Foundations of Plane Geometry

9780130479549

0130479543

This is a text for an upper level undergraduate course in plane geometry. It presents a unified account of the foundations of Euclidean and non-Euclidean planes. It proceeds from rather general axioms and yields a classification theorem for the three fundamental classical planes: Euclidean (or parabolic), spherical (or doubly elliptic), and hyperbolic. The treatment is careful, rigorous, and tightly focused, but it takes small and leisurely steps. I have used this approach for 15 years and have found it to be successful for our students, most of whom have been prospective secondary school mathematics teachers and have had little prior experience with abstraction and proof. The abstract exposition is grounded in concrete examples, including the coordinate Euclidean plane, the sphere, the Beltrami-Klein hyperbolic plane, the Minkowski plane, and the "gap" plane, which are presented early (in Chapter 1) and are cited often. The frequent comparison of different models is a strong motivation for the study of the relationships among various geometric properties and of why they hold or fail in particular contexts. (See, for instance, the discussion in Chapter 1 of the Exterior Angle Inequality for the Euclidean plane and its failure for the sphere.) The diversity of examples also justifies the study of concepts such as betweenness and separation, which the student might dismiss as obvious in the context of the Euclidean plane alone. An awareness of some bizarre examples helps to motivate the introduction of axioms as a way of eliminating pathology and of homing in on the fundamental models. Here are some remarks and suggestions about specific, chapters of the book: I think that it is important to spend a little time, but not too much, on Chapters 2, 3, and 4. Chapter 2 addresses our students' most common logical blunders and presents basic ideas about proofs. Its purpose is to enable students to understand our corrections of their logical errors throughout the semester, not to make them instant experts. (For example, if one is to find a model where a particular "If/then" statement fails, it is essential to know the general criterion for when such an implication is false.) Chapter 3 uses logical puzzles as a familiar way of gaining practice in creating and writing proofs, and Chapter 4 reviews the Least Upper Bound Property of the real numbers. The gradual introduction of the axioms and the development of some of their consequences takes up Chapters 5-13. New concepts are defined as soon as they make sense in context, and not necessarily before all axioms that relate to them have been introduced. This allows the occurrence of strange examples. For instance, "segment" and "ray" are defined in Chapter 6, and an example (the "Inside Out" model) is given wherein segments can have more than one set of endpoints and every point of a ray can be an endpoint. By studying and constructing such examples, students begin to understand that properties of a concept are not automatic, that any particular set of postulates has its strengths and its limitations, and that taking anything for granted is not a good idea. This understanding generally takes a few weeks to form; careful guidance and a little nurturing on the part of the instructor is usually needed, particularly in traversing the material of Chapters 6 through 10. Most of the axioms for coterminal rays are formulated (in Chapter 11) as exact analogs of previous axioms for collinear points. This analogy (duality) is invoked to establish instantly many properties of coterminal rays. The articulation of these properties helps to reinforce understanding of the previous results about collinear points and helps to justify to the student the time spent on those results. After the statement of axioms is completed in Chapter 13, the full list of assumptions is reviewed and commented on, and the ruler and protractor properties (which are theorems in our setup) are disc