Fractals

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Fractals
by Katie Burford, 2005
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What are Fractals?

• They're everywhere, those bright, weird,
  beautiful shapes called fractals. But what are
  they, really?
• Fractals are geometric figures, just like
  rectangles, circles and squares, but fractals
  have special properties that those figures do not
  have.
• There's lots of information on the Web about
  fractals, but most of it is either just pretty
  pictures or very high-level mathematics. Lets
  see if we can make it simple!

Info from http//math.rice.edu/lanius/frac/
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Why Study Fractals?

• They're New!
• Most math you study in school is old knowledge.
  For example, the geometry you study about
  circles, squares, and triangles was organized
  around 300 B.C. by a man named Euclid. Much of
  fractal geometry, however, is much newer.
  Research on fractals is being carried out right
  now by mathematicians. Have you ever thought
  about a career as a mathematician?
• You can understand them!
• Much research in mathematics is currently being
  done all over the world. Most of it is extremely
  complicated. Although we need to study and learn
  more before we can understand most modern
  mathematics, there's a lot about fractals that we
  can understand.

Info from http//math.rice.edu/lanius/frac/
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Properties of Fractals

• Self-Similarity
• Many figures that are not fractals are
  self-similar. Notice the figures below. Notice
• that the outline of the figure is a trapezoid.
  Now look inside at all the trapezoids that
• make up the larger trapezoid. This is an example
  of self similarity.
• To the left is the fractal named Sierpinski
  Triangle. Notice that the outline of the
  figure is an equilateral triangle. Now look
  inside at all the equilateral triangles. How
  many different sized triangles can you find?
  All of these are similar to each other and to the
  original triangle
• All fractals are self-similar

Info from http//math.rice.edu/lanius/frac/
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• Iterative Formation
• An iterative process is a repetitive operation.
  In a fractal it begins with a simple geometric
  shape then by iteration the figure gets more and
  more complicated.
• The iteration above will eventually create a
  famous fractal called The Koch Snowflake
• All Fractals are formed in this manner.

Info from http//math.rice.edu/lanius/frac/
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Fractals are used to model nature!
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• And to predict the change in the shore line
  after a major storm like Hurricane Katrina.

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Make your own Fractal!

• Step One -Draw an triangle. Connect the
  midpoints of each side. How many triangles do
  you now have? Shade out the triangle in the
  center. Think of this as cutting a hole in the
  triangle.
• Step Two -Connect the midpoints of the sides of
  the remaining triangles and shade the triangle in
  the center as before. Notice the three small
  triangles that also need to be shaded out in each
  of the three triangles on each corner - three
  more holes.
• Step Three -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 Four -Follow the above pattern and complete
  the Sierpinski Triangle
• Step Five Now try it again with another shape
  (not a triangle!)

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Cool Fractal Links

• http//www.bugman123.com/Fractals/Fractals.html
• http//fotofects.com/v2/adobe-photoshop/fractals.h
  tml
• http//www.fractalexperience.com/
• http//www.fbmn.fh-darmstadt.de/home/sandau/biofra
  ctals/abstract_sfi.html
• http//archive.ncsa.uiuc.edu/Edu/Fractal/Fractal_H
  ome.html
• http//sprott.physics.wisc.edu/fractals.htm
• http//www.math.umass.edu/mconnors/fractal/fracta
  l.html
• http//mathforum.org/alejandre/workshops/fractal/f
  ractal3.html
• http//www.ealnet.com/ealsoft/fracted.htm
• http//www.geom.uiuc.edu/java/LeapFractal/