Geometry Illuminated: An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry(MAA Textbooks)

Geometry Illuminated is an introduction to geometry in the plane, both Euclidean and hyperbolic. It is designed to be used in an undergraduate course on geometry, and as such, its target audience is undergraduate math majors. However, much of it should be readable by anyone who is comfortable with the language of mathematical proof. Throughout, the goal is to develop the material patiently. One of the more appealing aspects of geometry is that it is a very "visual" subject. This book hopes to takes full advantage of that, with an extensive use of illustrations as guides. Geometry Illuminated is divided into four principal parts. Part 1 develops neutral geometry in the style of Hilbert, including a discussion of the construction of measure in that system, ultimately building up to the Saccheri-Legendre Theorem. Part 2 provides a glimpse of classical Euclidean geometry, with an emphasis on concurrence results, such as the nine-point circle. Part 3 studies transformations of the Euclidean plane, beginning with isometries and ending with inversion, with applications and a discussion of area in between. Part 4 is dedicated to the development of the Poincar disk model, and the study of geometry within that model. While this material is traditional, Geometry Illuminated does bring together topics that are generally not found in a book at this level. Most notably, it explicitly computes parametric equations for the pseudosphere and its geodesics. It focuses less on the nature of axiomatic systems for geometry, but emphasizes rather the logical development of geometry within such a system. It also includes sections dealing with trilinear and barycentric coordinates, theorems that can be proved using inversion, and Euclidean and hyperbolic tilings.

Table of Contents:
Preface 0. Axioms and Models Part I: Neutral Geometry 1. The Axioms of Incidence and Order 2. Angles and Triangles 3. Congruence Verse I: SAS and ASA 4. Congruence Verse II: AAS 5. Congruence Verse III: SSS 6. Distance, Length, and the Axioms of Continuity 7. Angle Measure 8. Triangles in Neutral Geometry 9. Polygons 10. Quadrilateral Congruence Theorems Part II: Euclidean Geometry 11. The Axiom on Parallels 12. Parallel Projection 13. Similarity 14. Circles 15. Circumference 16. Euclidean Constructions 17. Concurrence I 18. Concurrence II 19. Concurrence III 20. Trilinear Coordinates Part III: Euclidean Transformations 21. Analytic Geometry 22. Isometrics 23. Reflections 24. Translations and Rotations 25. Orientation 26. Glide Reflections 27. Change of Coordinates 28. Dilation 29. Applications of Transformations 30. Area I 31. Area II 32. Barycentric Coordinates 33. Inversion 34. Inversion II 35. Applications of Inversion Part IV: Hyperbolic Geometry 36. The Search for a Rectangle 37. Non-Euclidean Parallels 38. The Pseudosphere 39. Geodesics on the Pseudosphere 40. The Upper Half Plane 41. The Poincaré disk 42. Hyperbolic Reflections 43. Orientation-Preserving Hyperbolic Isometries 44. The Six Hyperbolic Trigonometric Functions 45. Hyperbolic Trigonometry 46. Hyperbolic Area 47. Tiling Bibliography Index

About the Author :
Matthew Harvey is an Associate Professor of Mathematics at the University of Virginia's College at Wise, where he has taught since 2006. He graduated from the University of Virginia in 1995 with a BA in Mathematics, and from Johns Hopkins University in 2002 with a PhD in Mathematics.

Review :
…The author succeeds in elevating Euclidean geometry in particular to the level of advanced undergraduate study and in so doing presents it as a counterpart to a first analysis course. At the same time, some readers may prefer to move on to new ideas and results more quickly; to that end, Harvey gives an indication of several paths through the book in his preface." - Choice "To my mind, this book stands out from the crowd partly because of the way in which its imaginatively devised illustrations are used to stage the introduction of basic concepts and to guide the reader through various steps in a proof. It's not just a matter of pretty pictures, however, because Matthew Harvey's written commentary is easy going and yet mathematically precise. He has written a truly lovely book, which is now top of my reading list as an introduction to geometry at this level." - Peter Ruane