Fractal 8 ©2005 Aaron M. Cohen

 

                         

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 Fractals © Copyright Aaron M. Cohen
 Click each image for larger images

 

These fractals were generated by the program, ChaosPro
(© Martin Pfingstl), then graphically modified.

About FractalsThe fractal pictures on this site are images were generated*, captured, and graphically manipulated by Aaron Cohen. Fractals: the simple explanation . . .In your study of geometry, was there ever a way to describe or construct a picture of a coastline, mountain, or lightening?The development of fractals has enabled this, as well as producing breathtaking illustrations and magical, mystical "journeys."These journeys occur when a computer program generates a fractal and then provides the opportunity to zoom in to it.What one finds is an opportunity to zoom infinitely into the intricate detail, finding smaller copies of the whole.Three dimension fractal geometry has been used in the entertainment and animation industries to generate real-looking landscapes.A fractal is a mathematically-generated object that mimics, describes and mathematically replicates many real-world objects that do not have simple geometric shapes — such as clouds, mountains, turbulence, coastlines, snowflakes, lightening . . .For the more curious . . . a more complex definition:A fractal is a mathematically-generated object that, possesses infinite detail, and can be "broken up" and subdivided in parts, each of which, at different levels of magnification, is approximately a smaller copy of the whole.Thus, with the assistance of fractal-generating computer programs, you can keep zooming in — literally forever — to small areas of detail in the fractal. And you will find repetitions of the original.This is similar to holograms: if you cut a hologram into pieces, each contains a copy of the whole image.The discovery of fractals, and their rough or fragmented geometric shape, provided mathematics — for the first time — with a way to describe and mathematically replicate many real-world objects that do not have simple geometric shapes, such as clouds, mountains, turbulence, coastlines, snowflakes, lightening . . . These shapes are rough or irregular on all scales of length, and so appear to be 'broken up' in a radical way.

"The oldest standard example is a coastline ("How long is the coast of Britain?"), which when measured one kilometer at a time might turn out to be 5000 kilometers long, but when measured one meter at a time comes out to be, say, 12000 kilometers."
 (http://www.mrob.com/pub/muency/fractaldefinitionof.html)

Fractals describe "many situations which cannot be explained easily by classical geometry, and has often been applied in science, technology, and computer-generated art. The conceptual roots of fractals can be traced to attempts to measure the size of objects for which traditional definitions based on Euclidean geometry or calculus fail." **The term fractal was coined in 1975 by Benoît Mandelbrot, author of The Fractal Geometry of Nature" from the Latin, fractus, broken. The conceptual roots of fractals can be traced to attempts to measure the size of objects for which traditional definitions based on Euclidean geometry or calculus fail.Several definitions have been created over the years as mathematicians struggled with the complex properties of fractals.

** For further information:
   http://en.wikipedia.org/wiki/Fractal
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