Although easy to comprehend and fun to do, many geometric constructions defy completion with just a ruler and a compass. This book takes an intriguing look at the most famous of these "impossible" constructions.
In exploring ground rules, history, and angle trisection, the first part considers angle trisection and bird migration, constructed points, analytic geometry, algebraic classification of constructible numbers, fields of real numbers, cubic equations, and marked ruler, quadratix, and hyperbola (among other subjects). The second part treats nonconstructible regular polygons and the algebra associated with them; specifically, irreducibility and factorization, unique factorization of quadratic integers, finite dimensional vector spaces, algebraic fields, and nonconstructible regular polygons.
High school and college students as well as amateur mathematicians will appreciate this stimulating and provocative book, and its glimpses into the crucial role geometry plays in a wide range of mathematical applications.

Contenuti

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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