Some Functions and Their Graphs

Geometry is the study of form and shape. Our first encounter with it usually involves such figures as triangles, squares, and circles, or solids such as the cube, the cylinder, and the sphere. These objects all have finite dimensions of length, area, and volume—as do most of the objects around us. At first thought, then, the notion of infinity seems quite removed from ordinary geometry. That this is not so can already be seen from the simplest of all geometric figures—the straight line. A line stretches to infinity in both directions, and we may think of it as a means to go “far out” in a one-dimensional world. As we shall see, it was this simple idea that gave rise, around the middle of the nineteenth century, to one of the most profound revolutions in mathematical thought—the creation of non-Euclidean geometry.