Who Is Mandelbrot?

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Enjoy the Art of Mathematics
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the
Mandelbrot Set
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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 1970s, he devised the geometry of
  "fractals.
• The name fractal comes from the Latin word
  fractus, meaning "to break."

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What are
fractals?

• Fractals are shapes which have an overall
  structure that resembles each part of the
  structures smaller parts.

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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

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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.

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• 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

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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
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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.

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• 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 .

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• 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.

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• 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.

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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
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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/