Geometry: the line and the circle

Description

Geometry: The Line and the Circle is an undergraduate text with a strong narrative that is written at the appropriate level of rigor for an upper-level survey or axiomatic course in geometry. Starting with Euclid's Elements, the book connects topics in Euclidean and non-Euclidean geometry in an intentional and meaningful way, with historical context. The line and the circle are the principal characters driving the narrative. In every geometry considered which include spherical, hyperbolic, and taxicab, as well as finite affine and projective geometries these two objects are analyzed and highlighted. Along the way, the reader contemplates fundamental questions such as: What is a straight line? What does parallel mean? What is distance? What is area? There is a strong focus on axiomatic structures throughout the text. While Euclid is a constant inspiration and the Elements is repeatedly revisited with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean geometries are introduced very early to give the reader perspective on questions of axiomatics. Rounding out the thorough coverage of axiomatics are concluding chapters on transformations and constructibility. The book is compulsively readable with great attention paid to the historical narrative and hundreds of attractive problems. Cover Title Page Copyright Page Dedication Page Table of Contents Note to the Instructor Outline of the Book Designing a course using this text Note to the Reader Acknowledgements 1 The Line and the Circle 1.1 Introduction 1.2 Which came first? 1.3 What is a straight line, anyways? 2 Euclid’s Elements: Definitions and Axioms 2.1 The Elements 2.2 Definitions 2.3 Postulates and common notions 3 Book I of Euclid’s Elements: Neutral Geometry 3.1 Propositions I.1 through I.8 3.2 Propositions I.9 through I.15 3.3 Propositions I.16 through I.28 and I.31 4 Spherical Geometry 4.1 What is a straight line, anyways? - Part 2 4.2 Triangles in Spherical geometry 4.3 Euclid’s axioms viewed in Spherical geometry 4.4 Neutral geometry on the sphere 4.5 Area in Spherical geometry 4.6 Trigonometry for spherical triangles 4.7 Uniquely spherical constructions 5 Taxicab Geometry 5.1 Points, lines, angles, distances and circles 5.2 Euclid’s postulates in Taxicab geometry 5.3 Congruence schemes in Taxicab geometry 5.4 The rest of Neutral geometry 6 Hilbert and Gödel 6.1 Axiomatic systems 6.2 A Four Point geometry 6.3 Hilbert’s axioms for Euclidean plane geometry 6.4 Spherical and Taxicab geometries 6.5 Gödel and consistency 7 Book I: Non-Neutral Geometry 7.1 Parallel lines 7.2 Propositions I.32 and I.33 7.3 Area 7.4 Propositions I.34 through I.41 7.5 Propositions I.42 through I.46 7.6 The Pythagorean Theorem 8 Book II: Geometric Algebra 8.1 Proposition II.1 through II.10 8.2 Propositions II.11 through II.14 8.3 Quadrature on the sphere 9 Book VI: Similarity 9.1 Book V: Ratio and proportion 9.2 Similarity 9.3 A generalized Pythagorean Theorem 10 Book III: Circles 10.1 Definitions 10.2 Tangency 10.3 Arcs, chords and angles 10.4 Area Propositions: III.35 through III.37 10.5 The circumference of a circle & 𝜋 11 Book IV: Circles & Polygons 11.1 Definitions 11.2 Circles & triangles 11.3 Circles & squares 11.4 Circles & pentagons 11.5 Constructing regular polygons 11.6 The area of a circle & 𝜋 12 Models for the Hyperbolic Plane 12.1 Historical overview 12.2 Models of the hyperbolic plane 12.3 Arc length & distance in the half-plane model 13 Axiomatic Hyperbolic Geometry 13.1 Parallel lines 13.2 Omega triangles 13.3 Saccheri quadrilaterals 13.4 Hyperbolic area 14 Finite Geometries 14.1 Four Point geometry - Part 2 14.2 Fano’s plane 14.3 Projective geometry 14.4 Affine planes 14.5 Transforming afine into projective 14.6 Open problem in finite geometry 15 Isometries 15.1 Rigid motions or isometries 15.2 Reflections 15.3 Isometries of the Euclidean plane 15.4 Inversions in the Euclidean plane 15.5 Isometries of the hyperbolic plane 16 Constructibility 16.1 Four famous problems of antiquity 16.2 Constructible numbers 16.3 Four counterexamples 16.4 The limits of geometry Appendix A Euclid’s Definitions and Axioms A.1 Definitions A.2 Postulates A.3 Common notions Appendix B Euclid’s Propositions B.1 Book I B.2 Book II B.3 Book III B.4 Book IV B.5 Book VI Appendix C Visual Guide to Euclid’s Propositions C.1 Book I C.2 Book II C.3 Book III C.4 Book IV Appendix D Euclid’s Proofs D.1 Book I Appendix E Hilbert’s Axioms for Plane Euclidean Geometry Credits, Permissions and Acknowledgements Bibliography Notation Index Index