Foundations of Plane Geometry

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.

For junior/senior-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.

Features :

• Focus on one main topic--The axiomatic development of the absolute plane--which is pursued through a classification into Euclidean, hyperbolic, and spherical planes.
• Permits in-depth treatment at a pace that nurtures the growth of students' conceptual understanding.
• Theme of simultaneous study of different types of plane geometry--Followed throughout the book.
• Provides a fresh context for the principles of Euclidean geometry and motivates the usefulness of abstraction.
• Specific models are presented early--(Including the sphere, the Klein-Betrami hyperbolic model, and the "gap" plane in Chapter 1) and references them often.
• Gives meaning and depth to concepts such as betweenness and separation that can be dismissed as all too obvious in a strictly Euclidean context.
• Axioms for absolute plane geometry are gradually presented. (Chapters 5-13).
• Introduces students to axiomatic development without interrupting the mathematical progress of the main topic.
• Concepts defined as soon as they make sense in the general development--But not necessarily before all axioms that relate to them have been introduced. This often leads to some thought-provoking examples--e.g., when "segment" is defined in Chapter 6, an example (the "Inside-Out" model) and a result (Proposition 6.4) show that at this stage, "length" is an invariant of a segment but "endpoints" are not.
• Challenges students to consider familiar words in a new light.
• Unique approach to the standard sets of axioms--Facilitates incremental learning, e.g., the "Betweenness" and "Quadrichotomy" axioms for points are simply stated assumptions that permit the development of much of the theory of betweenness for collinear points even though there are many non-standard (even finite) models that satisfy these postulates. Most of the axioms for coterminal rays are formulated as exact analogs of previous axioms for collinear points. This analogy ("duality" ) is invoked to instantly establish many properties of coterminal rays. The articulation of these properties reinforces understanding of the previous results about collinear points.
• Promotes the recognition of analogy in theorems and proofs, an important part of the development of abstract reasoning skills.
• Original presentation of many concepts--e.g. The only polygons used in the development are triangles. The notions of rectangles and Saccheri quadrilaterals are confined to exercises. This leads to a novel proof of the theorem on angle sums of triangles in spherical planes (18.1). The separate introduction of halfplanes in Chapter 10 and fans of coterminal rays in Chapter 11 and the proof of their relationship in Theorem 12.2. The Crossbar Theorem (12.4) is then an almost immediate consequence.
• Facilitates deeper understanding via careful study of a relatively small number of concepts.
• The Ruler and Protractor Axioms (commonly used in secondary school geometry texts) as theorems--Presented after all the axioms have been introduced (Chapter 13)--and gives their proofs as structured exercises.
• Provides a context for prospective teachers to better understand aspects of the high school curriculum.
• Careful but not overly wordy explanations throughout the text--All proofs are written in detailed paragraph style. Informal explanations--and references to specific models--are interspersed frequently. Figures illustrate virtually all of the definitions and the key points of most proofs.
• Helps students to digest often challenging topics.
• Informal chapter on logic (Ch. 2)--Addresses students' most common mistakes and misconceptions and introduces basic ideas about proofs.
• Uses logical puzzles in Ch. 3 as a non-intimidating way of gaining practice in creating and writing proofs.
• Varied exercises at the end of each chapter--Some call for proofs of results that are stated in the book without proof; some are fairly immediate applications of the theorems in the text; some involve longer chains of reasoning and often include hints; and others develop concepts that are omitted from the text such as Saccheri quadrilaterals.
• The range of problems allows students of various levels of ability to engage the course.

0. The Question of Parallels.
1. Five Examples.
2. Some Logic.
3. Practice Proofs.
4. Set Terminology and Sets of Real Numbers.
5. An Axiom System for Plane Geometry: First Steps.
6. Betweenness, Segments and Rays.
7. Three Axioms for the Line.
8. The Real Ray Axiom and Its Consequences.
9. Antipodes and Opposite Rays.
10. Separation.
11. Pencils and Angles.
12. The Crossbar Theorem.
13. Side-Angle-Side.
14. Perpendiculars.
15. The Exterior Angle Inequality and Triangle Inequality.
16. Further Results on Triangles.
17. Parallels and the Diameter of the Plane.
18. Angle Sums of Triangles.
19. Parallels and Angle Sums.
20. Concurrence.
21. Circles.
22. Similarity.
Appendix I. Definitions and Assumptions from Book I of Euclid's Elements.
Appendix II. The Side-Angle-Side Axiom in the Hyperbolic Plane.
Bibliography.
Index.