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Ruler and the Round: Classic Problems in Geometric Constructions

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Ruler and the Round: Classic Problems in Geometric Constructions Cover

 

Synopses & Reviews

Publisher Comments:

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:

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.

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.

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.

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
 

Product Details

ISBN:
9780486425153
Author:
Kazarinoff, Nicholas D.
Publisher:
Dover Publications
Location:
Mineola, N.Y.
Subject:
General
Subject:
General science
Subject:
Geometry - General
Subject:
Geometry
Subject:
Problems, famous
Subject:
General Mathematics
Subject:
Famous problems
Subject:
Compass
Subject:
nonconstructible regular polygons
Subject:
Mathematics-Geometry and Trigonometry
Edition Description:
Trade Paper
Series Volume:
37
Publication Date:
20111031
Binding:
TRADE PAPER
Language:
English
Illustrations:
30 Figures + 1 ht (frontis)
Pages:
160
Dimensions:
8.5 x 5.38 in 0.27 lb

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Related Subjects

Humanities » Philosophy » General
Reference » Science Reference » General
Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Geometry » Geometry and Trigonometry

Ruler and the Round: Classic Problems in Geometric Constructions New Trade Paper
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Product details 160 pages Dover Publications - English 9780486425153 Reviews:
"Synopsis" by ,
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.
"Synopsis" by ,
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.

"Synopsis" by ,
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.
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