530 CHAPTER 8 Geometry Fig. 8.103(a) illustrates the fifth axiom of hyperbolic geometry. The diagram illustrates one way that, through the given point, two lines are drawn parallel to the given line. If we were to construct a triangle on a pseudosphere, the sum of the measures of the angles would be less than 180° (Fig. 8.103b). The theorem “The sum of the measures of the angles of a triangle is less than 180°” has been proven by means of the axioms of hyperbolic geometry. (a) Given line Given point Line 2 Line 1 (b) Figure 8.103 We have stated that the sum of the measures of the angles of a triangle is 180°, is greater than 180°, and is less than 180°. Which statement is correct? Each statement is correct in its own geometry. The many theorems based on the fifth postulate may differ in each geometry. It is important to realize that each theorem proved is true in its own geometry . No one system is the “best” system. Euclidean geometry may appear to be the one to use in the classroom, where the blackboard is flat. In discussions involving Earth as a whole, however, elliptical geometry may be the most useful, since Earth is a sphere. If the object under consideration has the shape of a saddle or pseudosphere, hyperbolic geometry may be the most useful. Did You Know? It’s All Relative m You can visualize Einstein’s theory by thinking of space as a rubber sheet pulled taut on which a mass is placed, causing the rubber sheet to bend. Albert Einstein’s general theory of relativity, published in 1916, approached space and time differently from our everyday understanding of them. Einstein’s theory unites the three dimensions of space with one of time in a four-dimensional space–time continuum. His theory dealt with the path that light and objects take while moving through space under the force of gravity. Einstein conjectured that mass (such as stars and planets) caused space to be curved. The greater the mass, the greater the curvature. To prove his conjecture, Einstein studied Riemann’s non-Euclidean geometry. Einstein believed that the trajectory of a particle in space represents not a straight line but the straightest curve possible, a geodesic. Einstein’s theory was confirmed by the solar eclipses of 1919 and 1922. Space–time is now thought to be a combination of three different types of curvature: spherical (described by Riemannian geometry), flat (described by Euclidean geometry), and saddleshaped (described by hyperbolic geometry). Fractal Geometry We are familiar with one-, two-, and three-dimensional figures. Many objects, however, are difficult to categorize as one-, two-, or three-dimensional. For example, how would you classify the irregular shapes we see in nature, such as a coastline, or the bark on a tree, or a mountain, or a path followed by lightning? For a long time mathematicians assumed that making realistic geometric models of natural shapes and figures was almost impossible, but the development of fractal geometry now makes it possible. Both images below were made by using fractal geometry. “The Great Architect of the universe now appears to be a great mathematician.” British physicist Sir James Jeans m Fractal images KristinaSh/Shutterstock Vladimir Caplinskij/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=