Fractals: Are They Just Mathematical Curiosities?
Joshua du Parc Braham
Introduction and Definition
The word fractal was first coined by Polish born mathematical superstar Benoit Mandelbrot in his 1975 work “les Objects Fractals”. This was later translated and released as “Fractals: Form, Chance and Dimension”. In this paper he outlined the basis of fractal geometry including fractal dimension, scaling and what a fractal actually is. A fractal is a shape or pattern of infinite complexity produced by an iterative process or function. Where you repeat the operation to the output just produced. A fractal has the property of self-similarity, where one section perfectly representing the whole pattern.
Examples of Fractals
Simple fractals can be made with two steps: a starting point and a generator. An example of this process is [ fig. 1 ] where we generate The Sierpinski Triangle . The starting point in this case is a filled equilateral triangle. The generator is to split an equilateral triangle up in to 4 equal smaller ones, removing the middle triangle. If you repeat this process ad infinitum, taking each triangle and splitting it up, over and over again, you end up with Sierpinski’s Triangle.
fig. 1:The Sierpinski Triangle
One of the best examples of a fractal being self-similar, which is where one part has an identical pattern as the whole, is the Koch Curve . Here you start with a line, and place an equilateral triangle in the centre where the length of the sides of the triangle is equal to a third of the length of the line. You proceed to repeat this process on all straight edges in each iteration.
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fig. 2: Forming the Koch Curve
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