Bibliographic record and links to related information available from the Library of Congress catalog.

Note: Contents data are machine generated based on pre-publication provided by the publisher. Contents may have variations from the printed book or be incomplete or contain other coding.

```Contents
1 An Invitation to Geometry 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 A Brief History of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Geometry on Surfaces: a first look . . . . . . . . . . . . . . . . . . . . . . . 7
2 The Plane and Complex Numbers 15
2.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Polar Form of a Complex Number . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Division of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Complex Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Transformations 29
3.1 Basic Transformations of C . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 The Extended Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 M¿obius Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 M¿obius transformations: A closer look . . . . . . . . . . . . . . . . . . . . . 63
4 Geometry 75
4.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 M¿obius Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Hyperbolic Geometry 85
5.1 The Hyperbolic Transformation Group . . . . . . . . . . . . . . . . . . . . . 85
5.2 Figures of Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Measurement in Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . 95
5.4 Area and triangle trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.5 The upper half plane model (optional) . . . . . . . . . . . . . . . . . . . . . 122
6 Elliptic Geometry 129
6.1 Antipodal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 Elliptic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3 Measurement in Elliptic Geometry . . . . . . . . . . . . . . . . . . . . . . . 140
6.4 Revisiting Euclid¿s Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7 Geometry on Surfaces 153
7.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2 Elliptic Geometry with curvature k > 0. . . . . . . . . . . . . . . . . . . . . 158
7.3 Hyperbolic geometry with curvature k < 0. . . . . . . . . . . . . . . . . . . 161
7.4 Observing curvature in a universe (optional) . . . . . . . . . . . . . . . . . . 165
7.5 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.6 Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.7 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8 Cosmic Topology 205
8.1 3-dimensional geometry and 3-manifolds . . . . . . . . . . . . . . . . . . . . 205
8.2 Cosmic crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.3 Circles in the sky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
```

Library of Congress Subject Headings for this publication:

Geometry.
Cosmic magnetic fields.
Topology.