Coordinate Geometry

Underlying everything we have done with dimensions is a basic framework called coordinate geometry or analytic geometry. Again and again we have encountered strings of numbers describing coordinates of a location or a shape. The identification of a point in space with a sequence of numbers is the basic connection between geometry and algebra. The fundamental relationships among points in the plane are mirrored in the relationships among pairs of numbers, while triples of numbers mirror the relationships among points in space. Geometric transformations like scaling and projecting correspond to transformations of the coordinate pairs or triples. Facts about geometry are translated into algebraic facts, and vice versa. The mathematics dealing with these transformations is called linear algrebra.

Unfortunately, this effective way of dealing with the mathematics of dimensions has also served to isolate many of its most beautiful results from a general readership. In this book, I have purposefully chosen to treat geometric subjects from what is called a synthetic point of view, using coordinate representation sparingly and not developing the algebraic aspects extensively.

The synthetic viewpoint dominated geometric studies from the time of the Egyptians and Greeks until the seventeenth century, when the invention of analytic geometry by Rene Descartes set the stage for the development of higher-dimensional coordinate geometry two centuries later. At first, mathematicians restricted their application of analytic geometry to numbers on the number line and to number pairs in the plane, but by the beginning of the nineteenth century, it was well understood that the algebra that worked to describe the number line and the coordinate plane also extended to three-dimensional space.

Today we treat such progressions very naturally. A theorem about objects in the plane, when expressed in coordinate form, often suggests a corresponding theorem in space -- instead of writing two coordinates, we simply write three. But if we write two or three, why not four? As the algebra is practically the same, the theorems about number pairs and number triples extend to yield formal theorems about manipulations of quadruples of numbers. In analytic geometry, the most powerful results occur when we express some geometric relationship in coordinate form, then manipulate the number pairs or triples algebraically, and finally reinterpret the effect of these transformations on the original points in the plane or in space. But what geometric interpretation can we give for the analogous manipulations of number quadruples? And what happens when we try to interpret abstract relationships among sequences of 5 or 11 or 26 coordinates?

For the most part, mathematicians concerned with higher dimensions have been content to make use of the formal statements of linear algebra, retaining the geometric vocabulary but abandoning the attempt to visualize the concepts in concrete terms. All this is beginning to change with the advent of modern graphics computers, which literally do not know what dimension they are in. If we enter a collection of number pairs, the computer will display them as points on a television screen. If we enter number triples, the computer will first of all replace each triple by a pair according to some rule, then display the point pairs. The methods used to determine the screen coordinates of a point come from linear algebra.