**Student Mathematical Library**

Volume: 64;
2012;
301 pp;
Softcover

MSC: Primary 51;
Secondary 01; 18

**Print ISBN: 978-0-8218-7571-1
Product Code: STML/64**

List Price: $51.00

Individual Price: $40.80

**Electronic ISBN: 978-0-8218-8788-2
Product Code: STML/64.E**

List Price: $48.00

Individual Price: $38.40

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#### Supplemental Materials

# Geometries

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*A. B. Sossinsky*

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.

Ultimately, the author makes the distinction between concrete
mathematical objects called “geometries” and the singular
“geometry”, which he understands as a way of thinking
about mathematics. Although the book does not address branches of
mathematics and mathematical physics such as Riemannian and
Kähler manifolds or, say, differentiable manifolds and conformal
field theories, the ideology of category language and transformation
groups on which the book is based prepares the reader for the study
of, and eventually, research in these important and rapidly developing
areas of contemporary mathematics.

#### Readership

Undergraduates interested in geometry.

#### Reviews & Endorsements

[A] very ambitious and pleasantly succinct text . . . Highly recommended.

-- CHOICE

#### Table of Contents

# Table of Contents

## Geometries

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface xiii14 free
- About Euclidean geometry 118 free
- Toy geometries and main definitions 3350
- Abstract groups and group presentations 5370
- Finite subgroups of 𝑆𝑂(3) and the platonic bodies 6784
- Discrete subgroups of the isometry group of the plane and tilings 85102
- Reflection groups and Coxeter geometries 99116
- Spherical geometry 109126
- The Poincaré disk model of hyperbolic geometry 125142
- The Poincaré half-plane model 143160
- The Cayley-Klein model 153170
- Hyperbolic trigonometry and absolute constants 163180
- History of non-Euclidean geometry 177194
- Projective geometry 185202
- “Projective geometry is all geometry” 203220
- Finite geometries 211228
- The hierarchy of geometries 229246
- Morphisms of geometries 241258
- Excerpts from Euclid’s “Elements” 255272
- Hilbert’s axioms for plane geometry 271288
- Answers & hints 283300
- Bibliography 297314
- Index 299316 free
- Back Cover Back Cover1322