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• 1 2016-05-03T16:15:58-07:00 What are Fractals? 6 plain 2016-05-17T22:31:22-07:00 Fractals are geometric objects that display self-similarity. The most basic fractal is a line, which, if you split into two pieces, will give you two lines that are exact replicas of the first at a specific magnification. Another basic fractal is the Cantor set which is produced by taking a line, splitting it into thirds, removing the middle third, and repeating the procedure on the remaining line segments infinitely. Although I won’t get into a heavy mathematical explanation here, fractals have been proven to have a fractional dimension rather than an integer Euclidean dimension. Line segments are a simple example, however, the more interesting examples are the more complex ones such as the Mandelbrot set. [1]
  
  When talking about fractals, Mandelbrot’s name is sure to arise in the conversation. Often referred to as “The Father of Fractal Geometery”, as he actually coined the term fractal. He developed a particularly beautiful fractal set based on a simple iteration applied to complex numbers. Here is an example of the Mandelbrot Set with the starting point for the iteration taken as the square root of five. [2]
  
  However, Mandelbrot acknowledges none other than Gottfried Leibniz as his inspiration. Many refer to Leibniz idea of self-similarity as one of the earliest mathematical predecessors of fractal geometery. Although Leibniz did not explicitly use the term fractal (since it did not come about until Mandelbrot), he did use his understanding of self-similarity in his theology.