Mandelbrot Fractals

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Mandelbrot Fractals

• Betsey Davis
• MathScience Innovation Center

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Benoit Mandelbrot

• largely responsible for the present interest in
  fractal geometry.
• He showed how fractals can occur in many
  different places in both mathematics and
  elsewhere in nature.
• Mandelbrot was born in Poland in 1924 into a
  family with a very academic tradition.

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Benoit Mandelbrot

• Sterling Professor of Mathematical
  SciencesMathematics DepartmentYale
  UniversityIBM Fellow Emeritus

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Lets start with Julia Sets

• Gaston Julia studied the iteration of polynomials
  and rational functions in the early twentieth
  century.
• If f(x) is a function, various behaviors can
  arise when f is iterated. Let's take, for
  example, the function
• f(x) x2 0.75.

http//aleph0.clarku.edu/djoyce/julia/julia.html
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Julia Sets

• We will iterate this function when initially
  applied to an initial value of x, say x  a0. Let
  a1 denote the first iterate f(a0), let a2 denote
  the second iterate f(a1), which equals f(f(a0)),
  and so forth. Then we'll consider the infinite
  sequence of iterates
• a0, a1 f(a0), a2 f(a1), a3 f(a2), ...

http//aleph0.clarku.edu/djoyce/julia/julia.html
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Julia Sets
http//aleph0.clarku.edu/djoyce/julia/julia.html

• It may happen that these values stay small or
  perhaps they don't, depending on the initial
  value a0. For instance, if we iterate our sample
  function f(x)  x2  0.75 starting with the
  initial value a0  1.0, we'll get the following
  sequence of iterates (easily computed with a
  handheld calculator)
• a0  1.0,
• a1 f(1.0) 1.02 0.75 0.25
• a2 f(0.25) 0.252 0.75 0.6875
• a3 f(0.6875) (0.6875)2 0.75 0.2773
• a4 f(0.2773) (0.2773)2 0.75 0.6731
• a5 f(0.6731) (0.6731)2 0.75 0.2970

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Julia Sets

• If you extend this table far enough, you'll see
  the iterates slowly approach the number 0.5. The
  iterates are above or below 0.5, but they get
  closer and closer to 0.5. In summary, when the
  initial value is a0  1.0, the iterates stay
  small, and, in particular, they approach 0.5.

http//aleph0.clarku.edu/djoyce/julia/julia.html
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Two things can happen

• In our example, they approach 0.5.
• So, one thing that can happen is that the value
  of f(x) approaches a limit but never exceeds it
• Another is that it can grow without bound

http//aleph0.clarku.edu/djoyce/julia/julia.html
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Two things can happen

• If value of f(x) approaches a limit but never
  exceeds it, it stays black
• oscillation back and forth creates bulbs
• If it grows without bound, and it is assigned a
  different color depending on when it breaks out
  (escapes)

http//aleph0.clarku.edu/djoyce/julia/julia.html
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Mandelbrot Sets

• Consider a whole family of functions
  parameterized by a variable. Although any family
  of functions can be studied, we'll look at the
  most studied family, that being the family of
  quadratic polynomials f(x) x2 - µ, where µ is a
  complex parameter. As µ varies, the Julia set
  will vary on the complex plane. Some of these
  Julia sets will be connected, and some will be
  disconnected, and so this character of the Julia
  sets will partition the µ-parameter plane into
  two parts.

http//aleph0.clarku.edu/djoyce/julia/julia.html
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Mandelbrot Sets

• Those values of µ for which the Julia set is
  connected is called the Mandelbrot set in the
  parameter plane. The boundary between the
  Mandelbrot set and its complement is often called
  the Mandelbrot separator curve. The Mandelbrot
  set is the black shape in the picture. This is
  the portion of the plane where x varies from -1
  to 2 and y varies between -1.5 and 1.5.

http//aleph0.clarku.edu/djoyce/julia/julia.html
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Mandelbrot Sets

• There are some surprising details in this image,
  and it's well worth exploring. The bulk of the
  Mandelbrot set is the black cardioid.
• A cardioid is a heart-shaped figure.

http//aleph0.clarku.edu/djoyce/julia/julia.html
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The period of this bulb is 5

• we include the spoke holding to the bulb
• numbers in this region repeat cycle in 5 steps

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Guess the period of this bulb

• 3

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Guess the period of this bulb

• 5

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Heres another zoom
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To Create your own

• Mandelbrot Explorer
• http//www.softlab.ece.ntua.gr/miscellaneous/mande
  l/mandel.html
• Julia and Mandelbrot Set Explorer
• http//aleph0.clarku.edu/djoyce/julia/explorer.ht
  ml