Week 8/ Fractal Geometry

This week in lecture Professor. Cartwright talked about fractals which I am really interested in. Fractals has strong connection with mathematics and geometry. It can be seen when an image be enlarged unlimited times and the enlarged image has the similar pattern as the original one. Fractals can be seen in our daily life such as trees, clouds, and mountains. Meanwhile, it is being used in many field such as animation and coding. The method of fractals saved people times of repeating the same action.

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Loren Carpenter, who worked at the Boeing Company was the first one who used the method of fractals to produce animation. He wanted to make an animation of a plane flying upon mountains. However, animation techniques back in 1980s could not allow him to do that. He used fractal method to break a triangle into four smaller triangles and did the same action again and again and finally got the moutain effect by using simple fractal method. Even though fractal are simple loops, it can create very complicate patterns.

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After working at Boeing Company, Carpenter left and joined Lucasfilm. He used similar techniques to create a whole new planet in Star Trek II.

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Benoit Mandelbrot also found the interesting pattern of formula 屏幕快照 2015-03-03 8.52.08 PM

f_c(z) = z^2 + c \,

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and keep repeating it we will get this pattern

屏幕快照 2015-03-03 8.54.48 PMWhen we enlarged this pattern we will get

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屏幕快照 2015-03-03 9.06.20 PM 屏幕快照 2015-03-03 9.06.27 PM 屏幕快照 2015-03-03 9.06.38 PM 屏幕快照 2015-03-03 9.06.44 PMEach branch looks self-similar to the original one. Fractals brought up the question of mathematical dimensions. Since fractals can be enlarged unlimited times, will dimensions can also do that? Maybe in the future scientists will discover not only 3 dimensions or 4 dimensions but n dimensions. I also thought about that our universe is also a fractal pattern. It can be enlarged from as big as the universe to as small as nanometer. We can also reduce our universe but we still do not know what is outside the universe. It could be infinity.  According to fractals, our brain cell could be a universe of another smaller “thing”, or the universe could be a brain cell of another bigger “thing”. There are way too many unknown of human and the universe.

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-Liwei Wang


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