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## Enjoy the Art of Mathematics

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**the**Mandelbrot Set**Who Is Mandelbrot?**• Benoit Mandelbrot • Mandelbrot was born in Poland in 1924. He studied mathematics in France under Gaston Julia and others. • Mandelbrot researched economic chaotic data for IBM. • In the 1970’s, he devised the geometry of "fractals.” • The name “fractal” comes from the Latin word fractus, meaning "to break."**What are fractals?**• Fractals are shapes which have an overall structure that resembles each part of the structure’s smaller parts.**Do fractals occur naturally?**• The term fractal can be used to describe the pattern in irregular shapes found in nature. • Plant growth patterns like broccoli • Waves and ripples in water • Mountain ranges • Snowflake edges • Cloud formations • Coastlines**What are fractals used for?**• Fractals are used to describe and predict the changes in ecosystems. • Fractals are used to distinguish the strengths of metals. • Fractals are used to describe occurrences in meteorology and astronomy, like galaxy clusters. • Fractals are used to create computer images of landscapes for landscape design and science fiction films.**Fractals, like the Mandelbrot Set, are used in the classroom**to encourage students to explore... • Patterns • Fraction division • and reduction • Scale and • magnification • Counting schemes • Coordinate systems • Integer arithmetic • Concept of Infinity**What is the Mandelbrot Set?**• The Mandelbrot Set is a beautiful pattern of fractal geometry. • The Mandelbrot Set is the set of all Julia sets combined. Julia Set Mandelbrot Set**How is the Mandelbrot Set calculated?**• Start with the quadratic function x2 + c, the simplest nonlinear function, where c is a constant. • Iterate (repeat) x2 + c by starting with a “seed” (any real or complex number used as a starting value). • Denote the seed by x0. Plug the seed x0 into the function x2 + c. • x1 = x02 + c.**Continue repeating this plugging-in operation, using the**computation results from one equation as the input for the next equation. x2 = x12 + c, x3 = x22 + c, x4 = x32 + c, x5 = x42 + c, This repeated process is called the orbit of x0 .**The values from these equations are represented by colors .**• Through computer imagery, the resulting beauty of these fractal shapes can be generated, such as the Mandelbrot Set.**The edges of the Mandelbrot Set can be magnified to reveal**the amazing detail of the pattern. • There is infinite depth in one finite point.**Enjoy the art of mathematicsthrough fractal geometry.Explore**the Mandelbrot Set.Create your own fractals on the web at: http://storm.shodor.org/eoe/mandy/index.html**References**• Devlin, K. (1998). Life By the Numbers. New York: Wiley & Sons. • Suplee, C. (1999). Physics In the 20th Century. New York: Abrams. • http://archive.ncsa.uiuc.edu/Edu/Fractal/ Fractal_Home.html • http://library.thinkquest.org/3288.html • www.deepleaf.com/fractal/