524 CHAPTER 8 Geometry a) Genus 0 b) Genus 1 c) Genus 2 d) Genus 3 or more Solution The answers are shown in the following columns. Strainer d) Genus 3 or more Grater Doughnut b) Genus 1 Coffee Cup Kettle c) Genus 2 Scissors Marble a) Genus 0 Drinking Glass 7 Topological Equivalence Someone once said that a topologist is a person who does not know the difference between a doughnut and a coffee cup. 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. If a doughnut is made of elastic material, it can be stretched, twisted, bent, shrunk, and distorted until it resembles a coffee cup with a handle, as shown in Fig. 8.96. Thus, the doughnut and the coffee cup are topologically equivalent. 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. A cup with a handle and a doughnut each have one hole and are of genus 1 (and are therefore topologically equivalent). Notice that the cup’s handle is considered a hole, whereas the opening at the rim of the cup is not considered a hole. For our purposes, we will consider an object’s opening a hole if you could pour liquid through the opening. For example, a typical bowling ball (see Fig. 8.97) has three openings in the surface into which you can put your fingers when preparing to roll the ball, but liquid cannot be poured through any of these openings. Therefore, a bowling ball has genus 0 and is topologically equivalent to a marble. Figure 8.96 If the line crosses the Jordan curve an even number of times, the point in question is outside the Jordan curve. If the line crosses the Jordan curve an odd number of times, the point in question is inside the Jordan curve. In the figure, the line connecting point A to a point outside the curve crosses the Jordan curve 10 times. Since 10 is an even number, point A is outside the Jordan curve. b) In the figure, the line connecting point B to a point outside the curve crosses the Jordan curve 7 times. Since 7 is an odd number, point B is inside the Jordan curve. 7 Now try Exercise 17 Example 2 Genus Name an object that has the given genus. Now try Exercise 25 Learning Catalytics Keyword: Angel-SOM-8.6 (See Preface for additional details.) Bowling Ball Figure 8.97 Instructor Resources for Section 8.6 in MyLab Math • Objective-Level Videos 8.6 • Animation: Four Color Challenge • PowerPoint Lecture Slides 8.6 • MyLab Exercises and Assignments 8.6
RkJQdWJsaXNoZXIy NjM5ODQ=