Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 1 AND

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1 Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 1 AND

2 Chapter 9 Geometry Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 2

3 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 6 - Slide 3

4 Section 6 Topology Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 4

5 Definitions The branch of mathematics called topology is sometimes referred to as rubber sheet geometry because it deals with bending and stretching of geometric figures. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 5

6 Möbius Band A Möbius strip, also called a Möbius band, is a one-sided, one-edged surface. Construct a Möbius band by taking a strip of paper, giving one end a half twist, and taping the two ends together. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 6

7 Properties of a Möbius Band It is one-edged. Place a felt marker on the edge and without removing the marker trace along the edge. Remarkably, the marker travels around the entire edge and ends where it begins! Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 7

8 Properties of a Möbius Band (continued) It is one-sided. Place a felt marker on the surface and without removing the marker trace along the surface. Remarkably, the marker travels around the entire surface and ends where it begins! Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 8

9 Properties of a Möbius Band (continued) Using scissors cut along the center of the length of the band. Remarkably, you end up with one larger band with three (half) twists! (Topologically the same as a Möbius band.) Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 9

10 Properties of a Möbius Band (continued) Using scissors make a small slit at a point about one-third of the width of the band. Cut along the length of the band keeping the same distance from the edge. Remarkably, you end up with one small Möbius band interlocked with one larger band with two (half) twists! Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 10

11 Klein Bottle The punctured Klein bottle resembles a bottle but only has one edge and one side. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 11

12 Klein Bottle Klein bottle, a one-sided surface, blown in glass by Alan Bennett. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 12

13 Klein Bottle Imagine trying to paint a Klein bottle. You start on the outside of the large part and work your way down the narrowing neck. When you cross the self-intersection, you have to pretend temporarily that it is not there, so you continue to follow the neck, which is now inside the bulb. As the neck opens up, to rejoin the bulb, you find that you are now painting the inside of the bulb! What appear to be the inside and outside of a Klein bottle connect together seamlessly since it is one-sided. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 13

14 Maps Mapmakers have known for a long time that regardless of the complexity of the map and whether it is drawn on a flat surface or a sphere, only four colors are needed to differentiate each country (or state) from its immediate neighbors. Thus, every map can be drawn by using only four colors, and no two countries with a common border will have the same color. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 14

15 Maps Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 15

16 Maps Mathematicians have show that on different surfaces, more colors may be needed. A Möbius band requires a maximum of six, while a torus (doughnut) requires a maximum of seven. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 16

17 Jordan Curves A Jordan curve is a topological object that can be thought of as a circle twisted out of shape. Like a circle it has an inside and an outside. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 17

18 Topological Equivalence Two geometric figures are said to be topologically equivalent if one figure can be elastically twisted, stretched, bent, or shrunk into the other figure without puncturing or ripping the original figure. The doughnut and coffee cup are topologically equivalent. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 18

19 In topology, figures are classified according to their genus. The genus of an object is determined by the number of holes that go through the object. Copyright 2009 Pearson Education, Inc. Chapter 9 Section 6 - Slide 19