The Foundations of Geometry and the Non-Euclidean Plane

This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non­ Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap­ ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten­ sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three­ and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref­ erence if necessary.

1. Equivalence Relations.- 1.1 Logic.- 1.2 Sets.- 1.3 Relations.- 1.4 Exercises.- Graffiti.- 2 Mappings.- 2.1 One-to-One and Onto.- 2.2 Composition of Mappings.- 2.3 Exercises.- Graffiti.- 3 The Real Numbers.- 3.1 Binary Operations.- 3.2 Properties of the Reals.- 3.3 Exercises.- Graffiti.- 4 Axiom Systems.- 4.1 Axiom Systems.- 4.2 Incidence Planes.- 4.3 Exercises.- Graffiti.- One Absolute Geometry.- 5 Models.- 5.1 Models of the Euclidean Plane.- 5.2 Models of Incidence Planes.- 5.3 Exercises.- Graffiti.- 6 Incidence Axiom and Ruler Postulate.- 6.1 Our Objectives.- 6.2 Axiom 1: The Incidence Axiom.- 6.3 Axiom 2: The Ruler Postulate.- 6.4 Exercises.- Graffiti.- 7 Betweenness.- 7.1 Ordering the Points on a Line.- 7.2 Taxicab Geometry.- 7.3 Exercises.- Graffiti.- 8 Segments, Rays, and Convex Sets.- 8.1 Segments and Rays.- 8.2 Convex Sets.- 8.3 Exercises.- Graffiti.- 9 Angles and Triangles.- 9.1 Angles and Triangles.- 9.2 More Models.- 9.3 Exercises.- Graffiti.- 10 The Golden Age of Greek Mathematics (Optional).- 10.1 Alexandria.- 10.2 Exercises.- 11 Euclid’S Elements (Optional).- 11.1 The Elements.- 11.2 Exercises.- Graffiti.- 12 Pasch’s Postulate and Plane Separation Postulate.- 12.1 Axiom 3: PSP.- 12.2 Pasch, Peano, Pieri, and Hilbert.- 12.3 Exercises.- Graffiti.- 13 Crossbar and Quadrilaterals.- 13.1 More Incidence Theorems.- 13.2 Quadrilaterals.- 13.3 Exercises.- Graffiti.- 14 Measuring Angles and the Protractor Postulate.- 14.1 Axiom 4: The Protractor Postulate.- 14.2 Peculiar Protractors.- 14.3 Exercises.- 15 Alternative Axiom Systems (Optional).- 15.1 Hilbert’s Axioms.- 15.2 Pieri’s Postulates.- 15.3 Exercises.- 16 Mirrors.- 16.1 Rulers and Protractors.- 16.2 MIRROR and SAS.- 16.3 Exercises.- Graffiti.- 17 Congruence and the Penultimate Postulate.- 17.1 Congruence for Triangles.- 17.2 Axiom 5: SAS.- 17.3 Congruence Theorems.- 17.4 Exercises.- Graffiti.- 18 Perpendiculars and Inequalities.- 18.1 A Theorem on Parallels.- 18.2 Inequalities.- 18.3 Right Triangles.- 18.4 Exercises.- Graffiti.- 19 Reflections.- 19.1 Introducing Isometries.- 19.2 Reflection in a Line.- 19.3 Exercises.- Graffiti.- 20 Circles.- 20.1 Introducing Circles.- 20.2 The Two-Circle Theorem.- 20.3 Exercises.- Graffiti.- 21 Absolute Geometry and Saccheri Quadrilaterals.- 21.1 Euclid’s Absolute Geometry.- 21.2 Giordano’s Theorem.- 21.3 Exercises.- Graffiti.- 22 Saccherfs Three Hypotheses.- 22.1 Omar Khayyam’s Theorem.- 22.2 Saccheri’s Theorem.- 22.3 Exercises.- Graffiti.- 23 Euclid’s Parallel Postulate.- 23.1 Equivalent Statements.- 23.2 Independence.- 23.3 Exercises.- Graffiti.- 24 Biangles.- 24.1 Closed Biangles.- 24.2 Critical Angles and Absolute Lengths.- 24.3 The Invention of Non-Euclidean Geometry.- 24.4 Exercises.- Graffiti.- 25 Excursions.- 25.1 Prospectus.- 25.2 Euclidean Geometry.- 25.3 Higher Dimensions.- 25.4 Exercises.- Graffiti.- Two Non-Euclidean Geometry.- 26 Parallels and the Ultimate Axiom.- 26.1 Axiom 6: HPP.- 26.2 Parallel Lines.- 26.3 Exercises.- Graffiti.- 27 Brushes and Cycles.- 27.1 Brushes.- 27.2 Cycles.- 27.3 Exercises.- Graffiti.- 28 Rotations, Translations, and Horolations.- 28.1 Products of Two Reflections.- 28.2 Reflections in Lines of a Brush.- 28.3 Exercises.- Graffiti.- 29 The Classification of Isometries.- 29.1 Involutions.- 29.2 The Classification Theorem.- 29.3 Exercises.- Graffiti.- 30 Symmetry.- 30.1 Leonardo’s Theorem.- 30.2 Frieze Patterns.- 30.3 Exercises.- Graffiti.- 31 HOrocircles.- 31.1 Length of Arc.- 31.2 Hyperbolic Functions.- 31.3 Exercises.- Graffiti.- 32 The Fundamental Formula.- 32.1 Trigonometry.- 32.2 Complementary Segments.- 32.3 Exercises.- Graffiti.- 33 Categoricalness and Area.- 33.1 Analytic Geometry.- 33.2 Area.- 33.3 Exercises.- Graffiti.- 34 Quadrature of the Circle.- 34.1 Classical Theorems.- 34.2 Calculus.- 34.3 Constructions.- 34.4 Exercises.- Hints and Answers.- Notation Index.