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The Fun Filled Fractal Phenomenon

A fractal is a type of geometric figure. It is generated by starting with a very simple pattern such as a triangle and, through the application of many repeated rules, adding to the figure to make it more complicated. Often, an input will be entered into a recursive function and it will yield an output. This output is then inserted back into the function as an input and the process is repeated infinitely. Fractals often exhibit self-similarity. This means that each small section of the fractal can be viewed as a reduced-scale replica of the whole. Some famous fractals include Sierpinski's triangle, Koch's snowflake and the length of a coastline. Fractals were brought to the public's attention by the work of French mathematician Benoit B. Mandelbrot in the 1970's. Mandelbrot discovered how to calculate fractal dimensions. The formula for fractal dimension is N=2D where N equals the number of copies of the original figure, which is calculated by doubling its size and D is the dimension. Mandelbrot named his creations fractals because each part is a fraction of the whole figure.

The Chaos Theory describes the complex and unpredictable motion of systems that are sensitive to their initial conditions. Chaotic systems follow precise laws but their irregular behavior can appear to be random to the casual observer. For example, weather is a chaotic system. If the rays of the sun bounce off the hood of a car in a certain way, causing a breeze, the breeze could blow a leave off a tree, which starts a series of additional events that could alter the weather in some other part of the world. Chaos can be related to fractals. In a fractal if one tiny change occurs in a repeated pattern, the entire fractal will change. The above picture is an example of a strange attractor that charts the trajectory of a system in chaotic motion. It is a fractal. The fractal exhibiting chaos is predictably unpredictable. This is because, in a chaotic system, it is predictable that there will be minute changes that will alter the entire shape.

Koch's snowflake, (above ) exhibits the concept of an infinite perimeter with a finite area. Koch's snowflake is created by dividing each of the sides of an equilateral triangle into three equal parts. Next, the center part of each side is taken out and replaced with two sides of equal length to that of the original centerpiece. This pattern is repeated infinitely. Each time the process is completed the perimeter gradually increases to infinity by increments of 4/3. However, the area of this snowflake is finite. If you draw a circle enclosing the original triangle that contains the vertices of the triangle, the area of the snowflake will never exceed the area of that circle no matter how many times its perimeter increases. Therefore, it has a finite area.

Fractals exhibit self-similarity. This is the concept that each small portion of the fractal can be viewed as a reduced-scale replica of the whole. For example, in Sierpinski's Triangle, each small triangle inside is similar to the large one on the outside.

A real life example of self-similarity is a tree. The tree has a trunk on which limbs grow. Branches grow from the limbs, and twigs grow from the branches, which is followed by sticks on the twigs and so on. The sticks growing on the twigs are just a smaller version of the twigs growing on the branches, which are a smaller version of the branches growing on the limbs, which are a smaller version of the limbs growing on the trees. Another example is a universe, which is composed of a collection of spinning ...

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