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Geometry with an Introduction to Cosmic Topology
Michael P. Hitchman
Contents
Index
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Contents
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Front Matter
Colophon
Preface
Acknowledgements
Dedication
1
An Invitation to Geometry
Introduction
A Brief History of Geometry
Geometry on Surfaces: A First Look
2
The Complex Plane
Basic Notions
Polar Form of a Complex Number
Division and Angle Measure
Complex Expressions
3
Transformations
Basic Transformations of \(\mathbb{C}\)
Inversion
The Extended Plane
Möbius Transformations
Möbius Transformations: A Closer Look
4
Geometry
The Basics
Möbius Geometry
5
Hyperbolic Geometry
The Poincaré Disk Model
Figures of Hyperbolic Geometry
Measurement in Hyperbolic Geometry
Area and Triangle Trigonometry
The Upper Half-Plane Model
6
Elliptic Geometry
Antipodal Points
Elliptic Geometry
Measurement in Elliptic Geometry
Revisiting Euclid's Postulates
7
Geometry on Surfaces
Curvature
Elliptic Geometry with Curvature \(k \gt 0\)
Hyperbolic Geometry with Curvature \(k \lt 0\)
The Family of Geometries \((X_k,G_k)\)
Surfaces
Geometry of Surfaces
Quotient Spaces
8
Cosmic Topology
Three-Dimensional Geometry and 3-Manifolds
Cosmic Crystallography
Circles in the Sky
Our Universe
Back Matter
A
List of Symbols
References
Index
Authored in PreTeXt
Front Matter
1
An Invitation to Geometry
2
The Complex Plane
3
Transformations
4
Geometry
5
Hyperbolic Geometry
6
Elliptic Geometry
7
Geometry on Surfaces
8
Cosmic Topology
Back Matter
