Geometry With an Introduction to Cosmic Topology

by Hitchman, Michael P.

• ISBN13: 9780763754570
• ISBN10: 0763754579
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-12-03
• Publisher: Jones & Bartlett Learning

Summary

The approach to geometry is that of the Erlangen Program. This approach draws on content and uses modes of thought that appear in advanced courses in the undergraduate mathematics curriculum.  , The use of constructions to prove theorems and motivate definitions gives a tangible and intuitive approach to the subject.  The constructive approach adopted in the text enables the students to connect more deeply with the material.  , Includes 164 illustrations to aid visualization, and several applets for manipulating figures in non-Euclidean geometry. , The exercises come in many varieties to help students practice different skills: intense computations (e.g., computing hyperbolic distances, the hyperbolic law of cosines), constructions (e.g., symmetry points, hyperbolic circles), proofs both big (e.g., proof of arc length invariance) and small (that a collection of transformations forms a group), open-ended exercises (do you think the pair separation histogram came from a toroidal universe or a Klein bottle universe?), applications of geometry and topology to cosmology (Sections 7.4, 7.6, 8.2), and investigative. , Chapter 8 includes a general discussion of two current research programs in cosmic topology: cosmic crystallography and circles-in-the-sky. Chapter 8 also includes a review of our understanding of the current state of the universe, a view that is based on significant observational discoveries over the last 10 years. 

Table of Contents

An Invitation to Geometry	p. 1
Introduction	p. 1
A Brief History of Geometry	p. 3
Geometry on Surfaces: A First Look	p. 8
The Plane and Complex Numbers	p. 15
Basic Notions	p. 15
Polar Form of a Complex Number	p. 18
Division of Complex Numbers	p. 21
Complex Expressions	p. 24
Transformations	p. 29
Basic Transformations of C	p. 29
Inversion	p. 39
The Extended Plane	p. 51
Mobius Transformations	p. 55
Mobius Transformations: A Closer Look	p. 64
Geometry	p. 75
The Basics	p. 76
Möbius Geometry	p. 82
Hyperbolic Geometry	p. 85
The Hyperbolic Transformation Group	p. 85
Figures of Hyperbolic Geometry	p. 91
Measurement in Hyperbolic Geometry	p. 95
Area and Triangle Trigonometry	p. 107
The Upper Half-Plane Model (optional)	p. 121
Elliptic Geometry	p. 129
Antipodal Points	p. 129
Elliptic Geometry	p. 133
Measurement in Elliptic Geometry	p. 141
Revisiting Euclid's Postulates	p. 150
Geometry on Surfaces	p. 153
Curvature	p. 153
Elliptic Geometry with Curvature k > 0	p. 157
Hyperbolic Geometry with Curvature k < 0	p. 161
Observing Curvature in a Universe (optional)	p. 165
Surfaces	p. 170
Geometry of Surfaces	p. 186
Quotient Spaces	p. 192
Cosmic Topology	p. 203
Three-Dimensional Geometry and 3-manifolds	p. 203
Cosmic Crystallography	p. 216
Circles in the Sky	p. 224
Final Remarks	p. 228
Bibliography	p. 233
Index	p. 235
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