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Full Description
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.
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
