Fractals

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Fractals

• Laura Bailey
• Trevelyan College

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• A fractal is a geometric object that is similar
  to itself on all scales. If you zoom in on a
  fractal object it will look similar or exactly
  like the original shape.
• This property is called self-similarity.

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• What does self similarity mean?

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• Exact self similarity
• This type of self similarity is common in
  mathematically defined fractals where the
  realities/constraints on structures by the
  physical world don't apply.

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The Koch Snowflake
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The Sierpinski Triangle
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http//home.t-online.de/home/lutz.tautenhahn/fract
al/sierpa.html
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• Approximate self similarity
• As you looks at the object at different scales
  you sees structures that are recognisably similar
  but not exactly so.

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The Mandelbrot fractal
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A Fern
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• Statistical self similarity
• Sometimes the self similarity isn't visually
  obvious but there may be numerical or statistical
  measures that are preserved across scales.

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Dimension

• The dimension of a fractal does not necessarily
  need to be an integer.
• This is a key and defining property of a fractal
  and is what gives fractals many of their unique
  properties for example, the Sierpinski triangle
  has an area of 0.

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Mandelbrots definition

• A FRACTAL is by definition a set for which the
  Hausdorff Besicovitch dimension strictly exceeds
  the topological dimension.

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Topological Dimension

• A set is n-dimensional if we need n independent
  variables to describe a neighborhood of any
  point.

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Hausdorff-Besicovitch dimension

• a dimension in which non integral values are
  permitted.
• If we take an object in dimension D and reduce
  its linear size by 1/r in each spatial direction,
  its measure (length, area, or volume) would
  increase to NrD times the original.

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• r the number of pieces that an edge is cut
  into.N the number of objects that result from
  cutting a parent object along each edge into r
  pieces.

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• Koch Curver 3, N 4, 4 3D, D ln4/ln3
  1.2619
• The Sierpinski Triangler 2, N 3, 3 2D, D
  1.58496

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Fractal Art
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Some Uses of Fractals

• To measure the lengths of coastlines
• To simulate turbulence
• To make scalable fonts
• To create excellent antennas
• To model DNA sequences
• To model crystal formations
• To model growth patterns
• In printer technology
• To resize objects on a computer
• To create landscape for films

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Fractal Representations of Mathematical concepts

• In Pascal's Triangle, colouring all Odd numbers
  black and Even numbers white produces the
  Sierpinski triangle.

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• Simple fractal trees can be used to
  geometrically represent number systems

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Binary Tree
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Fractal Music

• http//www.ocf.berkeley.edu/wwu/sounds/au/struct
  ure1.au

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Fractals in Nature

• Shells
• Cauliflowers
• Snowflakes
• Lung Structure
• Leaf Veins
• Clouds
• Trees
• Waterfalls
• Coastlines

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Conclusion