دانلود کتاب Symmetry: A Mathematical Exploration

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Preface Intended Audience Improvements in the Second Edition Advice to Instructors Acknowledgements Contents Introduction to Symmetry 1A Counting Symmetries 1B Objects 1C Rigid Motions 1D Bounded and Unbounded Objects (Revisited) 1E Elements of Mathematics: The Contrapositive Exercises The Algebra of Symmetry 2A Composition 2B Cayley Tables 2C Groups 2D Symmetry Groups 2E One Reflection Is Enough 2F An Improved Classification of Rigid Motions 2G Elements of Mathematics: The Counterexample Exercises The Classification Theorems 3A Rigid Equivalence 3B Bounded Objects 3C Border Patterns 3D Wallpaper Patterns (optional) 3E Elements of Mathematics: Equivalence (optional) Exercises Isomorphic Groups 4A The Definition of an Isomorphism 4B Isomorphism Examples 4C A Better Notation for Cyclic Groups 4D Elements of Mathematics: The Converse Exercises Subgroups and Product Groups 5A Subgroups 5B Generated Subgroups (optional) 5C Product Groups (optional) 5D Elements of Mathematics: Sets versus Lists Exercises Permutation Groups 6A Counting Permutations 6B Cycle Notation and Composition 6C Even and Odd Permutations 6D Symmetries and Permutations 6E Elements of Mathematics: Well-Defined 6F Elements of Mathematics: The Inductive Proof (optional) Exercises Symmetries of 3D Objects 7A Basics of 3D 7B Essentially Two-Dimensional Objects 7C A Trick for Counting Symmetries 7D The Tetrahedron 7E The Cube 7F The Dodecahedron 7G The Classification Theorem 7H Chirality 7I The Full Story (optional) Exercises The Five Platonic Solids 8A The Classification of Platonic Solids 8B Counting Their Parts 8C Duality 8D Euler´s Formula 8E The Euler Characteristic 8F The Platonic Solids Through the Ages Exercises Symmetry and Optimization 9A Minimal Surfaces 9B The Circle Wins 9C Elements of Mathematics: Proof by Contradiction Exercises What Is a Number? 10A Natural Numbers 10B Integers and Rational Numbers 10C Irrational Numbers 10D Real Numbers 10E Which Real Numbers Are Rational? 10F Real Numbers and Symmetry 10G Elements of Mathematics: Construction of Rational and Real Numbers Exercises Excursions in Numbers 11A How Many Prime Numbers Are There? 11B The Meaning of ``Same Size´´ 11C Are the Rational Numbers Countable? 11D Cantor´s Theorem Exercises Rigid Motions as Functions 12A Measuring Distance in Euclidean Space 12B Naming the Points on the Unit Circle 12C The Dot Product and Perpendicularity 12D Using the Dot Product to Find a Friend 12E Rigid Motions Are Functions Exercises Rigid Motions as Matrices 13A Matrix Computations 13B Representing Rigid Motions as Matrices 13C Orthogonal Matrices 13D Concluding Remarks Exercises Image Credits Index