Fractals

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Fractals
Benoît Mandelbrot, father of fractal geometry

• Jennifer Trinh

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Theyre SO BADASS!
Im badass too!
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Basic Idea

• Fractals are
• Self-similar (will go into details in a moment)
• Cannot be described accurately with Euclidean
  geometry (theyre complex)
• Have a higher Hausdorff-Besicovitch dimension
  than topological dimension (will go into details
  in a moment)
• Have infinite length or detail

Romanesco Broccoli
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With Euclidean geometry
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Exact Self-Similarity Koch Snowflake
Can be formed with L-systems
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Approximate Self-Similarity Mandelbrot Set
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Statistical Self-Similarity
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Hausdorff-Besicovitch Dimension Fractal
Dimension?

• relationship between the measured length and the
  ruler length is not linear, i.e. 1 dimensional
• The fractal/Hausdorff-Besicovitch dimension is d
  in the equation N Md, where N is the number of
  pieces left after an object is divided M times. 
  E.g., we divide the sides of a square into
  thirds, we have 9 total pieces left.  9 32, so
  the fractal dimension is 2.
• More formally seen as log(N(l)) log(c) - D
  log(l)
• Doesnt have to be an integer

Sierpinski Triangle
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Generating Fractals

• Escape-time fractals give each point a value and
  plug into a recursive function (Mandelbrot set
  consists of complex numbers such that
  x(n1)x(n)2 c does not go to infinity, like
  i they remain bounded). Depending on what a
  value does, that point gets a certain color,
  causes fractal picture
• Iterated function systems fixed geometric
  replacement
• Random fractals determined by stochastic
  processes (place a seed somewhere. Allow a
  particle to randomly travel until it hits the
  seed, then start a new randomly placed particle
  see here)

• Escape-time fractals

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Measuring Fractals

• Smaller and smaller rulers
• Box methods counting the number of
  non-overlapping boxes or cubes (went over in
  Kenkel)
• See Kenkel
• Lacunarity measuring how much space a fractal
  takes up (kind of like density). Another way to
  classify

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Sources

• http//tiger.towson.edu/gstiff1/fractalpage.htm
• http//www.fractal-animation.net/ufvp.html
• http//local.wasp.uwa.edu.au/pbourke/fractals/
• http//www.fractalus.com/info/layman.htm
• http//en.wikipedia.org/wiki/Fractal
• http//mathworld.wolfram.com/KochSnowflake.html

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