line
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Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 1
Chapter 9 Geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 2
WHAT YOU WILL LEARN • Transformational geometry, symmetry, and tessellations • The Mobius Strip, Klein bottle, and maps • Non-Euclidian geometry and fractal geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 3
Section 7 Non-Euclidean Geometry and Fractal Geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 4
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. 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. Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 5
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.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 6
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
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 7
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. Copyright © 2009 Pearson Education, Inc.
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.
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. Chapter 9 Section 7 - Slide 8
A Model for the Three Geometries
The term line is undefined. It can be interpreted differently in different geometries.
Euclidean Plane
Elliptical Sphere
Hyperbolic Pseudosphere Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 9
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.
Copyright © 2009 Pearson Education, Inc.
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. Chapter 9 Section 7 - Slide 10
Elliptical Geometry (continued)
If we were to construct a triangle on a sphere, the sum of its angles would be greater than 180º.
This sum of the measures of the angles varies with the area of the triangle and gets closer to 180º as the area decreases.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 11
Hyperbolic Geometry
Lines in hyperbolic geometry are represented by geodesics on the surface of a pseudosphere. A geodesic is the shortest and leastcurved arc between two points on the surface. Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 12
Hyperbolic Geometry (continued)
This illustrates one example of the fifth axiom: through the given point, two lines are drawn parallel to the given line.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 13
Hyperbolic Geometry (continued)
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.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 14
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. 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. Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 15
Koch Snowflake
Start with an equilateral triangle. Replace each edge with
.
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.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 16
Example: Fractal Tree
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 17
Example: Sierpinski Triangle
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 18
Example: Sierpinski Carpet
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 19
Example: Fractal Images
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 20
Examples Fractal Images
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 21
Chaos Theory
Fractal Geometry provides a geometric structure for chaotic processes in nature. The study of chaotic processes is called chaos theory.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 48
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 1
Chapter 9 Geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 2
WHAT YOU WILL LEARN • Transformational geometry, symmetry, and tessellations • The Mobius Strip, Klein bottle, and maps • Non-Euclidian geometry and fractal geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 3
Section 7 Non-Euclidean Geometry and Fractal Geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 4
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. 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. Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 5
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.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 6
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
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 7
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. Copyright © 2009 Pearson Education, Inc.
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.
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. Chapter 9 Section 7 - Slide 8
A Model for the Three Geometries
The term line is undefined. It can be interpreted differently in different geometries.
Euclidean Plane
Elliptical Sphere
Hyperbolic Pseudosphere Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 9
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.
Copyright © 2009 Pearson Education, Inc.
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. Chapter 9 Section 7 - Slide 10
Elliptical Geometry (continued)
If we were to construct a triangle on a sphere, the sum of its angles would be greater than 180º.
This sum of the measures of the angles varies with the area of the triangle and gets closer to 180º as the area decreases.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 11
Hyperbolic Geometry
Lines in hyperbolic geometry are represented by geodesics on the surface of a pseudosphere. A geodesic is the shortest and leastcurved arc between two points on the surface. Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 12
Hyperbolic Geometry (continued)
This illustrates one example of the fifth axiom: through the given point, two lines are drawn parallel to the given line.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 13
Hyperbolic Geometry (continued)
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.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 14
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. 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. Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 15
Koch Snowflake
Start with an equilateral triangle. Replace each edge with
.
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.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 16
Example: Fractal Tree
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 17
Example: Sierpinski Triangle
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 18
Example: Sierpinski Carpet
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 19
Example: Fractal Images
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 20
Examples Fractal Images
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 21
Chaos Theory
Fractal Geometry provides a geometric structure for chaotic processes in nature. The study of chaotic processes is called chaos theory.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 7 - Slide 48
