Synopses & Reviews
A number of geometric constructions, although easy to comprehend and fun to do, are nevertheless impossible to complete with just a ruler and a compass. This book discusses the most famous of these "impossible" constructions. Part I, in exploring ground rules, history and angle trisection, considers--among other subjects--angle trisection and bird migration, constructed points, analytic geometry, algebraic classification of constructible numbers, fields of real numbers, cubic equations, and marked ruler, quadratrix, and hyperbola. Part II 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. This stimulating and provocative book provides a fascinating glimpse of the crucial role geometry can play in a wide range of mathematical applications. Unabridged republication of
Ruler and the Round: or Angle Trisection and Circle Division, originally published by Prindle, Weber & Schmidt, Inc., Boston, 1970. 2 Appendices. References. Index. Numerous illustrations.
Synopsis
A number of geometric constructions are impossible to complete with just a ruler and a compass. This book discusses the most famous of these "impossible" constructions. Part I explores ground rules, history, and angle trisection. Part II treats nonconstructible regular polygons and the algebra associated with them. 1970 edition.
Synopsis
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.and#160;1970 edition.
Synopsis
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.
Table of Contents
Contents PART ONE. ANGLE TRISECTION
CHAPTER ONE. PROOF AND UNSOLVED PROBLEMS
1.1 Angle Trisection and Bird Migrationand#160;and#160;
1.2 Proofand#160;and#160;
1.3 Solved and Unsolved Problemsand#160;and#160;
1.4 Things to Comeand#160;and#160;
CHAPTER and#160;TWO. GROUND RULES AND THEIR
ALGEBRAIC INTERPRETATION 2.1 Constructed Pointsand#160;
2.2 Analytic Geometryand#160;and#160;
CHAPTER THREE. SOME HISTORY
CHAPTER FOUR. FIELDS
4.1 Fields of Real Numbersand#160;and#160;
4.2 Quadratic Fieldsand#160;
4.3 Iterated Quadratic Extensions of Rand#160;and#160; 4.4 Algebraic Classification of Constructible Numbersand#160;
CHAPTER FIVE. ANGLES, CUBES, AND CUBICS
5.1 Cubic Equationsand#160;and#160;
5.2 Angles of 20and#176;and#160;and#160;
5.3 Doubling a Unit Cube
5.4 Some Trisectable and Nontrisectable Anglesand#160;
5.5 Trisection with n Points Given CHAPTER SIX. OTHER MEANS 6.1 Marked Ruler, Quadratrix, and Hyperbola 6.2 Approximate Trisectionsand#160;
PART II. CIRCLE DIVISION
CHAPTER SEVEN. IRREDUCIBILITY AND
FACTORIZATION 7.1 Why Irreducibility?and#160;
7.2 Unique Factorizationand#160;
7.3 Eisenstein's Testand#160;
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 Dimensionand#160;
9.4 Bases for Iterated Quadratic Extensions of Rand#160;
CHAPTER TEN. ALGEBRAIC FIELDS
10.1 Algebraic Fields as Vector Spacesand#160;
10.2 The Last Link CHAPTER ELEVEN. NONCONSTRUCTIBLE REGULAR
POLYGONS 11.1 Construction of a Regular Pentagonand#160; 11.2 Constructibility of Regular Pentagons, a Second Viewand#160;
11.3 Irreducible Polynomials and Regular (2n + 1 )-gonsand#160;
11.4 Nonconstructible Regular Polygonsand#160;
11.5 Regular p"-gons 11.6 Squaring a Circleand#160; Appendix I Appendix II References Index and#160;