8.7 Non-Euclidean Geometry and Fractal Geometry 529 To understand the fifth axiom of the two non-Euclidean geometries, remember that the term line is undefined. Thus, a line can be interpreted differently in different geometries. A model for Euclidean geometry is a plane, such as a blackboard (Fig. 8.100a). A model for elliptical geometry is a sphere (Fig. 8.100b). A model for hyperbolic geometry is a pseudosphere (Fig. 8.100c). A pseudosphere is similar to two trumpets placed bell to bell. Obviously, a line on a plane cannot be the same as a line on either of the other two figures. (a) Plane (b) Sphere (c) Pseudosphere Figure 8.100 Profiles in Mathematics Grigori Perelman Grigori Perelman (1966– ) is a Russian mathematician whose major contributions in 2003–2004 to non-Euclidean geometry allowed him to prove a long-standing conjecture in topology. The Poincaré conjecture, first posed in 1904 by French mathematician Henri Poincaré, concerns the relationship between three-dimensional space and spheres. In addition to gaining fame for proving such an important idea, Perelman attained notoriety by refusing to accept two major recognitions for his accomplishment. When asked why he refused both the prestigious Fields Medal and the $1 million Millennium Prize, Perelman remarked: “I’m not interested in money or fame. I don’t want to be on display like an animal in a zoo. I’m not a hero of mathematics. I’m not even that successful; that is why I don’t want to have everybody looking at me.” In December of 2005, Perelman left his job at the Steklov Mathematical Institute in Moscow and has not published anything since. Many former colleagues believe that Perelman has completely stopped working on mathematics altogether. Elliptical Geometry A circle on the surface of a sphere is called a great circle if it divides the sphere into two equal parts. If we were to cut through a sphere along a great circle, we would have two identical pieces. If we interpret a line to be a great circle, then the two red curves in Fig. 8.101(a) are lines. Fig. 8.101(a) shows that the fifth axiom of elliptical geometry is true. Two great circles on a sphere must intersect; hence, there can be no parallel lines (Fig. 8.101a). If we were to construct a triangle on a sphere, the sum of its angles would be greater than 180° (Fig. 8.101b). The theorem “The sum of the measures of the angles of a triangle is greater than 180°” has been proven by means of the axioms of elliptical geometry. The sum of the measures of the angles varies with the area of the triangle and gets closer to 180° as the area decreases. (a) (b) Figure 8.101 Hyperbolic Geometry Lines in hyperbolic geometry are represented by geodesics on the surface of a pseudosphere. A geodesic is the shortest and least-curved arc between two points on a curved surface. Fig. 8.102 shows two different lines represented by geodesics, colored in red, on the surface of a pseudosphere. For simplicity of the diagrams, we show only one of the “bells” of the pseudosphere. (a) (b) Figure 8.102
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