Fractal by Zhixuan Li

1. Fractalby Zhixuan Li Applied Mathematics  Complex Systems  Fractals

2. History • The mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher Gottfried Leibniz considered recursive self-similarity • It was not until 1872 that a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable • In 1904, Helge von Koch, dissatisfied with Weierstrass ‘s abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch curve

3. What is a Fractal? • Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.

4. Important Characteristics of Fractals • They are recursive; that is, the process of their creation gets repeated indefinitely; • They are self-similar; that is, copies of the entire fractal may be found, in reduced form, within the fractal.

5. Box Fractal • It is a fractal arising from a construction similar to that of the Sierpinski carpet. It has applications including as compact antennas, particularly in cellular phones

6. Sierpinski carpet • The Sierpinski carpet is a plane fractal first described by WacławSierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions. Sierpiński demonstrated that this fractal is a universal curve, in that it has topological dimension one, and every other compact metric space of topological dimension 1 is homeomorphic to some subset of it.

7. Sierpinski triangle • The Sierpinski triangle is a fractal and attractive fixed set named after the Polish mathematician WacławSierpiński who described it in 1915.

8. Koch snowflake • A fractal, also known as the Koch island, which was first described by Helge von Koch in 1904. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely.

9. Hilbert curve • A Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891,as a variant of the space-filling curves discovered by Giuseppe Peano in 1890.

10. Mandelbrot set • The Mandelbrot set is a mathematical set of points whose boundary is a distinctive and easily recognizable two-dimensional fractal shape.

11. Fractals in nature • Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, crystals. Mountain ranges and so on.

15. reference • http://webecoist.momtastic.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/ • http://mathworld.wolfram.com/Fractal.html • http://mathworld.wolfram.com/MandelbrotSet.html • http://mathworld.wolfram.com/BoxFractal.html • http://mathworld.wolfram.com/SierpinskiCarpet.html • http://mathworld.wolfram.com/SierpinskiSieve.html • http://mathworld.wolfram.com/KochSnowflake.html