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Section 9.7
Non-Euclidean
Geometry and
Fractal
Geometry
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Non-Euclidean Geometry
Fractal Geometry
9.7-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Euclid’s Fifth Postulate
If a straight line falling on two
straight lines makes the interior
angles on the same side less than two
right angles, the two straight lines, if
produced indefinitely, meet on that
side on which the angles are less than
the two right angles.
9.7-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Euclid’s Fifth Postulate
The sum of angles A and B is less
than the sum of two right angles
(180º). Therefore, the two lines will
meet if extended.
9.7-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Playfair’s Postulate or
Euclidean Parallel Postulate
Given a line and a point not on the
line, one and only one line can be
drawn through the given point
parallel to the given line.
9.7-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Non-Euclidean Geometry
Euclidean geometry is geometry in a
plane.
Many attempts were made to prove the
fifth postulate.
These attempts led to the study of
geometry on the surface of a curved
object:
Hyperbolic geometry
Spherical, elliptical or Riemannian
geometry
9.7-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Non-Euclidean Geometry
A model may be considered a physical
interpretation of the undefined terms
that satisfies the axioms. A model may
be a picture or an actual physical
object.
9.7-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Fifth Axiom of Three Geometries
Euclidean
Given a line
and a point
not on the
line, one and
only one line
can be drawn
through the
given point
parallel to the
given line.
9.7-8
Elliptical
Given a line
and a point
not on the
line, no line
can be drawn
through the
given point
parallel to the
given line.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Hyperbolic
Given a line
and a point
not on the
line, two or
more lines
can be drawn
through the
given point
parallel to the
given line.
A Model for the Three Geometries
The term line is undefined.
It can be interpreted differently in
different geometries.
Euclidean
Elliptical
Plane
Sphere
Hyperbolic
Pseudosphere
9.7-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Elliptical Geometry
A circle on the surface of a sphere is
called a great circle if it divides the
sphere into two equal parts.
We interpret a line to be a great circle.
This shows the fifth axiom
of elliptical geometry to be
true. Two great circles on a
sphere must intersect;
hence, there can be no
parallel lines.
9.7-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Elliptical Geometry
If we were to construct a
triangle on a sphere, the
sum of its angles would be
greater than 180º.
The sum of the measures of the angles
varies with the area of the triangle and
gets closer to 180º as the area
decreases.
9.7-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Hyperbolic Geometry
Lines in hyperbolic
geometry are
represented by geodesics
on the surface of a
pseudosphere.
A geodesic is the shortest
and least-curved arc
between two points on
the surface.
9.7-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Hyperbolic Geometry
This illustrates one
example of the
fifth axiom:
through the given
point, two lines
are drawn parallel
to the given line.
9.7-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Hyperbolic Geometry
If we were to construct a
triangle on a pseudosphere,
the sum of the measures of
the angles of the triangle
would be less than 180º.
9.7-14
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Fractal Geometry
Many objects are difficult to categorize
as one-, two-, or three-dimensional.
Examples are: coastline, bark on a tree,
mountain, or path followed by lightning.
It’s possible to make realistic geometric
models of natural shapes using fractal
geometry.
9.7-15
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Fractal Geometry
Fractals have dimension between 1 and
2.
Fractals are developed by applying the
same rule over and over again, with the
end point of each simple step becoming
the starting point for the next step, in a
process called recursion.
9.7-16
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Koch Snowflake
Start with an equilateral triangle.
Whenever you see an edge
replace it with
.
9.7-17
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Koch Snowflake
The snowflake has infinite perimeter:
after each step, the perimeter is 4/3
times the perimeter of the previous
step.
If has finite area: 1.6 times the area of
the original equilateral triangle.
9.7-18
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For Example: The Fractal Tree
9.7-19
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For Example: Sierpinski Triangle
9.7-20
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For Example: Sierpinski Carpet
9.7-21
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For Example: Fractal Images
9.7-22
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Chaos Theory
Fractal Geometry provides a
geometric structure for chaotic
processes in nature.
The study of chaotic processes is
called chaos theory.
9.7-23
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Fractals
Potentially important role to play:
characterizing weather systems
providing insight into various physical
processes such as the occurrence of
earthquakes or the formation of
deposits that shorten battery life
9.7-24
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Fractals
Some scientists view fractal statistics
as a doorway to unifying theories of
medicine, offering a powerful glimpse
of what it means to be healthy.
9.7-25
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Fractals
Fractals lie at the heart of current
efforts to understand complex natural
phenomena.
Unraveling their intricacies could reveal
the basic design principles at work in
our world.
Until recently, there was no way to
describe fractals.
9.7-26
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Fractals
Today, we are beginning to see such
features everywhere.
Tomorrow, we may look at the entire
universe through a fractal lens
9.7-27
Copyright 2013, 2010, 2007, Pearson, Education, Inc.