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Contents 1 An Invitation to Geometry 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 A Brief History of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Geometry on Surfaces: a first look . . . . . . . . . . . . . . . . . . . . . . . 7 2 The Plane and Complex Numbers 15 2.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Polar Form of a Complex Number . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Division of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Complex Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Transformations 29 3.1 Basic Transformations of C . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 The Extended Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 M¿obius Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 M¿obius transformations: A closer look . . . . . . . . . . . . . . . . . . . . . 63 4 Geometry 75 4.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 M¿obius Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Hyperbolic Geometry 85 5.1 The Hyperbolic Transformation Group . . . . . . . . . . . . . . . . . . . . . 85 5.2 Figures of Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Measurement in Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . 95 5.4 Area and triangle trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.5 The upper half plane model (optional) . . . . . . . . . . . . . . . . . . . . . 122 6 Elliptic Geometry 129 6.1 Antipodal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 Elliptic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3 Measurement in Elliptic Geometry . . . . . . . . . . . . . . . . . . . . . . . 140 6.4 Revisiting Euclid¿s Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7 Geometry on Surfaces 153 7.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.2 Elliptic Geometry with curvature k > 0. . . . . . . . . . . . . . . . . . . . . 158 7.3 Hyperbolic geometry with curvature k < 0. . . . . . . . . . . . . . . . . . . 161 7.4 Observing curvature in a universe (optional) . . . . . . . . . . . . . . . . . . 165 7.5 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.6 Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.7 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8 Cosmic Topology 205 8.1 3-dimensional geometry and 3-manifolds . . . . . . . . . . . . . . . . . . . . 205 8.2 Cosmic crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.3 Circles in the sky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Library of Congress Subject Headings for this publication:
Geometry.
Cosmic magnetic fields.
Topology.