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OverviewFull Product DetailsAuthor: Gerard VenemaPublisher: Pearson Education (US) Imprint: Pearson Edition: 2nd edition Dimensions: Width: 20.60cm , Height: 1.40cm , Length: 25.00cm Weight: 0.650kg ISBN: 9780136020585ISBN 10: 0136020585 Pages: 416 Publication Date: 02 November 2011 Audience: College/higher education , Tertiary & Higher Education Replaced By: 9780136845348 Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of Contents1. Prologue: Euclid’s Elements 1.1 Geometry before Euclid 1.2 The logical structure of Euclid’s Elements 1.3 The historical significance of Euclid’s Elements 1.4 A look at Book I of the Elements 1.5 A critique of Euclid’s Elements 1.6 Final observations about the Elements 2. Axiomatic Systems and Incidence Geometry 2.1 The structure of an axiomatic system 2.2 An example: Incidence geometry 2.3 The parallel postulates in incidence geometry 2.4 Axiomatic systems and the real world 2.5 Theorems, proofs, and logic 2.6 Some theorems from incidence geometry 3. Axioms for Plane Geometry 3.1 The undefined terms and two fundamental axioms 3.2 Distance and the Ruler Postulate 3.3 Plane separation 3.4 Angle measure and the Protractor Postulate 3.5 The Crossbar Theorem and the Linear Pair Theorem 3.6 The Side-Angle-Side Postulate 3.7 The parallel postulates and models 4. Neutral Geometry 4.1 The Exterior Angle Theorem and perpendiculars 4.2 Triangle congruence conditions 4.3 Three inequalities for triangles 4.4 The Alternate Interior Angles Theorem 4.5 The Saccheri-Legendre Theorem 4.6 Quadrilaterals 4.7 Statements equivalent to the Euclidean Parallel Postulate 4.8 Rectangles and defect 4.9 The Universal Hyperbolic Theorem 5. Euclidean Geometry 5.1 Basic theorems of Euclidean geometry 5.2 The Parallel Projection Theorem 5.3 Similar triangles 5.4 The Pythagorean Theorem 5.5 Trigonometry 5.6 Exploring the Euclidean geometry of the triangle 6. Hyperbolic Geometry 6.1 The discovery of hyperbolic geometry 6.2 Basic theorems of hyperbolic geometry 6.3 Common perpendiculars 6.4 Limiting parallel rays and asymptotically parallel lines 6.5 Properties of the critical function 6.6 The defect of a triangle 6.7 Is the real world hyperbolic? 7. Area 7.1 The Neutral Area Postulate 7.2 Area in Euclidean geometry 7.3 Dissection theory in neutral geometry 7.4 Dissection theory in Euclidean geometry 7.5 Area and defect in hyperbolic geometry 8. Circles 8.1 Basic definitions 8.2 Circles and lines 8.3 Circles and triangles 8.4 Circles in Euclidean geometry 8.5 Circular continuity 8.6 Circumference and area of Euclidean circles 8.7 Exploring Euclidean circles 9. Constructions 9.1 Compass and straightedge constructions 9.2 Neutral constructions 9.3 Euclidean constructions 9.4 Construction of regular polygons 9.5 Area constructions 9.6 Three impossible constructions 10. Transformations 10.1 The transformational perspective 10.2 Properties of isometries 10.3 Rotations, translations, and glide reflections 10.4 Classification of Euclidean motions 10.5 Classification of hyperbolic motions 10.6 Similarity transformations in Euclidean geometry 10.7 A transformational approach to the foundations 10.8 Euclidean inversions in circles 11. Models 11.1 The significance of models for hyperbolic geometry 11.2 The Cartesian model for Euclidean geometry 11.3 The Poincaré disk model for hyperbolic geometry 11.4 Other models for hyperbolic geometry 11.5 Models for elliptic geometry 11.6 Regular Tessellations 12. Polygonal Models and the Geometry of Space 12.1 Curved surfaces 12.2 Approximate models for the hyperbolic plane 12.3 Geometric surfaces 12.4 The geometry of the universe 12.5 Conclusion 12.6 Further study 12.7 Templates APPENDICES A. Euclid’s Book I A.1 Definitions A.2 Postulates A.3 Common Notions A.4 Propositions B. Systems of Axioms for Geometry B.1 Filling in Euclid’s gaps B.2 Hilbert’s axioms B.3 Birkhoff’s axioms B.4 MacLane’s axioms B.5 SMSG axioms B.6 UCSMP axioms C. The Postulates Used in this Book C.1 The undefined terms C.2 Neutral postulates C.3 Parallel postulates C.4 Area postulates C.5 The reflection postulate C.6 Logical relationships D. Set Notation and the Real Numbers D.1 Some elementary set theory D.2 Properties of the real numbers D.3 Functions E. The van Hiele Model F. Hints for Selected Exercises Bibliography IndexReviewsAuthor InformationGerard Venema earned an A.B. in mathematics from Calvin College and a Ph.D. from the University of Utah. After completing his education, he spent two years in a postdoctoral position at the University of Texas at Austin and another two years as a Member of the Institute for Advanced Study in Princeton, NJ. He then returned to his alma mater, Calvin University, and has been a faculty member there ever since. While on the Calvin University faculty he also held visiting faculty positions at the University of Tennessee, the University of Michigan, and Michigan State University. He also spent two years as Program Director for Topology, Geometry, and Foundations in the Division of Mathematical Sciences at the National Science Foundation and nearly ten years as the Associate Secretary of the Mathematical Association of America. Venema is a member of the American Mathematical Society and the Mathematical Association of America. He is the author of two other books. One is an undergraduate textbook, Exploring Advanced Euclidean Geometry with GeoGebra, published by the Mathematical Association of America. The other is a research monograph, Embeddings in Manifolds, coauthored by Robert J. Daverman, that was published by the American Mathematical Society as volume 106 in its Graduate Studies in Mathematics series. In addition, Venema is author of over thirty research articles in geometric topology. Tab Content 6Author Website:Countries AvailableAll regions |
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