Ruler and the Round: Classic Problems in Geometric Constructions

by Nicholas D. Kazarinoff

Overview

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.

Product Details

ISBN-13:	9780486425153
Publisher:	Dover Publications
Publication date:	10/20/2011
Pages:	160
Product dimensions:	5.37(w) x 8.50(h) x (d)

Table of Contents

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