Meyer - Geometry and Its Applications 2e (AP, 2006).pdf - Geometry - Kuya

Preface

majors, especially prospective high school teachers. This second edition differs

from the first mainly in that errors have been corrected (many thanks to helpful

critics!) and we offer a more careful presentation of the axiomatic approach to

spherical geometry. Much of the content will be familiar to geometry instructors: a

solid introduction to axiomatic Euclidean geometry, some non-Euclidean geometry,

and a substantial amount of transformation geometry. However, we present some

important novelties: We pay significant attention to applications, we provide optional

dynamic geometry courseware for use with The Geometer’s Sketchpad , and we include

a chapter on polyhedra and planar maps. By extending the content of geometry

courses to include applications and newer geometry, such a course not only can

teach mathematical skills and understandings but can help students understand the

twenty-first century world that is unfolding around us. By providing software support

for discovery learning, we allow experiments with new ways of teaching and learning.

The intertwined saga of geometric theory and applications is modern as well

as ancient, providing a wonderful mathematical story that continues today. It is

a compelling story to present to students to show that mathematics is a seam-

less fabric, stretching from antiquity until tomorrow and stretching from theory

to practice. Consequently, one of our goals is to express the breadth of geometric

applications, especially contemporary ones. Examples include symmetries of artistic

patterns, physics, robotics, computer vision, computer graphics, stability of architec-

tural structures, molecular biology, medicine, pattern recognition, and more. Perhaps

surprisingly, many of these applications are based on familiar, long-standing geometric

T his book is for a first college-level geometry course and is suitable for mathematics

PREFACE

ideas — showing once again that there is no conflict between the timelessness and

modernity of good mathematics.

In recent years, high school instruction in geometry has become much less

extensive and much less rigorous in many school districts. Whatever advantage this

may bring at the high school level, it changes the way we need to instruct mathematics

students at the college level. In the first place, it makes instruction in geometry

that much more imperative for all students of mathematics. In addition, we cannot

always assume extensive familiarity with proof-oriented basic Euclidean geometry.

Consequently, we begin at the beginning, displaying a portion of classical Euclidean

geometry as a deductive system. For the most part, our proofs are in the style of

Euclid — which is to say that they are not as rigorous as they could be. We do

present a snapshot of some geometry done with full rigor so that students will have

exposure to that. In addition, we carefully discuss why full rigor is important in

some circumstances and why it is not always attempted in teaching, research, or

applications.

Except for Chapters 5 and 6 and parts of Chapter 7, this text requires little more

than high school mathematics. Nonetheless, students need the maturity to deal with

proofs and careful calculations. In Chapters 5 and 6 and parts of Chapter 7, we assume

a familiarity with vectors as commonly presented in multivariable calculus. Derivatives

also make a brief appearance in Chapter 5. Matrices are used in Chapter 6, but it is

not necessary to have studied linear algebra in order to understand this material. All

we assume is that students know how to multiply matrices and are familiar with the

associative law.

It would be foolish to pretend that this book surveys all of the major topics and

applications of geometry. For example, differential geometry is represented only by

one short section. I have tried to choose topics that would be most appealing and

accessible to undergraduates, especially prospective high school teachers.

A good deal of flexibility is possible in selecting a sequence of topics from which

to create a course. This book contains two approaches to geometry: the axiomatic

and the computational. When I am teaching mostly prospective teachers, I emphasize

the axiomatic (Chapters 1–4) and sprinkle in a little computational material from

Chapter 5 or 6. When I have mainly mathematics majors with applied interests and

others, such as computer science majors, I reverse the emphasis, concentrating on

Chapters 5, 6, and 7. I find Chapter 8 works well in either type of course.

There is a lot of independence among the chapters of the book. For example,

one might skip Chapters 1 through 3 since in only a few places in other chapters

(mainly Chapter 4) is there any explicit dependence on them. An instructor can

remind students of the relevant theorems as the need arises. Chapter 4 is not needed

for any of the other parts. Chapter 5 can be useful in preparation for Chapter 6 only

insofar as we often think of points as position vectors in Chapter 6. Chapter 7 relies

on one section of Chapter 2 and one section of Chapter 5. Chapter 8 is completely

PREFACE

independent of the other chapters. More detailed descriptions of prerequisites are

given at the start of each chapter.

In writing this book, I am aware of the many people and organizations that

have shaped my thoughts. I learned a good deal about applications of geometry at

the Grumman Corporation (now Northrop-Grumman) while in charge of a robotics

research program. Opportunities to teach this material at Adelphi University and

during a year spent as a visiting professor at the U.S. Military Academy at West Point

have been helpful. In particular, I thank my cadets and my students at Adelphi for

finding errors and suggesting improvements in earlier drafts. Thanks are due to the

National Science Foundation, the Sloan Foundation, and COMAP for involving me

in programs dedicated to the improvement of geometry at both the collegiate and

secondary levels. Finally, I wish to thank numerous individuals with whom I have been

in contact (for many years in some cases) about geometry in general and this book in

particular: Robert Bumcrot, Kevin Carmody, Donald Crowe, Sol Garfunkel, Marian

Gidea, Andrew Gleason, Greg Lupton, Joseph Malkevitch, John Oprea, Mohammed

Salmassi, Brigitte Selvatius, and Marie Vanisko.

Prof. Walter Meyer

Adelphi University

Supplements for the Instructor

Please contact Academic Press for the Instructor’s Manual , which contains a guide to

helping students use The Geometer’s Sketchpad , plus answers to even-numbered exercises.

Web content for students can be accessed following the instructions on p. xx of this

book.

Introduction

As the poet Edna St. Vincent Millay wrote, “Euclid alone has looked on beauty

bare.” But beyond beauty and logic, geometry also contains important tools for

applied mathematics. This should be no surprise, since the word geometry means “earth

measurement” in Greek. As just one example, we will illustrate the appropriateness

of this name by showing how geometry was applied by the ancient Greeks to measure

the circumference of the earth without actually going around it. But the story of

geometric applications is modern as well as ancient. The upsurge in science and

technology in the last few decades has brought with it an outpouring of new questions

for geometers. In this introduction we provide a sampler of the big ideas and important

applications that will be discussed in this book. 1

Individuals often have preferences, either for applications in contrast to theory

or vice versa. This is unavoidable and understandable. But the premise of this book

is that, whatever our preferences may be, it is good to be aware of how the two faces

of geometry enrich each other. Applications can’t proceed without an underlying

theory. And theoretical ideas, although they can stand alone, often surprise us with

unexpected applications. Throughout the history of mathematics, theory and appli-

cations have carried out an intricate dance, sometimes dancing far apart, sometimes

close. My hope is that this book gives a balanced picture of the dance at this time, in

the early years of a new millennium.

1 This introduction also appears in Perspectives on the Teaching of Geometry for the 21st Century , ed. V. Villani and

xiii

G eometry is full of beautiful theorems, and its logical structure can be inspiring.

xiv

INTRODUCTION

Axiomatic Geometry

Of all the marvelous abilities we human beings possess, nothing is more impressive

than our visual systems. We have no trouble telling circles apart from squares,

estimating sizes, noticing when triangles appear congruent, and so on. Despite this,

the earliest big idea in geometry was to achieve truth by proof and not by eye. Was

that really necessary or useful? These ideas are explored in Chapters 1 and 2.

Creating a geometry based on proof required some basic truths — which are

called axioms in geometry. Axioms are supposed to be uncontroversial and obviously

true, but Euclid seemed nervous about his parallel axiom. Other geometers caught

this whiff of uncertainty and, about 2000 years later, some were bold enough to

deny the parallel axiom. In doing this they denied the evidence of their own eyes

and the weight of 2000 years of tradition. In addition, they created a challenge for

students of this so-called “non-Euclidean geometry,” which asks them to accept axioms

and theorems that seem to contradict our everyday visual experience. According to

our visual experience, these non-Euclidean geometers are cranks and crackpots. But

eventually they were promoted to visionaries when physicists discovered that the

faraway behavior of light rays (physical examples of straight lines) is different from

the close-to-home behavior our eyes observe. Astronomers are working to make use

of this non-Euclidean behavior of light rays to search for “dark matter” and to foretell

the fate of the universe. These revolutionary ideas are explored in Chapter 3.

Rigidity and Architecture

If you are reading this indoors, the building you are in undoubtedly has a skeleton

of either wooden or steel beams, and your safety depends partly on the rigidity of

this skeleton (see Figure I.1b). Neither a single rectangle (Figure I.1a), nor a grid of

them, would be rigid if it had hinges where the beams meet. Therefore, when we

build frameworks for buildings, we certainly don’t put hinges at the corners — in

fact, we make these corners as strong as we can. But it is hard to make a corner

perfectly rigid, so we will think of the corners as a little bit like hinges that need to

be kept from flexing. If a rectangle is built of four sticks hinged together in such a

way that any flexing produces a four-sided figure in a plane, and if we add a diagonal

brace, then the rectangle can’t be flexed at all. Perhaps surprisingly, if we have a grid

of many rectangles, it is not necessary to brace every rectangle. The braced grid in

Figure I.1b turns out to be rigid even if every corner is hinged. In Section 2.3, we work

out a procedure for determining when a set of braces makes a grid of rectangles rigid

even though all corners are hinged.

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