526 CHAPTER 8 Geometry 23. 0 24. 3 Janifest/123RF 36. How many separate strips are obtained in Experiment 3 on page 521 One 37. How many separate strips are obtained in Experiment 4 on page 521 Two 38. Make a Möbius strip. Cut it one-third of the way from the edge, as in Experiment 4 on page 521. You should get two loops, one going through the other. Determine whether either (or both) of these loops is itself a Möbius strip. The smaller one is a Möbius strip; the larger one is not. 39. a) Take a strip of paper, give it one full twist, and connect the ends. Is the result a Möbius strip with only one side? Explain. No, it has an inside and an outside. b) Determine the number of edges, as in Experiment 1. Two c) Determine the number of surfaces, as in Experiment 2. Two d) Cut the strip down the middle. What is the result? Two strips, one inside the other 40. Take a strip of paper, make one whole twist and another half twist, and then tape the ends together. Test by a method of your choice to determine whether this has the same properties as a Möbius strip. No, it does not. Challenge Problems/Group Activities 41. Using clay (or glazing compound), make a doughnut. Without puncturing or tearing the doughnut, reshape it into a topologically equivalent figure, a cup with a handle. Answers will vary. 42. Using at most four colors, color the following map of the counties of New Mexico. Do not use the same color for any two counties that share a common border. Answers will vary. Hidalgo Luna Dona Ana Otero Eddy Lea Chaves Lincoln Socorro Sierra Grant Catron Cibola McKinley San Juan Rio Arriba Taos Colfax Union Harding Mora San Miguel Sandoval Los Alamos Santa Fe Torrance Guadalupe Quay Curry De Baca Valencia Roosevelt Bernalillo Recreational Mathematics 43. Topological Paper Constructions Using paper, scissors, and tape, perform the construction described in the Recreational Math box on page 523. Once you have completed the construction, cut along the dashed line as instructed. Set the result aside. 25. 5 26. 1 27. 3 28. 2 29. Larger than 5 30. Larger than 5 31. 4 32. SODA 0 33. Name at least three objects not mentioned in this section that have a) genus 0. Answers will vary. b) genus 1. Answers will vary. c) genus 2. Answers will vary. d) genus 3 or more. Answers will vary. 34. Use the result of Experiment 1 on page 521 to determine the number of edges on a Möbius strip. One 35. Use the result of Experiment 2 on page 521 to determine the number of surfaces on a Möbius strip. One
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