A B Sossinsky
140 x 216 mm
Year of Publishing
Territorial Rights
Universities Press

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.

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

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

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