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Full Description
Geometry: The Line and the Circle is an undergraduate text with a strong narrative that is written at the appropriate level of rigor for an upper-level survey or axiomatic course in geometry. Starting with Euclid's Elements, the book connects topics in Euclidean and non-Euclidean geometry in an intentional and meaningful way, with historical context.
The line and the circle are the principal characters driving the narrative. In every geometry considered--which include spherical, hyperbolic, and taxicab, as well as finite affine and projective geometries--these two objects are analyzed and highlighted. Along the way, the reader contemplates fundamental questions such as: What is a straight line? What does parallel mean? What is distance? What is area?
There is a strong focus on axiomatic structures throughout the text. While Euclid is a constant inspiration and the Elements is repeatedly revisited with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean geometries are introduced very early to give the reader perspective on questions of axiomatics. Rounding out the thorough coverage of axiomatics are concluding chapters on transformations and constructability. The book is compulsively readable with great attention paid to the historical narrative and hundreds of attractive problems.
Contents
The line and the circle
Euclid's Elements: Definitions and axioms
Book I of Euclid's Elements: Neutral geometry
Spherical geometry
Taxicab geometry
Hilbert and Godel
Book I: Non-Neutral geometry
Book II: Geometric algebra
Book VI: Similarity
Book III: Circles
Book IV: Circles & polygons
Models for the hyperbolic plane
Axiomatic hyperbolic geometry
Finite geometries
Isometries
Constructibility
Appendix A: Euclid's definitions and axioms
Appendix B: Euclid's propositions
Appendix C: Visual guide to Euclid's propositions
Appendix D: Euclid's proofs
Hilbert's axioms for plane Euclidean geometry
Credits, permissions and acknowledgments
Bibliography
Notation index
Index.
