Why is mandelbrot so important to us?
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It is a fractal, because no matter how far you zoom into it, you find intricate patterns that are similar to the whole thing. But it is a very interesting and tantalising fractal, because the patterns you find at different scales are only similar, not the same. There is actually infinite variety.
amberbamber116:
The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable.
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For at least two reasons.
Firstly, it is the best-known and perhaps the most dramatic demonstration of the fact that extremely simple rules can produce infinitely complicated results. This was very surprising when first discovered, since it was previously assumed that you needed a complicated system to produce complicated behaviour.
The rule for generating the Mandelbrot set is very simple indeed. You can write it in a couple of lines. Yet the result of applying that rule is a shape of staggering complexity that is endlessly surprising. It is a fractal, because no matter how far you zoom into it, you find intricate patterns that are similar to the whole thing. But it is a very interesting and tantalising fractal, because the patterns you find at different scales are only similar, not the same. There is actually infinite variety. It never repeats itself.
Secondly, it is beautiful. More than that: it is mind-blowingly beautiful in a way that seems inconceivable as the result of a mathematical formula, let alone a very simple formula. It looks like shellfish, like an abstract tapestry, like a psychedelic hallucination, like Indian wallpaper, like the decoration on an Arabian mosque.
It made us realise that if something that looks like that could arise from a few simple rules, then so could the beauty and complexity we see around us.
The discovery of the Mandelbrot set coincided with the advent of computers powerful enough to enable us to see it properly and appreciate its beauty. Previously it would simply not have been possible to calculate it in sufficient detail.
Firstly, it is the best-known and perhaps the most dramatic demonstration of the fact that extremely simple rules can produce infinitely complicated results. This was very surprising when first discovered, since it was previously assumed that you needed a complicated system to produce complicated behaviour.
The rule for generating the Mandelbrot set is very simple indeed. You can write it in a couple of lines. Yet the result of applying that rule is a shape of staggering complexity that is endlessly surprising. It is a fractal, because no matter how far you zoom into it, you find intricate patterns that are similar to the whole thing. But it is a very interesting and tantalising fractal, because the patterns you find at different scales are only similar, not the same. There is actually infinite variety. It never repeats itself.
Secondly, it is beautiful. More than that: it is mind-blowingly beautiful in a way that seems inconceivable as the result of a mathematical formula, let alone a very simple formula. It looks like shellfish, like an abstract tapestry, like a psychedelic hallucination, like Indian wallpaper, like the decoration on an Arabian mosque.
It made us realise that if something that looks like that could arise from a few simple rules, then so could the beauty and complexity we see around us.
The discovery of the Mandelbrot set coincided with the advent of computers powerful enough to enable us to see it properly and appreciate its beauty. Previously it would simply not have been possible to calculate it in sufficient detail.
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