The Mandelbrot Set

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The Mandelbrot Set

• By Matt Kaminski

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A Brief Summary

• The connectedness locus of the family of complex
  quadratic polynomials
• The set of complex numbers within the quadratic
  system, which is defined as Zm1 Zm2 c
• More specifically, the Mandelbrot set is the set
  of numbers for which Zm stays finite
• Research started by Adrian Douady and John H.
  Hubbard, established many fundamental properties
• Named after a mathematician, Benoit Mandelbrot
• Considered one of the most complicated and unique
  structures in the field of mathematics, yet is
  derived from such a simple definition
• It is a fractal, which means it contains small
  copies of itself
• Its picture was first drawn in 1978 by Brooks and
  Matelski
• Has aesthetic appeal, popular in the field of art

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Project

• The purpose of my project was to create a
  function in Matlab which plots the Mandelbrot set
  indicating which values of Zm stay finite and
  which go to infinity, mainly which stay finite
• The functions job is to create a matrix of a
  user-designated size consisting of points in c,
  both real and imaginary, apply it to the
  quadratic system, and plots the results.
• The graph will consist of the real part of c
  plotted on the x-axis and the imaginary part of c
  plotted on the y-axis, and can be bounded by the
  user
• The larger the matrix is, the more detailed the
  graph will look

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The Code

• this function graphs the Mandelbrot Set with
  an n x n matrix of points
• with an optional input to set the boundaries
  the graph will be plotted in.
• if this input is not made, the default boundary
  is -2,2 x -2,2,
• which is optimal viewing the Mandelbrot Set as
  a whole.
• function drawMandelbrot(n,boundary)
• if (nargin1)
• boundary -2,2,-2,2 set the default
  boundaries if only one argument is specified
• end
• a (boundary(2) - boundary(1))/n
• b (boundary(4) - boundary(3))/n
• X 0
• for x boundary(1)aboundary(2)
• X X 1
• Y 0
• for y boundary(3)bboundary(4)
• Y Y 1
• c x iy real and imaginary parts of
  c
• z 0 initial value of z
• m 0

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Results
5x5 matrix
10x10 matrix
25x25 matrix
50x50 matrix
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1000 x 1000 matrix
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Mandelbrot Set Images
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Julia and Dragon Sets
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The End

• Images taken from
• http//www.cps.unizar.es/jlsubias/
• http//www.sticksoftware.com/gallery/mandelbrot.ht
  ml