Math 123: General Course Outline
123. Foundations of Geometry. Lecture, three hours; discussion, one hour. Prerequisite: course 115A. Axioms and models, Euclidean geometry, Hilbert axioms, neutral (absolute) geometry, hyperbolic geometry, Poincare model, independence of parallel postulate.
The purpose of Math 123 is to study the classical geometries from an axiomatic perspective, with particular attention paid to Euclid's parallel postulate and to geometric systems that violate it. These systems are called Non-Euclidean Geometries. Among them, the Hyperbolic Geometry is the most important today. Here is some background.
In his Elements, Euclid (~365BC-~300BC) built his geometry using five axioms. The first 4 are:
(1) Any two points can be joined by a (straight) line.
(2) Any segment can be extended continuously in a (straight) line.
(3) Given any point and distance, there is a circle centered at the point with radius equal to the distance.
(4) All right angles are equal to each other. These are easily understood as Euclid gave them.
The fifth was less obvious, but was found to be equivalent to (5) Given a line L and a point P not on the line, there exists one and only one line which passes through P and is parallel to (i.e. does not intersect) L. Axiom (5), in this version, is called the Parallel Postulate (and also Playfair's Axiom).
From near the beginning, it seemed as if Euclid's 5th axiom might be a consequence of the first 4, but no proof was ever found. Finally, in the nineteenth century Bolyai, Gauss and Lobachevsky independently put the question to rest by showing that a new geometry, Hyperbolic Geometry, satisfies the first 4 axioms but not the 5th. Thus, one of the goals of Math 123 is to study the concept of a "geometry" and to illustrate the implementation of this concept in examples.
The course can be useful for prospective secondary school teachers, in that it illustrates how a mathematical structure can be built upon an axiom system, and how the Euclidean geometry that is traditionally studied in the schools is only one of many possible "geometries".
Math 123 is a flexible course, and it is taught quite differently by different instructors. For example, some instructors may approach the course primarily through the classical axiom systems, while others may take the Kleinian approach according to which geometries are classified by their symmetry groups.