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| Preface | |
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| The Euclidean Plane | |
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| Approaches to Euclidean Geometry | |
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| Isometries | |
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| Rotations and Reflections | |
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| The Three Reflections Theorem | |
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| Orientation-Reversing Isometries | |
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| Distinctive Features of Euclidean Geometry | |
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| Discussion | |
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| Euclidean Surfaces | |
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| Euclid on Manifolds | |
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| The Cylinder | |
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| The Twisted Cylinder | |
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| The Torus and the Klein Bottle | |
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| Quotient Surfaces | |
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| A Nondiscontinuous Group | |
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| Euclidean Surfaces | |
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| Covering a Surface by the Plane | |
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| The Covering Isometry Group | |
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| Discussion | |
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| The Sphere | |
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| The Sphere S[superscript 2] in R[superscript 3] | |
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| Rotations | |
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| Stereographic Projection | |
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| Inversion and the Complex Coordinate on the Sphere | |
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| Reflections and Rotations as Complex Functions | |
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| The Antipodal Map and the Elliptic Plane | |
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| Remarks on Groups, Spheres and Projective Spaces | |
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| The Area of a Triangle | |
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| The Regular Polyhedra | |
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| Discussion | |
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| The Hyperbolic Plane | |
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| Negative Curvature and the Half-Plane | |
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| The Half-Plane Model and the Conformal Disc Model | |
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| The Three Reflections Theorem | |
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| Isometries as Complex Functions | |
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| Geometric Description of Isometries | |
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| Classification of Isometries | |
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| The Area of a Triangle | |
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| The Projective Disc Model | |
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| Hyperbolic Space | |
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| Discussion | |
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| Hyperbolic Surfaces | |
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| Hyperbolic Surfaces and the Killing-Hopf Theorem | |
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| The Pseudosphere | |
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| The Punctured Sphere | |
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| Dense Lines on the Punctured Sphere | |
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| General Construction of Hyperbolic Surfaces from Polygons | |
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| Geometric Realization of Compact Surfaces | |
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| Completeness of Compact Geometric Surfaces | |
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| Compact Hyperbolic Surfaces | |
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| Discussion | |
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| Paths and Geodesics | |
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| Topological Classification of Surfaces | |
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| Geometric Classification of Surfaces | |
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| Paths and Homotopy | |
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| Lifting Paths and Lifting Homotopies | |
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| The Fundamental Group | |
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| Generators and Relations for the Fundamental Group | |
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| Fundamental Group and Genus | |
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| Closed Geodesic Paths | |
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| Classification of Closed Geodesic Paths | |
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| Discussion | |
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| Planar and Spherical Tessellations | |
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| Symmetric Tessellations | |
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| Conditions for a Polygon to Be a Fundamental Region | |
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| The Triangle Tessellations | |
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| Poincare's Theorem for Compact Polygons | |
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| Discussion | |
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| Tessellations of Compact Surfaces | |
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| Orbifolds and Desingularizations | |
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| From Desingularization to Symmetric Tessellation | |
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| Desingularizations as (Branched) Coverings | |
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| Some Methods of Desingularization | |
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| Reduction to a Permutation Problem | |
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| Solution of the Permutation Problem | |
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| Discussion | |
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| References | |
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| Index | |