An intriguing look at the "impossible" geometric constructions (those that defy completion with just a ruler and a compass), this book covers angle trisection and circle division. 1970 edition.
An intriguing look at the "impossible" geometric constructions (those that defy completion with just a ruler and a compass), this book covers angle trisection and circle division. 1970 edition.
Contents PART ONE. ANGLE TRISECTION CHAPTER ONE. PROOF AND UNSOLVED PROBLEMS 1.1 Angle Trisection and Bird Migration 1.2 Proof 1.3 Solved and Unsolved Problems 1.4 Things to Come CHAPTER TWO. GROUND RULES AND THEIR ALGEBRAIC INTERPRETATION 2.1 Constructed Points 2.2 Analytic Geometry CHAPTER THREE. SOME HISTORY CHAPTER FOUR. FIELDS 4.1 Fields of Real Numbers 4.2 Quadratic Fields 4.3 Iterated Quadratic Extensions of R 4.4 Algebraic Classification of Constructible Numbers CHAPTER FIVE. ANGLES, CUBES, AND CUBICS 5.1 Cubic Equations 5.2 Angles of 20° 5.3 Doubling a Unit Cube 5.4 Some Trisectable and Nontrisectable Angles 5.5 Trisection with n Points Given CHAPTER SIX. OTHER MEANS 6.1 Marked Ruler, Quadratrix, and Hyperbola 6.2 Approximate Trisections PART II. CIRCLE DIVISION CHAPTER SEVEN. IRREDUCIBILITY AND FACTORIZATION 7.1 Why Irreducibility? 7.2 Unique Factorization 7.3 Eisenstein's Test CHAPTER EIGHT. UNIQUE FACTORIZATION OF QUADRATIC INTEGERS CHAPTER NINE. FINITE DIMENSIONAL VECTOR SPACES 9.1 Definitions and Examples 9.2 Linear Dependence and Linear Independence 9.3 Bases and Dimension 9.4 Bases for Iterated Quadratic Extensions of R CHAPTER TEN. ALGEBRAIC FIELDS 10.1 Algebraic Fields as Vector Spaces 10.2 The Last Link CHAPTER ELEVEN. NONCONSTRUCTIBLE REGULAR POLYGONS 11.1 Construction of a Regular Pentagon 11.2 Constructibility of Regular Pentagons, a Second View 11.3 Irreducible Polynomials and Regular (2n + 1 )-gons 11.4 Nonconstructible Regular Polygons 11.5 Regular p"-gons 11.6 Squaring a Circle Appendix I Appendix II References Index
Contents PART ONE. ANGLE TRISECTION CHAPTER ONE. PROOF AND UNSOLVED PROBLEMS 1.1 Angle Trisection and Bird Migration 1.2 Proof 1.3 Solved and Unsolved Problems 1.4 Things to Come CHAPTER TWO. GROUND RULES AND THEIR ALGEBRAIC INTERPRETATION 2.1 Constructed Points 2.2 Analytic Geometry CHAPTER THREE. SOME HISTORY CHAPTER FOUR. FIELDS 4.1 Fields of Real Numbers 4.2 Quadratic Fields 4.3 Iterated Quadratic Extensions of R 4.4 Algebraic Classification of Constructible Numbers CHAPTER FIVE. ANGLES, CUBES, AND CUBICS 5.1 Cubic Equations 5.2 Angles of 20° 5.3 Doubling a Unit Cube 5.4 Some Trisectable and Nontrisectable Angles 5.5 Trisection with n Points Given CHAPTER SIX. OTHER MEANS 6.1 Marked Ruler, Quadratrix, and Hyperbola 6.2 Approximate Trisections PART II. CIRCLE DIVISION CHAPTER SEVEN. IRREDUCIBILITY AND FACTORIZATION 7.1 Why Irreducibility? 7.2 Unique Factorization 7.3 Eisenstein's Test CHAPTER EIGHT. UNIQUE FACTORIZATION OF QUADRATIC INTEGERS CHAPTER NINE. FINITE DIMENSIONAL VECTOR SPACES 9.1 Definitions and Examples 9.2 Linear Dependence and Linear Independence 9.3 Bases and Dimension 9.4 Bases for Iterated Quadratic Extensions of R CHAPTER TEN. ALGEBRAIC FIELDS 10.1 Algebraic Fields as Vector Spaces 10.2 The Last Link CHAPTER ELEVEN. NONCONSTRUCTIBLE REGULAR POLYGONS 11.1 Construction of a Regular Pentagon 11.2 Constructibility of Regular Pentagons, a Second View 11.3 Irreducible Polynomials and Regular (2n + 1 )-gons 11.4 Nonconstructible Regular Polygons 11.5 Regular p"-gons 11.6 Squaring a Circle Appendix I Appendix II References Index
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