**About the Book**

The book is an innovative modern exposition of geometry, or rather, of geometries; it is the first textbook in which Felix Klein’s Erlangen program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal--although some, of course, remain more equal than others. The author concentrates on several of the more distinguished and beautiful ones, which include what he terms “toy geometries”, the geometries of platonic bodies, discrete geometries, and classical continuous geometries. The text is based on first-year semester course lectures delivered at the Independent University of Moscow in 2003 and 2006. It is by no means a formal algebraic or analytic treatment of geometric topics, but rather, a highly visual exposition containing upwards of 200 illustrations. The reader is expected to possess a familiarity with elementary Euclidean geometry, albeit those lacking this knowledge may refer to a compendium in Chapter 0. Per the author’s predilection, the book contains very little regarding the axiomatic approach to geometry (save for a single chapter on the history of non-Euclidean geometry), but two appendices provide a detailed treatment of Euclid’s and Hilbert’s axiomatics. Perhaps the most important aspect of this course is the problems, which appear at the end of each chapter, and are supplemented with answers at the conclusion of the text. By analyzing and solving these problems, the reader will become capable of thinking and working geometrically, much more so than by simply learning the theory.

**Table of Contents**Preface

Chapter 0. About Euclidean Geometry

Chapter 1. Toy Geometries and Main Definitions

Chapter 2. Abstract Groups and Group Presentations

Chapter 3. Finite Subgroups of SO(3) and the Platonic Bodies

Chapter 4. Discrete Subgroups of the Isometry Group of the Plane and Tilings

Chapter 5. Reflection Groups and Coxeter Geometries

Chapter 6. Spherical Geometry

Chapter 7. The Poincare Disk Model of Hyperbolic Geometry

Chapter 8. The Poincare Half-Plane Model

Chapter 9. The Cayley–Klein Model

Chapter 10. Hyperbolic Trigonometry and Absolute Constants

Chapter 11. History of Non-Euclidean Geometry

Chapter 12. Projective Geometry

Chapter 13. “Projective Geometry Is All Geometry”

Chapter 14. Finite Geometries

Chapter 15. The Hierarchy of Geometries

Chapter 16. Morphisms of Geometries

Appendix A. Excerpts from Euclid’s “Elements” Postulates of Book I

Appendix B. Hilbert’s Axioms for Plane Geometry I

Bibliography

Index

**Contributors (Author(s), Editor(s), Translator(s), Illustrator(s) etc.)**

**A B Sossinsky** is Professor at the Moscow Center for Continuous Mathematical Education, Independent University of Moscow, Moscow, Russia.