Fractals: the Koch Snowflake

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  • Topic: Fractal, Recursion, Mandelbrot set
  • Pages : 18 (5711 words )
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  • Published : December 28, 2012
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Executive Summary
"Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does the lightning travel in a straight line." - Benoit Mandelbrot
Beautiful patterns surround us. You can see them on trees, clouds, on bodies of water. You can even see them on plants, on animals and on our very skin. The very tips of our fingers prove just that. There is also no doubt that patterns are just as mysterious as they are beautiful. In fact, there are some patterns that are so perfect that they self-replicate. To be technical, some patterns are fractal in nature. Fractal or not, patterns give us something more to admire and wonder about. Introduction

Fractals never fail to fascinate. If you aren't just gazing at their unearthly beauty, you ponder the mathematics behind them... and then you can't help but wonder how such prosaic, unsensational mathematical formulae can give rise to such intricacy. What is it that makes it possible for (to some) a short, ugly equation to generate the exuberant beauty of the Mandelbrot set? Or is it all just in the way our brains are wired? Fractals are objects with infinite lengths that occupy finite volumes, resulting in a "fractional dimension" that is not 1-, 2-, or 3-D, but a combination of all three, depending on its spatial configuration. The Koch snowflake is the repetitive procedure of dividing the image into three equal parts and replacing the middle piece with two similar pieces. Hypothesis

Fractals mimic nature. (true or false)
This is the basic belief of fractals, and a common concept among those who study fractals. In nature, symmetry is often remarked upon. To mimic is to be similar in to a certain object, and in this case, of a lesser proportion. Thus, we would like to propose that fractals may mimic nature.

1. A curve or geometric figure, each part of which has the same statistical character as the whole. 2. Any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size The principle of fractals is the repetition of a geometric shape on different scales. The original shape would be repeated on a smaller scale over and over again for infinite times. This infinite repetition should not be able to occur practically although the theory is possible. Fractals may be the same at every scale, or nearly the same at different scales. In other words, fractals are the same at first glance as from when one takes a closer look, making them characteristically self-similar patterns. Fractals rarely consist of any straight lines, typically possessing the spiral and Mandelbrot. [pic]

The Mandelbrot set.

When geometric fractals are split into parts, each is essentially a copy of the whole, except at a reduced size. The definition of fractals, however, excludes trivial self-similarity, but includes the idea of a detailed pattern repeating itself. In short, a fractal is an object or quantity that displays self-similarity on all scales. The object need not exhibit precisely the same arrangement at all scales, but the same "type" of structures appears on all scales. A fractal would be infinite in two ways. It is infinite in the macro sense, being able to extend to infinitely large values of the coordinates, outwards in all directions from the centre. It is also infinite in the micro sense, as it theoretically has infinite granularity, allowing one to zoom in without limit to show ever finer detail, an equivalent to ever decreasing the spatial range. A fractal is a never-ending pattern. Fractals are created by the repetition of a simple process in an on-going feedback loop, forming a complex looking shape. They are infinitely complex patterns that are self-similar across different scales. Driven by recursion, fractals are images of dynamic systems. Abstract fractals - such as the Mandelbrot Set - can be...
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