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/ |
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