Fractal Geometry

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Fractal Geometry

• Fractals are important because they reveal a new
  area of mathematics directly relevant to the
  study of nature.
• - Ian Stewart

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Euclidean Geometry

• Triangles

• Circles

• Squares

• Rectangles

• Trapezoids

• Pentagons

• Hexagons

• Octagons

• Cyclinders

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Can nature be described in terms of Euclidean
Geometry?
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Try drawing nature using Euclidean Geometry

• A tree using cylinders???
• A mountain range using triangles and pyramids???
• Clouds using circles???
• Leaves???
• Rocks???
• Humans and animals with rectangles and circles???

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Look outside Do you see any shapes in Euclidean
Geometry?
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If so, they were more than likely man made. For
example
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The point is this

• Our world is fashioned with rough edges and
  non-uniform shapes.
• Euclidean geometry describes ideal shapes which
  rarely occur in nature.
• Why then do we even bother with Euclidean
  geometry?

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1. Historically

• Plato believed he could explain nature with five
  regular solid forms.
• Astronomers believed that our orbit around the
  sun was circular.

• Scientists now know that Platos shapes are
  particles and waves.
• Astronomers now know that our orbit is elliptical
  not circular.

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2. The mathematics is relatively easy.

• Perimeter
• Area
• Surface area
• Volume
• Relationships between two shapes
• For example Square vs. shape of Mississippi

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3. Men and women consider a house with smooth
edges and uniform shapes more beautiful than a
house with rough edges and non-uniform shapes.

• Why is that nature is more beautiful with rough
  edges and non-uniform shapes and man made objects
  are less beautiful with rough edges and
  non-uniform shapes? And vice versa?

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Fractals Defined.

• Geometry of irregular shapes which are
  characterized by infinite detail, infinite
  length, and the absence of smoothness.
• Lets first see what a fractal is not.

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A rectangle is not a fractal.

• When we look through a microscope at the
  rectangle do we see any new details.

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• The teacher may want to put fractal pictures here.