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Put Simply, Fractals Are the Images Produced When Particular Complex Functions Are Infinitely Iterated

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Put Simply, Fractals Are the Images Produced When Particular Complex Functions Are Infinitely Iterated Put simply, Fractals are the Images produced when particular complex functions are infinitely iterated. Although they have yet to be defined formally they are often described and defined by the properties they possess: A fractal is a complex shape which, when viewed in finer and finer detail (infinite magnification), shows itself to be constructed of ever smaller parts, similar to the original. The idea of fractals was first discovered by Karl Weierstrass in 1872. Although the term “fractal” would not be used for a long time, the graph of the Weierstrass function was one of the very first to be discovered to have the property of self similarity at infinitesimally small intervals on the graph. However, this graph was a side-product since the function was designed originally to prove a statement by Riemann that a real valued function that is continuous everywhere can be differentiable nowhere. Here is the formula and subsequent graph of the Weierstrass Function: (The close-up illustrated by the red circle shows a magnification of a small region of the graph. The image produced is clearly a scaled version of the overall graph, showing the self-similarity property.) In 1915 Waclaw Sierpinski produced a space filling curve (Aptly named the Sierpinski Curve) whose geometric description produces a fractal. This curve is responsible for the addition of another description used to describe fractals: Shapes that have finite area bounded by an infinite perimeter. Sierpinski’s curve can be described by graphs but is most effectively and most famously shown by the Sierpinski Triangle shown here: Whose area is obviously bounded above by the area of the largest triangle but whose perimeter length is described by the exponential growth function: The above triangle clearly produces self similarity at any level. Another description for how to produce the above triangle is as follows: 1) Start with an equilateral triangle colored completely black 2) Mark the midpoint of each side; connect the points to produce a smaller equilateral triangle contained in the first and color this triangle white. 3) For each black triangle repeat step (1) Further analysis of the Sierpinski triangle brings us to another startling discovery with yet another definition for a fractal. We define the following: A point has no dimensions - no length, no width, no height; A line has one dimension - length. It has no width and no height, but infinite length; A plane has two dimensions - length and width, no depth. Space, a huge empty box, has three dimensions, length, width, and depth, extending to infinity in all three directions. We then analyze what will happen when we double each of the parameters for the different shapes. A line, for example, will produce to equal lines when its one parameter (the length) is doubled. A square, however, will produce 4 exact copies of itself when the two parameters (length and width) are doubled. A cube will likewise produce 8 exact copies when the length, width and height are each doubled. We now organize this information into a simple table. This is illustrated below: Figure Dimension No. of Copies Line Segment 1 2 = 21 Square 2 4 = 22 Cube 3 8 = 23 Doubling Similarity d n = 2d The question then arises: What will happen when a fractal, like the Sierpinski Triangle, is similarly doubled. By taking the triangle and double the length of each side, we produce the following image. Clearly the triangle then produces 3 copies of itself. But let’s look at this fact closer. Using the dimension formula calculated above, we set the number of copies, n, to be 3. Thus the dimension, d, is the solution d to the equation 3 = 2 . This function has a simple yet counterintuitive answer: d=log23 which leads the fractal to have dimension approximately equal to 1.585! Thus fractals have the property of being fractionally dimensional. Perhaps the most famous and most thoroughly studied fractal is the one produced by Benoit Mandelbrot in 1975. Mandelbrot is primarily responsible for the now commonly used term “fractal,” which he derived from the Latin root word “fractus” which means broken or fractured. The Mandelbrot 2 set is defined by a recursive function: zn+1 = zn + c. In other words, any point c in the complex plane that, upon an infinite iteration in the function defined before, stays bounded below a given constant, belongs to the Mandelbrot set. By graphing on the complex plane we produce the graph: where the black points represent the points in the Mandelbrot set. Clearly the entire area of the black region is bounded by the square image it is contained in, thus giving it a finite area. However, upon closer zoom on each region of the perimeter of this graph, we find the function to be infinitely complex in shape giving rise to an infinite perimeter. Better graphs of this function illustrate this point. By once again making points contained in the Mandelbrot set black, but coloring the points just outside the set in a gradient, we produce some beautiful and telling graphs. In the following graph we color the graph based on the number of iterations of the function necessary for the point to escape to infinity. The slower the point escapes, the brighter the spot to allow for maximum contrast. The zoom towards the neck of this graph (named the Seahorse region) produces the self similarity more clearly. This video also illustrates this point beautifully http://www.youtube.com/watch?v=WAJE35wX1nQ Interestingly, the Mandelbrot set also has some striking properties. The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters c for which the dynamics changes abruptly under small changes of c. This is illustrated below: Aside from their apparent beauty and complexity, fractals have become a very promising field of research for more practical uses. One such use is for the compression and decompression of images as well as image zoom. This concept utilizes the idea of self similarity to deconstruct an image or to zoom effectively without losing too much information. Two images are posted below: the left picture has been blown up using fractal zoom, while the second image is a regular zoom from Jpeg compression. Look at that monkey. .
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