Geometry with an Introduction to Cosmic Topology

Description

Motivated by questions in cosmology, the open-content text Geometry with an Introduction to Cosmic Topology (GCT) uses Möbius transformations to develop hyperbolic, elliptic, and Euclidean geometries - three possibilities for the global geometry of the universe. The text, written for students who have taken vector calculus, also explores the interplay between the shape of a space and the type of geometry it admits. GCT is suitable for a semester course in non-Euclidean geometry or as a guide to independent study, with over 200 exercises and several essays on topics including the history of geometry, parallax and curvature, and research aimed at determining the shape of the universe. More about the text can be found in the Preface. Previously published in 2009, the author has made this updated and revised edition freely available, thanks in large part to the PreTeXt Project. Preface Acknowledgements An Invitation to Geometry Introduction A Brief History of Geometry Geometry on Surfaces: A First Look The Complex Plane Basic Notions Polar Form of a Complex Number Division and Angle Measure Complex Expressions Transformations Basic Transformations of C Inversion The Extended Plane Möbius Transformations Möbius Transformations: A Closer Look Geometry The Basics Möbius Geometry Hyperbolic Geometry The Poincaré Disk Model Figures of Hyperbolic Geometry Measurement in Hyperbolic Geometry Area and Triangle Trigonometry The Upper Half-Plane Model Elliptic Geometry Antipodal Points Elliptic Geometry Measurement in Elliptic Geometry Revisiting Euclid's Postulates Geometry on Surfaces Curvature Elliptic Geometry with Curvature k >0 Hyperbolic Geometry with Curvature k