Chapter 9 Geometry © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 9: Geometry 9.1 Points, Lines, Planes, and Angles 9.2 Curves, Polygons, and Circles 9.3 Perimeter, Area, and Circumference 9.4 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem 9.5Space Figures, Volume, and Surface Area 9.6Transformational Geometry 9.7 Non-Euclidean Geometry, Topology, and Networks 9.8 Chaos and Fractal Geometry

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 9-7 Non-Euclidean Geometry, Topology, and Networks

© 2008 Pearson Addison-Wesley. All rights reserved Non-Euclidean Geometry, Topology, and Networks Euclid’s Postulates and Axioms The Parallel Postulate (Euclid’s Fifth Postulate) The Origins of Non-Euclidean Geometry Topology Networks

© 2008 Pearson Addison-Wesley. All rights reserved Postulates and Axioms To some Greek writers, postulates were truths about a particular field, while axioms were general truths. Today “axiom” is used in either case.

© 2008 Pearson Addison-Wesley. All rights reserved Euclid’s Postulates 1. Two points determine one and only one straight line. 2. A straight line extends indefinitely far in either direction. 3. A circle may drawn with any given center and any given radius. 4. All right angles are equal. 5. Given a line k and a fixed point P not on the line, there exists one and only one line m through P that is parallel to k (known as Playfair’s axiom).

© 2008 Pearson Addison-Wesley. All rights reserved Euclid’s Axioms 6. Things equal to the same thing are equal to each other. 7. If equals are added to equals, the sums are equal. 8. If equals are subtracted from equals, the remainders are equal. 9. Figures that can be made to coincide are equal. 10. The whole is greater than any of its two parts.

© 2008 Pearson Addison-Wesley. All rights reserved The Parallel Postulate (Euclid’s Fifth Postulate) In its original form, it states that if two lines (k and m below) are such that a third line, n, intersects them so that the sum of the two interior angles (A and B) on one side of line n is less than (the sum of) two right angles, then the two lines, if extended far enough, will meet on the same side of n that has the sum of the interior angles less than (the sum of) two right angles. k m n A B

© 2008 Pearson Addison-Wesley. All rights reserved Non-Euclidean Geometry A system of geometry in which the fifth postulate is changed is called a non-Euclidean geometry.

© 2008 Pearson Addison-Wesley. All rights reserved Lobachevskian Geometry Replaces the fifth postulate with: Through a point P off a line k, at least two different lines can be drawn parallel to k. k m n P

© 2008 Pearson Addison-Wesley. All rights reserved Lobachevskian Geometry The sum of the measures of the angles in any triangle is less than 180°. Triangles of different sizes can never have equal angles so similar triangles do not exist. This geometry can be represented as a surface called a pseudosphere (see below). A B

© 2008 Pearson Addison-Wesley. All rights reserved Riemannian Geometry “Lines” in this geometry are really great circles, or circles whose centers are at the center of a sphere.

© 2008 Pearson Addison-Wesley. All rights reserved Riemannian Geometry The sides of a triangle drawn on a sphere would be arcs of great circles. And, in Riemannian geometry, the sum of the measures of the angles in any triangle is more than 180°. Replaces the fifth postulate with: Through a point P off a line k, no line can be drawn that is parallel to k.

© 2008 Pearson Addison-Wesley. All rights reserved Topology Topology studies figures that could be stretched, bent, or otherwise distorted without tearing or scattering. Topological questions concern the basic structure of objects rather than size or arrangement. A typical topological question has to do with the number of holes in an object.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Topological Equivalence Decide if a doughnut and an unzipped coat are topologically equivalent. Solution No, since a doughnut has one hole in it and the coat has two (the sleeve openings). The doughnut can not be twisted and stretched into the shape of the coat without tearing a new hole in it.

© 2008 Pearson Addison-Wesley. All rights reserved Genus Figures can be classified according to their genus – that is, the number of cuts that can be made without cutting the figures into two pieces. The genus of an object is the number of holes in it. See the next slide for examples.

© 2008 Pearson Addison-Wesley. All rights reserved Genus

© 2008 Pearson Addison-Wesley. All rights reserved Networks A network is a diagram showing the various paths (or arcs) between points (called vertices, or nodes). A network can be thought of as a set of arcs and vertices. An odd vertex is one where an odd number of paths meet and an even vertex is one where an even number of paths meet. Transversability is concerned with crossing each path in a network only once.

© 2008 Pearson Addison-Wesley. All rights reserved Results on Vertices and Traversability 1. The number of odd vertices of any network is even (That is, a network must have 2n odd vertices, where n = 0, 1, 2, 3,….) 2. A network with no odd vertices or exactly two odd vertices can be traversed. In the case of exactly two, start at one odd vertex and end at the other. 3. A network with more than two odd vertices cannot be transversed.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Deciding Whether Networks are Transversable Decide whether the network below is transversable. A B C D E Solution Because there are more than two odd vertices (there are 5), this network can not be transversed.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Applying Transversability to a Floor Plan Decide if it is possible to travel through the house (floor plan below) going through each door once. ABC DE

© 2008 Pearson Addison-Wesley. All rights reserved Example : Applying Transversability to a Floor Plan Solution Rooms A, C, and D have an even number of doors, while rooms B and E have an odd number. With the rooms as vertices, we have exactly two odd vertices It is possible to travel through each door exactly once. ABC DE