Geometry: From Euclid to Knots / Edition 1

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Overview

The main purpose of this book is to inform the reader about the formal, or axiomatic, development of Euclidean geometry. It follows Euclid's classic text Elements very closely, with an excellent organization of the subject matter, and over 1,000 practice exercises provide the reader with hands-on experience in solving geometrical problems. Providing a historical perspective about the study of plane geometry, this book covers such topics as other geometries, the neutral geometry of the triangle, non-neutral Euclidean geometry, circles and regular polygons, projective geometry, symmetries, inversions, informal topology, graphs, surfaces, and knots and links.
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Editorial Reviews

From The Critics
The main purpose of this book is to provide prospective high school mathematics teachers with geometric background. The first chapter introduces non-Euclidean geometries, and four core chapters are devoted to an axiomatic development of Euclidean geometry, following Euclid's Elements very closely. Later chapters complement this with an exposition of transformation geometry, and final chapters examine more recent geometric discoveries, including homeomorphism and isotopy, graph theory, the topology of surfaces, and knot theoretical invariants. Chapter exercises are included. Stahl is affiliated with the University of Kansas. Annotation c. Book News, Inc., Portland, OR
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Product Details

  • ISBN-13: 9780130329271
  • Publisher: Pearson
  • Publication date: 7/31/2002
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 480
  • Product dimensions: 8.58 (w) x 8.30 (h) x 0.89 (d)

Table of Contents

Preface xi
1 Other Geometries: A Computational Introduction 1
1.1 Spherical Geometry 1
1.2 Hyperbolic Geometry 9
1.3 Other Geometries 21
2 The Neutral Geometry of the Triangle 29
2.1 Introduction 29
2.2 Preliminaries 34
2.3 Propositions 1 through 28 46
2.4 Postulate 5 Revisited 81
3 Nonneutral Euclidean Geometry 87
3.1 Parallelism 87
3.2 Area 99
3.3 The Theorem of Pythagoras 112
3.4 Consequences of the Theorem of Pythagoras 119
3.5 Proportion and Similarity 122
4 Circles and Regular Polygons 133
4.1 The Neutral Geometry of the Circle 133
4.2 The Nonneutral Euclidean Geometry of the Circle 141
4.3 Regular Polygons 150
4.4 Circle Circumference and Area 155
4.5 Impossible Constructions 165
5 Toward Projective Geometry 177
5.1 Division of Line Segments 177
5.2 Collinearity and Concurrence 184
5.3 The Projective Plane 191
6 Planar Symmetries 197
6.1 Translations, Rotations, and Fixed Points 197
6.2 Reflections 203
6.3 Glide Reflections 210
6.4 The Main Theorems 216
6.5 Symmetries of Polygons 219
6.6 Frieze Patterns 223
6.7 Wallpaper Designs 229
7 Inversions 249
7.1 Inversions as Transformations 249
7.2 Inversions to the Rescue 257
7.3 Inversions as Hyperbolic Motions 261
8 Symmetry in Space 271
8.1 Regular and Semiregular Polyhedra 271
8.2 Rotational Symmetries of Regular Polyhedra 283
8.3 Monstrous Moonshine 290
9 Informal Topology 297
10 Graphs 307
10.1 Nodes and Arcs 307
10.2 Traversability 310
10.3 Colorings 316
10.4 Planarity 319
10.5 Graph Homeomorphisms 328
11 Surfaces 335
11.1 Polygonal Presentations 335
11.2 Closed Surfaces 348
11.3 Operations on Surfaces 360
11.4 Bordered Surfaces 369
12 Knots and Links 381
12.1 Equivalence of Knots and Links 381
12.2 Labelings 387
12.3 The Jones Polynomial 395
Appendix A A Brief Introduction to The Geometer's Sketchpad 407
Appendix B Summary of Propositions 411
Appendix C George D. Birkhoff's Axiomatization of Euclidean Geometry 417
Appendix D The University of Chicago School Mathematics Project's Geometrical Axioms 419
Appendix E David Hilbert's Axiomatization of Euclidean Geometry 423
Appendix F Permutations 427
Appendix G Modular Arithmetic 431
Solutions and Hints to Selected Problems 435
Bibliography 449
Index 451
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Preface

Organization

The main purpose of this book is to provide prospective high school mathematics teachers with the geometric background they need. Its core, consisting of Chapters 2 to 5, is therefore devoted to a fairly formal (that is, axiomatic) development of Euclidean geometry. Chapters 6, 7, and 8 complement this with an exposition of transformation geometry. The first chapter, which introduces teachers-to-be to non-Euclidean geometries, provides them with a perspective meant to enhance their appreciation of axiomatic systems. The much more informal Chapters 9 through 12 are meant to give students a taste of some more recent geometric discoveries.

The development of synthetic Euclidean geometry begins by following Euclid's Elements very closely. This has the advantage of convincing students that they are learning "the real thing." It also happens to be an excellent organization of the subject matter. Witness the well-known fact that the first 28 propositions are all neutral. These subtleties might be lost on the typical high school student, but familiarity with Euclid's classic text must surely add to the teacher's confidence and effectiveness in the classroom. I am also in complete agreement with the sentiments Todhunter displayed in the previous excerpt: No other system of teaching geometry is better than Euclid's, provided, of course that his list of propositions is supplemented with a sufficient number of exercises. Occasionally, though, because some things have changed over two millennia, or else because of errors in the Elements, it was found advantageous either to expound both the modern and ancient versions in parallel or else to partways with Euclid.

In order to convince prospective teachers of the need to prove "obvious" propositions, the axiomatic development of Euclidean geometry is preceded by the informal description of both spherical and hyperbolic geometry. The trigonometric formulas of these geometries are included in order to lend numerical substance to these alternate and unfamiliar systems. The part of the course dedicated to synthetic geometry covers the standard material about triangles, parallelism, circles, ratios, and similarity; it concludes with the classic theorems that lead to projective geometry. These lead naturally to a discussion of ideal points and lines in the extended plane. Experience indicates that the nonoptional portions of the first five chapters can be completed in about three quarters of a one semester course. During that time the typical weekly homework assignment called for about a dozen proofs.

Chapters 6 and 7 are concerned mostly with transformation geometry and symmetry. The planar rigid motions are completely and rigorously classified. This is followed by an informal discussion of frieze patterns and wallpaper designs. Inversions are developed formally and their utility for both Euclidean and hyperbolic geometry is explained.

The exposition in Chapters 8 through 12 is informal in the sense that few proofs are either offered or required. Their purpose is to acquaint students with some of the geometry that was developed in the last two centuries. Care has been taken to supply a great number of calculational exercises that will provide students with hands-on experience in these advanced topics.

The purpose of Chapter 8 is threefold. First there is an exposition of some interesting facts, such as Euler's equation and the Platonic and Archimedian solids. This is followed by a representation of the rotational symmetries of the regular solids by means of permutations, a discussion of their symmetry groups, and a visual definition of isomorphism. Both of these discussions aim to develop the prospective teacher's ability to visualize three dimensional phenomena. Finally comes the tale of Monstrous Moonshine.

Chapter 9 consists of a short introduction to the notions of homeomorphism and isotopy. Chapter 10 acquaints students with some standard topics of graph theory: traversability, colorability, and planarity. The topology of surfaces, both of the closed and the bordered varieties, is the subject of Chapter 11. Algorithms are described for identification of the topological type of any bordered surface. Two knot theoretic invariants are described in Chapter 12, including the recently discovered Jones polynomial.

Exercises

In Chapters 2 to 5 exercises are listed following every two or three propositions. This helps the professor select appropriate homework assignments and eliminates some of the guesswork for students. The great majority of these exercises call for straightforward applications of the immediately preceding propositions. On the other hand, the chapter review exercises provide problems for which the determination of the applicable propositions does require thought. In the remaining and less formal chapters the exercises appear at the end of each section. Each chapter concludes with a list of review problems. Solutions and/or hints to selected exercises are provided at the end of the book.

The exercises that are interspersed with the propositions are of four types. There are relational and constructive propositions in whose answers the students should adhere to the same format that is used in the numbered propositions. The third type of exercises has to do with the alternate spherical, hyperbolic, and taxicab geometries; in these the appropriate response usually consists of one or two English sentences. The fourth, and last, type of exercises calls for the use of some computer program, and these are marked with a (C).

In the other chapters, namely Chapter 1 and Chapters 6 through 12, exercises are listed at the end of each section. The emphasis in these is on the algorithmic aspect of geometry. They mostly require the straightforward, albeit nontrivial, application of the methods expounded in the text.

Notation and Conventions

Chapters 2 to 5 of this text present most of the content of Book I and selected topics from Books, II, III, IV, and VI of The Elements. In addition to the conventional labeling of propositions by chapter, section, and number these propositions are also identified by a parenthesized roman numeral, number that pinpoints their appearance in Euclid's book. For example, the Theorem of Pythagoras is listed as Proposition 3.3.2(L47). Exercises are identified in a similar manner,: Exercise 5.313.6 is the sixth exercise in group B of the third section in Chapter 5. The symbol (C) is used to distinguish exercises that call for the use of computer applications.

Read More Show Less

Introduction

Organization

The main purpose of this book is to provide prospective high school mathematics teachers with the geometric background they need. Its core, consisting of Chapters 2 to 5, is therefore devoted to a fairly formal (that is, axiomatic) development of Euclidean geometry. Chapters 6, 7, and 8 complement this with an exposition of transformation geometry. The first chapter, which introduces teachers-to-be to non-Euclidean geometries, provides them with a perspective meant to enhance their appreciation of axiomatic systems. The much more informal Chapters 9 through 12 are meant to give students a taste of some more recent geometric discoveries.

The development of synthetic Euclidean geometry begins by following Euclid's Elements very closely. This has the advantage of convincing students that they are learning "the real thing." It also happens to be an excellent organization of the subject matter. Witness the well-known fact that the first 28 propositions are all neutral. These subtleties might be lost on the typical high school student, but familiarity with Euclid's classic text must surely add to the teacher's confidence and effectiveness in the classroom. I am also in complete agreement with the sentiments Todhunter displayed in the previous excerpt: No other system of teaching geometry is better than Euclid's, provided, of course that his list of propositions is supplemented with a sufficient number of exercises. Occasionally, though, because some things have changed over two millennia, or else because of errors in the Elements, it was found advantageous either to expound both the modern and ancient versions in parallel or else to part wayswith Euclid.

In order to convince prospective teachers of the need to prove "obvious" propositions, the axiomatic development of Euclidean geometry is preceded by the informal description of both spherical and hyperbolic geometry. The trigonometric formulas of these geometries are included in order to lend numerical substance to these alternate and unfamiliar systems. The part of the course dedicated to synthetic geometry covers the standard material about triangles, parallelism, circles, ratios, and similarity; it concludes with the classic theorems that lead to projective geometry. These lead naturally to a discussion of ideal points and lines in the extended plane. Experience indicates that the nonoptional portions of the first five chapters can be completed in about three quarters of a one semester course. During that time the typical weekly homework assignment called for about a dozen proofs.

Chapters 6 and 7 are concerned mostly with transformation geometry and symmetry. The planar rigid motions are completely and rigorously classified. This is followed by an informal discussion of frieze patterns and wallpaper designs. Inversions are developed formally and their utility for both Euclidean and hyperbolic geometry is explained.

The exposition in Chapters 8 through 12 is informal in the sense that few proofs are either offered or required. Their purpose is to acquaint students with some of the geometry that was developed in the last two centuries. Care has been taken to supply a great number of calculational exercises that will provide students with hands-on experience in these advanced topics.

The purpose of Chapter 8 is threefold. First there is an exposition of some interesting facts, such as Euler's equation and the Platonic and Archimedian solids. This is followed by a representation of the rotational symmetries of the regular solids by means of permutations, a discussion of their symmetry groups, and a visual definition of isomorphism. Both of these discussions aim to develop the prospective teacher's ability to visualize three dimensional phenomena. Finally comes the tale of Monstrous Moonshine.

Chapter 9 consists of a short introduction to the notions of homeomorphism and isotopy. Chapter 10 acquaints students with some standard topics of graph theory: traversability, colorability, and planarity. The topology of surfaces, both of the closed and the bordered varieties, is the subject of Chapter 11. Algorithms are described for identification of the topological type of any bordered surface. Two knot theoretic invariants are described in Chapter 12, including the recently discovered Jones polynomial.

Exercises

In Chapters 2 to 5 exercises are listed following every two or three propositions. This helps the professor select appropriate homework assignments and eliminates some of the guesswork for students. The great majority of these exercises call for straightforward applications of the immediately preceding propositions. On the other hand, the chapter review exercises provide problems for which the determination of the applicable propositions does require thought. In the remaining and less formal chapters the exercises appear at the end of each section. Each chapter concludes with a list of review problems. Solutions and/or hints to selected exercises are provided at the end of the book.

The exercises that are interspersed with the propositions are of four types. There are relational and constructive propositions in whose answers the students should adhere to the same format that is used in the numbered propositions. The third type of exercises has to do with the alternate spherical, hyperbolic, and taxicab geometries; in these the appropriate response usually consists of one or two English sentences. The fourth, and last, type of exercises calls for the use of some computer program, and these are marked with a (C).

In the other chapters, namely Chapter 1 and Chapters 6 through 12, exercises are listed at the end of each section. The emphasis in these is on the algorithmic aspect of geometry. They mostly require the straightforward, albeit nontrivial, application of the methods expounded in the text.

Notation and Conventions

Chapters 2 to 5 of this text present most of the content of Book I and selected topics from Books, II, III, IV, and VI of The Elements. In addition to the conventional labeling of propositions by chapter, section, and number these propositions are also identified by a parenthesized roman numeral, number that pinpoints their appearance in Euclid's book. For example, the Theorem of Pythagoras is listed as Proposition 3.3.2(L47). Exercises are identified in a similar manner,: Exercise 5.313.6 is the sixth exercise in group B of the third section in Chapter 5. The symbol (C) is used to distinguish exercises that call for the use of computer applications.

Read More Show Less

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  • Anonymous

    Posted November 6, 2006

    excellent book

    Everybody has heard about Euclid and his geometry, but who knows exactly the subtlety of it? Saul Stahl's book is a very interesting presentation of the subject. Moreover, it is presented in a very pedagogical way so that even the beginner in geometry will be given methods to solve geometry problems. I therefore would recommend this book to all students for apprenticeship. Finally, since it is written in the book that at the times of Euclid the knowledge of this geometry was compulsary to all politicians, those of today would be well inspired to read it and to train their minds to such a way of reasoning which is at the basis of the progress of human kind.

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