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    Default Theory of Fractals

    Part I


    Some Brief History:

    Coined by Benoit B. Mandelbrot from his studies in complexity and chaos theory the theory of fractals had huge impact in Chemistry, Fluid-Mechanics and Physiology. (Other great mathematicians such as Leibniz (recursive self-similarity and fractal exponents), Weierstrass ("Weierstrass Function"), Klein and Poincaré ("self-inverse fractals"), Cantor (Cantor's sets) and many others were developing some of the early concepts of what is known by "Theory of Fractals").

    The main difference between Fractal Geometry and the classical Euclidean Geometry is that fractals works with objects in non-integer dimensions.

    What is a geometric fractal afterall?

    Generally (as often cited as a description made by Mandelbrot):
    "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"
    .

    The Sierpinski Triangle can give a better visualization of this concept:

    http://www.evl.uic.edu/luc/488/diagrams/sierpinski.gif


    Fractals found in nature

    Snowflake

    http://local.wasp.uwa.edu.au/~pbourk...cdim/exact.gif

    Romanesco

    http://www.fourmilab.ch/images/Roman...mages/Scr1.jpg

    Cauliflower

    http://4.bp.blogspot.com/_EKjPKeIQhM...auliflower.jpg

    Fern

    http://therealsasha.files.wordpress....rn-fractal.jpg


    Some initiators and generators

    To guarantee self-similarity is to build a shape and apply the same process in smaller scales over and over again. Example:

    Sierpinski Triangle

    http://www.digitalartforall.com/wp-c...sierpinski.png

    Koch Curve

    http://www.vanderbilt.edu/AnS/psycho...hop/Fig4.7.GIF

    Remember the "snowflakes"?



    The Theory of Fractals can be applied in a wide range of technologies, for example:


    Digital imaging;

    Computer and video game design;

    Seismology;

    Signal and image compression;

    Medicine (classification of histopathology slides);

    Technical analysis; and

    Coastline Complexity.


    Back soon.
    Last edited by Yautja_BR; 2012-04-05 at 22:31.

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