Introduction to Geometries

Until the last century, Euclidean geometry was the only known system of geometry dealing with measurement and concepts of congruence, parallelism, and perpendicularity. Early in that century another system was discovered, which differed from Euclidean geometry in its basic assumption regarding parallel lines. This new geometry, which became known as hyperbolic geometry, stimulated the study of a number of different so-called non-Euclidean geometries.

This course will provide a rigorous treatment of the fundamentals of plane geometry (Euclidean, spherical, elliptic, and hyperbolic) from an analytic point of view. The formal prerequisites are minimal. We will assume some familiarity with linear algebra and all other necessary background will be developed as needed.

The course serves several purposes, the most obvious of which is to acquaint students with certain geometrical facts, namely, the classical results of plane Euclidean and non-Euclidean geometry. In this capacity, it serves as an appropriate background for teachers of high school geometry. The second purpose is to provide students not only with facts and an understanding of the structure of the classical geometries, but also with a number of computational techniques for geometric investigation. The third purpose is to provide a link between classical geometry and modern geometry, with the aim of preparing students for further study in mathematics.

The text for the course will be Patrick J. Ryan,Euclidean and Non-Euclidean Geometry: An Analytic Approach(Cambridge University Press).

The prerequisite is MATH2021.03/2022.03 (formerly MATH2000.06) or MATH2221.03/2222.03 (formerly MATH2220.06). A degree credit exclusion is AK/MATH3550.06.

Quizzes and class tests will comprise 60% of the final grade and the final exam 40%.