Preface                                            vii
  Chapter 1.  The Euclidean Plane                  1  (20)
    1.1  Approaches to Euclidean Geometry          1  (1)
    1.2  Isometries                                2  (3)
    1.3  Rotations and Reflections                 5  (4)
    1.4  The Three Reflections Theorem             9  (2)
    1.5  Orientation-Reversing Isometries          11 (3)
    1.6  Distinctive Features of Euclidean         14 (4)
    Geometry
    1.7  Discussion                                18 (3)
  Chapter 2.  Euclidean Surfaces                   21 (24)
    2.1  Euclid on Manifolds                       21 (1)
    2.2  The Cylinder                              22 (3)
    2.3  The Twisted Cylinder                      25 (1)
    2.4  The Torus and the Klein Bottle            26 (3)
    2.5  Quotient Surfaces                         29 (4)
    2.6  A Nondiscontinuous Group                  33 (1)
    2.7  Euclidean Surfaces                        34 (2)
    2.8  Covering a Surface by the Plane           36 (3)
    2.9  The Covering Isometry Group               39 (2)
    2.10  Discussion                               41 (4)
  Chapter 3.  The Sphere                           45 (30)
    3.1  The Sphere S^(2) in R^(3)                 45 (3)
    3.2  Rotations                                 48 (2)
    3.3  Stereographic Projection                  50 (2)
    3.4  Inversion and the Complex Coordinate      52 (4)
    on the Sphere
    3.5  Reflections and Rotations as Complex      56 (4)
    Functions
    3.6  The Antipodal Map and the Elliptic        60 (3)
    Plane
    3.7  Remarks on Groups, Spheres and            63 (2)
    Projective Spaces
    3.8  The Area of a Triangle                    65 (2)
    3.9  The Regular Polyhedra                     67 (2)
    3.10  Discussion                               69 (6)
  Chapter 4.  The Hyperbolic Plane                 75 (36)
    4.1  Negative Curvature and the Half-Plane     75 (5)
    4.2  The Half-Plane Model and the Conformal    80 (5)
    Disc Model
    4.3  The Three Reflections Theorem             85 (3)
    4.4  Isometries as Complex Functions           88 (4)
    4.5  Geometric Description of Isometries       92 (4)
    4.6  Classification of Isometries              96 (3)
    4.7  The Area of a Triangle                    99 (2)
    4.8  The Projective Disc Model                 101(4)
    4.9  Hyperbolic Space                          105(3)
    4.10  Discussion                               108(3)
  Chapter 5.  Hyperbolic Surfaces                  111(24)
    5.1  Hyperbolic Surfaces and the               111(1)
    Killing-Hopf Theorem
    5.2  The Pseudosphere                          112(1)
    5.3  The Punctured Sphere                      113(5)
    5.4  Dense Lines on the Punctured Sphere       118(4)
    5.5  General Construction of Hyperbolic        122(4)
    Surfaces from Polygons
    5.6  Geometric Realization of Compact          126(3)
    Surfaces
    5.7  Completeness of Compact Geometric         129(1)
    Surfaces
    5.8  Compact Hyperbolic Surfaces               130(2)
    5.9  Discussion                                132(3)
  Chapter 6.  Paths and Geodesics                  135(28)
    6.1  Topological Classification of Surfaces    135(3)
    6.2  Geometric Classification of Surfaces      138(2)
    6.3  Paths and Homotopy                        140(3)
    6.4  Lifting Paths and Lifting Homotopies      143(2)
    6.5  The Fundamental Group                     145(2)
    6.6  Generators and Relations for the          147(6)
    Fundamental Group
    6.7  Fundamental Group and Genus               153(1)
    6.8  Closed Geodesic Paths                     154(2)
    6.9  Classification of Closed Geodesic Paths   156(4)
    6.10  Discussion                               160(3)
  Chapter 7.  Planar and Spherical Tessellations   163(22)
    7.1  Symmetric Tessellations                   163(4)
    7.2  Conditions for a Polygon to Be a          167(5)
    Fundamental Region
    7.3  The Triangle Tessellations                172(6)
    7.4  Poincare's Theorem for Compact Polygons   178(4)
    7.5  Discussion                                182(3)
  Chapter 8.  Tessellations of Compact Surfaces    185(18)
    8.1  Orbifolds and Desingularizations          185(4)
    8.2  From Desingularization to Symmetric       189(1)
    Tessellation
    8.3  Desingularizations as (Branched)          190(4)
    Coverings
    8.4  Some Methods of Desingularization         194(2)
    8.5  Reduction to a Permutation Problem        196(2)
    8.6  Solution of the Permutation Problem       198(3)
    8.7  Discussion                                201(2)
References                                         203(4)
Index                                              207