Items related to Foundations of Plane Geometry
Synopsis
For undergraduate level courses in Geometry. Ideal for students who may have little previous experience with abstraction and proof, this text provides a rigorous and unified - yet straightforward and accessible - exposition of the foundations of Euclidean, hyperbolic, and spherical geometry. Unique in approach, it combines an extended theme - the study of a generalized absolute plane from axioms through classification into the three fundamental classical planes - with a leisurely development that allows ample time for students' mathematical growth. It is purposefully structured to facilitate the development of analytic and reasoning skills and to promote an awareness of the depth, power and subtlety of the axiomatic method in general, and of Euclidean and non-Euclidean plane geometry in particular. *Focuses on one main topic - The axiomatic development of the absolute plane - which is pursued through a classification into Euclidean, hyperbolic, and spherical planes. *The theme of simultaneous study of different types of Plane geometry. *Presents the axioms for absolute plane geometry gradually. *Forces students to consider familiar words in a new light. *Unique approach to the standard sets of axioms. *States and discusses the Ruler and Protractor Axioms (commonly used in secondary school geometry texts) as theorems. *Informal chapter on logic. *10 to 20 exercises varied in length and difficulty, at the end of each chapter.
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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 discussed.
The general theory that continues through Chapters 13-19 includes criteria for congruence of triangles, perpendicularity, the Exterior Angle Inequality (to the extent that it is true), the Triangle Inequality, angle sums of triangles, and parallel lines. It culminates in the classification theorem (19.4) mentioned previously.
Chapters 20 and 21 study concurrence and circles, respectively, in the general context, and Chapter 22 treats similarity in a Euclidean plane. Appendix I reproduces a list of Euclid's definitions and assumptions, which are referred to several times in the text. Appendix II contains a derivation of formulas for angle measure in the Beltrami-Klein model, as well as a complete proof that this model satisfies the Side-Angle-Side congruence axiom.
I have found it possible in most semesters, with careful planning, to cover Chapters 0-19 and to treat in detail the proofs in all but the last two or three of these chapters. I have used the material in Chapters 20-22 and Appendix II for independent study projects for honors students. An exceptionally well-prepared class would be able to skip Chapters 2-4 and cover the entire book in a semester.
Whatever their prior level of preparation and mathematical maturity, I believe that most junior- and senior-level mathematics majors, particularly those who plan to teach high school mathematics, will benefit from a careful study of this book. They will gain an awareness of some rather surprising properties of hyperbolic and spherical geometry, understand better the relationships among some familiar Euclidean properties, and discover some unfamiliar Euclidean properties as well. But even more important, they will develop their abilities to understand abstract and rigorous arguments, to solve nontrivial problems, and to create and articulate reasoned and coherent proofs. Many students have told me that gaining and using such skills is a source of much enjoyment and satisfaction.
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