A regular hexagon is rotated about a centrally located

Spherical geometry or elliptic geometry is actually a model of Riemannian geometry, named in honor of Georg F. B. Riemann 1826–1866, the German mathematician responsible for the next postulate. The Reimannian Postulate is not numbered in this book, because it does not characterize Euclidean geometry. To understand the Reimannian Postulate, consider a sphere Figure 2.54 containing line 艎 and point P not on 艎. Any line drawn through point P must intersect 艎 in two points. To see this develop, follow the frames in Figure 2.55, which depict an attempt to draw a line parallel to 艎 through point P. the surface for study, mathematicians use a saddle-like surface known as a hyperbolic paraboloid. See Figure 2.56. A line 艎 is the intersection of a plane with this surface. Clearly, more than one plane can intersect this surface to form a line containing P that does not intersect 艎. In fact, an infinite number of planes intersect the surface in an infinite number of lines parallel to 艎 and containing P. Table 2.5 compares the three types of geometry. a P Figure 2.54 b P Consider the natural extension to Riemannian geometry of the claim that the shortest distance between two points is a straight line. For the sake of efficiency and common sense, a person traveling from New York City to London will follow the path of a line as it is known in spherical geometry. As you might guess, this concept is used to chart international flights between cities. In Euclidean geometry, the claim suggests that a person tunnel under the earth’s surface from one city to the other. A second type of non-Euclidean geometry is attributed to the works of a German, Karl F. Gauss 1777–1855, a Russian, Nikolai Lobachevski 1793–1856, and a Hungarian, Johann Bolyai 1802–1862. The postulate for this system of non-Euclidean geometry is as follows: This form of non-Euclidean geometry is termed hyperbolic geometry. Rather than using the plane or sphere as P a Small part of surface of the sphere Figure 2.55 b Line through P “parallel” to on larger part of surface P c Line through P shown to intersect on larger portion of surface P P d All of line and the line through P shown on entire sphere P Figure 2.56 RIEMANNIAN POSTULATE Through a point not on a line, there are no lines parallel to the given line. LOBACHEVSKIAN POSTULATE Through a point not on line, there are infinitely many lines parallel to the given line.