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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
Geometry with an Introduction to Cosmic Topology
Michael P. Hitchman
Mathematics Department
Linfield College
McMinnville, Oregon, USA
2018 Edition
compiled October, 2020
Colophon
Preface
Acknowledgements
Dedication
