#### Thesis Title

A Method of Coordinatizing an Affine Geometry

Summer 1974

#### Degree

Master of Science in Mathematics

Mathematics

James Downing

Mathematics

#### Abstract

The union of geometry and algebra was initiated in an appendix to Discours de la methode, written by Rene Descartes. It was the first work to use a coordinate system for the study of geometry. This first treatment of analytic geometry assumed that the points of a line in a Euclidean space are in one-to-one correspondence with the real numbers. Many modern treatments of analytic geometry take a different approach. A somewhat similar but more general examination of the problem of coordinatizing geometries was published in Transactions of the American Mathematical Society in 1943. This recent work, which dealt with coordinatizing projective planes, was primarily due to the mathematician, Marshall Hall, Jr. Although Hall's paper dealt with projective geometry, many mathematicians have since utilized his methods for coordinatizing the affine plane. Unlike the original analytic geometry of Descartes, in which it was assumed that the points of a line in a Euclidean space are in one-to-one correspondence with the real numbers, the mathematicians of today approach the problem of coordinatizing geometry in a more rigorous sense. For example, in coordinatizing an affine plane, the points on a line of the plane are associated with an abstract set, and the algebraic properties imposed on the abstract set are derived. One such treatment is contained in the text, A Modern View of Geometry, by Leonard M. Blumenthal. He uses Hall's methods to demonstrate that the points of any affine plane with the Pappus property are in one-to-one correspondence with the set of all ordered pairs of elements of a field. Irving Kaplansky notes that a vector space, which may be described a n-tuples of the elements of a field, may be used to further demonstrate the union of algebra and geometry. He asserts that an affine plane is nothing more than a two-dimensional vector space. The purpose of this thesis is to bring together the works of Blumenthal and Kaplansky. A vector approach is used to demonstrate that an affine plane augmented by the Pappus propety is a two-dimensional vector space, and a two-dimensional vector space is an augmented affine plane. The coordinate system developed may not lend itself as well to algebraic descriptions of geometric elements, but it avoids the sometimes confusing ternary operator necessary to other methods.