Fractal Geometry
Fractals are important because they
reveal a new area of mathematics
directly relevant to the study of nature.
- Ian Stewart
Euclidean Geometry
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Triangles
Circles
Squares
Rectangles
Trapezoids
Pentagons
Hexagons
Octagons
Cyclinders
Can nature be described in terms
of Euclidean Geometry?
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???
Look outside… Do you see any
shapes in Euclidean Geometry?
If so, they were more than likely
man made. For example…
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?
1. Historically…
• Plato believed he
• Scientists now know
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could explain nature
with five regular solid
forms.
Astronomers believed
that our orbit around
the sun was circular.
that Plato’s shapes
are particles and
waves.
Astronomers now
know that our orbit is
elliptical not circular.
2. The mathematics is relatively
easy.
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Perimeter
Area
Surface area
Volume
Relationships between two shapes
For example: Square vs. shape of Mississippi
3. Men and women consider a
house with smooth edges and
uniform shapes more beautiful than
a house with rough edges and nonuniform 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?
Fractals Defined.
• Geometry of irregular shapes which are
characterized by infinite detail, infinite
length, and the absence of smoothness.
• Let’s first see what a fractal is not.
A rectangle is not a fractal.
• When we look through a microscope at
the rectangle do we see any new details.
• The teacher may want to put fractal
pictures here.