Fractals

1
Fractals

• Siobhán Rafferty

2
What Are Fractals?

• a set of points whose fractal dimension exceeds
  its topological dimension
• A self-similar geometrical shape that includes
  the same pattern, scaled down and rotated and
  repeated.

3
The Koch Curve

• The Koch Curve is a famous example of a Fractal
  published by Niels Fabien Helge von Koch in 1906
• Stage 0 is a straight line segment
• Stages 1 - infinity are produced by repeating
  stage1 along every line segment of the previous
  stage.

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The Koch Snowflake

• The perimeter of each stage is 1.33 x the
  perimeter of the pervious stage.
• When we repeat the stages to infinity the
  perimeter is infinite.
• Most geometrical shapes have an Area Perimeter
  Relationship. This does not hold with Fractals
• An infinite perimeter encloses a finite area.

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Dimensions

• Fractals have non-integer dimensions that can be
  calculated using logarithms.
• If the length of the edges on a cube is
  multiplied by 2, 8 of the old cubes would fit
  into the new curve.
• Log8/Log2 3, a cube is 3 dimensional.
• Similarly for a fractal of size P, made of
  smaller units (size p), the number of units (N)
  that fits into the larger object is equal to the
  size ratio (P/p) raised to the power of d
• D Log(N)/Log(P/p)

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Dimensions, an example.

• Each line in stage 1 is made up of lines 3cm long
  (P3)
• There are 12 line segments
• Stage 2 has lines of length 1cm (p1)
• It has 48 line segments (N 48/12 4)
• d log 4/ log 3 1.2619

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Benoit Mandelbrot

• Born in 1924 and currently a mathematics
  professor at Yale University
• The Mandelbrot Set
• x²c, where c is a complex number.
• X1 0² c
• X2 c² c
• X3 (c² c)² c

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The Mandelbrot Set

• If it takes very few iterations for the
  iterations to become very large and tend to
  infinity then the value c is marked in red.
• Numbers are marked on the set following the light
  spectrum orange, yellow, green, blue, indigo,
  violet in order of those tending to infinity at
  different rates.
• The values shown in black do not escape to
  infinity
• The result is a fractal!