8.7 Non-Euclidean Geometry and Fractal Geometry 527 Consider the following question: Given a line l and a point P not on the line l, how many lines can you draw through P that are parallel to l ? Non-Euclidean Geometry and Fractal Geometry SECTION 8.7 LEARNING GOALS Upon completion of this section, you will be able to: 7 Describe non-Euclidean geometry including elliptical and hyperbolic geometry. 7 Describe fractal geometry. P l You may answer that only one line may be drawn through P parallel to l. This answer would be correct provided the setting of the problem is in a plane and not on the surface of a curved object. The study of this question led to the development of several new branches of geometry. It is now believed these branches of geometry, taken together, can be used to accurately represent space. In this section, we will study the geometry of surfaces other than the geometry of the plane. Why This Is Important While Euclidean geometry works well for most applications we encounter on Earth, scientists believe that non-Euclidean geometry is necessary to model the universe. Einstein’s General Theory of Relativity is based on a theory that space is curved. The non-Euclidean geometry that we discuss in this section is also curved and provides a better scientific basis for studying both time and space. Additionally, non-Euclidean geometry is used in many technology applications including computer storage, medical imaging, and many of our smartphone apps. Origins of Non-Euclidean Geometry In Section 8.1, we stated that postulates or axioms are statements to be accepted as true. In his book Elements, Euclid’s fifth postulate was, “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, In this exercise we will construct another interesting surface. We begin by constructing a “cross” shape from two strips of paper, as shown below, using scissors and tape. Note the red dashed line and the green dashed line and the ends of the strips labeled A or B. A A B B Next, using tape connect the two ends labeled A without twisting the ends. Then, connect the two ends labeled B by giving one end a half twist. The strip that connects the B ends should resemble a Möbius strip. Finally, cut the object first along the green dashed line and then along the red dashed line. Compare the result with that from the construction on page 523. What do you notice? Answers will vary. Research Activities 44. Counties in Your State Print out a map of the state you live in that shows the outline of all of the counties. Using at most four colors, color the map so that no two counties that share a common border are the same color. 45. Paul Bunyan The short story Paul Bunyan versus the Conveyor Belt (1947) by William Hazlett Upson focuses on a conveyor belt in the shape of a Möbius strip. The story can be found in several books that include mathematical essays. Read Upson’s short story and write a 200-word description of what Paul Bunyan does to the conveyor belt. Confirm the outcome of the story by repeating Paul’s actions with a paper Möbius strip. NASA images/Shutterstock
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