You start with one triangle. [Level Zero.] Then you mark the midpoint 
on each of the three sides and draw a line from Midpoint 1 to Midpoint 
2 to Midpoint 3. You will have the original larger triangle, and 
inside it will be four smaller triangles, three pointing up and one 
down. [Level One.]

You can see this process at Cynthia Lanius' Web unit on fractals:

  http://math.rice.edu/~lanius/fractals/   

Here the triangles have been shaded in, and are black.

Levels One, Two, etc. are called "iterations" - repetitions of the 
same process, where the output of one level becomes the input for the 
next.

Fractals can be made using different functions. Sierpinski's Triangle 
is made by continually dividing a triangle into other triangles, on 
and on and on. Koch's snowflake - see Cynthia Lanius' page at

  http://math.rice.edu/~lanius/frac/koch.html   

- is made by starting with a large equilateral triangle, making a 
star, dividing one side of the triangle into three parts, taking out 
the middle section and replacing it with two lines the same length as 
the section you removed, doing this to all three sides of the 
triangle... and then doing it all again... and again... and again... 
as you keep dividing and dividing the perimeter of the figure, 
although the area of the interior is finite, the perimeter is 
infinite!

An amazing list of links has been compiled by Chaffey High School, and 
you can even listen to fractal music:

  http://www.chaffey.org/fractals/   

On the Quiddity Design Team's page, "What is a Fractal,"

  http://www.dsoe.com/people/hoyle/fractal.html   

they say: "Often fractals are self-similar, that is, they have the 
property that each small portion of the fractal can be viewed as a 
reduced-scale replica of the whole."


In the 1970s, Benoit Mandelbrot discovered fractal geometry and 
adopted a more abstract definition of dimension than that used in 
Euclidean geometry. He stated that when measuring the size of a 
fractal, its dimension must be used as an exponent. Thus fractals are 
not one- or two- or three- (or any other whole number-) dimensional, 
but must be handled mathematically as if they have fractional 
dimension.

Here are some more sites where you can to read about fractals and/or 
find links to other Web pages about them:

Chopping Broccoli
  http://www.glenbrook.k12.il.us/gbsmat/fractals/fractals.html   

The sci.fractals FAQ
  http://www.mta.ca/~mctaylor/sci.fractals-faq/   

The Math Forum's page of fractal links
  http://mathforum.org/library/browse/static/topic/fractals.html   

Have fun surfing!

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