Fractals

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Fractals
Melinda Stacknick
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What Are Fractals?

• Geometric figures that model structures in nature
  (ferns and clouds)
• Objects that are NOT formed by squares or
  triangles
• Have special properties
• Studied beginning in the late 1800s
• Benoit Mandelbrot led research on fractals

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Benoit Mandelbrot

• Showed how fractals can occur in mathematics and
  nature
• Did not agree with conventional mathematics
  
  education
• Took a geometrical approach to mathematics
• Worked at IBM where he developed the first
  computer program to print graphics
• Wrote The Fractal Geometry of Nature in 1982

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Mandelbroit Set

• Z1 z02 z0
• Z2 z12 z0
• Z3 z22 z0

The points diverge to infinity.
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Fractals in Nature

• Waterline
• Branches of trees
• Ferns
• Snail shells

If you measured the waterline using a mile long
ruler and again using a shorter ruler would you
get the same measurements?
No, shorter rulers would be a more accurate
measure, and you could keep measuring using
shorter and shorter rulers.
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Fractal Properties

• Self-similarity
• Fractal dimension
• Iterative formation

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Self-similarity

• Copies of the same shape
• Get smaller and smaller

How many equilateral triangles are in
Sierpinskis triangle?
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Make Sierpinskis Triangle
Step 1 On triangular grid paper, draw an
equilateral triangle with sides of 2 triangle
lengths each. Connect the midpoints of each
side. Shade out the center triangle.
Step 2 Draw another equilateral triangle with
sides of 4 triangle lengths each. Connect the
midpoints of the sides and shade the triangle in
the center as before. Also shade out the other
three small triangles.
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Step 3 Draw an equilateral triangle with sides
of 8 triangle lengths each. Follow the same
procedure as before, making sure to follow the
shading pattern. You will have 1 large, 3 medium,
and 9 small triangles shaded.
Step 4 Follow the pattern and complete the
Sierpinski Triangle.
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Fractal Dimension

• Fractals are formed by an infinite number of
  steps
• Shapes keep getting smaller and smaller

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Iterative Formation

• Repeating the same operation creates a more
  complicated figure
• Fractal snowflake example

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Koch Snowflake

• Step 1
• Step 2
• Step 3

• Step 4
• Step 5

Start with a large equilateral triangle. Divide
one side of the triangle into three equal
parts and remove the middle section. Replace
it with two lines the same length as the
section you removed. Do this to all three
sides of the triangle. Repeat.