Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

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

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

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

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. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. Elliptical Geometry

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

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

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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Koch Snowflake Start with an equilateral triangle. Whenever you see an edge replace it with

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. For Example: The Fractal Tree

Copyright 2013, 2010, 2007, Pearson, Education, Inc. For Example: Sierpinski Triangle

Copyright 2013, 2010, 2007, Pearson, Education, Inc. For Example: Sierpinski Carpet

Copyright 2013, 2010, 2007, Pearson, Education, Inc. For Example: Fractal Images

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Chaos Theory Fractal Geometry provides a geometric structure for chaotic processes in nature. The study of chaotic processes is called chaos theory

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Fractals Today, we are beginning to see such features everywhere. Tomorrow, we may look at the entire universe through a fractal lens