Orient BlackSwan

About the BookAbout the AuthorTable of Contents

Elementary geometry provides the foundation of modern geometry. For the most part, the standard introductions end at the formal Euclidean geometry of high school. Agricola and Friedrich revisit geometry, but from the higher viewpoint of university mathematics. Plane geometry is developed from its basic objects and their properties and then moves to conics and basic solids, including the Platonic solids and a proof of Euler's polytope formula. Particular care is taken to explain symmetry groups, including the description of ornaments and the classification of isometries by their number of fixed points. Complex numbers are introduced to provide an alternative, very elegant approach to plane geometry. The authors then treat spherical and hyperbolic geometries, with special emphasis on their basic geometric properties.

This largely self-contained book provides a much deeper understanding of familiar topics, as well as an introduction to new topics that complete the picture of two-dimensional geometries. For undergraduate mathematics students the book will be an excellent introduction to an advanced point of view on geometry. For mathematics teachers it will be a valuable reference and a source book for topics for projects.
The book contains over 100 figures and scores of exercises. It is suitable for a one-semester course in geometry for undergraduates, particularly for mathematics majors and future secondary school teachers.

Ilka Agricola: Humboldt-Universität zu Berlin, Berlin, Germany,
Thomas Friedrich: Humboldt-Universität zu Berlin, Berlin, Germany

• Preface to the English Edition    6
• Preface to the German Edition    8

• Chapter 1. Introduction: Euclidean space    14
o Exercises    19

• Chapter 2. Elementary geometrical figures and their properties    22
o §2.1. The line    22
o §2.2. The triangle    32
o §2.3. The circle    58
o §2.4. The conic sections    76
o §2.5. Surfaces and bodies    90
o Exercises    102

• Chapter 3. Symmetries of the plane and of space    112
o §3.1. Affine mappings and centroids    112
o §3.2. Projections and their properties    118
o §3.3. Central dilations and translations    121
o §3.4. Plane isometries and similarity transforms    127
o §3.5. Complex description of plane transformations    140
o §3.6. Elementary transformations of the space E[sup(3)]    144
o §3.7. Discrete subgroups of the plane transformation group    152
o §3.8. Finite subgroups of the spatial transformation group
o Exercises    169

• Chapter 4. Hyperbolic geometry    180
o §4.1. The axiomatic development of elementary geometry 180
o §4.2. The Poincaré model    187
o §4.3. The disc model    196
o §4.4. Selected properties of the hyperbolic plane    198
o §4.5. Three types of hyperbolic isometries    202
o §4.6. Fuchsian groups    207
o Exercises    217

• Chapter 5. Spherical geometry    222
o §5.1. The space S[sup(2)]    222
o §5.2. Great circles in S[sup(2)]    224
o §5.3. The isometry group of [sup(2)]    228
o §5.4. The Möbius group of S[sup(2)]    229
o §5.5. Selected topics in spherical geometry    231
o Exercises    239

• Bibliography    242

• List of Symbols    248

• Index