Continuous symmetry : from Euclid to Klein

the Fundamental Idea Of Geometry Is That Of Symmetry. With That Principle As The Starting Point, Barker And Howe Begin An Insightful And Rewarding Study Of Euclidean Geometry. The Primary Focus Of The Book Is On Transformations Of The Plane. The Transformational Point Of View Provides Both A Path For Deeper Understanding Of Traditional Synthetic Geometry And Tools For Providing Proofs That Spring From A Consistent Point Of View. As A Result, Proofs Become More Comprehensible, As Techniques Can Be Used And Reused In Similar Settings. The Approach To The Material Is Very Concrete, With Complete Explanations Of All The Important Ideas, Including Foundational Background. The Discussions Of The Nine-point Circle And Wallpaper Groups Are Particular Examples Of How The Strength Of The Transformational Point Of View And The Care Of The Authors' Exposition Combine To Give A Remarkable Presentation Of Topics In Geometry. This Text Is For A One-semester Undergraduate Course On Geometry. It Is Richly Illustrated And Contains Hundreds Of Exercises. Cover Title Copyright Contents Instructor Preface Student Preface Acknowledgments Chapter I. Foundations of Geometry in the Plane I.1. The Real Numbers I.2. The Incidence Axioms I.3. Distance and the Ruler Axiom I.4. Betweenness I.5. The Plane Separation Axiom I.6. The Angular Measure Axioms I.7. Triangles and the SAS Axiom I.8. Geometric Inequalities I.9. Parallelism I.10. The Parallel Postulate I.11. Directed Angle Measure and Ray Translation I.12. Similarity I.13. Circles I.14. Bolzano's Theorem I.15. Axioms for the Euclidean Plane Chapter II. Isometries in the Plane: Products of Reflections II.1. Transformations in the Plane II.2. Isometries in the Plane II.3. Composition and Inversion II.4. Fixed Points and the First Structure Theorem II.5. Triangle Congruence and Isometries Chapter III. Isometries in the Plane: Classification and Structure III.1. Two Reflections: Translations and Rotations III.2. Glide Reflections III.3. The Classification Theorem III.4. Orientation III.5. Groups of Transformations III.6. The Second Structure Theorem III.7. Rotation Angles Chapter IV. Similarities in the Plane IV.1. Elementary Properties of Similarities IV.2. Dilations as Similarities IV.3. The Structure of Similarities IV.4. Orientation and Rotation Angles IV.5. Fixed Points for Similarities Chapter V. Conjugacy and Geometric Equivalence V.l. Congruence and Geometric Equivalence V.2. Geometric Equivalence of Transformations: Conjugacy V.3. Geometric Equivalence under Similarities V.4. Euclidean Geometry Derived from Transformations Chapter VI. Applications to Plane Geometry VI.1. Symmetry in Early Geometry VI.2. The Classical Coincidences VI.3. Dilation by Minus Two around the Centroid VI.4. Reflections, Light, and Distance VI.5. Fagnano's Problem and the Orthic Triangle VI.6. The Fermat Problem VI.7. The Circle of Apollonius Chapter VII. Symmetric Figures in the Plane VII.1. Symmetry Groups VII.2. Invariant Sets and Orbits VII.3. Bounded Figures in the Plane Chapter VIII. Frieze and Wallpaper Groups VIII.1. Point Groups and Translation Subgroups VIII.2. Frieze Groups VIII.3. Two-Dimensional Translation Lattices VIII.4. Wallpaper Groups Chapter IX. Area, Volume, and Scaling IX.1. Length of Curves IX.2. Area of Polygonal Regions: Basic Properties IX.3. Area and Equidecomposability IX.4. Area by Approximation IX.5. Area and Similarity IX.6. Scaling and Dimension References Index A B C D E F G H I J K L M N O P Q R S T U V W Back Cover "The fundamental idea of geometry is that of symmetry. With that principle as the starting point, Barker and Howe begin an insightful and rewarding study of Euclidean geometry." "The primary focus of the book is on transformations of the plane. The transformational point of view provides both a path for deeper understanding of traditional synthetic geometry and tools for providing proofs that spring from a consistent point of view. As a result, proofs become more comprehensible, as techniques can be used and reused in similar settings." "The approach to the material is very concrete, with complete explanations of all the important ideas, including foundational background. The discussions of the nine-point circle and wallpaper groups are particular examples of how the strength of the transformational point of view and the care of the authors' exposition combine to give a remarkable presentation of topics in geometry." "This text is for a one-semester undergraduate course on geometry. It is richly illustrated and contains hundreds of exercises."--Jacket