© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 10 Geometry

© 2010 Pearson Prentice Hall. All rights reserved Beyond Euclidean Geometry

© 2010 Pearson Prentice Hall. All rights reserved. Objective 1.Gain an understanding of some of the general ideas of other kinds of geometries. 3

© 2010 Pearson Prentice Hall. All rights reserved. The Geometry of Graphs (Graph Theory) The Swiss mathematician Leonhard Euler (1707 – 1783) proved that it was not possible to stroll through the city of Kőnigsberg, Germany by crossing each of 7 bridges exactly once. His solution opened up a new kind of geometry called graph theory. 4

© 2010 Pearson Prentice Hall. All rights reserved. Graph Definitions Vertex is a point. Edge is a line segment or curve that starts and ends at a vertex. Graph consists of vertices and edges Odd vertex has an odd number of attached edges. Even vertex has an even number of attached edges. 5

© 2010 Pearson Prentice Hall. All rights reserved. Rules of Traversability A graph is traversable if it can be traced without lifting the pencil from the paper and without tracing an edge more than once. 1.A graph with all even vertices is traversable. One can start at any vertex and end where one began. 2.A graph with two odd vertices is traversable. One must start at either of the odd vertices and finish at the other. 3.A graph with more than two odd vertices is not traversable. 6

© 2010 Pearson Prentice Hall. All rights reserved. Example 1: To Traverse or Not to Traverse? Is this graph traversable? Solution Begin by determining if each vertex is even or odd. This graph has two odd vertices, by Euler’s second rule, it is traversable. Describe the path to traverse it. Solution By Euler’s second rule, start at one of the odd vertices and finish at the other. 7

© 2010 Pearson Prentice Hall. All rights reserved. Topology The Study of Shapes Objects are classified according to the number of holes in them, called their genus. Genus is the number of cuts that can be made in the object without cutting the object in two pieces. Topologically Equivalent objects have the same genus. The topology of knots is used to identify viruses and how they invade our cells. 8

© 2010 Pearson Prentice Hall. All rights reserved. Examples of Topological Equivalency The three shapes below have the same genus: 0. No complete cuts can be made without cutting these objects into two pieces. A doughnut and a coffee cup have a genus of 1. 9

© 2010 Pearson Prentice Hall. All rights reserved. Klein Bottle The figures below show the transformation into the figure called the Klein bottle. Because the inside surface loops back on itself to merge with the outside, it has neither an outside nor an inside. It passes through itself without the existence of a hole, which is impossible in three-dimensional space. It only exists when generated on a computer. 10

© 2010 Pearson Prentice Hall. All rights reserved. Comparing the Three Systems of Geometry (Euclidean and non-Euclidean) Euclidean Geometry Euclid (300 B.C.) Hyperbolic Geometry Lobachevsky, Bolyai (1830) Elliptic Geometry Riemann(1850) Given a point not on a line, there is one and only one line through the point parallel to the given line. Given a point not on a line, there are an infinite number of lines through the point that do not intersect the given line. There are no parallel lines Geometry is on a plane: Geometry is on a pseudosphere:Geometry is on a sphere: 11

© 2010 Pearson Prentice Hall. All rights reserved. Comparing the Three Systems of Geometry (Euclidean and non-Euclidean) Euclidean Geometry Euclid (300 B.C.) Hyperbolic Geometry Lobachevsky, Bolyai (1830) Elliptic Geometry Riemann(1850) The sum of the measures of the angles of a triangle is 180 . The sum of the measures of the angles of a triangle is less than 180 . The sum of the measures of the angles of a triangle is greater than 180 . 12

© 2010 Pearson Prentice Hall. All rights reserved. Fractal Geometry Developed by Benoit Mandelbrot using computer programming. Geometry of natural shapes. Self-similarity is the quality of smaller versions of an object appearing in the object itself. Iteration is the process of repeating a rule again and again to create a self-similar fractal. 13