Fractals
Fractals
Introduction
Fractals are geometric patterns that when repeated at increasingly smaller scales they produce irregular shapes and surfaces. All fractals have a feature of ‘self-similarity’. A set is self-similar if it can be broken into arbitrary small pieces, each of which is a small copy of the entire set, for fractals the pattern reproduced must be detailed (Nuhfer 2006). Self-similarity may be demonstrated as exact self-similarity meaning the fractal is identical at all scales a fractal that demonstrates exact self-similarity is the Koch Snowflake. Other fractals exhibit quasi self-similarity. This is when fractals approximate the same pattern at different scales, they contain small copies of the entire fractal in altered or degenerate forms, and an example of this is the Mandelbrot set (Fractal 2009). Also, fractal curves are ‘nowhere differentiable’ meaning that the gradient of the curve can never be found; because of this fractals cannot be measured in traditional ways (Turner 1998). I find it interesting to note that many phenomena in nature have fractal features including clouds, mountains, fault lines and coastlines. There are also a range of mathematical structures that are fractals including, Sierpinski triangle, Koch snowflake, Peano curve and the Mandelbrot set (Mandelbrot 1977). This investigation aims to explore a method in which to create the central region of the Mandelbrot set. I am fascinated with the Mandelbrot set as the equation used to generate points which create the image is relatively uncomplicated yet the outcome is highly detailed and to me a beautiful piece of mathematics. I was interested in using Microsoft Excel as a tool to generate the fractal as oppose to any other generator available because I am familiar with Excel. I was curious to see how far I could take this program which I primarily only use for creating things such as linear, quadratic, logarithmic and exponential graphs.
Theory of Constructing the Mandelbrot
Introduction
Fractals are geometric patterns that when repeated at increasingly smaller scales they produce irregular shapes and surfaces. All fractals have a feature of ‘self-similarity’. A set is self-similar if it can be broken into arbitrary small pieces, each of which is a small copy of the entire set, for fractals the pattern reproduced must be detailed (Nuhfer 2006). Self-similarity may be demonstrated as exact self-similarity meaning the fractal is identical at all scales a fractal that demonstrates exact self-similarity is the Koch Snowflake. Other fractals exhibit quasi self-similarity. This is when fractals approximate the same pattern at different scales, they contain small copies of the entire fractal in altered or degenerate forms, and an example of this is the Mandelbrot set (Fractal 2009). Also, fractal curves are ‘nowhere differentiable’ meaning that the gradient of the curve can never be found; because of this fractals cannot be measured in traditional ways (Turner 1998). I find it interesting to note that many phenomena in nature have fractal features including clouds, mountains, fault lines and coastlines. There are also a range of mathematical structures that are fractals including, Sierpinski triangle, Koch snowflake, Peano curve and the Mandelbrot set (Mandelbrot 1977). This investigation aims to explore a method in which to create the central region of the Mandelbrot set. I am fascinated with the Mandelbrot set as the equation used to generate points which create the image is relatively uncomplicated yet the outcome is highly detailed and to me a beautiful piece of mathematics. I was interested in using Microsoft Excel as a tool to generate the fractal as oppose to any other generator available because I am familiar with Excel. I was curious to see how far I could take this program which I primarily only use for creating things such as linear, quadratic, logarithmic and exponential graphs.
Theory of Constructing the Mandelbrot