Thinking Geometrically

Thinking GeometricallyA Survey of Geometries

AMS / MAA TEXTBOOKS VOL 26

Thomas Q. Sibley

Thinking Geometrically

A Survey of Geometries

c© 2015 byThe Mathematical Association of America (Incorporated)

Library of Congress Control Number: 2015936100

Print ISBN: 978-1-93951-208-6

Electronic ISBN: 978-1-61444-619-4

Printed in the United States of America

Current Printing (last digit):10 9 8 7 6 5 4 3 2 1

Thinking Geometrically

A Survey of Geometries

Thomas Q. SibleySt. John’s University

Published and distributed by

The Mathematical Association of America

10.1090/text/026

Council on Publications and CommunicationsJennifer J. Quinn, Chair

Committee on BooksFernando Gouvea, Chair

MAA Textbooks Editorial BoardStanley E. Seltzer, Editor

Matthias BeckRichard E. Bedient

Otto BretscherHeather Ann Dye

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AckermanCombinatorics: A Guided Tour, David R. MazurCombinatorics: A Problem Oriented Approach, Daniel A. MarcusComplex Numbers and Geometry, Liang-shin HahnA Course in Mathematical Modeling, Douglas Mooney and Randall SwiftCryptological Mathematics, Robert Edward LewandDifferential Geometry and its Applications, John OpreaDistilling Ideas: An Introduction to Mathematical Thinking, Brian P. Katz and Michael StarbirdElementary Cryptanalysis, Abraham SinkovElementary Mathematical Models, Dan KalmanAn Episodic History of Mathematics: Mathematical Culture Through Problem Solving, Steven

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Non-Euclidean Geometry, H. S. M. CoxeterNumber Theory Through Inquiry, David C. Marshall, Edward Odell, and Michael StarbirdOrdinary Differential Equations: from Calculus to Dynamical Systems, V. W. NoonburgA Primer of Real Functions, Ralph P. BoasA Radical Approach to Lebesgue’s Theory of Integration, David M. BressoudA Radical Approach to Real Analysis, 2nd edition, David M. BressoudReal Infinite Series, Daniel D. Bonar and Michael Khoury, Jr.Thinking Geometrically: A Survey of Geometries, Thomas Q. SibleyTopology Now!, Robert Messer and Philip StraffinUnderstanding our Quantitative World, Janet Andersen and Todd Swanson

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Contents

Preface xv

1 Euclidean Geometry 11.1 Overview and History........................................................................... 1

1.1.1 The Pythagoreans and Zeno .......................................................... 21.1.2 Plato and Aristotle ...................................................................... 41.1.3 Exercises for Section 1.1 .............................................................. 5

1.2 Euclid’s Approach to Geometry I: Congruence and Constructions................... 101.2.1 Congruence............................................................................... 111.2.2 Constructions ............................................................................ 121.2.3 Equality of Measure .................................................................... 151.2.4 The Greek Legacy....................................................................... 161.2.5 Exercises for Section 1.2 .............................................................. 171.2.6 Archimedes............................................................................... 25

1.3 Euclid’s Approach II: Parallel Lines ......................................................... 261.3.1 Exercises for Section 1.3 .............................................................. 30

1.4 Similar Figures.................................................................................... 331.4.1 Exercises for Section 1.4 .............................................................. 37

1.5 Three-Dimensional Geometry................................................................. 421.5.1 Polyhedra.................................................................................. 421.5.2 Geodesic Domes ........................................................................ 461.5.3 The Geometry of the Sphere.......................................................... 481.5.4 Exercises for Section 1.5 .............................................................. 511.5.5 Buckminster Fuller ..................................................................... 591.5.6 Projects for Chapter 1.................................................................. 591.5.7 Suggested Readings .................................................................... 65

2 Axiomatic Systems 672.1 From Euclid to Modern Axiomatics ......................................................... 67

2.1.1 Overview and History.................................................................. 672.1.2 Axiomatic Systems ..................................................................... 682.1.3 A Simplified Axiomatic System..................................................... 712.1.4 Exercises for Section 2.1 .............................................................. 73

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viii Contents

2.2 Axiomatic Systems for Euclidean Geometry .............................................. 752.2.1 SMSG Postulates........................................................................ 762.2.2 Hilbert’s Axioms ........................................................................ 772.2.3 Exercises for Section 2.2 .............................................................. 782.2.4 David Hilbert............................................................................. 81

2.3 Models and Metamathematics................................................................. 822.3.1 Exercises for Section 2.3 .............................................................. 882.3.2 Kurt Godel................................................................................ 942.3.3 Projects for Chapter 2.................................................................. 942.3.4 Suggested Readings .................................................................... 96

3 Analytic Geometry 973.1 Overview and History........................................................................... 98

3.1.1 The Analytic Model .................................................................... 983.1.2 Exercises for Section 3.1 .............................................................. 1003.1.3 Rene Descartes .......................................................................... 104

3.2 Conics and Locus Problems ................................................................... 1043.2.1 Exercises for Section 3.2 .............................................................. 1103.2.2 Pierre de Fermat......................................................................... 114

3.3 Further Topics in Analytic Geometry........................................................ 1143.3.1 Parametric Equations................................................................... 1143.3.2 Polar Coordinates ....................................................................... 1163.3.3 Barycentric Coordinates............................................................... 1183.3.4 Other Analytic Geometries............................................................ 1203.3.5 Exercises for Section 3.3 .............................................................. 121

3.4 Curves in Computer-Aided Design .......................................................... 1263.4.1 Exercises for Section 3.4 .............................................................. 131

3.5 Higher Dimensional Analytic Geometry.................................................... 1333.5.1 Analytic Geometry in Rn .............................................................. 1333.5.2 Regular Polytopes....................................................................... 1373.5.3 Exercises for Section 3.5 .............................................................. 1403.5.4 Gaspard Monge.......................................................................... 1453.5.5 Projects for Chapter 3.................................................................. 1453.5.6 Suggested Readings .................................................................... 149

4 Non-Euclidean Geometries 1514.1 Overview and History........................................................................... 152

4.1.1 The Advent of Hyperbolic Geometry............................................... 1534.1.2 Models of Hyperbolic Geometry .................................................... 1554.1.3 Exercises for Section 4.1 .............................................................. 1584.1.4 Carl Friedrich Gauss.................................................................... 160

4.2 Properties of Lines and Omega Triangles .................................................. 1614.2.1 Omega Triangles ........................................................................ 1644.2.2 Exercises for Section 4.2 .............................................................. 1674.2.3 Nikolai Lobachevsky and Janos Bolyai ............................................ 169

Contents ix

4.3 Saccheri Quadrilaterals and Triangles....................................................... 1694.3.1 Exercises for Section 4.3 .............................................................. 1734.3.2 Omar Khayyam.......................................................................... 1744.3.3 Giovanni Girolamo Saccheri.......................................................... 175

4.4 Area, Distance, and Designs................................................................... 1764.4.1 Exercises for Section 4.4 .............................................................. 183

4.5 Spherical and Single Elliptic Geometries................................................... 1854.5.1 Exercises for Section 4.5 .............................................................. 1894.5.2 Georg Friedrich Bernhard Riemann................................................. 1894.5.3 Projects for Chapter 4.................................................................. 1904.5.4 Suggested Readings .................................................................... 192

5 Transformational Geometry 1955.1 Overview and History........................................................................... 195

5.1.1 Exercises for Section 5.1 .............................................................. 2005.2 Isometries .......................................................................................... 201

5.2.1 Classifying Isometries ................................................................. 2035.2.2 Congruence and Isometries........................................................... 2085.2.3 Klein’s Definition of Geometry ...................................................... 2095.2.4 Exercises for Section 5.2 .............................................................. 2095.2.5 Felix Klein................................................................................ 212

5.3 Algebraic Representation of Transformations............................................. 2135.3.1 Isometries................................................................................. 2175.3.2 Exercises for Section 5.3 .............................................................. 220

5.4 Similarities and Affine Transformations .................................................... 2235.4.1 Similarities................................................................................ 2235.4.2 Affine Transformations ................................................................ 2265.4.3 Iterated Function Systems............................................................. 2285.4.4 Exercises for Section 5.4 .............................................................. 2315.4.5 Sophus Lie................................................................................ 234

5.5 Transformations in Higher Dimensions; Computer-Aided Design ................... 2355.5.1 Isometries of the Sphere............................................................... 2355.5.2 Transformations in Three and More Dimensions................................ 2395.5.3 Computer-Aided Design and Transformations ................................... 2415.5.4 Exercises for Section 5.5 .............................................................. 243

5.6 Inversions and the Complex Plane ........................................................... 2455.6.1 Exercises for Section 5.6 .............................................................. 2525.6.2 Augustus Mobius........................................................................ 2545.6.3 Projects for Chapter 5.................................................................. 2545.6.4 Suggested Readings .................................................................... 259

6 Symmetry 2616.1 Overview and History........................................................................... 262

6.1.1 Exercises for Section 6.1 .............................................................. 264

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6.2 Finite Plane Symmetry Groups ............................................................... 2676.2.1 Exercises for Section 6.2 .............................................................. 270

6.3 Symmetry in the Plane.......................................................................... 2726.3.1 Frieze Patterns ........................................................................... 2736.3.2 Wallpaper Patterns...................................................................... 2766.3.3 Exercises for Section 6.3 .............................................................. 2826.3.4 M. C. Escher ............................................................................. 288

6.4 Symmetries in Higher Dimensions........................................................... 2886.4.1 Finite Three-Dimensional Symmetry Groups .................................... 2886.4.2 The Crystallographic Groups......................................................... 2906.4.3 General Finite Symmetry Groups ................................................... 2916.4.4 Exercises for Section 6.4 .............................................................. 2916.4.5 H. S. M. Coxeter......................................................................... 293

6.5 Symmetry in Science............................................................................ 2946.5.1 Chemical Structure ..................................................................... 2946.5.2 Quasicrystals............................................................................. 2966.5.3 Symmetry and Relativity.............................................................. 2976.5.4 Exercises for Section 6.5 .............................................................. 2996.5.5 Marjorie Senechal....................................................................... 304

6.6 Fractals ............................................................................................. 3046.6.1 Exercises for Section 6.6 .............................................................. 3106.6.2 Benoit Mandelbrot...................................................................... 3126.6.3 Projects for Chapter 6.................................................................. 3126.6.4 Suggested Readings .................................................................... 314

7 Projective Geometry 3177.1 Overview and History........................................................................... 318

7.1.1 Projective Intuitions .................................................................... 3197.1.2 Exercises for Section 7.1 .............................................................. 322

7.2 Axiomatic Projective Geometry .............................................................. 3277.2.1 Duality..................................................................................... 3317.2.2 Perspectivities and Projectivities..................................................... 3327.2.3 Exercises for Section 7.2 .............................................................. 3337.2.4 Jean Victor Poncelet .................................................................... 336

7.3 Analytic Projective Geometry................................................................. 3377.3.1 Cross Ratios .............................................................................. 3397.3.2 Conics ..................................................................................... 3407.3.3 Exercises for Section 7.3 .............................................................. 3427.3.4 Julius Plucker ............................................................................ 346

7.4 Projective Transformations .................................................................... 3467.4.1 Exercises for Section 7.4 .............................................................. 351

7.5 Subgeometries .................................................................................... 3547.5.1 Hyperbolic Geometry as a Subgeometry .......................................... 3557.5.2 Single Elliptic Geometry as a Subgeometry ...................................... 3587.5.3 Affine and Euclidean Geometries as Subgeometries............................ 358

Contents xi

7.5.4 Exercises for Section 7.5 .............................................................. 3597.5.5 Arthur Cayley............................................................................ 361

7.6 Projective Space.................................................................................. 3617.6.1 Perspective and Computer-Aided Design.......................................... 3627.6.2 Subgeometries of Projective Space ................................................. 3677.6.3 Exercises for Section 7.6 .............................................................. 3687.6.4 Projects for Chapter 7.................................................................. 3707.6.5 Suggested Readings .................................................................... 372

8 Finite Geometries 3738.1 Overview and History........................................................................... 373

8.1.1 Exercises for Section 8.1 .............................................................. 3768.1.2 Leonhard Euler .......................................................................... 3778.1.3 Rev. Thomas Kirkman ................................................................. 378

8.2 Affine and Projective Planes................................................................... 3788.2.1 Affine Planes............................................................................. 3788.2.2 Projective Planes. ....................................................................... 3828.2.3 Exercises for Section 8.2 .............................................................. 384

8.3 Design Theory .................................................................................... 3858.3.1 Error-correcting Codes ................................................................ 3898.3.2 Exercises for Section 8.3 .............................................................. 3918.3.3 Sir Ronald A. Fisher.................................................................... 393

8.4 Finite Analytic Geometry ...................................................................... 3948.4.1 Finite Analytic Planes.................................................................. 3958.4.2 Ovals in Finite Projective Planes .................................................... 3988.4.3 Finite Analytic Spaces ................................................................. 3998.4.4 Exercises for Section 8.4 .............................................................. 4018.4.5 Projects for Chapter 8.................................................................. 4048.4.6 Suggested Readings .................................................................... 407

9 Differential Geometry 4099.1 Overview and History........................................................................... 409

9.1.1 Exercises for Section 9.1 .............................................................. 4119.2 Curves and Curvature ........................................................................... 413

9.2.1 Exercise for Section 9.2................................................................ 4209.2.2 Sir Isaac Newton ........................................................................ 423

9.3 Surfaces and Curvature ......................................................................... 4249.3.1 Surfaces ................................................................................... 4249.3.2 Curvature.................................................................................. 4269.3.3 Surfaces of Revolution................................................................. 4289.3.4 Exercises for Section 9.3 .............................................................. 429

9.4 Geodesics and the Geometry of Surfaces................................................... 4329.4.1 Geodesics ................................................................................. 4329.4.2 Geodesics on Surfaces of Revolution............................................... 4359.4.3 Arc Length on Surfaces................................................................ 436

xii Contents

9.4.4 The Gauss-Bonnet Theorem......................................................... 4399.4.5 Higher Dimensions.................................................................... 4399.4.6 Exercises for Section 9.4............................................................. 4419.4.7 Albert Einstein ......................................................................... 4449.4.8 Projects Chapter 9 ..................................................................... 4459.4.9 Suggested Readings ................................................................... 446

10 Discrete Geometry 44710.1 Overview and Explorations .................................................................... 448

10.1.1 Distances between Points ............................................................ 44810.1.2 Triangulations .......................................................................... 44910.1.3 The Art Gallery Problem............................................................. 44910.1.4 Tilings .................................................................................... 45010.1.5 Voronoi Diagrams ..................................................................... 45110.1.6 Exercises for Section 10.1 ........................................................... 45310.1.7 Paul Erdos ............................................................................... 457

10.2 Points and Polygons ............................................................................. 45710.2.1 Distances between Points ............................................................ 45710.2.2 Triangulations .......................................................................... 45910.2.3 The Art Gallery Problem............................................................. 46210.2.4 Exercises for Section 10.2 ........................................................... 465

10.3 Tilings .............................................................................................. 46910.3.1 Exercises for Section 10.3 ........................................................... 47410.3.2 Branko Grunbaum..................................................................... 477

10.4 Voronoi Diagrams................................................................................ 47710.4.1 Exercises for Section 10.4 ........................................................... 48110.4.2 Projects for Chapter 10 ............................................................... 48310.4.3 References............................................................................... 485

11 Epilogue 48711.1 Topology ........................................................................................... 48811.2 Henri Poincare .................................................................................... 489

A Definitions, Postulates, Common Notions, and Propositions from Book Iof Euclid’s Elements 491A.1 Definitions ......................................................................................... 491A.2 The Postulates (Axioms) ....................................................................... 492A.3 Common Notions ................................................................................ 492A.4 The Propositions of Book I .................................................................... 492

B SMSG Axioms for Euclidean Geometry 497

C Hilbert’s Axioms for Euclidean Plane Geometry 499

Contents xiii

D Linear Algebra Summary 503D.1 Vectors.............................................................................................. 503D.2 Matrices ............................................................................................ 504D.3 Determinants ...................................................................................... 505D.4 Properties of Matrices........................................................................... 506D.5 Eigenvalues and Eigenvectors................................................................. 506

E Multivariable Calculus Summary 509E.1 Vector Functions ................................................................................. 509E.2 Surfaces ............................................................................................ 509E.3 Partial Derivatives................................................................................ 510

F Elements of Proofs 511F.1 Direct Proofs ...................................................................................... 511F.2 Proofs by Contradiction......................................................................... 512F.3 Induction Proofs.................................................................................. 512F.4 Other Remarks on Proofs....................................................................... 513

Answers to Selected Exercises 515

Acknowledgements 549

Index 551

Preface

I begin to understand that while logic is a mostexcellent guide in governing our reason, it doesnot, as regards stimulation to discovery, comparewith the power of sharp distinction which belongsto geometry.

— Galileo Galilei (1564–1642)

Geometry combines visual delights and powerful abstractions, concrete intuitions and generaltheories, historical perspective and contemporary applications, and surprising insights andsatisfying certainty. In this textbook, I try to weave together these facets of geometry. I alsowant to convey the multiple connections that topics in geometry have with each other and thatgeometry has with other areas of mathematics. The connections link chapters together withoutsacrificing the survey nature of the whole text.

Geometric thinking fuses reasoning and intuition in a characteristic fashion. The enduringappeal and importance of geometry stem from this synthesis. Mathematical insight is as hard formathematics students to develop as is the skill of proving theorems. Geometry is an ideal subjectfor developing both, leading to deeper understanding. However, geometry texts for mathematicsmajors often emphasize proofs over visualization, whereas some texts for mathematics educationmajors focus on intuition instead. This book strives to build both and so geometrical thinkingthroughout the text, the exercises, and the projects. The dynamic software now available providesone valuable way for students to build their intuition and prepare them for proofs. Thus exercisesbenefiting from technology join hands-on explorations, proofs, and other types of problems.This book builds on the momentum of the NCTM Standards, the calculus reform movement,the Common Core State Standards (CCSS), and the ongoing discussion of how to help studentsinternalize mathematical concepts and thinking. (We abbreviate the National Council of Teachersof Mathematics throughout as NCTM.)

There are two natural audiences for a geometry course at the college level—mathematicsmajors and future secondary mathematics teachers. Of course, the two audiences overlap consid-erably. In many states, however, requirements for secondary mathematics education majors candiffer substantially from those for traditional mathematics majors. This book seeks to serve bothaudiences, partially by using a survey format so that instructors can choose among topics. Inaddition, those sections matching CCSS expectations and NCTM recommendations for teacherpreparation assume less mathematical sophistication, although they have plenty of problems

xv

xvi Preface

and projects to challenge more advanced undergraduates. In particular, in those sections vitalfor teacher preparation, earlier exercises require less sophistication than later ones. A latersubsection of this preface discusses possible course choices.

It has been a treat to revisit the material in this text, re-envision the problems and projects,and add new ones. I have also enjoyed adding the material in Chapters 9 and 10. In the processI realized again how geometric my thinking is and how much I enjoy sharing the elegance andexcitement I experience in geometry. I hope that some of my enthusiasm shows through.

Geometric IntuitionEveryday speech equates intuitive with easy and obvious. However, psychology research con-firms what mathematicians have always understood: people build their own intuitions throughreflection on their experiences. My students often describe this process as learning to think in anew geometry.

As Galileo’s quote introducing this preface suggests, geometry has for centuries been anideal place for developing mathematical intuition. Since the advent of analytic geometry (at theend of Galileo’s life) mathematicians have repeatedly turned geometric insights into algebraicformulations. The applicability, efficiency, simplicity, and power of algebra have reasonably lededucators from middle school through graduate school to emphasize algebraic representations.In my view, the success of algebra has so focused the curriculum that students’ geometricalthinking often lags far behind. My text tries to correct that imbalance without neglecting thepower of algebra. The NCTM calls for high school students to develop geometric intuition andunderstanding. Similarly the CCSS expects students to build geometrical thinking in a varietyof ways.

Throughout this book I seek to help students develop their geometrical intuition. Visualiza-tion is an important part of this effort, and the hundreds of figures in the book provide an obviousmeans to this end. Many of the more than 750 exercises ask students to draw or create their ownfigures and models. In addition, I have included many exercises and projects requesting studentsto explore and conjecture new ideas, as well as explain or prove unusual properties. I advocatehaving students use dynamic geometry software, such as Geometer’s Sketchpad or Geogebra,to explore geometric ideas. While some texts explicitly incorporate such software, I don’t wantto tie my book to one program. However, in my teaching experience students gain different andoften more insight working with physical models than from manipulating computer models. SoI strongly encourage instructors to give students physical models to use for as many topics asthe class time, budget, and their creativity allows. I provide several suggestions in the text andin the projects.

I hope the text’s explanations are clear and provide new insights, but I know that studentsneed to reflect on the text and the exercises. I also hope that students find the many non-routineproblems and projects challenging, but solvable with effort; such challenges enrich intuition.

The Role of ProofsSince Euclid, over two thousand years ago, proofs have had a central place in mathemat-ical thought. Non-mathematicians often think the value of mathematics is restricted to itsamazing ability to calculate “answers.” Certainly, many applications of mathematics rely on

Preface xvii

computational power. However, people applying mathematics want to know more than thatthere is an answer—after all, an astrologer gives answers. We want to know that the answersare valid. While confidence in much of science depends on experiments, it also often dependson mathematical models. The models make explicit assumptions about how some aspect of theworld behaves and recasts them in mathematical terms. So applications of mathematics requirethat someone—a mathematician—actually prove the results that others use.

However, the need for someone to have proved any given result doesn’t lead to a need forevery mathematics student to prove every result. Most people acknowledge the value of honingstudents’ ability to reason critically, and mathematical proofs certainly contribute greatly to thatskill. Educators actively debate how much students need to prove and at what level of rigor,although all agree that the answers depend on the level of the student. The amount of proof inhigh school courses now varies greatly across the United States. Still, the NCTM and CCSScall for high school students to do a certain amount of work with proofs. It follows that highschool mathematics teachers need substantial background and facility in proof. I have writtenthis text for mathematics majors and future high school mathematics teachers, and I think thereis a range of proof experiences both audiences need. All these students need facility in makinggood arguments in a mathematical context, something I ask for repeatedly throughout the book.At a minimum, that means they need to make their assumptions explicit and use clear reasoningleading from them to their conclusions. I think it also means that they should be introduced tomore formal proofs and axiomatic systems, although I don’t think that should be the primaryfocus of an entire course at this level.

In Chapter I, I employ a fairly informal focus on proof to fit that chapter’s goals. One goalis to include enough coverage of the content of Euclidean geometry for students who have nothad a solid year-long high school geometry course (and provide a review for others, as needed).Another goal is to develop students’ ability to prove substantive results in a context where theyalready have a comfortable intuition. Hence in that chapter I haven’t made explicit the manysubtleties and assumptions discussed in Chapter 2. Instead, I ask students to build on Euclid’stheorems so that they can prove results that aren’t instantly apparent, although they shouldbe plausible. Chapter 2, which looks more carefully at axiomatics, makes axiomatic systemsexplicit and builds up theorems carefully from the axioms. The level of proof in later chapterslies in between the informality of Chapter 1 and the axiomatically based proofs in Chapter 2,depending on the chapter. In most chapters the content is less familiar and the mathematicsmore sophisticated than in Chapter 1. Thus the value of proofs in them also connects with thegoal of increasing geometric intuition.

Proofs at the college level are written in paragraph form, unlike in high school geometry,where two-column proofs appear frequently. The range of argumentation appropriate at thecollege level can make the two-column format artificial and overly restrictive. A good proofought to help the reader understand why the theorem is correct, as well as make the correctness ofthe reasoning clear, and two-column proofs can sacrifice understanding for clarity of reasoning.Appendix F gives an introduction to proof techniques used in this text.

General NotationProofs end with the symbol �. Examples end with the symbol ♦. Exercises or parts of exerciseswith answers or partial answers in the back of the book are marked with an asterisk (*).

xviii Preface

Definitions italicize the word being defined. We use the abbreviation b.c.e. (before the commonera) for dates predating the common era, which started somewhat more than 2000 years ago.Dates in the common era will not have the abbreviation c.e. added to them.

PrerequisitesIn general, students need the maturity of Calculus I and II, although only Chapter 9 and Section2.4 use calculus extensively. Chapter 9 needs some content from Calculus III as well. Appendix Esummarizes the material from Calculus III used in Chapter 9. Of course, additional mathematicalmaturity and familiarity with proofs will help throughout the book. Sections 3.3 and 3.5 andChapter 9 require an understanding of vectors. Sections 5.3, 5.4, 5.5, 7.3, 7.4, 7.5, 7.6, and 8.4depend on a more extensive understanding of linear algebra. Appendix D summarizes the linearalgebra material needed in the text. Although Chapter 6 builds on concepts from Chapter 5, itdoesn’t require linear algebra. Section 5.6 makes use of complex numbers and their arithmetic.Chapters 5, 6, and 7 discuss groups and Section 8.4 discusses finite fields, but don’t assume anyprior familiarity with the concepts.

ExercisesLearning mathematics centers on doing mathematics, so problems are the heart of any math-ematics textbook. A number of exercises appear in the text and are meant to be done whilereading that material. Far more appear at the end of each section. I hope that both studentsand instructors enjoy spending time pondering, solving, discussing, and even extending theproblems. They should make lots of diagrams and, when relevant, physical and computer mod-els. The problems include routine and non-routine ones, traditional proofs and computations,hands-on experimentation, conjecturing, and more. Exercises or parts of exercises with answersor partial answers in the back of the book are marked with an asterisk (*).

ProjectsToo often textbooks and courses shift to a new topic just when students are ready to make theirown connections. And geometry is a particularly fertile area for such connections. Projectsencourage extending ideas discussed in the text and appear at the end of each chapter. Theyinclude essay questions, paper topics, and more extended and open-ended problems. Many ofthe projects benefit from group efforts. The most succinct projects, of the form “investigate. . . ,” are leads for paper topics.

HistoryGeometry reveals the rich influences over the centuries between areas of mathematics andbetween mathematics and other fields. Students in geometry, even more than other areas ofmathematics, benefit from historical background. The introductory sections of each chapterseek to link the material of the chapter to a broader context and to students’ general knowledge.The biographies give additional historical perspective and add a personal flavor to some ofthe work discussed in the text. One common thread I found in reading about these geometers

Preface xix

was the importance of intuition and visual thinking. As a student I sometimes questioned mymathematical ability because I needed to visualize and construct my own intuitive understanding,instead of grasping abstract ideas directly from a text or a lecture. Now I realize that far greatermathematicians than I built on intuition and visualization for their abstract insights, proofs, andtheories. Perhaps this understanding will help the next generation as well.

Chapter ContentEach chapter starts with an overview, including a discussion of the relevant history, and endswith projects and a list of suggested readings. Geometry is blessed, more than other areas ofmathematics, with many wonderful and accessible expository writings as well as texts. (The vastnumber of web sites, software, and other media devoted to geometry surpass my ability to view,let alone recommend a helpful selection. Further, any printed list would quickly be outdated.So, while I do not make suggestions, I encourage instructors to find media that support theircourses.)

1. Euclidean Geometry. Most of this chapter considers plane geometry and follows the leadof the ancient Greeks’ approach, especially Euclid’s synthesis. (Appendix A gives thedefinitions, axioms, and propositions of Book I of Euclid’s Elements.) Since high schoolgeometry courses include much of Euclid’s emphases, this approach adds context to a teacherpreparation course. Students’ preparation varies greatly, so instructors should adjust theirpace and coverage according to how much this material is review for the students. All theexercises in Section 1.1 and many of the others have Greek or pre-Greek roots. The three-dimensional material has a more modern focus, considering polyhedra, including geodesicdomes, and the sphere. The material on the sphere is a useful transition into a study ofnon-Euclidean geometry, although it is presented as part of Euclidean geometry.

2. Axiomatic Systems and Models. The first section introduces axiomatic systems and investi-gates simple ones. The next section considers a high school axiomatic system for Euclideangeometry and Hilbert’s axiomatization. (Appendices B and C give the axiomatic systems.)I chose to use the SMSG axioms, one of the “ancestors” of all high school axioms systems,rather than try to choose among contemporary ones. (SMSG is an abbreviation for theSchool Mathematics Study Group.) The final section explores models and metamathemat-ics. Instructors wishing to include more experience with axiomatic systems and models caninclude material from Chapter 8.

3. Analytic Geometry. While high school students use analytic geometry, they often don’tunderstand it and often don’t see many of the traditional topics. And, although calculustexts include topics such as parametric equations and polar coordinates, instructors oftenleave them out for lack of time. In addition to these topics, later sections discuss Beziercurves in computer aided design and geometry in three and more dimensions.

4. Non-Euclidean Geometry. The bulk of the chapter develops hyperbolic geometry axiomat-ically. In addition to typical axioms, we assume the first twenty-eight of Euclid’s theo-rems, which also hold in hyperbolic geometry. By assuming them we can use familiarapproaches to focus on how this geometry differs from Euclidean geometry. Models helpillustrate the concepts and theorems. The final section considers spherical and single ellipticgeometry.

xx Preface

5. Transformational Geometry. The first two sections develop the key ideas of transformationsand plane isometries without linear algebra. The CCSS strongly emphasize transformationsin high school geometry, so at least these first two sections are vital for teacher preparation.Students who have already studied Chapter 4 can consider the corresponding theorems inhyperbolic and spherical geometries. (See Project 22.) The next three sections use linearalgebra to delve into isometries more deeply, and into similarities, affine transformations,and transformations in higher dimensions. The final section investigates inversions usingcomplex numbers and relates to the Poincare disk model of hyperbolic geometry and isnot used elsewhere in the text. Appendix D covers the linear algebra needed for this andsubsequent chapters.

6. Symmetry. While this material uses concepts from Chapter 5, it doesn’t depend on linearalgebra. Students find this material accessible, compared with some of Chapter 5, and gaininsight into the power of the transformational approach, including for applications.

7. Projective Geometry. Projective geometry historically and pedagogically provides a cap-stone unifying Chapters 1, 4, and 5. The first two sections briefly develop it intuitively andaxiomatically. Later sections use linear algebra extensively and provide connections withcomputer graphics and the special theory of relativity.

8. Finite Geometry. Since the late nineteenth century geometers have drawn important insightsabout traditional geometry from the study of simplified finite systems. Section 8.2 discussesthe most important of these, finite affine and projective planes, axiomatically. Section 8.3generalizes the material to balanced incomplete block designs. The final section exploresanalytic models of finite affine and projective planes and spaces over the fields Zp, theintegers (mod p), where p is prime.

9. Differential Geometry. Differential geometry deserves an entire undergraduate semestercourse, but many schools can’t offer it. I think students in a survey course benefit from anintroduction to this vital area of geometry. I try to convey here its geometric insight andintroduce some key geometric ideas—curvature and geodesics, and I endeavor to minimizethe machinery of multivariable calculus. The chapter connects differential geometry to Eu-clidean, spherical, and hyperbolic geometries. Students need little more from multivariablecalculus than a familiarity with parametric equations, partial derivatives, and cross products.Appendix E covers the needed multivariable material.

10. Discrete Geometry. This relatively new and rapidly growing area focuses on problems,especially ones that remain unsolved. Therefore I organized the chapter around problems.Students are encouraged to explore them in the first section. Subsequent sections developthem more fully, with relevant theorems and more in-depth problems.

Course SuggestionsThis text supports a variety of approaches to geometry and different levels of coverage of thematerial. Many of the sections benefit from more than one class period, especially to enablestudents to present problems or projects. The entire book would require a full year geometrycourse to cover, a luxury few mathematics department can offer.

A. Teacher Preparation. To meet the goals of the CCSS and of the NCTM for teacher prepa-ration, a course should include at least Euclidean geometry (Chapter 1), axiomatic systems

Preface xxi

and models (Chapter 2), transformational geometry (Chapter 5, except Section 5.6), andsome of non-Euclidean geometry (Chapter 4). As time, interest and student backgroundindicate, topics from analytic geometry (Chapters 3) and symmetry (Chapter 6) are valuablesupplements for future teachers.

B. Historical Survey. Chapters 1, 2, 4, 5, and 7 provide an understanding of the importanthistorical sweep of geometry through the nineteenth and early twentieth centuries. In 1800there was just Euclidean geometry (Chapter 1). Geometrical thinking expanded enormously,including non-Euclidean geometry (Chapter 4), transformational geometry (Chapter 5), andprojective geometry (Chapter 7), among others. The transformations of projective geome-try provided a vital unification of geometric thought, both historically and pedagogically.Because of these advances, mathematicians realized the need for a careful investigation ofproofs, theories, and models (Chapter 2).

C. Euclidean Geometry. Chapters 1, 2, 3, 5, and, as time permits, topics from Chapters 6, 9,and 10. If the class needs little review of Euclidean geometry, instructors could interleaveChapters 1 and 10 together at the start of the class.

D. Transformational Geometry. Chapters 5, 6, and 7 and as much of Chapters 1 and 2 as needed.E. Axiomatic Systems and Models. Chapters 1, 2, 4, 8, and Sections 3.1, 7.1, 7.2, 7.3.F. Topics. Instructors of courses for mathematics majors have fewer constraints than those

teaching mathematics education majors and so can choose topics more freely. Students’background and interest will suggest different options. Students with a weaker backgroundwill benefit from Chapters 1, 2, 3, 5, and 6. Chapters 4, 7, 8, 9, and 10 can stretch betterprepared students in different ways.

Dependence and Links Between ChaptersI have tried to keep chapters as independent as reasonable. Students with a decent geometryunderstanding from high school will have adequate Euclidean and analytic geometry backgroundfor all chapters except Chapter 4, which depends explicitly on Chapter 1. The basic conceptsof axiomatic systems and models from Chapter 2 appear in Chapters 4, 7, and 8. Chapter 5 isa prerequisite for Chapters 6 and 7. Sections 5.4 and 6.6 consider aspects of fractals. Sections6.5 and 7.6 briefly consider aspects of the special theory of relativity, and Section 9.4 toucheson the general theory of relativity. Section 9.3 refers to Chapter 4.

A number of exercises (denoted with #) connect with other material.Section 1.1 See Example 1 of Section 10.3 for another proof of Theorem 1.1.2 (the

Pythagorean theorem).# 1.2.9 anticipates Theorem 4.3.1.# 1.2.10 asks students to prove the converse of the Pythagorean theorem.# 1.2.14 anticipates Section 4.4.# 1.2.15 Compare this approach with #3.1.8. Section 10.4 uses this result.# 1.2.17 is used in a number of later sections.# 1.2.23, the law of cosines is used in # 3.3.12, # 3.3.21, #3.5.17, #3.5.18, and #10.3.14.# 1.2.25 is referred to in Section 3.3.Playfair’s axiom appears in Sections 1.3 and 2.2.# 1.3.8 (d) Compare this approach with # 3.1.4 (a).

xxii Preface

Section 1.5 Euler’s formula (Theorem 1.5.1) is used in Section 10.4. Project 14 in Chapter 3asks students to investigate this formula in higher dimensions.

Section 1.5 Shortest paths on a sphere are explored more in # 9.1.6 and in Section 9.4.Section 1.5 Theorem 1.5.3 relates to Theorem 4.1.1 and their generalization Theorem 9.4.3.# 1.5.35 anticipates concepts in Sections 2.1 and 2.3.Chapter 1, Project 6 is used in Project 11 of Chapter 10.Chapter 1, Project 7 relates to the art gallery theorem in Chapter 10.Many axiomatic systems in Section 2.1 are developed further in Section 2.3. Here are the

pairings: Subsection 2.1.3 and # 2.3.6, # 2.1.7 and # 2.3.8, #2.1.8 and #2.3.9, #2.1.9 and #2.3.11,# 2.1.10 and # 2.3.12, # 2.1.12 and # 2.3.13.

# 2.1.12 and # 2.3.13 anticipate projective planes, discussed in more detail in Section 8.2.Section 2.3 Example 3 (taxicab geometry) is used in # 3.3.21 , in Project 23 of Chapter 5,

and in Sections 10.1 and 10.4.#3.1.6 and #3.1.7 relate to Section 2.3.# 3.1.9 and # 3.1.10 relate to the first fundamental form in Chapter 9.# 3.1.14 and # 3.1.15 introduce complex numbers and their arithmetic, used in Section 5.6.Section 3.3 Parametric equations are used extensively in Chapter 9.#3.3.17 to #3.3.22 relate to Section 2.3.# 3.5.14 connects with Project 2 in Chapter 9.# 3.5.17 and # 3.5.18 relate to #10.3.14.Section 4.1 The pseudosphere is discussed in #9.3.16.Section 4.1 Theorem 4.1.1 connects with Theorem 1.5.3 and the generalization Theo-

rem 9.4.3.Chapter 4, Project 1—Compare with Chapter 5, Project 1 and Chapter 6, Project 3.# 5.610 (b)—Compare with Section 7.1 # 14.# 5.6.14 relates the Poincare model and the half plane model of Section 4.1.Section 6.3 relates to tilings in Sections 10.1 and 10.3.# 7.5.7 connects with relativistic velocities in Section 6.5 and Lorentz transformations in

Section 7.6.Chapter 7, Project 8 connects Euclidean isometries in Section 5.2 with hyperbolic

isometries.# 8.1.8 investigates an axiomatic system and its models.

AcknowledgmentsThis text is based on my book The Geometric Viewpoint: A Survey of Geometries, publishedby Addison Wesley in 1998. First, the acknowledgments from the earlier text: I appreciatethe support and helpful suggestions that many people made to improve this book, startingwith those students who studied from the rough versions. My previous editors, Marianne Leppand Jennifer Albanese, and my production supervisor, Kim Ellwood, provided much neededdirection, encouragement and suggestions for improvement. I am grateful to the reviewers,whose insightful and careful critiques helped me greatly. They are: Bradford Findell, Universityof New Hampshire; Yvonne Greenleaf, Rivier College; Daisy McCoy, Lyndon State University;Jeanette Palmiter, Portland State University; and Diana Venters, University of North Carolina,

Preface xxiii

Charlotte. They pointed out many mistakes and unclear passages; any remaining faults are,naturally, my full responsibility.

I particularly want to thank two people involved in the earlier text. Paul Krueger not onlypainstakingly made many of the fine illustrations that are an integral part of this book. Healso pondered with me over the years the nature of geometric thinking and the connections ofgeometry with other fields. Connie Gerads Fournelle helped greatly with her valuable perspectiveon the text, both as a student using an early version and as my assistant writing answers to selectedproblems.

I want to thank a number of people who have helped in my rewriting this book. First of all,Prof. Peiyi Zhao at St. Cloud State University and Prof. Sue Hagen at Virginia Tech and theirstudents were kind enough to use an early version of this rewrite in their classes and provideme feedback. I thank my students who also used an early version. Prof. Peter Haskell and Prof.Nick Robbins of Virginia Tech kindly provided feedback on Chapter 9. Prof. Ezra Brown, alsoof Virginia Tech, provided advice for Chapter 8. Prof. Frank Farris of Santa Clara University,pushed me to rewrite this text and provided enthusiastic support over the years. I am grateful toSt. John’s University for funding the sabbatical making this rewrite possible. Thanks go to theMathematics Department at Virginia Tech for welcoming me, encouraging me, and providingthe materials and space needed in the long rewriting process. I appreciate the extensive andcheerful secretarial support Suzette Ehlinger and Gail Schneider have provided. I am gratefulto the editorial staff at the Mathematical Association of America for their hard work: BeverlyRuedi, Carol Baxter, the copy editor, and editors Stan Selzer and Zaven Karien provided thetechnical support and positive encouragement needed to make this book a reality. Finally I thankmy wife, Jennifer Galovich, for her unswerving support and love throughout the years of thewriting and now rewriting.

If you have comments or suggestions for improvement, please contact me by e-mail [email protected].

Thomas Q. SibleyMathematics Department

St. John’s UniversityCollegeville, MN 56321-3000

Answers to Selected Exercises

Section 1.11.1.1. Throughout replace “even” by “a multiple of 3” and “2” by “3.” In the parentheses

consider the cases p = 3k + 1 and p = 3k + 2. For√

4 note that if p2 is a multiple of4, p need only be a multiple of 2, not 4.

1.1.2. There is no simple error, but Zeno’s reasoning does point out the difficulty of reasoningabout infinite processes based on finite steps.

1.1.3. Difference is ( 89 20)2(20) − π (10)2(20) ≈ 37.8 (units)3, error is 37.8

π (10)2(20) ≈ 0.6%,

( 89 2r )2 ≈ πr2 reduces to π ≈ 3.1604938.

1.1.5. 1.4146296 ≈ √2, 42.42638 ≈ 30

√2.

1.1.7. The man walks 300 units. Let x be how far the wizard flew up and z the diagonal of thewizard’s flight. Then x + z = 300 and z2 = 2002 + (x + 100)2. So x = 50.

1.1.9. (a) Let l and w be the length and width. Then l + w = 6.5, lw = 7.5 and l2 − 6.5l −7.5 = 0. So l = 5, w = 1.5.

1.1.11. (a) n(n + 1), where n is the number of rows.(b) Divide oblong into two triangles, one upside down. Thus

∑ni=1 i = n(n + 1)/2.

(c) Pentagonal numbers: 1, 5, 12, 22, etc. Hexagonal numbers: 1, 6, 15, 28, etc.(d) Pentagonal numbers: (3n2 − n)/2. Hexagonal numbers: 2n2 − n

1.1.13. (a) Add arithmetic mean to itself to get a + b. Multiply the geometric mean by itselfto get a · b. In geometric terms, the square with side

√ab has the same area as a

rectangle with sides a and b.(b) b = √

a(a + b) or b2 − ab − a2 = 0. Then b = a( 1+√5

2 ) ≈ 1.618a.

1.1.15. (a) 72◦, 54◦, 108◦, 36◦, 108◦, 72◦, 72◦, 36◦.(b) Use angles. �AB H ∼ �D AB ∼ �AFG.(c) If AB = 1, BC = 1 and GC = 1, and AC = AG + GC = 1 + x .(d) Since �AB D ∼ �BGC , 1+x

1 = 1x and x2 + x − 1 = 0. So x = −1+√

52 or 1 + x =

(1 +√5)/2.

1.1.17. Angles show similar triangles �ADC and �AC B. So ca = a

y , cb = b

x . So cy = a2 and

cx = b2. Also x + y = c, giving c2 = c(x + y) = a2 + b2.

515

516 Answers to Selected Exercises

Section 1.21.2.1. From the construction use SSS (I-8) to show �B AF ∼= �B E F and so ∠AB F ∼=

∠E B F .

1.2.2. By construction and SSS (I-8) �AB D ∼= �G E H ∼= �G E I and so ∠AB D ∼=∠G E H ∼= ∠G E I .

1.2.3. Use the Pythagorean theorem.

1.2.4. (a) 1, 2, 3, 9, 10, 11, 12, 22, 23, 31, 42, 44, 45, 46.(b) 4, 5, 6, 8, 13, 15, 26, 29, 32–38, 43, 47, 48.(c) SAA, AAA, AAS, SSA(d) SAA is equivalent to AAS and ASA using Theorem 1.1.1. AAA is not a congru-

ence theorem—but shows similar triangles. Neither ASS nor SSA are congruencetheorems. Consider �P Q R and �P QT with R between P and T and P Q = 52,QT = 25, PT = 31.5 and P R = 16.5. Let V be between R and T with PV = 24and use the converse of the Pythagorean theorem.

1.2.6. Suppose in �ABC that AB ∼= AC . Now ∠B AC ∼= ∠C AB. Then by SAS (I-4),�ABC ∼= �AC B. Hence ∠ABC ∼= ∠AC B.

1.2.8. Suppose that AC ⊥ B E , AE ⊥ C F and AD bisects ∠E AC .

(a) In �ADF and �ADB, ∠AF D ∼= ∠AB D and ∠F AD ∼= ∠B AD. By Theorem1.1.1 (I-32) we also have ∠ADF ∼= ∠AB F . Since AD ∼= AD, by ASA (I-26), wehave �ADF ∼= �ADB.

(b) From part (a) we have AF ∼= AB. By ASA (I-26), �AFC ∼= �AB E .

1.2.11. (a) For a + b on a line construct adjacent segments of lengths a and b. For a − b, makethe segments overlapping with a common endpoint.

(b) By similar triangles, P QP R = P S

PT .

(c) If P Q = 1, P R = a, and P S = b, then PT = a · b. If P Q = 1, PT = a, andP R = b, then P S = a/b.

1.2.13. (a) Pick any A on a circle with center O and construct B and C on the circle so thatAB = AC = AO . Construct D and E on the circle so that B D = C E = AB. Then�ADE is equilateral.

(b) Continue from (a) to construct F on the circle so that DF = B D. Then AB DF ECis a regular hexagon.

(c) Pick any point A on a circle with center O and construct the diameter AC throughO . Construct diameter B D perpendicular to AC at O . Then ABC D is a square.

1.2.15. (a) Let �ABC be any triangle and G be the intersection of the perpendicular bisectorG D of AB and the perpendicular bisector G E of BC . Then �ADG ∼= �B DGand �B EG ∼= �C EG by SAS (I-4). Then AG ∼= BG and BG ∼= CG.

(b) (Continuing from part (a).) Let F be the midpoint of AC . Then �AFG ∼= �C FGby SSS (I-8). Thus ∠AFG ∼= ∠C FG. Together they make a straight angle, sothey are each right angles, FG ⊥ AC and FG is the perpendicular bisector ofAC .


1.2.18. Let m∠ABC = p and m∠BC A = q. Use isosceles triangles, Theorem 1.1.1 and alge-bra to show that 3p = q and p = (180/7)◦ ≈ 25.7◦.

1.2.20. m∠QT S = 180◦ − x . m∠RQT = |(180◦ − x) − 90◦| = |90◦ − x |.1.2.23. (a) c2 = a2 + b2 − 2a · x . Note that x = b cos(C).

(b) Let y be the length of AD and use the Pythagorean theorem.(c) No because the perpendicular from A does not intersect between B and C . So

B D = x + a and c2 = a2 + b2 + 2ax .

Section 1.31.3.1. (a) 55◦. Reasoning: By I-29, m(∠FG D) = 35◦. From Theorem 1.1.1, m(∠E FG) =

180◦ − m(∠F DG) − m(∠FG D) = 180◦ − 90◦ − 35◦ = 55◦.(b) (i) k ‖ m by I-30.

(ii) k ‖ m by I-27(iii) k ⊥ m Use Playfair’s axiom and I-29.

(c) A rectangle is a parallelogram with at least one right angle. A square is a paral-lelogram with at least one right angle and at least two congruent adjacent sides. Arhombus is a parallelogram with at least two congruent adjacent sides.

1.3.3. (a) By I-29 ∠R P Q ∼= ∠P RS. Since P R ∼= P R and P Q ∼= RS, by SAS, �P Q R ∼=�RSQ. �

(b) From part (a), ∠S ∼= ∠Q and ∠R P Q ∼= ∠P RS. From Theorem 1.1.1, m(∠S) +m(∠S RQ) = 180◦. Then from I-28, P S ‖ Q R and P Q RS is a parallelogram. �

1.3.5. (a) Proof. Suppose that ABC D is a parallelogram with diagonal AC . By I-29∠C AB ∼= ∠AC D and ∠AC B ∼= ∠C AD. Since AC ∼= AC , by ASA (I-26)�ABC ∼= �C D A. The two triangles have the same area and together they have thearea of ABC D. So each have half of the area of ABC D. �

1.3.6. (a) Proof. Assume the given relationships. Use I-23 to construct an angle ∠P E Hcongruent to ∠AB E with P on the same side of B E as D. By I-28 P E ‖ AB. ByPlayfair’s axiom, there is only one parallel to AB through E . So P must be on

←→DE

and so ∠AB E ∼= ∠DE H . �

1.3.7. Proof. Let ABC D be a parallelogram with (∠ABC) = 90◦. Since AB ‖ C D and byI-29 m∠ABC + m∠BC D = 180◦, m∠BC D = 90◦. The other angles are similar. �

1.3.9. (b) Proof. Let ABC D be a kite with AB ∼= BC and C D ∼= AD. Draw B D. By SSS�B AD ∼= �BC D and so ∠B AD ∼= ∠BC D. �

(c) Let O be intersection of←→B D and

←→AC . Show �C B O ∼= �AB O . No.

1.3.11. (a) Use the four vertices of a tetrahedron.(b) Use the points (0, 0), (1, 0), (2, 0) and (3, 0).(c) Use the “bow tie” with vertices (0, 0), (0, 1), (2, 0) and (2, 1), in that order.

1.3.13. (a) Proof. Suppose that Q is the midpoint of P R and S and T are on the sameside of P R with P S ∼= QT and QS ∼= RT . By SSS �P QS ∼= �Q RT . Then∠S P Q ∼= ∠T Q R and these are corresponding angles, showing P S ‖ QT .


(b) (continuing the proof) By �P QS ∼= �T SQ, ST ∼= P Q. Further, ∠P QS ∼=∠T SQ. So by I-27 ST ‖ P Q.

Section 1.41.4.1. For k, ai and bi nonzero, kai = bi if and only if a1 · b2 = a1 · ka2 = ka1 · a2 = a2 · b1

if and only if a1/a2 = ka1/ka2 = b1/b2 if and only if a1/b1 = 1/k = a2/b2.

1.4.2. Let �ABG be a right triangle with AB = 13, AG = 12 and BG = 5. Let C be on AGwith AC = 8.25 and F be on

−→AG with AF = 15.75. Pick A = D and B = E .

1.4.3. For �ABC let AB be the longest side. Construct rectangle AB E F with base AB andheight AF equal to the height C D of �ABC . Then AB E F contains �ABC . ConstructD on AB so that C D ⊥ AB. Then �ADC ∼= �C F A and �B DC ∼= �C E B. Hencethe area of �ABC is half of the area of AB E F .

1.4.4. (a) 2 : 1 because �AF B ∼ �C F E and AB = 2 · C E .(b) 4 : 1 by Theorem 1.4.5.(c) 3 : 1. Let G be the midpoint of AB. Then �ADE has half the area of ADEG and

so 14 of area of ABC D. Similarly for �BC E . Hence ADE B has 3

4 of the area ofADE B.

(d) 3 : 4 as in part (c).

1.4.6. ratio a : b. Note that the large circle has radius a + b and the regions are made fromsemicircles.

1.4.8. First construct �D P Q congruent to �ABC based on Exercise 1.2.2. Draw the parallelto P Q through E . Let F be the intersection of this parallel with

−−→DQ. Then �DE F ∼

�ABC .

1.4.10. (d) is similar since it halves both the x and y-values and in general, y = sin(x) is similarto y = 1

k sin(kx).

1.4.12. Use Theorem 1.4.4 to show �P SU ∼ �P Q R ∼ �SQT ∼ �U T R. Use Theorem1.4.3 to show �T U S ∼ �P Q R.

1.4.13. (a) Proof. By Exercise 1.2.17, the measure of an inscribed angle of a given arc ishalf the measure of the corresponding central angle. Thus inscribed angles of thesame arc of a circle are congruent. So ∠S P R ∼= ∠SQ R and ∠P SQ ∼= ∠P RQ. ByTheorem 1.4.2 �P ST ∼ �Q RT . So PT/ST = QT/RT . Cross multiply to getPT · RT = QT · ST . �

1.4.14. Construct E F ‖ AB with F on BC . Because B F E D is a parallelogram, Exercise 1.3.8implies B F = DE . As with the argument in the proof, B F , FC , and BC are in thesame proportions with AE , EC , and AC .

1.4.19. Assume two similar polygons P and P ′ with a ratio of k can be divided into matched sim-ilar triangles T1 ∼ T ′

1, T2 ∼ T ′2, . . . , Tn ∼ T ′

n . These triangles also have the ratio k. Thenthe Area(P) = ∑n

i=1Area(Ti ) =∑n

i=1 k2Area(T ′i ) = k2

∑ni=1Area(Ti ) = k2Area(P ′).


1.4.20. (a) Note that f (kz/k) = f (z), so f (x/k) stretches out f (x) by a factor of k in thex-direction.

(b) Use integration by substitution where u = xk .

(c) A similar volume has k3 times the original volume.

1.4.23. (a) This definition may seem stronger because it considers all diagonals as well assides and so all angles formed by diagonals and sides as well as angles bounded byadjacent sides.

(b) Let A1 A2 A3 A4 and A′1 A′

2 A′3 A′

4 be two squares with sides of length s and ks, respec-tively. Their diagonals have length s

√2 and ks

√2, respectively. All corresponding

angles are 90◦ or 45◦.(c) A square and a rhombus.(d) A rectangle and a square.(e) All sides and angles of a regular polygon and so their corresponding diagonals are

congruent to each other.

Section 1.51.5.2. Replace B by πr2.

1.5.3. Use any interior point of the polyhedron as the vertex of pyramids whose bases are thefaces of the polyhedron.

1.5.4. Tetrahedron: V = 4, E = 6, F = 4; cube: V = 8, E = 12, F = 6; octahedron: V = 6,E = 12, F = 8; dodecahedron: V = 20, E = 30, F = 12; icosahedron: V = 12, E =30, F = 20.

1.5.6. 360 − 5(68.86) = 15.7 and 360 − 2(60) − 4(55.57) = 17.72.

1.5.7. Use proportions.

1.5.8. Possible angles are 60, 90, and 120.

1.5.9. (α + β + γ − 180)(π/180)r2.

1.5.10. Similar triangles have the same angle sum and so the same area.

1.5.11. (a) The volume is lwh. Each pyramid has volume 13 lwh.

(b) The diagonal of the base is perpendicular to the edge of the height(c) Not as easily. Let �ABC and �A′B ′C ′ be the two triangles and use pyramids

ABC A′, A′B ′C ′C and A′B B ′C .

1.5.13. (a) V = 2n, E = 3n, F = n + 2.

1.5.15. (a) 1/2,√

2/2,√

3/2.(b) 2

√3/(3π ) ≈ 37%.

(c) π/6 ≈ 52%(d) A tetrahedron, volume 1/3.

1.5.17. (c) If an edge has length l, the volume is 2l3√

2 − 8 16√

2l3 = 4

3 l3√

2.

(d) l, l/√

2, l√

23 .


1.5.19. (a) Two pyramids glued together. V = n + 2, E = 3n, F = 2n.(c) Tetrahedron is self-dual, the cube and octahedron are duals of each other, the

dodecahedron and the icosahedron are duals of each other.(d) Let the subscript P indicate the original polyhedron and D its dual. Then VP =

FD, EP = ED, FP = VD .

1.5.23. (b) Use Exercise 1.5.11.(b) for the points (1, b, 0) and (0, 1, b) to find b = (1 + √5)/2.

1.5.25. Find parallel diagonals of adjacent pentagons and their opposites.

1.5.27. (a) Use symmetry: Each angle is 90◦ and each side is the same length.(b) 195◦ = 13π/12 radians.(c) The area of �AC F is 1/48 of the sphere’s area, πr2/12.

1.5.29. (a) πr/2.(b) πr/2.(c) The ratio of the lengths of AN and AP equals the ratio of the measures of angles

∠AB N and ∠AB P .

1.5.33. (a) 42 = 12 + 30: the original 12 vertices plus one for each original edge. 720/42 ≈17.14◦. Use Exercise 1.5.6 to find (12 · 15.7 + 30 · 17.72)/42 = 17.14.

(b) 92 = 12 + 20 + 2 · 30: the original 12 vertices plus one for each original face andtwo for each original edge.

Section 2.12.1.1. P Q is the set of points R so that R is on the line

←→P Q and additionally R is P , R is Q

or R is between P and Q.

2.1.2. Undefined terms: 1, 2, 4, 5, 7. Assuming the terms used are defined, reasonable defini-tions are 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23. (We don’t define some of theseterms this way.)

2.1.3. (a) For a contradiction, suppose B(A)C . Then by axiom (i), C(A)B, which we showedwas false. Hence not B(A)C .

(b) Suppose for a contradiction that A = B. Then from A(B)C we’d have B(A)C andfrom each of these and axiom (i) C(B)A and C(A)B. These contradict axiom (ii),showing A = B.

2.1.5. (a) Let S and T be convex sets and P , Q and R points with P and R in S ∩ T . SoP, R ∈ S. Because S is convex, Q ∈ S. Similarly, Q ∈ T and so q ∈ S ∩ T .

2.1.7. (a) Use axiom (iii) with s = t .(b) Consider cases: If a → b and a → c, does b → c?

2.1.9. (a) Suppose for a contradiction that x is both an apple and an orange. By axiom (iv) xlikes some y. Now use axioms (ii), (iii) and (v).

(b) 4(c) 8


Section 2.2(2.2.1.)

2.2.2. (a) See Exercise 2.1.1(b) See the definition in Section 2.1.(c) Ray

−→P Q contains all points R on line

←→P Q so that in postulates 3 and 4 R corresponds

with a non-negative number, provided P corresponds with 0 and Q correspondswith 1.

(e) Use the idea of postulate 9.

2.2.3. (a) Use Euclid’s definition 10 for a right angle and show its measure must be 90◦.

2.2.5. (a) Use Exercise 2.2.3 part (a) and postulate 12.

2.2.6. By postulate 5 there are at least 2 points in the plane, say, O and X . By Exercise 2.2.5there is exactly one line k in this plane perpendicular to

←→O X through O . By postulate

3 there is a point Y on k so that the distance from O to Y equals the distance fromO to X . By postulate 4 we can choose this common distance to be 1. For any point Aof this plane, construct the parallel m to k and call its intersection with

←→O X the point

Ax . Define Ay similarly on k. Give A the coordinates (a, b), where a is the numbercorresponding to Ax and b is the number corresponding to Ay using postulate 4.

2.2.9. (a) The interior of �ABC is the intersection of the interiors of the angles ∠ABC ,∠BC A and ∠C AB.

(b) Yes. Use Exercise 2.1.5.

2.2.10. (a) 3 points, 3 lines.(b) Two points lying on two distinct lines contradict I-2.

2.2.12. (a) I-1, I-2, I-3 (first part), II-1, II-2, II-3, III-1, III-2, III-3, V-1, V-2.(b) On every line there exist two points, say P0 and P1 by I-3. By II-2 there exists a

point P2 with P1 between P0 and P2. By II-1 these are all distinct. Similarly thereare P−1 and P0.5 different from P0 and P1 with P0 between P−1 and P1 and withP0.5 between P0 and P1. By II-3 P2, P−1 and P0.5 must also differ from each other.

(c) In III-1 let A = P0, B = P1 and A′ = P1. Then B ′ = P2 for the ray from P1 notincluding P0. Repeat with A′ = P2.

2.2.14. (e) Note that we have ∠AB D ∼= ∠A′B ′C ′. Use SAS.

Section 2.32.3.1. Yes.

2.3.2. (a)−→AB = {B, C, E} = −→

AC .−→AG = {DFG}. Yes.

(c) I-1, I-2, first sentence of I-3, II-1, II-2, II-3.

2.3.3. (b) III-1, III-2, III-3, III-4.(c) Intersections: a point, two points, a line segment, two line segments, an entire circle.


2.3.5. (a) 5b, 8, 10, 21, 22.(b) If we suitably alter the definition of angle to avoid rays, only 1 and 3 (ii) fail.

2.3.6. (a) Euclidean line.(c) Use part (b) and: For (i) 3 points A, B, C with just the relations A(B)C , B(C)A

and C(B)A. For (ii) 3 points with all possible relations X (Y )Z . For (iii) the emptyset with no relations. For (iv) two points with no relations.

2.3.8. (a) a → b, a → c, b → c, b → d, c → d, c → e, d → e, d → a, e → a, e → b.(b) First model: ai → ai+1 and ai → ai+2, where addition is (mod 6). Second model:

ai → ai+1 and ai → ai−2, where addition and subtraction are (mod 6). In the firstmodel, a1 and a2 both beat a3. However, in the second model no two teams beat thesame team.

(e) With 6 teams, ai → ai+1, where addition is (mod 6) and a1 → a4, a2 → a4, a3 →a6, a4 → a6, a5 → a2, a6 → a2.

2.3.10. (a) An infinite chessboard.(b) A cube.

2.3.12. (a) On a set with at least 4 points, interpret a circle as any subset with exactly 3 points.(d) Without loss of generality the points are A, B, C, and D. For axioms (i) and (iii)

to hold there must be one circle with exactly three points on it. Without loss ofgenerality let {A, B, C} be a circle. Now D must be on a circle with any 2 of theothers, but we can’t have {A, B, C, D} as a circle because of axiom (ii). Hence eachof the sets {A, B, D}, {A, C, D}, and {B, C, D} must be circles.

2.3.14. (c) Use contradiction and axiom (iii).(e) Let Q be any point. By part (d) there is a line j not on Q. Use axiom (iv).(g) For at least two parallels to any line h: By parts (d), h has at least one point P not

on it. Of the three points on h at least one, say, Q is on a line with P . Let R be thethird point on the line with P and Q. Then h has parallels through P and R. Forexactly two parallels, suppose there were another and find a contradiction.

2.3.18. (a) II-2, II-4 (and II-4′), III-1, IV-1, V-1, V-2.(b) III-1, III-4, V-1, V-2.(c) V-2.

Section 3.13.1.1. slope: −a/b, intercept: −c/b

3.1.2. Perpendicular lines have the form bx − ay plus any constant.

3.1.3. (a) A quarter circle.

(c) x2 + y2 = (12

)2.

3.1.5. (b) If the vertices are C = (0, 0), A = (d, 0), and B = (e, f ), then cosC =e/√

e2 + f 2.


3.1.7. (a) {(x, y) : (x − a)2 + (y − b)2 = r2}.(c) {(x, y) : (x − a)2 + (y − b)2 < r2}.

3.1.9. Distance formula:√

k2(x1 − x2)2 + j2(y1 − y2)2,Circle: ((x − a)/k)2 + ((y − b)/j)2 = r2.

3.1.11. (c) u = ±b. Calculus gives a root at 0 as well.

3.1.13. (a) y = x2, y = x − x2, y = x4, y = ex , y = ln x(b) Convex functions: y = x2, y = x4, y = ex . Convex functions are “concave up” as

well as enclose convex regions. Concave functions: y = x − x2, y = lnx .(c) Consider the second derivative.

3.1.14. (a) The parallelogram law holds for the addition of vectors.(b) They are reflections in the real axis.(c) The distance a point is from the origin.

Section 3.23.2.4. (a) The line that is the perpendicular bisector of the 2 points.

(b) A great circle on the sphere.(c) The plane that is the perpendicular bisector of the 2 points. Parts (a) and (b) are the

intersections of this plane with their respective domains.(d) A point, two points, and a line perpendicular to the plane determined by the three

points.

3.2.6. (c) A hyperbola.

3.2.8. (a) Parabola.(b) Ellipse.(c) Point (2, 3), degenerate ellipse.(d) Hyperbola with asymptotes y = x and y = 2x .(e) Ellipse.

3.2.10. For k = 1, the locus is x = 0. For k = 1, it is x2 + y2 − (2k/(1 − k))x = 0.

3.2.11. (c) 2xy − 1 = 0, ac − b2 = −1 < 0, determinant is 1.(d) Consider, for example, the points (1, 0.5) and (1/

√2, 1/

√2).

(e) y = 0 and x = 0. Use limits.

3.2.12. (b) y − x = 0 and x = 0, x2 − xy + 1 = 0, ac − b2 = −1/4, determinant is −1/4.

Section 3.33.3.1. t = 0, ±1, vertical tangent.

3.3.2. Mirror reflection over the y-axis. It is rotated by an angle of c.

3.3.3. The y-axis becomes the tangent (θ = ±π/2) so the curve looks more like two adjacentcircles.


3.3.4. The algebra needed to find the intersection involves only the operations +, − , ×, and÷, so rational values remain rational.

3.3.5. (a) They give the same graph, but a point on the second curve goes twice as quickly.(d) They give the same graph, but a point traverses the second curve k times as quickly.

3.3.6. (b) As t → −1−, the point goes to (∞,−∞). As t → −1+, the point goes to (−∞,∞).(c) As t → ±∞, the values go to 0 and the point approaches the origin.

3.3.8. (a) (− sin t, cos t) and (−2 sin t, 2 cos t). The second is twice as long as the first becausea point on it is moving twice as quickly.

(b) (− sin t, 2 cos 2t). Both t = π/2 and t = 3π/2 give the same point (0, 0), wherethe curve crosses itself.

(c) (cos t − t sin t, sin t + t cos t). Directions: (1, 2π ) and (1, 4π ), lengths:√1 + 4π2 ≈ 6.36 and

√1 + 16π2 ≈ 12.61.

(d) Same point ( f (c), g(c)), length of second is k times as large as for the first.

3.3.10. (a) It is a sort of spiral on the surface of a cone.(b) It is a sort of spiral on the surface of a sphere.

3.3.12. (d) x4 + 2x2 y2 + y2 − x2 + y2 = 0.(e) x4 + 2x2 y2 + y2 − 2xy = 0.

3.3.16. (a) Center of square(b) Center of square, (a, a, 0.5 − a, 0.5 − a)(c) (a, a,−a, 1 − a).(d) c = d.(e) (−4,−3).

3.3.19. (a) (x, 90◦), (x,−90◦).(b) Circles of latitude, great circles of longitude.(c) Spirals from the south pole to the north pole. A given spiral intersects each circle

of latitude at the same angle.(d) I-1, I-3, I-4.

3.3.21. (a) They have the same x- or y-coordinate. Taxicab distance is greater by at most afactor of

√2.

(b) Squares at a 45◦ angle to the axes.

Section 3.43.4.3. a = p/2 − q + r/2, b = −3p/2 + 2q − r/2, c = p

3.4.4. (a) 3x3 − 5x2 + x + 1.(b) For example, x(t) = 3t − 6t2 + 4t3 and y(t) = 1 + 3t − 9t2 + 5t3.

3.4.5. (a) x(t) = 3t − 3t2, y(t) = 3t − 6t2 + 4t3.(b) x ′(t) = 3 − 6t , y′(t) = 3 − 12t + 12t2. x ′(0.5) = 0 = y′(0.5).(c) The point has velocity 0 at t = 0.5, so it can switch directions smoothly then.


3.4.7. (b) (2, 2), y = 3(c) For example, use P0 = (4, 0), P1 = (5, 0), P2 = (4, 1), and P3 = (5, 3).(d) For example, use P0 = (5, 3), P1 = (6, 5), P2 = (−1,−2), and P3(0, 0).

3.4.9. (a) T3(x) = x − x3/3!(b) T5(x) = x − x3/3! + x5/5!, etc.(c) sin′(−π ) = −1, sin′(0) = 1, sin′(π ) = −1. Left half: P0 = (−π, 0), P1 = (−π +

a,−a), P2 = (−b,−b), and P3 = (0, 0), where a > 0 and b > 0. Right half: P0 =(0, 0), P1 = (a, a), P2 = (π − b, b), and P3 = (π, 0), where a > 0 and b > 0.

3.4.10. (a) (n3 + 6n2 + 6n + 1)/3(b) 36(c) 36w

Section 3.53.5.1. At α = 1, the point is −→s and at α = 0 the point is

−→t . Since it is a linear combination

of 1 variable, we get a line. At α = 0 = β, we are at−→t , at α = 0, β = 1 we are at −→s

and at α = 1, β = 0, we are at −→r . A linear combination of two variables gives a plane.

3.5.3. They differ in just one coordinate from (1, 1, 1, 1). They differ in 2, 3 or all 4 coordinates.

3.5.4.√

2 for edges and 2 for diagonals

3.5.5. (a) α(1, 2, 4) + β(2, 0, 0).(b) 2y − z = 0.

3.5.7. (a) α(1,−2, 1).(b) They intersect at (3, 3, 3).(c) skew.(d) −→w is a scalar multiple of −→u .

3.5.9. (a) Draw mi parallel to ki with mi going through Pi .(b) 7.(c) (3, 5, 3).(d) Neither.(e) both projections are parallel. The projections intersect at points with the same

common coordinate. The projections intersect at points with different commoncoordinates.

3.5.13. (a) unit sphere(b) ellipsoid(c) paraboloid(d) hyperboloid of one sheet.

3.5.15. (c) (1/√

5, 0,−2/√

5) and (−1/√

5, 0, 2/√

5).(d) x − 2z = 0.

3.5.17. C = arccos(1/3) ≈ 1.23096 radians or 70.5288◦.


3.5.19. (a) Let e/v represent edges per vertex, and so on.

Dimension 1 2 3 4

e/v 1 2 3 4f/v 0 1 3 6c/v 0 0 1 4f/e 0 1 2 3c/e 0 0 1 3c/f 0 0 1 2v 2 4 8 16e 1 4 12 32f 0 1 6 24c 0 0 1 8

(b) For dimension n, e/v = n, f/v = n(n − 1)/2, f/e = n − 1, c/e = (n − 1)(n −2)/2, c/ f = n − 2, v = 2n , e = n2n−1.

3.5.21.

(a)

Dimension 1 2 3 4

e/v 1 2 3 4f/v 0 1 3 6c/v 0 0 1 4f/e 0 1 2 4c/e 0 0 1 4c/f 0 0 1 2v 2 3 4 5e 1 3 6 10f 0 1 4 10c 0 0 1 5

(b) For dimension n, if n is big enough, e/v = n, f/v = n(n − 1)/2, f/e = n − 1,c/e = (n − 1)(n − 2)/2, c/ f = n − 2, v = n + 1, e = (n + 1)n/2.

3.5.23. (a) tetrahedron {3, 3}, cube {4, 3}, octahedron {3, 4}, dodecahedron {5, 3}, icosahedron{3, 5}.

Section 4.14.1.1. Area is proportional to how far the angle sum falls short of π in hyperbolic geometry

and how far it exceeds π in spherical geometry. Replace π by 180◦.

4.1.3. Infinitely many. See Theorem 4.2.1.

4.1.4. (a) Intersections at (0.8,±0.6).(d) angles of 69.4◦, 18.4◦, and 18.4◦.

4.1.5. II-4, III-4, and IV-1.


4.1.6. (a) y =√

5 − (x − 3)2.(d) For example, y =

√100 + (c + 4)2 − (x − c)2, for −11.5 ≤ c ≤ 3.5.

(f) (5, 0).(g) They are tangent.

4.1.8. (a) intersections: (±3.2, 2.4).(b) (±5, 3), ≈ 1.03 radians ≈ 59◦.

Section 4.24.2.3. (a) All can occur.

(b) There are 4 common sensed parallels.

4.2.5. Proof. By definition, the angle of parallelism is the smaller of the two angles a sensedparallel makes with the perpendicular and so is less than or equal to 90◦. However,if the angle of parallelism were 90◦, the two sensed parallels would be the same line,contradicting the characteristic axiom.

By I-28 two lines with a common perpendicular can’t intersect and so are ultraparal-lel or sensed parallel. Since the angle of parallelism is acute, lines with a commonperpendicular are ultraparallel. �

4.2.7. Euclid’s parallel lines are ultraparallel in hyperbolic geometry.

4.2.9. ∠C B� is bigger than ∠C A�.

4.2.10. It is less than 360◦.

Section 4.34.3.1. and 4.3.2. rectangle.

4.3.3. The angle sum of the “summit angles” of a rectangle is 180◦.

4.3.4. Use symmetry.

4.3.7. (a) Assume that X is on−→DC with DX ∼= B A. Then AB DX is a Saccheri quadrilateral.

Theorem 4.3.1 and Corollary 4.2.3 show←→AX is ultraparallel to B D. Because

←→AC

is a sensed parallel it must be under←→AX .

(b) Part (a) implies that C is between D and X . So the perpendiculars from one sensedparallel

←→AC to another

←→B D get shorter as they “approach” the omega point.

4.3.11. The angle sum of an n-gon is less than 180(n − 2)◦. Proof. For n = 3 use Theorem3.3.3. For the induction step suppose for n = k the angle sum is less than 180(k − 2)◦

and we have a convex polygon with n = k + 1 sides. Divide this polygon into a triangleand a k-gon and apply the induction hypothesis and Theorem 3.3.3. This proof holdsfor nonconvex polygons, although one must show one can always divide the k + 1-goninto a triangle and an k-gon.


Section 4.44.4.1. Consider three congruent triangles �ABC , �B AD and �E FG, where �E FG is dis-

joint from the other two and these other two overlap only on the side AB. By postulate 19R = �ABC ∪ �E FG has the same area as the quadrilateral AC B D. However, R hasone more side E F then AC B D has since we don’t include AB twice in the quadrilateral.So the side E F must have area 0. Also, E F is made of the union of the point E and therest of the segment. Since the union has area 0, so does the point E .

4.4.5. The maximum defect a triangle can have is 180◦, so K = k × 180◦ in Theorem 4.1.1.

4.4.6. The area of �AB1 Bi is finite even as i → ∞, so the area of �ABi Bi+1 approaches 0 asi→ ∞.

4.4.8. Kn = (n − 2)K .

Section 4.54.5.1. See Exercise 4.5.6. (a).

4.5.4. They are congruent right angles.

4.5.5. (a) From the proof of Theorem 4.5.1 common perpendiculars have the same length.Apply SSS.

(b) Use Theorem 4.5.1 and SAS.

4.5.6. (a) Consider two doubly right triangles with different included sides.

4.5.7. In spherical geometry, the corresponding theorem says that all lines perpendicular to agiven line intersect in two antipodal points. Otherwise, the same argument applies.

4.5.9. Mimic the proof of Theorem 3.5.2.

4.5.10. (a) Half of a sphere, 2πr2.

Section 5.15.1.1. Circles with centers on y = 0, the line of reflection.

5.1.2. Solve x = y + 2 and y = 2 − x to find the only solution. All points on these circles arethe same distance from the fixed point. No line is stable. ρ does not switch orientation.

5.1.3. Mirror reflection over y = 2 − x . No, it is a reflection over y = x − 2.

5.1.5. (a) Both formulas give functions. To show one-to-one and onto, solve y = α(x) andy = β(x) for x to see that the choice of x in each is unique.

(b) (2x − 1)3, 2x3 − 1.(c) For α, 0, ±1; for β, 0.5.(d) For α, 1; for β, 3

√0.5.

(e) 3√

x , 0.5x + 0.5.(g) 3

√0.5x + 0.5, 0.5 3

√x + 0.5.


5.1.7. (a) Rotation of 180◦ around (2, 3).(b) Solve (4 − x, 6 − y) = (a, b) to find the unique solution.(c) inverse is θ , fixed point is (2, 3) and stable lines are y = mx + 3 − 2m and x = 2.

5.1.9. (a) Rotation of 180◦ around (1, 2).(b) Mirror reflection over y = x + 1.(c) Dilation by a factor of 2 about (0, 0).(d) Dilation by a factor of 0.5 about (2,−2).

Section 5.25.2.1. Isometries preserve distance and a circle is the set of points a fixed distance from its

center.

5.2.2. Each point of k is fixed, so the entire line is fixed and hence stable. If m ⊥ k, each pointon m goes to another point on m, so m is stable.

5.2.3. The identity is a translation of length 0 and a rotation of an angle of 0.

5.2.4. If the translation part of the glide reflection is the identity, we have a mirror reflection.Points on the line of reflection slide along the line and points off of this line switch sides.So no point is fixed by a glide reflection that isn’t a mirror reflection. The sliding aspectof a glide reflection means only lines parallel to the line of reflection have a chanceof being stable. But other lines parallel to this line switch sides from the reflectionpart.

5.2.5. (a) (0, 1), (−3, 4), (−y − 1, x + 2), rotation of 90◦ around (−1.5, 0.5).(b) (−3, 1), (0, 4), (x − 2, y + 2), translation of 2 to the left and 2 up.(c) (−2, 0), (1, 3), (y − 1, x + 1), mirror reflection over y = x + 1.(d) (−2, 2), (1,−1), (y − 1,−x + 1), rotation of 270◦ around (0, 1).

5.2.7. (a) (0, 2), (0, 3), (1, 4).(b) (4, 2), (5, 2), (6, 3).(c) τ ◦ μ = μ ◦ τ

(d) composition is the translation τ ◦ τ taking (x, y) to (x + 4, y + 4).

5.2.9. (a) μx (x, y) = (x,−y), μy(x, y) = (−x, y), μ3(x, y) = (−y + 1,−x + 1).(b) μy ◦ μx (x, y) = (−x,−y), rotation of 180◦ around (0, 0).(c) μ3 ◦ μx (x, y) = (y + 1,−x + 1), a rotation of 270◦ around (1, 0). μx ◦ μ3(x, y) =

(−y + 1, x − 1), a rotation of 90◦ around (1, 0).(d) μ3 ◦ μy ◦ μx (x, y) = (y + 1, x + 1), glide reflection along y = x .

5.2.11. (a) μm fixes points on m, namely A and B. �AC D ∼= �μk ◦ μm(A)C D so Aμk ◦μm(A) = 2AC = 2d. Similarly for B.

(b) Use SSS. Similar to (a).(c) Compose them.

5.2.13. Note that d(P, Q) + d(Q, R) = d(P, R) just when Q is between P and R and thatisometries preserve distance.


5.2.15. τ 1 ◦ τ 2 = τ 2 ◦ τ 1. Let P and Q be any points. Then Pτ 2(P)‖Qτ 2(Q),τ 2(P)τ 1 ◦ τ 2(P)‖τ 2(Q)τ 1 ◦ τ 2(Q), and ∠τ 1 ◦ τ 2(P)τ 2(P)P ∼= ∠τ 1 ◦ τ 2(Q)τ 2(Q)Q.Because τ 1 and τ 2 are translations, corresponding sides are congruent. So by SASPτ 1 ◦ τ 2(P) = Qτ 1 ◦ τ 2(Q). Thus P Qτ 1 ◦ τ 2(Q)τ 1 ◦ τ 2(P) is a parallelogram andτ 1 ◦ τ 2 is a translation.

5.2.19. (a) V consists of translations, 180◦ rotations, vertical and horizontal mirror and glidereflections. Proof. Let l be a vertical line. ι(l) = l is vertical, so ι ∈ V. If α, β ∈ V,β(l) and so α(β(l)) are vertical, so α ◦ β ∈ V. Suppose α ∈ V and k the verticalline so that α(k) = l. Then α−1(l) = k, so α−1 ∈ V.

Section 5.3

5.3.3. The matrix

[a bc d

]becomes

⎡⎣a b 0c d 00 0 1

⎤⎦.

5.3.8. (a)

⎡⎣1 0 b0 1 00 0 1

⎤⎦.

(b)

⎡⎣1 0 00 −1 00 0 1

⎤⎦.

(c)

⎡⎣1 0 b0 −1 00 0 1

⎤⎦.

(d)

⎡⎣0 1 01 0 00 0 1

⎤⎦.

(e)

⎡⎣0 1 b1 0 b0 0 1

⎤⎦.

5.3.9. (a)

⎡⎣cos(30) − sin(30) 0sin(30) cos(30) 0

0 0 1

⎤⎦.

(b)

⎡⎣cos(30) − sin(30) 3.5 − √3

sin(30) cos(30) 2 − 3√

3/20 0 1

⎤⎦.

5.3.10. A is a rotation, B is a translation, C is a mirror reflection, D is a glide reflection, E is arotation.

5.3.12. (b) c = 2 f .

5.3.16. (a)

⎡⎣−1 0 00 −1 00 0 1

⎤⎦ .


(b)

⎡⎣−1 0 2u0 −1 2v

0 0 1

⎤⎦ .

(c) A direct isometry, a rotation of 180◦.(d) Lines through (u, v, 1).(e) Translations twice the distance between the centers. You get the inverse translation.

5.3.20. a = ±1, b = 0, c free, d = 0, e = ±1, f = 0.

Section 5.45.4.2. The inverse of a contraction mapping is not a contraction mapping.

5.4.3. (a) (−2,−4, 1), [m,−1, 2m − 4], and [1, 0, 2].(b) (2, 2, 1), no stable lines.

5.4.4. direct:

⎡⎣0 −4 −114 0 −70 0 1

⎤⎦ and indirect:

⎡⎣0 4 134 0 −70 0 1

⎤⎦.

5.4.5. (a) (1, 0, 1), (0, 2, 1), (−4, 0, 1), and (0,−8, 1). Spiral.(b) (1, 1, 1), (−2, 2, 1), and (−4,−4, 1). Yes.(c) M is a rotation by θ and a scaling by r , S is a rotation by θ/2 and a scaling by

√r .

(d) For C use a rotation by θ/3 and a scaling by 3√

r and for N use a rotation by θ/nand a scaling by n

√r .

5.4.7. (a)

⎡⎣ 2 2 −2−2 2 40 0 1

⎤⎦ and

⎡⎣ 2 −2 −2−2 −2 40 0 1

⎤⎦.

(b)

⎡⎣−3 4 2−4 −3 110 0 1

⎤⎦ and

⎡⎣−3 −4 10−4 3 50 0 1

⎤⎦.

5.4.13. Follow Exercise 5.3.17 with the following substitutions.

(a) a = r cos θ , e = −r cos θ , and b = d = r sin θ.

(c) and (d) In (a) set a = e = r cos θ , b = −r sin θ , and d = r sin θ .

5.4.15. (a) fixed points: (x, x, 1), stable line: [1,−1, 0].(b) (0.5, 0.5, 1), [−1,−1, 1].

5.4.20 (c) This IFS fractal is the part of Figure 5.32 on the x-axis.

5.4.21. The 4 corners of the square, (0, 0, 1), (1, 0, 1), (0, 1, 1), and (1, 1, 1), must go to points

in that square. For

⎡⎣a b cd e f0 0 1

⎤⎦, the following must be between 0 and 1: c, a + c, b + c,

a + b + c, f , d + f , e + f , and d + e + f .


Section 5.55.5.3. No essential change for translation from Section 5.2. For glide reflection, replace “line

k” with “plane P.”

5.5.5. No essential change for translation from Section 5.2. An (n + 1) × (n + 1) matrix with1s on the main diagonal, any numbers in the far right column (except for a 1 in the lowerright corner) and 0s elsewhere.

5.5.7. (a) cos(180◦) = −1 gives the upper two diagonal entries. Note that (0, 0, z, 1) is mappedto itself.

(b)

⎡⎢⎢⎣1 0 0 00 −1 0 00 0 −1 00 0 0 1

⎤⎥⎥⎦,

⎡⎢⎢⎣−1 0 0 00 1 0 00 0 −1 00 0 0 1

⎤⎥⎥⎦, rotation of 180◦ around the z-axis.

The other of these two matrices.(d) Rotations of 120◦ around opposite vertices of the cube.

(e) For example,

⎡⎢⎢⎣1 0 0 00 0 −1 00 1 0 00 0 0 1

⎤⎥⎥⎦ is a 90◦ rotation around the x-axis and

⎡⎢⎢⎣0 1 0 00 0 1 01 0 0 00 0 0 1

⎤⎥⎥⎦ is a 120◦ rotation around the axis through (1, 1, 1, 1) and

(−1,−1,−1, 1).

(f)

⎡⎢⎢⎣1 0 0 00 cos θ − sin θ 00 sin θ cos θ 00 0 0 1

⎤⎥⎥⎦.

5.5.9. (a)

⎡⎢⎢⎣cos θ 0 − sin θ 0

0 1 0 ksin θ 0 cos θ 0

0 0 0 1

⎤⎥⎥⎦ (or with the − on the other sin θ , depending on the

orientation of the axes).(b) Yes. These rotation and translation matrices commute.(c) a translation of 6 in the y-direction.

5.5.11. (a) (x, y,−x + 2, 1) which are points on the plane x + z − 2 = 0 or [1, 0, 1,−2], thefixed plane and the family [1, b,−1, d], mirror reflection.

(c) (x, x, x, 1), which are points on a line, family of parallel planes [1, 1, 1, d], rotationof 120◦.

5.5.13. (a) A 4 × 4 affine matrix where the upper left 3 × 3 submatrix is r times an orthogonalmatrix.

(b) The dilation affects only the upper left 3 × 3 submatrix, multiplying each of itsentries by r .


(c) r3, the ratio of the volumes of the image of an object and the origin.(d) (0, 0,−2, 1), [1,−1, 1, 2].

Section 5.65.6.1. Circle x2 + y2 = r2 inverts to x2 + y2 = 1/r2.

5.6.2. d(O, P) · d(O, νc(P)) = d(O, νc(P)) · d(O, P).

5.6.3. Case 1. r = 0 gives q = ps/r and f (z) = p. Case 2. r = 0. Either p = 0 or s = 0.For p = 0, f (z) = q/s. For s = 0, f is undefined. As z → s/r , the denominator off (z) goes to 0. As z → ∞ the limit makes the constant terms irrelevant. To convertthe formulas of Theorem 5.6.4, first ignore the complex conjugate to rewrite r2/(z −w) + w as (r2 + wz − w2)/(z − w) = (wz + r2 − w2)/(z − w), which is a Mobiustransformation. Now apply complex conjugates.

5.6.4. ν(A) = (0,−2), ν(B) = (1, 1), ν(C) = C , ν(D) = (2, 4).

5.6.5. (a) νD(1, 0) = (1, 0), νC (νD(1, 0)) = (4, 0), νC (1, 0) = (4, 0) and νD(νC (1, 0)) =(0.25, 0). No.

(b) νD(0, 3) = (0, 1/3), νC (νD(0, 3)) = (0, 12), νC (0, 3) = (0, 4/3) and νD(νC

(0, 3)) = (0, 3/4).

5.6.7. (a) Center (0, 0) and radius 2, center (−1, 0) and radius√

10.(b) Center must be on the line through (4, 0) and (1, 0). All x-axis except points between

x = 1 and x = 4.(c) r = √

c2 − 5c + 4, where c is the x-coordinate of the center.

5.6.9. (a) x = 1.6 and x = 2.5.

5.6.12. (a) Translation.(b) If r > 0, a dilation by a ratio of r ; if r < 0, a dilation and a rotation of 180◦. A

rotation. A composition of a dilation and a rotation.(c) A mirror reflection over the real axis.(d) Use parts (a), (b), and (c) and composition.

5.6.13. (a) 2/(z + 2i) − 2i .(b) 4/(z + 3i) − 3i .(c) E has center −1.5i and radius 0.5.(e) −0.6i .

5.6.15. (b) The equator. Mirror reflection over the equator.

Section 6.16.1.1. Jelly fish drift in any (horizontal) direction, so there is no natural front/back axis. Single

cellular organisms suspended in water can move in all directions, so spherical symmetrymay be advantageous.

6.1.2. Rotations of multiples of 60◦ and mirror reflections over 6 lines through the center.


6.1.3. (a) No symmetry: F, G, J, L, P, Q, R; Vertical symmetry: A, M, T, U, V, W, Y; Horizontalsymmetry: B, C, D, E, K; Rotational symmetry: N, S, Z; All of the above symmetries:H, I, O, X.

(b) Some examples: Vertical (written vertically): TOMATO; Horizontal: CHOICE,BEDECK; Rotation: MOW.

(c) A familiar example: MADAM, I’M ADAM. In a palindrome we switch the orderof the letters, but we don’t reflect the individual letters.

6.1.5. Iranian: rotations of 0◦, 120◦, and 240◦ and 3 mirror reflections. Byzantine: rotationsof 0◦, 90◦, 180◦, and 270◦ and 4 mirror reflections. Afghani: rotations of 0◦, 60◦, 120◦,180◦, 240◦, and 300◦.

6.1.7. Left figure: color preserving include rotations of multiples of 120◦ and 3 mirror reflec-tions, including vertical; color switching include rotations of 60◦, 180◦ and 300◦ and 3mirror reflections, including horizontal.

6.1.9. (a) Mexican: translations, 180◦ rotations, vertical mirror reflections, and horizontalglide reflections. Chinese: translations, 180◦ rotations, vertical and horizontal mirrorreflections, and horizontal glide reflections.

6.1.11. (a) 25%, 25%. Use rotations.(b) 33.3%, 33.3%.

Section 6.26.2.1. D6, C4.

6.2.2. Dn .

6.2.3. (a) Gothic: C3, Islamic: D10, Gothic: C2.(b) equilateral: D3, isosceles: D1, scalene: C1.(c) General: C1, parallelogram: C2, kite and isosceles trapezoid: D1, rectangle and

rhombus: D2, and square: D4.

6.2.5. (a) Rotations of 0◦, 90◦, 180◦, and 270◦. Yes, C4.(b) Rotations of 45◦, 135◦, 225◦, and 315◦. No, closure and identity fail.(c) Yes, C8.

6.2.6. (a) Triangular prism: 12; square prism: 16.(b) 4n.(c) Tetrahedron: 24, cube: 48, octahedron: 48, dodecahedron: 120, and icosahedron:

120.

Section 6.36.3.2. If τ is the smallest translation to the right, all other translations are of the form τ n , for

some n. If n > 0, τ n represents a translation of n to the right. If n < 0, it is a translationof n to the left. τ 0 is the identity.


6.3.3. (a) p2mm.(b) p211.(c) p11g.(d) p2mg.(e) p11m.(f) p1m1.

6.3.5. (a) pg.(b) p2.(c) p3.(d) cm.(e) p6m.(f) cmm.

6.3.9. Use a sequence of numbers to denote the sequence of n-gons around any vertex. Thenthe options are (a) {6, 6, 6}, {4, 4, 4, 4}, and {3, 3, 3, 3, 3, 3}.

6.3.11. p4g, pgg.

6.3.15. (a) τ n(x, y) = (x + n, y).(b) ν(x, y) = (x,−y).(d) μd (x, y) = (2d − x, y).

Section 6.46.4.1. A cube is a special type of a square prism, which is a special type of a rectangular box.

A shape has all the symmetries of a more general shape. 8,16,48. Rotations of 180◦

around the x-, y-, and z-axes, mirror reflections over the xy-, xz-, and yz-planes, theidentity and the central symmetry.

6.4.3. 4n. Dnh . All symmetries of these prisms are symmetries of a prism with a regular 2n-gonas a base.

6.4.5. ζ =⎡⎣−1 0 0

0 −1 00 0 −1

⎤⎦.

6.4.6. (a) Identity, 9 rotations of 90◦, 180◦, and 270◦ around the centers of opposite faces, 8rotations of 120◦ and 240◦ around opposite vertices, and 6 rotations of 180◦ aroundcenters of opposite edges; mirror reflections over 6 planes through opposite edgesand 3 planes between opposite faces. Switch roles of faces and vertices to matchcube and octahedron symmetries. 15 rotatory reflections.

6.4.11. The groups T, W, and P each have one of the Archimedean solids. The groups W andP each have 5 of them.

Section 6.56.5.1. p6m.

6.5.2. 180◦ rotation through a vertical or horizontal axis going midway between adjacentatoms.


6.5.4. (a) D2.(b) D1, D1, and C2.

6.5.7. (a) D6.(b) D1, D1, and D2 ( for cyclic arrangement ClHHClHH).(c) D2/D1 (for HHHClClCl), D1/C1 (for HHClHClCl), and D6/D3 (for HClHClHCl).

6.5.9. (a) p31m.(b) p6m.

6.5.12. (a) 0.9945.(b) 0.2273.

Section 6.66.6.1. The matrices have a scaling factor of r = 1/3, which corresponds with Koch’s method.

We need four matrices because there are four smaller copies of the original.

6.6.6. (a) ln 2/ ln 2 = 1.(b) ln 3/ ln 2 ≈ 1.585.(c) ln 2/ ln 3 ≈ 0.631.(d) ln 5/ ln 3 ≈ 1.465.

6.6.8. (a) ln 6/ ln 2 ≈ 2.585.(b) ln 13/ ln 3 ≈ 2.335.(c) ln 26/ ln 3 ≈ 2.966.

6.6.9. (a) 2n−1/(3n).(b) 1/3 + 2/9 + 4/27 + · · · = 1. Length left is 0.(d) 2n−1/(kn), 1/k + 2/k2 + 4/k3 + · · · = 1/(k − 2). Length left is (k − 3)/(k − 2).

6.6.12. (a) Estimates may range from d ≈ 1.38 to d ≈ 1.51. (The number of segments ofvarious lengths can easily vary, significantly altering the estimates of d.)

Section 7.17.1.2. Larger.

7.1.5. (a) Let T be the intersection of←→P P ′ and

←→Q Q′.

7.1.8. 0.6. Different constructions should give approximately the same point.

7.1.10. (a) In the definition of a harmonic set of points, interchange point and line and replacequadrangle with quadrilateral.

(b) m⊥l, m is the other bisector of j and k.

7.1.11. (a) T1 = (0, a), T2 = ( a2 , a

2 ).(d) y = (1 − 2a)x/(2 − a) + a/(2 − a).


7.1.14. (a) V , E , and D.

7.1.15. (a) y = −w.(b) midpoint. at infinity

Section 7.27.2.1. Define X−n to be the point such that H (X X0, Xn X−n).

7.2.3. Two distinct lines have exactly one point on them. There are at least four lines withno three on the same point. Every two distinct points have at least one line on bothpoints.

7.2.4. Dual of 7.2.1: Two distinct points have exactly one line on both points. Every point hasat least four distinct lines on it. If H (pq, rs), then H (pq, sr ). 7.2.2: If pq//rs, thenqp//rs, qp//sr , rs//qp, sr//pq, and sr//qp. If a, b, c, and d are distinct concurrentlines, then exactly one of the following holds: ab//cd, ac//bd, or ad//bc. 7.2.3: If x p

and xq are determined and p = q, then x p and xq are distinct.

7.2.5. Each individual perspectivity preserves these properties, so their composition does.

7.2.6. A perspectivity with respect to a line k is a mapping of the lines ui on one point to thelines vi on another point so that vi is the image of ui if and only if ui , vi and k areconcurrent. 7.2.6: A perspectivity preserves harmonic sets of lines and the relation ofseparation. That is, if a perspectivity from k maps the concurrent lines p, q, r , and sto the concurrent lines p′, q ′, r ′, and s ′ and H (pq, rs), then H (p′q ′, r ′s ′). Similarly,if pq//rs, then p′q ′//r ′s ′. 7.2.7: A projectivity of the lines on a point is completelydetermined by three lines on the original point and their images.

7.2.7. (a) Xa is between Xb and Xc.(b) Xa is the midpoint of Xb and Xc.(c) j and k form two pairs of vertical angles. l is in one pair and m is in the other

pair.

7.2.8. (b) m = −1.(e) Yes.

7.2.12. (a) Given the construction for H (0 1, a a2a−1 ), draw parallel lines to these through

the corresponding points 0, k, ak and ak2a−1 . By Theorem 1.4.1 the corresponding

triangles are similar. Hence the sixth new line must go through ak2a−1 .

(b) In part (a) let k = 1n+1 and a = n+1

n .

7.2.15. (a) s < 0 or 1 < s. 0 < t < r < 1. Yes because t satisfies the conditions r satisfies.

7.2.17. (b) For example, (vii) becomes If pq//rs, then p, q, r , and s are distinct, concurrentlines, pq//sr and rs//pq.

7.2.18. (b) In a complete quadrilateral with lines t1, t2, t3, and t4, the diagonal lines are the three

lines on the “opposite” points:←−−−−−−→(t1 · t2)(t3 · t4),

←−−−−−−−−→(t1 · t3) · (t2 · t4), and

←−−−−−−−→(t1 · t4)(t2 · t3).


Section 7.37.3.1. (x, 2x, 0), [0, 0, c].

7.3.2. Two points are two Euclidean lines through the origin O , which determine a uniqueEuclidean plane through O , i.e. a line. Consider the points of the x-axis, the y-axis,the z-axis, and the line through O and (1, 1, 1). Any two Euclidean planes through Ointersect in a Euclidean line.

7.3.3. Note that three of the four terms in R(a, b, c, d) are negative.

7.3.6. y = x2 and y = 1/x are functions. [1, 0 − b] intersects x2 − yz = 0 in (b, b2, 1) and(0, 1, 0). It intersects xy − z2 = 0 in (b, 1/b, 1) and (0, 1, 0). (0, 1, 0). [0, 1, 0]. (1, 0, 0).

7.3.7. (±1, 1) and (±√2, 2).

7.3.8. (a) (1, m, 0). Parallel lines “meet” at infinity.(b) [m,−1,−mp + q]. This is the line through (p, q, 1) with slope m.

7.3.10. (b) {A, C} separate {B, E} and {D, E}; {A, D} separate {B, C} and {B, E}; {D, E}separate {B, C}.

(c) H (AC, DE).(d) P = B, Q = C , R = E , S = D, T = A.

7.3.12. (a) 1 < x < 2, x = 1.6.

7.3.14. (a) (4, 2, 1), (−12, 0, 1), (12, 12, 1).(b) For A = (4, 2, 1) and B = (−12, 0, 1), k = [1,−8, 12], C = (24/7, 24/7, 1), and

D = (7.2, 2.4, 1).

7.3.15. (a) S = (10,−9, 1), T = (1, 3, 1), U = (2.5, 1, 1), line: [−4,−3, 13].

7.3.20. (a) −xz + xy + yz = 0, y − z = 0, and x + z = 0. (1, 0, 0) and (0, 1, 0).

7.3.22. (b) In nonhomogeneous coordinates: y = 1, y = −1, x = 1, x = −1, y = x , and y =−x .

(c) Euclidean circle with center (0, 0, 1) and radius√

2.(e) Euclidean hyperbola.

7.3.23. (a) −x2 + yz = 0.(b) (x0, x2

0 , 1), y = 2x0x − x20 , [2x0,−1,−x2

0 ].

Section 7.4

7.4.2. (a)

[a b0 d

], where a = 0;

[a 0c d

], where d = 0; and

[a bc d

], where a + b = c + d.

(b)

⎡⎣a b c0 e f0 h i

⎤⎦, where a = 0;

⎡⎣a 0 cd e fg 0 i

⎤⎦, where e = 0;

⎡⎣a b 0d e 0g h i

⎤⎦, where i = 0;

and

⎡⎣a b cd e fg h i

⎤⎦, where a + b + c = d + e + f = g + h + i .


7.4.3. (a)

[3 01 2

], 2

3 goes to 34 and H (0 1, 3

234 ).

7.4.4. (a)

⎡⎣3 −1 04 4 21 1 2

⎤⎦.

7.4.7. (a)

⎡⎣a 0 00 a 00 0 i

⎤⎦. Dilation fixing the origin.

(b) The corresponding sides of �ABC and �A′B ′C ′ are parallel.

7.4.10. (a)

⎡⎣2 0 00 −0.5 10 −0.5 −1

⎤⎦, 4x2 − 2yz = 0.

(c) y = 2x2, a parabola.

7.4.12. (a)

⎡⎣ 1 0 00 1 0

−w 0 1

⎤⎦,

⎡⎣1 0 −w

0 1 00 0 1

⎤⎦, and (1 + 2w)x2 + y2 − 2xz = 0.

(b) For w = 0.5, 2x2 + y2 − 2x = 0.(d) For w > −0.5 and w = 0, the image is a Euclidean ellipse.(f) For w = 0.5, [2, 0,−2] and [0.5, 1,−1].

7.4.13. (a) [−4,−3, 5] and [0,−1,−1].

7.4.14. (a) tangent at A: [1,−1, 0], at B : [−1,−1, 0], where A = (1, 1, 1), B = (−1, 1, 1),k = [0,−1, 1].

(b) S = (0, 0.5, 1), T = (0, 1, 0), Q = (0, 1, 1).

Section 7.57.5.1. Note that Xx · X−x = (1 − x2)I .

7.5.2. Compare h-inner product with the regular inner product.

7.5.3. Consider the bottom row.

7.5.6. (a) S = (−0.8, 0.6, 1), T = (0.8, 0.6, 1), dH ((0, 0, 1), (0.5, 0, 1)) = |log(1/3)| ≈0.477, and dH ((0, 0.6, 1), (0.5, 0.6, 1)) = |log(3/13)| ≈ 0.637.

7.5.8. (d) The x-axis. Rotations of 180◦.

7.5.9. Yb =⎡⎣√

1 − b2 0 00 1 b0 b 1

⎤⎦. Transposes of each other.


Section 7.67.6.1. h-inner product of two vectors (p, q, r, s) and (t, u, v, w) is (p, q, r, s) ·h (t, u, v, w) =

pt + qu + rv − sw. They are h-orthogonal if this product is 0. The h-length of(p, q, r, s) is (p, q, r, s) ·h (p, q, r, s).

7.6.2. (a) [0, 0, 0, 1].(b) [1, 1, 1,−5].

7.6.3. (a) Four distinct points (a, b, c, d), (e, f, g, h), (i, j, k, l), and (m, n, o, p), in P3 are

coplanar if and only if the determinant

∣∣∣∣∣∣∣∣a e i mb f j nc g k od h l p

∣∣∣∣∣∣∣∣ = 0.

7.6.4. (a)

⎡⎢⎢⎣2 0 0 04 −1 0 00 −2 2 00 0 4 1

⎤⎥⎥⎦.

7.6.5. (a) By a quadric surface we mean a symmetric invertible 3 × 3 matrix. Two suchmatrices represent the same conic if and only if one is the multiple of the otherby some real number λ = 0. A point P is on a quadric surface S if and only ifPT S P = 0.

(b) x2 + y2 + z2 − t2 = 0,

⎡⎢⎢⎣1 0 0 00 1 0 00 0 1 00 0 0 −1

⎤⎥⎥⎦.

7.6.7. (a) n ≤ 3.(b) n ≤ 4.

7.6.8. (a) The change to −0.25 brings the vanishing points closer to the origin, moving theapparent position of the viewer closer to the cube. Vx = (2, 1.73), Vy = (−3.46, 1),and Vz = (0,−3.46).

7.6.12. (a)

⎡⎣1 0 00 1 02 2 −1

⎤⎦.

(b) The image of (x, 0, 1) is (x, 0, 2x − 1) = (x/(2x − 1), 0, 1), which is on the x-axison either side of [0, 1]. Note that for 0 < x < 1/2, 2x − 1 < 0. The diagonal ispointwise fixed.

Section 8.1

8.1.1. (a)A1 B2 C3C2 A3 B1B3 C1 A2

.

8.1.2. (a) Shift two to the right and one down. Shift one right and two down.


8.1.3. 1 2 3, 4 5 6, 7 8 9; 1 4 7, 2 5 8, 3 6 9;1 6 8, 2 4 9, 3 5 7; and 1 5 9, 2 6 7, 3 4 8. Four days.

8.1.5. (b) 4k + 1.

8.1.8. (a) A pentagon, 5 points.

Section 8.28.2.3. All lines have infinitely many points on them and each point is on infinitely many

lines.

8.2.4. In spherical geometry two lines intersect in two points, whereas in single elliptic geom-etry and projective geometry two lines intersect in exactly one point.

8.2.6. (b) The points of the plane are the girls. Each day corresponds to a family of parallellines with points on one of those lines representing girls in the same row for thatday.

8.2.8. (b) Suppose k‖l and m intersects k. If m didn’t intersect l, there would be two parallelsto l through the intersection of k and m, contradicting axiom (iii).

8.2.10. (a) 4, 4.

8.2.16. (a) (ii)(c) The dual of Theorem 7.2.1 (i) is weaker than axiom (i). Parts (ii) and (iii) of Theorem

7.2.1 are duals of each other. The dual of part (iv) of Theorem 7.2.1 is weaker thanpart (v).

8.2.17. (a) A triangle.

Section 8.38.3.1. Since k = 3, each variety must be on a block with two other varieties, giving an odd

number of varieties.

8.3.3. (a) 5 varieties with each pair forming a block. b = 10, r = 4.

8.3.4. (a) b = v(v − 1)/6, r = (v − 1)/2.

8.3.5. (a) r ≥ k.(b) r = k = n + 1, v = b = k2 − k + 1 = n2 + n + 1.

8.3.6. (a) For k = 3 and λ = 1, the first equation of Theorem 7.3.1 gives v = 2r + 1, so v isodd.

8.3.7. Rotate {1, 2, 5, 7} to obtain the lines.


8.3.13. (a)

1 0 1 1 1 0 0 0 1 0 00 1 0 1 1 1 0 0 0 1 00 0 1 0 1 1 1 0 0 0 11 0 0 1 0 1 1 1 0 0 00 1 0 0 1 0 1 1 1 0 00 0 1 0 0 1 0 1 1 1 00 0 0 1 0 0 1 0 1 1 11 0 0 0 1 0 0 1 0 1 11 1 0 0 0 1 0 0 1 0 11 1 1 0 0 0 1 0 0 1 00 1 1 1 0 0 0 1 0 0 1

. 5 and 2.

Section 8.48.4.1. There is a nonzero constant k so that ka = a′ and kb = b′. [0, 1, 0] has points of the

form (x, 0, 1) and [0, 1, 2] has points of the form (x, 3, 1). No point is on both.

8.4.2. There are only 4 points on a line in PZ23.

8.4.3. (0, 0, 1), (1, 1, 1), (2, 4, 1), (3, 4, 1), and (4, 1, 1). Additional projective point: (0, 1, 0).

8.4.4. (a) [1, 2, 1].(b) (2, 0, 1).(c) [2, 2, 1].(d) [−2,−1, 4], yes. (5,−6, 1), yes. [2, 2,−8], yes.

8.4.6. (a) Switches x- and y-coordinates. Similar to mirror reflection.(b) Rotation of 90◦ around (1, 1, 1) for both planes.

8.4.8. (a) For x = b, the tangent is y = 2bx − b2 or [2b,−1,−b2].(b) Each line is tangent. For example, for x = 0, [0, 4, 0] at (0, 0, 1) and for x = 3,

[1, 4, 1] at (3, 4, 1).

8.4.9. Many answers are possible. (a) [2, 2, 2] has 8 points.

(b) Consider [2, 1, 0] and [0, 1, 0].

8.4.10. (a) (2, 0, 1).(b) [3, 1, 4].(d) (3, 0, 1) is the only choice for S.

8.4.12. (a) [3, 4, 2], [1, 4, 2], and (0, 3, 4).

8.4.16. (a) Fixed points of the form (x, x, z), which are on the line [1, 4, 0]. The other stablelines are [4, 4, c] for any c. This matches the mirror reflection over y = x .

8.4.18. (a) (1, 0, 1), (2, 0, 1), (0, 1, 1), and (0, 2, 1).


8.4.20. x2 + y2 = 1 has (0, 1, 1), (0, 4, 1), (1, 0, 1), and (4, 0, 1). x2 + 4y = 0 has (1, 1, 1),(4, 1, 1), (2, 4, 1), (3, 4, 1), and (0, 0, 1). x2 + 3y2 = 1 has (1, 0, 1), (4, 0, 1), (2, 2, 1),(2, 3, 1), (3, 2, 1), and (3, 3, 1).

8.4.21. (a) [1, 1, 1, 4].

8.4.22. (a) 775, 806.

8.4.23. (a) (2, 2, 1, 2).

Section 9.19.1.1. (a) So the bending can match, x = 0, x2 + (y − b)2 = b2, or x2 + y2 − 2by = 0 for

b ≥ 0.(b) y = b −√

b2 − x2, b = 0.5 is the radius.

9.1.2. (a) x2 + (y − b)2 = (b − 1)2 for b ≤ 1, b = 0, radius =1.

9.1.6. (a) A and B are on a line of longitude (xz-plane) at the same latitude determined by c,where c = 0 gives the equator and c = π/2 gives the north pole.

(b) 1.(c) cos(c).(d) d(c) = π cos(c), a half circle.(e) D(c) = π − 2c.(f) At c = 0, both give a half circle of radius 1. At c = π/2, A = B, so the distance

is 0.

Section 9.29.2.1. x ′(t) and y′(t) can’t both be 0 so

∥∥c′(t)∥∥ =

√x ′(t)2 + y′(t)2 > 0.

9.2.2. (b) x = π/2 is the line of symmetry. (x − π/2)2 + y2 = 1.

9.2.4. (a) r (x) = (1 + e2x )3/2/ex .(b) y = −x + 1, r (0) = 2

√2 ≈ 2.828, center is (−2, 3), (x + 2)2 + (y − 3)2 = 8.

(c) minimum at x = 0.5 ln(0.5) ≈ −0.3, which has a radius of approx. 2.598.

9.2.7. (a) Near x = ±1.(b) κ(0) = 0.125, κ(1) = 14/(51.5) ≈ 1.2522.

9.2.8. (a) At an inflection point we expect f ′′(x) to be 0, and κ(x) can’t be smaller than 0.Radius is infinite.

9.2.9. (a)−−→c′(t) = (2t, 1 − 2t) and

−−→c′′(t) = (2,−2) give

−−→c′(t) · −−→c′′(t) = 8t − 2.

(b) κ(0) = 2, κ(0.25) = 4√

2 ≈ 5.656.

9.2.10. (b) Maximum curvature of 2 at t = π/2 + wπ , minimum curvature of 0.25 at t = wπ .(c) Maximum at π/4 + wπ/2, minimum at wπ/2, crosses when t = π/2 + wπ .


9.2.13. (a) Definitions A smooth space curve is a function −→c from a real interval into R3, writ-

ten−→c(t) = (x(t), y(t), z(t)) so that x ′, y′, z′, x ′′, y′′,and z′′ exist and are continuous

and in addition, x ′(t), y′(t), and z′(t) are not simultaneously all 0.(b) circle of longitude.(c) κ(t) = 1.

Section 9.39.3.4. Nu = (− sin(u) cos(v), cos(u) cos(v), 0) and Nv = (− cos(u) sin(v),− sin(u) sin(v),

cos(v)).

9.3.9. (a) These are all of the points at a distance of R from the z-axis.

9.3.10. (a) sx = (1, 0,∓x/√

R2 − x2 − y2), sy = (0, 1,∓y/√

R2 − x2 − y2), which are notorthogonal.

9.3.11. (a) At points with u = kπ and v = mπ , positive curvature. At points with u = π/2 +kπ and v = π/2 + mπ , negative curvature.

9.3.15. (a) s(u, v) = (cos(u) cos(v), sin(u) cos(v), a sin(v)).(b) su = (− sin(u) cos(v), cos(u) cos(v), 0) and sv = (− cos(u) sin(v),− sin(u) sin(v),

a cos(v)).

(d) Nu =(

−a sin(u) cos(v)√a2 cos2(v)+sin2(v)

, a cos(u) cos(v)√a2 cos2(v)+sin2(v)

, 0

)= a√

a2 cos2(v)+sin2(v)su .

9.3.16. (a) su = (−(r cos(v) + R) sin(u), (r cos(v) + R) cos(u), 0) andsv = (−r cos(u) sin(v),−r sin(u) sin(v), r cos(v)).

9.3.18. (a) s(u, v) = (cos(u)√

1 + v2, sin(u)√

1 + v2, v).(c) s(u, v) = (cos(u)

√v2 − 1, sin(u)

√v2 − 1, v). Curvature is always positive.

Section 9.49.4.2. E = 1 = G, F = 0. For these values ds2 = du2 + dv2 gives Euclidean distance from

the Pythagorean theorem.

9.4.3. su = (− f (v) cos(u), f (v) sin(u), 0), sv = ( f ′(v) cos(u), f ′(v) sin(u), g′(v)), E(u,

v) = f (v)2, F(u, v) = 0, and G(u, v) = f ′(v)2 + g′(v)2 = 1.

9.4.5. For v0 = 0, sin(v0) = 0, so N (t) = (cos(t), sin(t), 0) = −T ′(t).

Section 10.110.1.1. Vertices of a square, a rectangle, a parallelogram, {(0, 0), (1, 0), (0, 1), (2, 3)},

{(0, 0), (1, 0), (0, 2), (2, 3)}. The circle with center A going through B and the cir-cle with center B going through A intersect in two points C and D. However, thedistances AB and C D are unequal.

10.1.2. Yes.


10.1.3. Draw all the diagonals from one vertex.

10.1.4. Label the vertices from A to J going counterclockwise from the lower right. Placeguards at A, B, and E . Extend C D, F E , G H , and J I to where they intersect AB atK , L , M, and N , respectively. There must be a guard in each of �BC K , L FG M , and�AJ N in order to see the points C , G, and J .

10.1.5. Triangle, square, hexagon. The vertex angles must divide 360◦.

10.1.6. Rotate �ABC 180◦ about the midpoint of AB to get �B AC ′. Then AC BC ′ is aparallelogram and we can tile the plane as a slanted checkerboard with copies ofAC BC ′ and hence with copies of �ABC .

10.1.7. Region A is between regions B and D and between regions C and E . Hence theperpendicular bisectors of A with each of these regions shrink the regions B, C , Dand E more than the bisectors for B and D or for C and E .

10.1.8. The plane is infinite, so at least one of finitely many regions has to be infinite to coverit all.

10.1.9. (a) No: two equilateral �s back to back.

10.1.10. (a) 3.(b) 3.(c) 3, 4.(d) 3, 4, 5, 6.

10.1.11. (a) 3: {1, 2, 3, 4}, 4: {1, 2, 3, 5}, 5: {1, 2, 4, 7}, 6: {1, 2, 4, 8}.10.1.13. D2(5) = 2, D2(6) = 3 = D2(7), D2(8) = 4.

10.1.15. (a) 2.(b) 5.

10.1.16. (a) 2.(b) 5.

10.1.17. (a) One of the diagonals is interior. Place a guard on this diagonal.(b) a hexagon.

10.1.18. (a) 2 guards at opposite vertices.(b) 3 guards at vertices that are not all adjacent.

10.1.20. All five give monohedral tilings.

10.1.21. The vertex angles of two octagons and a square add to 360◦.

Section 10.210.2.1. Two well chosen guards at vertices or elsewhere suffice.

10.2.3. Suppose Dd (n) is given. Let V = {v1, . . . , vn, vn+1} be any set of n + 1 vertices ind dimensions. Then {v1, . . . , vn} has at least Dd (n) distances, forcing V to have at


least that many. Hence Dd (n) ≤ Dd (n + 1). Now suppose W = {w1, . . . , wn} is a setof n points in d dimensions with exactly Dd (n) distances. We can embed W in d + 1dimensions and still have the same number of distances. So Dd+1(n) ≤ Dd (n).

10.2.4. (a) n(b) Suppose we have n numbers 1, 2, . . . , k, where all adjacent differences are 1 or 2.

Then we get all distances from 1 to k − 1. The largest k can be is 2n − 2, so wecan get up to 2n − 3 different distances.

10.2.6. (a) 3.(b) 9.

10.2.7. (a) n(r ): 7, 19, 37, 61; d(r ): 3, 8, 15, 23.

10.2.9. (b) n(n − 1)/2, n. (Note: n(n − 1)/2 − n = n(n − 3)/2.) The polygon is convex soevery segment except outside edges is interior and so a diagonal.

10.2.10. (a) 2, 3 or 4 diagonals.

10.2.11. (a) 2, 3, 4, 5.

10.2.12. (b) 1, 2, 3, 5.

10.2.13. (a) One guard must be in �AB H to see B and one must be in �E FG to see E .(b) “Square” the left peak: Make AB vertical and add in B ′ above C and level with B.

10.2.17. (b) 2, 3, 4, 5.

10.2.18. (a) 3 guards suffice. Note: There is just one corner of 270◦, so essentially just onedesign.

10.2.19. (a) �n/2 . Position the guards to cover the outside. Any one of them also covers theinside.

Section 10.310.3.1. Color the big squares with two colors as with a checkerboard. Use the third color for

the small squares. Since two adjacent big squares also are adjacent to a small square,they must each be a different color.

10.3.2. Use each polygonal tile as the base of a prism with rectangular sides of, say, heighth, perpendicular to the plane of the tiling. Stack layers of these prisms on top of oneanother to fill space.

10.3.4. No. Each angle is 120◦ so every vertex must have 3 polygons meeting at it. If a shortedge of one polygon is matched with a long edge, there will be a 60◦ angle. So matchingedges must be the same length. However at each vertex each polygon has a long and ashort edge, so this is impossible.

10.3.6. (a) Suppose the measures of the angles are A, B, C , A, B, and C , consecutively.The angle sum is 720◦ so A + B + C = 360 and any the angle sum of any threeconsecutive angles is 360.


10.3.8. (a) The angle sum is 1080◦ = x · 90◦ + (8 − x)270◦ and so x = 6.(b) By symmetry the 270◦ angles are opposite.

10.3.9. (b) Any two lengths will work. The polygon with the shorter edge is surrounded bypolygons with the longer edge.

10.3.11. (a) 2 colors.(c) 3 colors. Use two alternating colors for the triangles and the third color for the

hexagons.

10.3.12. (a) Connect midpoints of each edge to create four similar triangles to the bigger one.(One can do a similar construction with n2 smaller triangles.)

Section 10.410.4.1. The boundaries are two parallel lines. Three parallel lines.

10.4.2. All but the ones with P−1,1 and P1,−1. All but P1,1 and P−1,−1.

10.4.3. Suppose part of the perpendicular bisector of the sites A and B forms part of theboundary of the region surrounding B. Then the point C so that B is the midpoint ofA and C is another site and the perpendicular bisector of the sites C and B forms partof the boundary of the region surrounding B. Thus these sides are centrally symmetricwith respect to B. This holds for all sides of the region of B and so all regions.

10.4.4. The tiling is made of squares.

10.4.5. The regions are regular hexagons.

10.4.7. Sites D, E , F, and G, where D is the intersection of the angle bisectors of �ABCand E , F, and G are the mirror reflections of D in the three sides of the triangle.

10.4.8. (a) Sites: vertices of equilateral triangle �ABC . Voronoi vertex, D, is the center of�ABC and the edges are rays from D perpendicular to the sides of �ABC .

(b) Use A, B, C, and D from part (a) as sites. The Voronoi diagram has an equilateraltriangle and three rays, giving 3 vertices and 6 edges.

10.4.10. (a) k + 1 sites with k at the vertices of a regular k-gon and the last one at the centerof the k-gon.

(b) For example when k = 3, 6 sites at (±1, 0), (±2,√

3), and (±2,−√3).

10.4.14. (a) Yes, the sites are at the centers of the triangles and lie at the vertices of a tiling byregular hexagons and form part of a lattice.

10.4.15. (a) The ray of points (x, a+b2 ) for x < 0, the ray (x, b−a

2 ) for x > a and the segmentbetween (0, a+b

2 ) and (a, b−a2 ).

10.4.17. (a) A great circle is the perpendicular bisector.(b) Three half circles through the poles at angles of 120◦.

Acknowledgements

We gratefully acknowledge the permissions we received to use the following.

Figure 1.0, courtesy the Estate of R. Buckminster Fuller.

Figure 1.51, from Galileo, Two New Sciences, copyright 1954 by Dover Publications, NewYork, reprinted with permission.

Chapter 2. Example 1, from Dubnov, Mistakes in Geometric Proofs, copyright 1963 by D.C.Heath, Lexington, MA, reprinted with permission.

Figure 3.0, St. Joseph Government Center, courtesy of Murray A. Mack, HMA Architects.

Figures 4.0, 4.39, and 4.40, courtesy of Douglas Dunham.

Figure 4.5, courtesy of Caren Diefenderfer.

Figure 5.1, from Wade, Geometric Patterns and Borders, copyright 1982 by Nostrand ReinholdCo., New York, reprinted with permission.

Figure 5.3, from Thompson, On Growth and Form, copyright 1942 by Cambridge UniversityPress, New York, reprinted with permission.

Figure 5.40, courtesy of Murray Mack.

Figures 6.0, 6.1, 6.3, 6.6, 6.12, 6.16, 6.17, 6.18, 6.20, 6.23, 6.24, 6.25, 6.26, 6.28, 6.29, fromWade, Geometric Patterns and Borders, copyright 1982 by Nostrand Reinhold Co., New York,reprinted with permission.

Figure 6.9, from Bentley and Humphrey, Snow Crystals, copyright 1962, by Dover Publications,New York, reprinted with permission.

Figures 6.19 and 6.27, from Crowe and Washburn, “Groups and geometry in the ceramic artof San Ildefonso,” Algebra, Groups and Geometries, copyright 1985 by Hadronic Press, PalmHarbor, FL, reprinted with permission.

Figure 6.31, courtesy of David Paul Lange, O.S.B.

Figure 6.37, from Holden and Morrison, Crystals and Crystal Growing, copyright 1982 by MITPress, Cambridge, MA, reprinted with permission.

549

550 Acknowledgements

Figure 6.39, from Peterson, The Mathematical Tourist: Snapshots of Modern Mathematics,p. 208. Copyright 1988 by Ivars Peterson. Used by permission of Henry Holt and Company,LLC. All rights reserved.

Figure 6.48, from Mandelbrot, The Fractal Geometry of Nature, p. 265. Copyright 1977, 1982,1983 by Benoit B. Mandelbrot. Used by permission of Henry Holt and Company, LLC. Allrights reserved.

Figures 6.51 and 6.53, courtesy of U.S. Geological Survey.

Figure 6.52, from Moore and Persaud, The Developing Human: Clinically Oriented Embryol-ogy, p. 248. Copyright 2003 by Elsevier, reprinted with permission.

Figures 7.0 and 7.1, from Art Resource, reprinted with permission.

Figure 10.1, from Johnson, The Ghost Map, copyright 2006 by Penquin Books, New York,courtesy of UCLA Snow Site.

Figure 11.1, courtesy of David Paul Lange, O.S.B.

Appendix B, from School Mathematics Study Group, Geometry, copyright 1961 by Yale Uni-versity Press, New Haven, reprinted with permission.

Appendix C, from Hilbert, The Foundations of Geometry, 2nd ed., translated E. Townsend,copyright 1921 by Open Court, Peru, IL, reprinted with permission.

Index

TermsAAS, 11, 493 (I-26)absolute conic, 355, 358absolute quadric surface, 367acceleration vector, 509Achilles and the Tortoise, 4affine geometry, 358affine matrix, 215

plane, 215three-dimensional, 235n-dimensional, 241

affine plane, 379ff, 396affine space, 397, 400affine transformation, 215, 243, 358alternate interior angles, 27, 494alternate exterior angles, 31, 494altitude of a triangle, 41analysis, 11analytic geometry, 98analytic projective geometry, 337analytic model, 98angle, 3, 49, 70, 500angle bisector, 13angle defect for polyhedra, 46angle of parallelism, 162angle of two great circles, 49angle-angle-side, 11, 493 (I-26)angle-side-angle, 11, 493 (I-26)angle sum, 3antipodal points, 49antiprism, 51, 291apex, 42arc length, 438Archimedean axiom, 501Archimedean solid, 63, 289area, 36, 176arithmetic mean, 7art gallery problem, 449, 462

art gallery theorem, 463ASA, 11, 493 (I-26)arithmetic meanasymptote, 108, 341atoms, 294axiom, 69axiom of linear completeness, 501axiomatic system, 68axioms, projective geometry, 327ffaxis of rotation, 237

Babylonian mathematics, 2balanced incomplete block design,

386barycentric coordinates, 118, 366base of a pyramid, 42base of a Saccheri quadrilateral, 169b.c.e. (before the common era), xviiibetween, 366Bezier curve, 126ffBIBD, 386bilateral symmetry, 262binormal, 433bipyramid, 53bisector, 12block, 386bonds, 294Brianchon’s theorem, 336Bruck-Ryser theorem, 381

CAD, 126ff, 241, 362cases, 514Catalan number, 461Cavalieri’s principle, 56, 498cell, 144center of inversion, 246central angle, 21central symmetry, 221, 243

551

552 Index

centroid of a triangle, 41chaos, 489characteristic axiom, hyperbolic geometry, 153characteristic equation, 241chemical structure, 294circle, 79circular points at infinity, 358circumcenter of a triangle, 20circumscribed circle, 20closure, 200code, 390codeword, 390collinear, 29, 328collineation, 347, 362, 398color group, 281coloring, 473color preserving group, 281color preserving symmetry, 271, 282color switching symmetry, 271, 282color symmetry, 271, 282combinations, 448combinatorics, 375commensurable, 3Common Core State Standards (CCSS), xv, xvi,

xx, 196, 208complete (axiomatic system), 86complete quadrangle, 320complete quadrilateral, 321complex conjugate, 103, 249complex numbers, 103–104, 248ffcomposition of functions, 198computer-aided design, 126ff, 241, 362conclusion, 511concurrent, 20, 328conformal, 252congruent, 3, 11, 76, 166conic, 104ff, 341, 370conic surface, 135consistent, 83constructible angle, 25constructible number, 24constructible polygon, 19constructions, 12continuity axiom (projective), 330contraction mapping, 230contradiction, 512converse, 27convex, 7, 366convex hull, 464convex set of points, 458correlation, 371corresponding angles, 31, 494counterexample, 513

cross polytope, 139cross product, 503, 509cross ratio, 339cross section, 510crystallographic group, 290crystallographic restriction, 277crystals, 290, 294cube, 51cuboctahedron, 52curvature of a curve, 415curvature of a surface, 154, 427curve, 509cyclic group, 267cycloid, 115

decomposition of a figure, 17defect of a hyperbolic triangle, 178definition, 69degenerate conic, 108, 341derivative vector, 509Desargues’ theorem, 322, 370Descartes’ formula, 46descriptive geometry, 136design theory, 385ffdeterminant, 505diagonal points, 335diagrams in proofs, 70, 71diameter of a circle, 79diameter of an ellipse, 107differential geometry, 409ffdihedral angle, 140dihedral group, 267dilation, 225direct isometry, 206direct proof, 511directrix, 106discrete geometry, 448ffdiscrete pattern, 273distance, 99, 134, 214, 355dodecahedron, 55dot product, 503, 509double elliptic geometry, 185doubling the cube, 16dual, 92, 144, 331, 362, 382dual of a polyhedron, 53duality, 331ff, 362, 382dynamical systems, 201, 489

ear of a polygon, 462edges, 19, 42Egyptian mathematics, 2eigenvalue, 219, 506eigenvector, 219, 506

Index 553

Elements, The, 10, 491ellipse, 105ellipsoid, 135equal (functions), 197equal (in measure), 15equivalence relation, 259, 380equivalent polygons, 177Erlanger programm, 209, 212error-correcting code, 389Euclidean geometry, 1ff, 358Euclidean isometry, 202, 241Euler’s formula, 44example, 513excess of a spherical triangle, 51exterior of a conic, 353externally tangent circles, 40

faces, 42ff (“following” for page references), 51field, 39415-schoolgirl problem, 375fifth postulate, 28, 492finite geometry, 373fffirst fundamental form, 437first stellation, 55fixed point, 197, 347flow chart, wallpaper groups, 279foci, focus, 106fortress problem, 454fortress theorem, 464fractal, 228, 230, 304fffractal curve, 309fractal dimension, 307fractal surface, 309frequency, 47frieze pattern, 273fundamental theorem, projective geometry, 333

Gauss-Bonnet theorem, 439Gaussian curvature of a surface, 427general theory of relativity, 409, 440generate, 274generic, 451geodesic, 155, 409, 433geodesic dome, 47geometric mean, 7geometry (definition), 209glide reflection, 206global positioning system, 440golden ratio, 8GPS, 440great circle, 48Greek mathematics, 2ff

group, 200guard point, 62

h-inner product, 357h-length, 357h-orthogonal, 357half plane, 76half plane model, hyperbolic geometry, 157, 253Hamming distance, 390harmonic (music), 326harmonic set of lines, 325harmonic set of points, 320Hausdorff dimension, 305helix, 116Hilbert’s axioms, 77ff, 499homogeneous coordinates, 337homogeneous second degree equation, 340horocycle, 191, 258horolation, 258hyperbola, 105hyperbolic distance, 182, 355hyperbolic geometry, 152ff, 247, 355hyperbolic glide reflection, 258hyperbolic isometry, 250, 355hyperbolic translation, 258, 356hyperboloid of one sheet, 136hyperboloid of two sheets, 135hypercube, 137hypercycle, 258hyperplane, 134, 241hypothesis, 511

icosahedral group, 289icosahedron, 54ideal line, 318ideal point, 318identity function, 199identity matrix, 505if and only if, 514IFS, 228, 230image, 197incenter, 21incidence matrix, 390incommensurable, 3independent, 84indirect isometry, 206induction proof, 512inscribed angle, 21inscribed circle, 21inscribed polygon, 19interior angles on the same side, 31, 494interior of a conic, 353, 355inverse function, 199

554 Index

inverse matrix, 505inversion, 246inversive line, 246inversive plane, 246invertible matrix, 505irrational, 3isomer, 299isometry, 202

Euclidean plane, 202Euclidean n-dimensional, 241spherical, 235, 241

isometric, 212isomorphic, 87isosceles, 17iterated function system, 228, 230iteration, 304

Kirkman’s 15 schoolgirl problem, 375kite, 32Klein model, hyperbolic geometry, 157

latitude, 49, 424law of cosines, 23law of the lever, 26lemniscate, 117length, 3, 237, 503length of a vector, 418, 509lightlike (relativity), 303, 367limiting curve, 191line, 3, 99, 133, 216, 337, 362line conic, 345line segment, 3, 366linear algebra, 503linear combination, 504linear transformation, 504locus problem, 105logically equivalent, 27longitude, 49, 424Lorentz transformation, 297, 368lunes, 8, 49

manifold, 439matrices, matrix, 504matrix multiplication, 504mean curvature, 445median of a triangle, 41, 100metamathematics, 83Michelson-Morley experiment, 297midline, 273midpoint, 18minimal surface, 445Minkowski geometry, 367

minor of a matrix, 505mirror reflection, 203, 237Mobius transformation, 250model, 82mod (n), modulo, 395monohedral tile, 450motif, 264

Napoleon’s theorem, 145National Council of Teachers of Mathematics

(NCTM), xv, xxn-gon, 19non-orientable surface, 188, 365norm of a vector, 509normal, 414normal line, 415

oblate, 430oblong number, 7octahedral group, 289octahedron, 53omega point, 165, 355omega triangle, 165on, 99, 134, 216, 337, 341one-point perspective, 323operational definition, 69opposite interior angles, 29orbit of a point, 269orbit-stabilizer theorem, 269order, affine plane, 379order, field, 395order, projective plane, 383orientable, 188, 488orientation, 196oriented point, 365oriented projective geometry, 365origin, 133orthocenter, 41orthogonal, 237, 503, 509orthogonal circles, 154, 247orthogonal matrix, 241orthogonal polygon, 454orthonormal basis, 237, 504osculating circle, 414oval, 399

Pappus’ theorem, 336parabola, 106paraboloid, 135parallel, 26, 134, 379, 492parallel postulate, 26, 492, 498parallelogram, 29

Index 555

parametric equations, 114partial derivative, 510Pascal’s theorem, 336Pasch’s axiom, 77, 499Penrose tiling, 296pentamino, 483perpendicular, 12, 143perpendicular bisector, 12perspective (art), 318perspective (computers), 362ffperspective from a line, 321perspective from a point, 321perspectivity, 320Pick’s theorem, 145plane, 134, 238, 362plane curve, 509plane tiling, 450platonic solid, 44Playfair’s axiom, 28, 498, 501Poincare conjecture, 489Poincare model, hyperbolic geometry, 157,

247point, 99, 133, 214, 235, 337, 362polar, 252, 326polar coordinates, 116pole, 252, 326polygon, 19polyhedra, polyhedron, 42polytope, 137postulate, 12, 28, 76power of a point, 39principle of mathematical induction, 512prism, 43product of matrices, 505projection mapping, 242projective geometry, 318ffprojective plane, 382ff, 397projective space, 362ff, 399projective transformation, 346ffprojectivity, 333, 347prolate, 430proof, 70proof by contradiction, 512properties of matrices, 506proportional, 33pseudosphere, 156, 431pyramid, 42Pythagorean theorem, 3, 495 (I-47)

quadric surface, 368quadrilateral, 29quasicrystal, 296

radians, 50radius, 79radius of curvature, 415ratio of proportionality, 33rational line, 120rational numbers, 120rational point, 120ray, 70, 500real numbers, 82rectangle, 30rectangular box, 51recursive, 461regular polygon, 19regular polyhedron, 44regular polytope, 137regular star figure, 62regular star polygon, 61regular tiling, 484relatively consistent, 83relativity theory

special, 263, 297Galilean, 297general, 409, 440

reproducing an angle, 13rep-tile, 476rhombus, 22Richardson’s equation (dimension), 308right angle, 12rotary reflection, 237rotation, 203, 237

Saccheri quadrilateral, 169SAS, 11, 492 (I-4), 498, 500scalar, 503scalar multiple, 133scaling ratio, 223Schonflies’ notation, 289screw motion, 240segment, 3, 499self-dual, 53, 362self-similarity, 304semi-regular tiling, 484sensed parallel lines, 162separation (projective geometry), 329separation axiom (or postulate), 76, 77, 498,

500separation axiom (projective geometry), 329set, 401shape operator, 427shear, 227side-angle-side, 11, 492 (I-4)side-side-side, 11, 492 (I-8)

556 Index

signed curvature, 427similar, 34, 41similarity, 223simplex, simplices, 139simply connected, 488single elliptic geometry, 185ff, 358site, 453smooth plane curve, 415smooth surface, 426SMSG postulates, 76ff, 497spacelike (relativity), 303, 368sphere, 48, 424spherical cap, 58spherical excess, 51spherical geometry, 185ffspherical isometry, 235spherical triangle, 50spheroid, 430spiral of Archimedes, 418spline, 130square, 30square matrix, 504squaring the circle, 16SSS, 11, 493 (I-8)stabilizer of a point, 268stable, 197, 238, 347standard basis, 504statistical design theory, 374statistical self-similarity, 305Steiner quadruple system, 392Steiner triple system, 388stereographic projection, 254straight angle, 12straightedge and compass constructions, 12subgeometry, 354ff, 367subgroup, 274subspace, 504sum, 133summit of Saccheri quadrilateral, 169supplementary angles, 79surface, 426ff, 509surface of revolution, 428symmetric design, 391symmetric group, 291symmetric matrix, 504symmetries of a prism, 289symmetry, 264symmetry group, 264synthetic, 11

tangent, 15, 341, 399tangent plane, 510tangram, 61

taxicab geometry, 84, 125tetrahedral group, 289tetrahedron, 42tetromino, 455theorem, 7036-officer problem, 374three body problem, 489three-point perspective, 323tile, 450tilings, 450fftimelike (relativity), 303, 368topology, 488torsion, 410, 420, 445torus, 425total angle defect, 46tractrix, 431transcendental, 17transformation, 197transformation group, 200translation, 203transpose, 237, 504transversals, 26trapezoid, 39triangle, 3triangle inequality, 493 (I-20)triangular number, 7triangulation, 449, 459trilinear plot, 118trisecting an angle, 16two-point perspective, 323

ultraparallel lines, 162undefined term, 69unit normal, 426, 510unit tangent vector, 419, 509

vanishing point, 318variety, 386vector, 133, 503vector function, 509velocity vector, 509vertex, vertices, 42, 203vertical angles, 79, 493 (I-15)volume, 43Voronoi diagram, 453Voronoi edge, 478Voronoi region, 453Voronoi vertex, 478

wallpaper group flow chart, 279wallpaper pattern, 273

Zeno’s paradoxes, 4

Index 557

People(biography pages in bold)Archimedes, 25Aristotle, 5, 474

Barnsley, Michael, 228, 230Beltrami, Eugenio, 156, 161, 424Bernoulli, Jakob, 116, 410Bernoulli, Johann, 377Bezier, Pierre Etiene, 126Bolyai, Janos, 153, 169Bolyai, W., 17Bonnet, Pierre Ossian, 439Bravais, Auguste, 263, 288Brianchion, Charles Julien, 336

Cayley, Arthur, 318, 354, 361Cavalieri, Bonaventura, 56Chvatal, Vaclav, 450Clairaut, Alexis-Claude, 410Coxeter, H. S. M., 180, 263, 288, 293Crowe, Don, 280

da Vinci, Leonardo, 267, 318Dehn, Max, 17Desargues, Girard, 318, 322, 336Descartes, Rene, 46, 98, 104, 423, 477, 487Dirichlet, Peter Gustave Lejeune, 477Durer, Albrecht, 263, 318, 319

Eddington, Sir Arthur, 409, 440Einstein, Albert, 155, 297, 411, 444Eratosthenes, 8Erdos, Paul, 449, 457, 459, 466Escher, M. C., 180, 288, 293Euclid, 10ff, 318, 487Eudoxus, 4Euler, Leonhard, 46, 98, 114, 374, 377, 410

Fano, Gino, 374, 382Fedorov, Vyatseglav, 278Fermat, Pierre de, 98, 114, 487Fisher, Sir Ronald A., 374, 385, 393Fisk, Steve, 462Francesca, Piero della, 318Fuller, Buckminster, 1, 46, 59, 293

Galileo Galilei, 36, 297, 423Galois, Evariste, 263Gauss, Carl Friedrich, 19, 153, 160, 410, 424,

437, 439Gerwien, P., 17

Godel, Kurt, 83, 94Grunbaum, Branko, 477

Hamilton, William Rowan, 361Hausdorff, Felix, 305Helmholtz, Hermann, 235Hessel, J. H. C., 263, 288, 290Hilbert, David, 77, 81

Jordan, Camille, 263

Kant, Immanuel, 152Kelvin, Lord, 378Kepler, Johannes, 267, 283, 318, 423Khayyam, Omar, 169, 174Kirkman, Rev. Thomas, 375, 378, 388Klein, Felix, 185, 196, 209, 212, 263, 278, 318,

346, 354Koch, Helge von, 304

Lagrange, Joseph Louis, 263Leibniz, Gottlieb Wilhelm, 409, 423Lie, Sophus, 196, 234, 263Lindemann, Ferdinand, 16Lobachevsky, Nikolai, 153, 169Lorentz, Hendrik, 263, 297, 489

Mandelbrot, Benoit, 304, 312, 488Minkowski, Hermann, 299Mobius, Augustus, 118, 137, 195, 250, 254, 263,

318, 487Monge, Gaspard, 136, 145, 318, 336

Nasir Eddin, 60Newton, Sir Isaac, 297, 409, 423, 430

Oresme, Nicole, 98

Pappus, 336Pascal, Blaise, 114, 318Penrose, Sir Roger, 296Perelman, Grigori, 489Plato, 4Plucker, Julius, 149, 254, 318, 346, 388Poincare, Henri, 212, 247, 263, 297, 488Poncelet, Jean Victor, 318, 336Pythagoras, 2Pythagoreans, 2

Richardson, Lewis, 308Riemann, Georg, 154, 185 189, 411, 439

Saccheri, Giovanni, 152, 175

558 Index

Schlafli, Ludwig, 137Senechal, Margorie, 304Snow, John, 447, 477Steiner, Jacob, 346, 388

Tait, Peter Guthrie, 378Thales, 33, 511Theaetetus, 4

Van Staudt, Karl, 318Viete, Francois, 98Voronoi, Georgy, 477

Wantzel, Pierre, 16Washburn, Dorothy, 280Wiles, Andrew, 114

Zeno, 4

NotationAB, length of AB, 3←→AB, line on A and B, 3−→AB, ray from A through B, 70AB, segment between A and B, 3∠ABC , angle with vertex at B, 3�ABC , triangle with vertices A, B, and C , 3−−−−→(a, b, c) · −−−−−→(d, e, f ), dot product, 338, 503, 509a + bi , complex conjugate, 103, 249α−1, inverse of α, 199AF2, affine plane over F, 396AFd , affine space over F, 400

B(t0), binormal, 433

cm, etc. wallpaper groups, 279C, complex numbers, 249C#, extended complex numbers, 248Cn , cyclic group, 267−→c(t), curve, 415, 509

‖−→c(t)‖, norm (length) of−→c(t), 418, 503, 509−−→

c′(t), derivative (tangent) of−→c(t), 418, 509−−→

c′′(t), acceleration vector, 509

Dd (n), minimum number of distances, 449det(M), determinant of M , 505Dn , dihedral group, 267, 289Dnh , symmetry group of a prism, 289∂x∂u , partial derivative, 510

F, field, 395

〈g1, g2, . . . , gn〉, the subgroup generated byg1, g2, . . . , gn , 274

·h , h-inner product, 357H (P Q, RS), harmonic set, 320, 339H ( jk, lm), harmonic set, lines, 325

I , identity matrix, 505

k · l, intersection of lines, 328κ(p, q), curvature of a surface, 427κ(t), curvature of a curve, 416

M , shape operator, 427M−1, inverse of M , 216, 505MT , transpose of M , 504m∠ABC , measure of ∠ABC , 21

N, the natural numbers, 512!n", the greatest integer less than or equal to n,

459�n , the least integer greater than or equal to n,

467n!, n factorial, 290(n

k

), combinations of n, k at a time, 448

N (u, v), unit normal vector, 426, 510

P, icosahedral group, 289P , oriented point, 365p111, etc. frieze pattern groups, 276PF2, projective plane over F, 397PFd , projective space over F, 399pg, etc. wallpaper groups, 279P Q//RS, separation, 329, 339

Q, the rational numbers, 120

R, the real numbers, 82R(P, S, U, W ), cross ratio, 339r (x), radius of curvature, 416

Sn , symmetric group, 290su , partial derivative of surface, 426, 510s(u, v), surface, 426

T, tetrahedral group, 289T (t), unit tangent vector, 419, 509T (n), triangulations of a convex n-gon,

461

(v, k, λ), parameters of a BIBD, 385V or (S), Voronoi diagram of S, 453−→v ×−→w , cross product, 503, 509

Index 559

W, octahedral group, 289

Xx , hyperbolic translation, 356

Zn , integers modulo n, 395

� end of a proof, xvii♦ end of an example, xvii

* exercise answered in back of book,xvii

∼=, congruent, 3⊥, perpendicular, 12‖, parallel, 26, 379∼, similar, 34⊕, velocity addition in relativity, 298∞, added point for inversive plane, 246

This is a well written and comprehensive survey of college geometry that would

serve a wide variety of courses for both mathematics majors and mathematics

education majors. Great care and attention is spent on developing visual insights

and geometric intuition while stressing the logical structure, historical develop-

ment, and deep interconnectedness of the ideas. Students with less mathemat-

ical preparation than upper-division mathematics majors can successfully study

the topics needed for the preparation of high school teachers.

There is a multitude of exercises and projects in those chapters developing all

aspects of geometric thinking for these students as well as for more advanced

students. These chapters include Euclidean Geometry, Axiomatic Systems and

Models, Analytic Geometry, Transformational Geometry, and Symmetry. Topics

in the other chapters, including Non-Euclidean Geometry, Projective Geometry,

Finite Geometry, Differential Geometry, and Discrete Geometry, provide a

broader view of geometry. The different chapters are as independent as possible,

while the text still manages to highlight the many connections between topics.

The text is self-contained, including appendices with the material in Euclid’s first

book and a high school axiomatic system as well as Hilbert’s axioms. Appendices

give brief summaries of the parts of linear algebra and multivariable calculus

needed for certain chapters. While some chapters use the language of groups,

no prior experience with abstract algebra is presumed. The text will support an

approach emphasizing dynamical geometry software without being tied to any

particular software.

An instructor’s manual for this title is available electronically. Please send email

to [email protected] for more information.

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