Foundations of Geometry (Subscription)

This is the eBook of the printed book and may not include any media, website access codes, or print supplements that may come packaged with the bound book. Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers—and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra. This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites. MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict Cover 1 Copyright Page 2 Title Page 3 Contents 7 Preface 11 Acknowledgments 18 1 Prologue: Euclid’s Elements 19 1.1 Geometry before Euclid 19 1.2 The logical structure of Euclid’s Elements 20 1.3 The historical significance of Euclid’s Elements 21 1.4 A look at Book I of the Elements 22 1.5 A critique of Euclid’s Elements 25 1.6 Final observations about the Elements 27 2 Axiomatic Systems and Incidence Geometry 32 2.1 The structure of an axiomatic system 32 2.2 An example: Incidence geometry 34 2.3 The parallel postulates in incidence geometry 38 2.4 Axiomatic systems and the real world 40 2.5 Theorems, proofs, and logic 42 2.6 Some theorems from incidence geometry 50 3 Axioms for Plane Geometry 53 3.1 The undefined terms and two fundamental axioms 54 3.2 Distance and the Ruler Postulate 55 3.3 Plane separation 64 3.4 Angle measure and the Protractor Postulate 69 3.5 The Crossbar Theorem and the Linear Pair Theorem 73 3.6 The Side-Angle-Side Postulate 80 3.7 The parallel postulates and models 84 4 Neutral Geometry 87 4.1 The Exterior Angle Theorem and existence of perpendiculars 88 4.2 Triangle congruence conditions 92 4.3 Three inequalities for triangles 95 4.4 The Alternate Interior Angles Theorem 100 4.5 The Saccheri-Legendre Theorem 102 4.6 Quadrilaterals 106 4.7 Statements equivalent to the Euclidean Parallel Postulate 109 4.8 Rectangles and defect 115 4.9 The Universal Hyperbolic Theorem 122 5 Euclidean Geometry 124 5.1 Basic theorems of Euclidean geometry 125 5.2 The Parallel Projection Theorem 128 5.3 Similar triangles 130 5.4 The Pythagorean Theorem 132 5.5 Trigonometry 134 5.6 Exploring the Euclidean geometry of the triangle 136 6 Hyperbolic Geometry 149 6.1 Basic theorems of hyperbolic geometry 151 6.2 Common perpendiculars 156 6.3 The angle of parallelism 158 6.4 Limiting parallel rays 162 6.5 Asymptotic triangles 167 6.6 The classification of parallels 170 6.7 Properties of the critical function 173 6.8 The defect of a triangle 178 6.9 Is the real world hyperbolic? 182 7 Area 185 7.1 The Neutral Area Postulate 185 7.2 Area in Euclidean geometry 191 7.3 Dissection theory 198 7.4 Proof of the dissection theorem in Euclidean geometry 199 7.5 The associated Saccheri quadrilateral 203 7.6 Area and defect in hyperbolic geometry 208 8 Circles 213 8.1 Circles and lines in neutral geometry 213 8.2 Circles and triangles in neutral geometry 218 8.3 Circles in Euclidean geometry 224 8.4 Circular continuity 230 8.5 Circumference and area of Euclidean circles 233 8.6 Exploring Euclidean circles 240 9 Constructions 246 9.1 Compass and straightedge constructions 246 9.2 Neutral constructions 249 9.3 Euclidean constructions 252 9.4 Construction of regular polygons 253 9.5 Area constructions 257 9.6 Three impossible constructions 260 10 Transformations 263 10.1 Properties of isometries 264 10.2 Rotations, translations, and glide reflections 270 10.3 Classification of Euclidean motions 278 10.4 Classification of hyperbolic motions 281 10.5 A transformational approach to the foundations 283 10.6 Similarity transformations in Euclidean geometry 288 10.7 Euclidean inversions in circles 293 11 Models 305 11.1 The Cartesian model for Euclidean geometry 307 11.2 The Poincaré disk model for hyperbolic geometry 308 11.3 Other models for hyperbolic geometry 314 11.4 Models for elliptic geometry 319 12 Polygonal Models and the Geometry of Space 321 12.1 Curved surfaces 322 12.2 Approximate models for the hyperbolic plane 333 12.3 Geometric surfaces 339 12.4 The geometry of the universe 344 12.5 Conclusion 349 12.6 Further study 349 12.7 Templates 354 APPENDICES 362 A: Euclid’s Book I 362 A.1 Definitions 362 A.2 Postulates 364 A.3 Common Notions 364 A.4 Propositions 364 B: Systems of Axioms for Geometry 368 B.1 Hilbert’s axioms 369 B.2 Birkhoff’s axioms 371 B.3 MacLane’s axioms 372 B.4 SMSG axioms 373 B.5 UCSMP axioms 375 C: The Postulates Used in this Book 378 C.1 Criteria used in selecting the postulates 378 C.2 Statements of the postulates 380 C.3 Logical relationships 381 D: The van Hiele Model 383 E: Set Notation and the Real Numbers 384 E.1 Some elementary set theory 384 E.2 Axioms for the real numbers 385 E.3 Properties of the real numbers 386 E.4 One-to-one and onto functions 388 E.5 Continuous functions 389 F: Hints for Selected Exercises 390 Bibliography 397 Index 400 A 400 B 400 C 401 D 402 E 402 F 402 G 403 H 403 I 403 K 403 L 404 M 404 N 404 O 404 P 404 Q 405 R 405 S 405 T 406 U 406 V 407 W 407 Z 407 Synopsis: Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers-and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra. This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites