Thinking Geometrically a Survey of Geometries

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Thinking Geometrically A Survey of Geometries

Thomas Q. Sibley

Thinking Geometrically

A Survey of Geometries c 2015 by The 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): 10987654321 10.1090/text/026

Thinking Geometrically

A Survey of Geometries

Thomas Q. Sibley St. John’s University

Published and distributed by The Mathematical Association of America Council on Publications and Communications Jennifer J. Quinn, Chair Committee on Books Fernando Gouvea,ˆ Chair MAA Textbooks Editorial Board Stanley E. Seltzer, Editor Matthias Beck Richard E. Bedient Otto Bretscher Heather Ann Dye Charles R. Hampton Suzanne Lynne Larson John Lorch Susan F. Pustejovsky MAA TEXTBOOKS Bridge to Abstract Mathematics, Ralph W. Oberste-Vorth, Aristides Mouzakitis, and Bonita A. Lawrence Calculus Deconstructed: A Second Course in First-Year Calculus, Zbigniew H. Nitecki Calculus for the Life Sciences: A Modeling Approach, James L. Cornette and Ralph A. Ackerman Combinatorics: A Guided Tour, David R. Mazur Combinatorics: A Problem Oriented Approach, Daniel A. Marcus Complex Numbers and Geometry, Liang-shin Hahn A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological Mathematics, Robert Edward Lewand Differential Geometry and its Applications, John Oprea Distilling Ideas: An Introduction to Mathematical Thinking, Brian P.Katz and Michael Starbird Elementary Cryptanalysis, Abraham Sinkov Elementary Mathematical Models, Dan Kalman An Episodic History of Mathematics: Mathematical Culture Through Problem Solving, Steven G. Krantz Essentials of Mathematics, Margie Hale Field Theory and its Classical Problems, Charles Hadlock Fourier Series, Rajendra Bhatia Game Theory and Strategy, Philip D. Straffin Geometry Illuminated: An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry, Matthew Harvey Geometry Revisited, H. S. M. Coxeter and S. L. Greitzer Graph Theory: A Problem Oriented Approach, Daniel Marcus An Invitation to Real Analysis, Luis F. Moreno Knot Theory, Charles Livingston Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem, Al Cuoco and Joseph J. Rotman The Lebesgue Integral for Undergraduates, William Johnston Lie Groups: A Problem-Oriented Introduction via Matrix Groups, Harriet Pollatsek Mathematical Connections: A Companion for Teachers and Others, Al Cuoco Mathematical Interest Theory, Second Edition, Leslie Jane Federer Vaaler and James W. Daniel Mathematical Modeling in the Environment, Charles Hadlock Mathematics for Business Decisions Part 1: Probability and Simulation (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Business Decisions Part 2: Calculus and Optimization (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Secondary School Teachers, Elizabeth G. Bremigan, Ralph J. Bremigan, and John D. Lorch The Mathematics of Choice, Ivan Niven The Mathematics of Games and Gambling, Edward Packel Math Through the Ages, William Berlinghoff and Fernando Gouvea Noncommutative Rings, I. N. Herstein Non-Euclidean Geometry, H. S. M. Coxeter Number Theory Through Inquiry, David C. Marshall, Edward Odell, and Michael Starbird Ordinary Differential Equations: from Calculus to Dynamical Systems, V.W. Noonburg A Primer of Real Functions, Ralph P. Boas A Radical Approach to Lebesgue’s Theory of Integration, David M. Bressoud A Radical Approach to Real Analysis, 2nd edition, David M. Bressoud Real Infinite Series, Daniel D. Bonar and Michael Khoury, Jr. Thinking Geometrically: A Survey of Geometries, Thomas Q. Sibley Topology Now!, Robert Messer and Philip Straffin Understanding our Quantitative World, Janet Andersen and Todd Swanson

MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-240-396-5647 Contents

Preface xv

1 Euclidean Geometry 1 1.1 Overview and History...... 1 1.1.1 The Pythagoreans and Zeno ...... 2 1.1.2 Plato and Aristotle...... 4 1.1.3 Exercises for Section 1.1...... 5 1.2 Euclid’s Approach to Geometry I: Congruence and Constructions...... 10 1.2.1 Congruence...... 11 1.2.2 Constructions...... 12 1.2.3 Equality of Measure...... 15 1.2.4 The Greek Legacy...... 16 1.2.5 Exercises for Section 1.2...... 17 1.2.6 Archimedes...... 25 1.3 Euclid’s Approach II: Parallel Lines...... 26 1.3.1 Exercises for Section 1.3...... 30 1.4 Similar Figures...... 33 1.4.1 Exercises for Section 1.4...... 37 1.5 Three-Dimensional Geometry...... 42 1.5.1 Polyhedra...... 42 1.5.2 Geodesic Domes ...... 46 1.5.3 The Geometry of the Sphere...... 48 1.5.4 Exercises for Section 1.5...... 51 1.5.5 Buckminster Fuller ...... 59 1.5.6 Projects for Chapter 1...... 59 1.5.7 Suggested Readings ...... 65

2 Axiomatic Systems 67 2.1 From Euclid to Modern Axiomatics...... 67 2.1.1 Overview and History...... 67 2.1.2 Axiomatic Systems...... 68 2.1.3 A Simpliﬁed Axiomatic System...... 71 2.1.4 Exercises for Section 2.1...... 73

vii viii Contents

2.2 Axiomatic Systems for Euclidean Geometry...... 75 2.2.1 SMSG Postulates...... 76 2.2.2 Hilbert’sAxioms...... 77 2.2.3 Exercises for Section 2.2...... 78 2.2.4 David Hilbert...... 81 2.3 Models and Metamathematics...... 82 2.3.1 Exercises for Section 2.3...... 88 2.3.2 Kurt Godel.¨ ...... 94 2.3.3 Projects for Chapter 2...... 94 2.3.4 Suggested Readings ...... 96

3 Analytic Geometry 97 3.1 Overview and History...... 98 3.1.1 The Analytic Model ...... 98 3.1.2 Exercises for Section 3.1...... 100 3.1.3 Rene´ Descartes...... 104 3.2 Conics and Locus Problems ...... 104 3.2.1 Exercises for Section 3.2...... 110 3.2.2 Pierre de Fermat...... 114 3.3 Further Topics in Analytic Geometry...... 114 3.3.1 Parametric Equations...... 114 3.3.2 Polar Coordinates...... 116 3.3.3 Barycentric Coordinates...... 118 3.3.4 Other Analytic Geometries...... 120 3.3.5 Exercises for Section 3.3...... 121 3.4 Curves in Computer-Aided Design...... 126 3.4.1 Exercises for Section 3.4...... 131 3.5 Higher Dimensional Analytic Geometry...... 133 3.5.1 Analytic Geometry in Rn ...... 133 3.5.2 Regular Polytopes...... 137 3.5.3 Exercises for Section 3.5...... 140 3.5.4 Gaspard Monge...... 145 3.5.5 Projects for Chapter 3...... 145 3.5.6 Suggested Readings ...... 149

4 Non-Euclidean Geometries 151 4.1 Overview and History...... 152 4.1.1 The Advent of Hyperbolic Geometry...... 153 4.1.2 Models of Hyperbolic Geometry...... 155 4.1.3 Exercises for Section 4.1...... 158 4.1.4 Carl Friedrich Gauss...... 160 4.2 Properties of Lines and Omega Triangles ...... 161 4.2.1 Omega Triangles ...... 164 4.2.2 Exercises for Section 4.2...... 167 4.2.3 Nikolai Lobachevsky and Janos´ Bolyai...... 169 Contents ix

4.3 Saccheri Quadrilaterals and Triangles...... 169 4.3.1 Exercises for Section 4.3...... 173 4.3.2 Omar Khayyam...... 174 4.3.3 Giovanni Girolamo Saccheri...... 175 4.4 Area, Distance, and Designs...... 176 4.4.1 Exercises for Section 4.4...... 183 4.5 Spherical and Single Elliptic Geometries...... 185 4.5.1 Exercises for Section 4.5...... 189 4.5.2 Georg Friedrich Bernhard Riemann...... 189 4.5.3 Projects for Chapter 4...... 190 4.5.4 Suggested Readings ...... 192

5 Transformational Geometry 195 5.1 Overview and History...... 195 5.1.1 Exercises for Section 5.1...... 200 5.2 Isometries...... 201 5.2.1 Classifying Isometries...... 203 5.2.2 Congruence and Isometries...... 208 5.2.3 Klein’s Definition of Geometry...... 209 5.2.4 Exercises for Section 5.2...... 209 5.2.5 Felix Klein...... 212 5.3 Algebraic Representation of Transformations...... 213 5.3.1 Isometries...... 217 5.3.2 Exercises for Section 5.3...... 220 5.4 Similarities and Affine Transformations...... 223 5.4.1 Similarities...... 223 5.4.2 Affine Transformations...... 226 5.4.3 Iterated Function Systems...... 228 5.4.4 Exercises for Section 5.4...... 231 5.4.5 Sophus Lie...... 234 5.5 Transformations in Higher Dimensions; Computer-Aided Design ...... 235 5.5.1 Isometries of the Sphere...... 235 5.5.2 Transformations in Three and More Dimensions...... 239 5.5.3 Computer-Aided Design and Transformations...... 241 5.5.4 Exercises for Section 5.5...... 243 5.6 Inversions and the Complex Plane...... 245 5.6.1 Exercises for Section 5.6...... 252 5.6.2 Augustus Mobius...... ¨ 254 5.6.3 Projects for Chapter 5...... 254 5.6.4 Suggested Readings ...... 259

6 Symmetry 261 6.1 Overview and History...... 262 6.1.1 Exercises for Section 6.1...... 264 x Contents

6.2 Finite Plane Symmetry Groups ...... 267 6.2.1 Exercises for Section 6.2...... 270 6.3 Symmetry in the Plane...... 272 6.3.1 Frieze Patterns...... 273 6.3.2 Wallpaper Patterns...... 276 6.3.3 Exercises for Section 6.3...... 282 6.3.4 M. C. Escher ...... 288 6.4 Symmetries in Higher Dimensions...... 288 6.4.1 Finite Three-Dimensional Symmetry Groups ...... 288 6.4.2 The Crystallographic Groups...... 290 6.4.3 General Finite Symmetry Groups ...... 291 6.4.4 Exercises for Section 6.4...... 291 6.4.5 H. S. M. Coxeter...... 293 6.5 Symmetry in Science...... 294 6.5.1 Chemical Structure...... 294 6.5.2 Quasicrystals...... 296 6.5.3 Symmetry and Relativity...... 297 6.5.4 Exercises for Section 6.5...... 299 6.5.5 Marjorie Senechal...... 304 6.6 Fractals...... 304 6.6.1 Exercises for Section 6.6...... 310 6.6.2 Benoit Mandelbrot...... 312 6.6.3 Projects for Chapter 6...... 312 6.6.4 Suggested Readings ...... 314

7 Projective Geometry 317 7.1 Overview and History...... 318 7.1.1 Projective Intuitions...... 319 7.1.2 Exercises for Section 7.1...... 322 7.2 Axiomatic Projective Geometry...... 327 7.2.1 Duality...... 331 7.2.2 Perspectivities and Projectivities...... 332 7.2.3 Exercises for Section 7.2...... 333 7.2.4 Jean Victor Poncelet...... 336 7.3 Analytic Projective Geometry...... 337 7.3.1 Cross Ratios...... 339 7.3.2 Conics...... 340 7.3.3 Exercises for Section 7.3...... 342 7.3.4 Julius Plucker...... ¨ 346 7.4 Projective Transformations...... 346 7.4.1 Exercises for Section 7.4...... 351 7.5 Subgeometries ...... 354 7.5.1 Hyperbolic Geometry as a Subgeometry ...... 355 7.5.2 Single Elliptic Geometry as a Subgeometry ...... 358 7.5.3 Afﬁne and Euclidean Geometries as Subgeometries...... 358 Contents xi

7.5.4 Exercises for Section 7.5...... 359 7.5.5 Arthur Cayley...... 361 7.6 Projective Space...... 361 7.6.1 Perspective and Computer-Aided Design...... 362 7.6.2 Subgeometries of Projective Space ...... 367 7.6.3 Exercises for Section 7.6...... 368 7.6.4 Projects for Chapter 7...... 370 7.6.5 Suggested Readings ...... 372

8 Finite Geometries 373 8.1 Overview and History...... 373 8.1.1 Exercises for Section 8.1...... 376 8.1.2 Leonhard Euler ...... 377 8.1.3 Rev.Thomas Kirkman...... 378 8.2 Afﬁne and Projective Planes...... 378 8.2.1 Afﬁne Planes...... 378 8.2.2 Projective Planes...... 382 8.2.3 Exercises for Section 8.2...... 384 8.3 Design Theory...... 385 8.3.1 Error-correcting Codes ...... 389 8.3.2 Exercises for Section 8.3...... 391 8.3.3 Sir Ronald A. Fisher...... 393 8.4 Finite Analytic Geometry...... 394 8.4.1 Finite Analytic Planes...... 395 8.4.2 Ovals in Finite Projective Planes ...... 398 8.4.3 Finite Analytic Spaces ...... 399 8.4.4 Exercises for Section 8.4...... 401 8.4.5 Projects for Chapter 8...... 404 8.4.6 Suggested Readings ...... 407

9 Differential Geometry 409 9.1 Overview and History...... 409 9.1.1 Exercises for Section 9.1...... 411 9.2 Curves and Curvature...... 413 9.2.1 Exercise for Section 9.2...... 420 9.2.2 Sir Isaac Newton...... 423 9.3 Surfaces and Curvature...... 424 9.3.1 Surfaces ...... 424 9.3.2 Curvature...... 426 9.3.3 Surfaces of Revolution...... 428 9.3.4 Exercises for Section 9.3...... 429 9.4 Geodesics and the Geometry of Surfaces...... 432 9.4.1 Geodesics...... 432 9.4.2 Geodesics on Surfaces of Revolution...... 435 9.4.3 Arc Length on Surfaces...... 436 xii Contents

9.4.4 The Gauss-Bonnet Theorem...... 439 9.4.5 Higher Dimensions...... 439 9.4.6 Exercises for Section 9.4...... 441 9.4.7 Albert Einstein...... 444 9.4.8 Projects Chapter 9 ...... 445 9.4.9 Suggested Readings...... 446

10 Discrete Geometry 447 10.1 Overview and Explorations...... 448 10.1.1 Distances between Points ...... 448 10.1.2 Triangulations ...... 449 10.1.3 The Art Gallery Problem...... 449 10.1.4 Tilings...... 450 10.1.5 Voronoi Diagrams ...... 451 10.1.6 Exercises for Section 10.1...... 453 10.1.7 Paul Erdos...... ˝ 457 10.2 Points and Polygons ...... 457 10.2.1 Distances between Points ...... 457 10.2.2 Triangulations ...... 459 10.2.3 The Art Gallery Problem...... 462 10.2.4 Exercises for Section 10.2...... 465 10.3 Tilings...... 469 10.3.1 Exercises for Section 10.3...... 474 10.3.2 Branko Grunbaum...¨ ...... 477 10.4 Voronoi Diagrams...... 477 10.4.1 Exercises for Section 10.4...... 481 10.4.2 Projects for Chapter 10...... 483 10.4.3 References...... 485

11 Epilogue 487 11.1 Topology ...... 488 11.2 Henri Poincare...... ´ 489

A Deﬁnitions, Postulates, Common Notions, and Propositions from Book I of Euclid’s Elements 491 A.1 Deﬁnitions...... 491 A.2 The Postulates (Axioms)...... 492 A.3 Common Notions...... 492 A.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 503 D.1 Vectors...... 503 D.2 Matrices...... 504 D.3 Determinants...... 505 D.4 Properties of Matrices...... 506 D.5 Eigenvalues and Eigenvectors...... 506

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

F Elements of Proofs 511 F.1 Direct Proofs...... 511 F.2 Proofs by Contradiction...... 512 F.3 Induction Proofs...... 512 F.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 most excellent guide in governing our reason, it does not, as regards stimulation to discovery, compare with the power of sharp distinction which belongs to geometry. — Galileo Galilei (1564–1642)

Geometry combines visual delights and powerful abstractions, concrete intuitions and general theories, historical perspective and contemporary applications, and surprising insights and satisfying certainty. In this textbook, I try to weave together these facets of geometry. I also want to convey the multiple connections that topics in geometry have with each other and that geometry has with other areas of mathematics. The connections link chapters together without sacriﬁcing the survey nature of the whole text. Geometric thinking fuses reasoning and intuition in a characteristic fashion. The enduring appeal and importance of geometry stem from this synthesis. Mathematical insight is as hard for mathematics students to develop as is the skill of proving theorems. Geometry is an ideal subject for developing both, leading to deeper understanding. However, geometry texts for mathematics majors often emphasize proofs over visualization, whereas some texts for mathematics education majors focus on intuition instead. This book strives to build both and so geometrical thinking throughout the text, the exercises, and the projects. The dynamic software now available provides one valuable way for students to build their intuition and prepare them for proofs. Thus exercises beneﬁting 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 students internalize mathematical concepts and thinking. (Weabbreviate the National Council of Teachers of Mathematics throughout as NCTM.) There are two natural audiences for a geometry course at the college level—mathematics majors and future secondary mathematics teachers. Of course, the two audiences overlap consid- erably. In many states, however, requirements for secondary mathematics education majors can differ substantially from those for traditional mathematics majors. This book seeks to serve both audiences, partially by using a survey format so that instructors can choose among topics. In addition, those sections matching CCSS expectations and NCTM recommendations for teacher preparation 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 vital for teacher preparation, earlier exercises require less sophistication than later ones. A later subsection 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 process I realized again how geometric my thinking is and how much I enjoy sharing the elegance and excitement I experience in geometry. I hope that some of my enthusiasm shows through.

Geometric Intuition Everyday speech equates intuitive with easy and obvious. However, psychology research con- firms what mathematicians have always understood: people build their own intuitions through reflection on their experiences. My students often describe this process as learning to think in a new geometry. As Galileo’s quote introducing this preface suggests, geometry has for centuries been an ideal place for developing mathematical intuition. Since the advent of analytic geometry (at the end of Galileo’s life) mathematicians have repeatedly turned geometric insights into algebraic formulations. The applicability, efficiency, simplicity, and power of algebra have reasonably led educators from middle school through graduate school to emphasize algebraic representations. In my view, the success of algebra has so focused the curriculum that students’ geometrical thinking often lags far behind. My text tries to correct that imbalance without neglecting the power of algebra. The NCTM calls for high school students to develop geometric intuition and understanding. Similarly the CCSS expects students to build geometrical thinking in a variety of 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 obvious means to this end. Many of the more than 750 exercises ask students to draw or create their own figures and models. In addition, I have included many exercises and projects requesting students to explore and conjecture new ideas, as well as explain or prove unusual properties. I advocate having 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 want to tie my book to one program. However, in my teaching experience students gain different and often more insight working with physical models than from manipulating computer models. So I strongly encourage instructors to give students physical models to use for as many topics as the class time, budget, and their creativity allows. I provide several suggestions in the text and in the projects. I hope the text’s explanations are clear and provide new insights, but I know that students need to reflect on the text and the exercises. I also hope that students find the many non-routine problems and projects challenging, but solvable with effort; such challenges enrich intuition.

The Role of Proofs Since Euclid, over two thousand years ago, proofs have had a central place in mathematical thought. Non-mathematicians often think the value of mathematics is restricted to its amazing ability to calculate “answers.” Certainly, many applications of mathematics rely on Preface xvii computational power. However, people applying mathematics want to know more than that there is an answer—after all, an astrologer gives answers. We want to know that the answers are valid. While confidence in much of science depends on experiments, it also often depends on mathematical models. The models make explicit assumptions about how some aspect of the world behaves and recasts them in mathematical terms. So applications of mathematics require that 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 for every mathematics student to prove every result. Most people acknowledge the value of honing students’ ability to reason critically, and mathematical proofs certainly contribute greatly to that skill. 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 in high school courses now varies greatly across the United States. Still, the NCTM and CCSS call for high school students to do a certain amount of work with proofs. It follows that high school mathematics teachers need substantial background and facility in proof. I have written this text for mathematics majors and future high school mathematics teachers, and I think there is a range of proof experiences both audiences need. All these students need facility in making good 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 reasoning leading from them to their conclusions. I think it also means that they should be introduced to more formal proofs and axiomatic systems, although I don’t think that should be the primary focus 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 goal is to include enough coverage of the content of Euclidean geometry for students who have not had 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 they already have a comfortable intuition. Hence in that chapter I haven’t made explicit the many subtleties and assumptions discussed in Chapter 2. Instead, I ask students to build on Euclid’s theorems so that they can prove results that aren’t instantly apparent, although they should be plausible. Chapter 2, which looks more carefully at axiomatics, makes axiomatic systems explicit and builds up theorems carefully from the axioms. The level of proof in later chapters lies 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 mathematics more sophisticated than in Chapter 1. Thus the value of proofs in them also connects with the goal 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 the college level can make the two-column format artificial and overly restrictive. A good proof ought to help the reader understand why the theorem is correct, as well as make the correctness of the 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 Notation Proofs end with the symbol . Examples end with the symbol ♦. Exercises or parts of exercises with 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 common era) 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.

Prerequisites In general, students need the maturity of Calculus I and II, although only Chapter 9 and Section 2.4 use calculus extensively. Chapter 9 needs some content from Calculus III as well. Appendix E summarizes the material from Calculus III used in Chapter 9. Of course, additional mathematical maturity and familiarity with proofs will help throughout the book. Sections 3.3 and 3.5 and Chapter 9 require an understanding of vectors. Sections 5.3, 5.4, 5.5, 7.3, 7.4, 7.5, 7.6, and 8.4 depend on a more extensive understanding of linear algebra. Appendix D summarizes the linear algebra material needed in the text. Although Chapter 6 builds on concepts from Chapter 5,it doesn’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 any prior familiarity with the concepts.

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

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

History Geometry reveals the rich influences over the centuries between areas of mathematics and between mathematics and other fields. Students in geometry, even more than other areas of mathematics, benefit from historical background. The introductory sections of each chapter seek 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 of the 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 my mathematical 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 greater mathematicians than I built on intuition and visualization for their abstract insights, proofs, and theories. Perhaps this understanding will help the next generation as well.

Chapter Content Each chapter starts with an overview, including a discussion of the relevant history, and ends with projects and a list of suggested readings. Geometry is blessed, more than other areas of mathematics, with many wonderful and accessible expository writings as well as texts. (The vast number 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 their courses.) 1. Euclidean Geometry. Most of this chapter considers plane geometry and follows the lead of the ancient Greeks’ approach, especially Euclid’s synthesis. (Appendix A gives the definitions, axioms, and propositions of Book I of Euclid’s Elements.) Since high school geometry courses include much of Euclid’semphases, this approach adds context to a teacher preparation course. Students’ preparation varies greatly, so instructors should adjust their pace and coverage according to how much this material is review for the students. All the exercises 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 geodesic domes, and the sphere. The material on the sphere is a useful transition into a study of non-Euclidean geometry, although it is presented as part of Euclidean geometry. 2. Axiomatic Systems and Models. The first section introduces axiomatic systems and investigates simple ones. The next section considers a high school axiomatic system for Euclidean geometry 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 the School Mathematics Study Group.) The final section explores models and metamathematics. Instructors wishing to include more experience with axiomatic systems and models can include material from Chapter 8. 3. Analytic Geometry. While high school students use analytic geometry, they often don’t understand it and often don’t see many of the traditional topics. And, although calculus texts include topics such as parametric equations and polar coordinates, instructors often leave them out for lack of time. In addition to these topics, later sections discuss Bezier´ curves in computer aided design and geometry in three and more dimensions. 4. Non-Euclidean Geometry. The bulk of the chapter develops hyperbolic geometry axiomatically. In addition to typical axioms, we assume the first twenty-eight of Euclid’s theorems, which also hold in hyperbolic geometry. By assuming them we can use familiar approaches to focus on how this geometry differs from Euclidean geometry. Models help illustrate the concepts and theorems. The final section considers spherical and single elliptic geometry. xx Preface

5. Transformational Geometry. The first two sections develop the key ideas of transformations and plane isometries without linear algebra. The CCSS strongly emphasize transformations in 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 in hyperbolic and spherical geometries. (See Project 22.) The next three sections use linear algebra to delve into isometries more deeply, and into similarities, affine transformations, and transformations in higher dimensions. The final section investigates inversions using complex numbers and relates to the Poincare´ disk model of hyperbolic geometry and is not used elsewhere in the text. Appendix D covers the linear algebra needed for this and subsequent chapters. 6. Symmetry. While this material uses concepts from Chapter 5, it doesn’t depend on linear algebra. Students find this material accessible, compared with some of Chapter 5, and gain insight 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 and axiomatically. Later sections use linear algebra extensively and provide connections with computer graphics and the special theory of relativity. 8. Finite Geometry. Since the late nineteenth century geometers have drawn important insights about traditional geometry from the study of simplified finite systems. Section 8.2 discusses the most important of these, finite affine and projective planes, axiomatically. Section 8.3 generalizes the material to balanced incomplete block designs. The final section explores analytic models of finite affine and projective planes and spaces over the fields Zp,the integers (mod p), where p is prime. 9. Differential Geometry. Differential geometry deserves an entire undergraduate semester course, but many schools can’t offer it. I think students in a survey course benefit from an introduction to this vital area of geometry. I try to convey here its geometric insight and introduce some key geometric ideas—curvature and geodesics, and I endeavor to minimize the machinery of multivariable calculus. The chapter connects differential geometry to Eu- clidean, spherical, and hyperbolic geometries. Students need little more from multivariable calculus 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 develop them more fully, with relevant theorems and more in-depth problems.

Course Suggestions This text supports a variety of approaches to geometry and different levels of coverage of the material. Many of the sections beneﬁt from more than one class period, especially to enable students to present problems or projects. The entire book would require a full year geometry course 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 preparation, 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), and some of non-Euclidean geometry (Chapter 4). As time, interest and student background indicate, topics from analytic geometry (Chapters 3) and symmetry (Chapter 6) are valuable supplements for future teachers. B. Historical Survey. Chapters 1, 2, 4, 5, and 7 provide an understanding of the important historical sweep of geometry through the nineteenth and early twentieth centuries. In 1800 there was just Euclidean geometry (Chapter 1). Geometrical thinking expanded enormously, including non-Euclidean geometry (Chapter 4), transformational geometry (Chapter 5), and projective geometry (Chapter 7), among others. The transformations of projective geometry provided a vital uniﬁcation of geometric thought, both historically and pedagogically. Because of these advances, mathematicians realized the need for a careful investigation of proofs, 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 interleave Chapters 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 background will beneﬁt from Chapters 1, 2, 3, 5, and 6. Chapters 4, 7, 8, 9, and 10 can stretch better prepared students in different ways.

Dependence and Links Between Chapters I have tried to keep chapters as independent as reasonable. Students with a decent geometry understanding from high school will have adequate Euclidean and analytic geometry background for all chapters except Chapter 4, which depends explicitly on Chapter 1. The basic concepts of axiomatic systems and models from Chapter 2 appear in Chapters 4, 7, and 8. Chapter 5 is a prerequisite for Chapters 6 and 7. Sections 5.4 and 6.6 consider aspects of fractals. Sections 6.5 and 7.6 brieﬂy consider aspects of the special theory of relativity, and Section 9.4 touches on 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’sformula (Theorem 1.5.1) is used in Section 10.4. Project 14 in Chapter 3 asks 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.

Acknowledgments This text is based on my book The Geometric Viewpoint: A Survey of Geometries, published by Addison Wesley in 1998. First, the acknowledgments from the earlier text: I appreciate the support and helpful suggestions that many people made to improve this book, starting with those students who studied from the rough versions. My previous editors, Marianne Lepp and Jennifer Albanese, and my production supervisor, Kim Ellwood, provided much needed direction, encouragement and suggestions for improvement. I am grateful to the reviewers, whose insightful and careful critiques helped me greatly. They are: Bradford Findell, University of 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 only painstakingly made many of the ﬁne illustrations that are an integral part of this book. He also pondered with me over the years the nature of geometric thinking and the connections of geometry with other ﬁelds. Connie Gerads Fournelle helped greatly with her valuable perspective on the text, both as a student using an early version and as my assistant writing answers to selected problems. 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 their students were kind enough to use an early version of this rewrite in their classes and provide me 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, also of 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 to St. John’s University for funding the sabbatical making this rewrite possible. Thanks go to the Mathematics Department at Virginia Tech for welcoming me, encouraging me, and providing the materials and space needed in the long rewriting process. I appreciate the extensive and cheerful secretarial support Suzette Ehlinger and Gail Schneider have provided. I am grateful to the editorial staff at the Mathematical Association of America for their hard work: Beverly Ruedi, Carol Baxter, the copy editor, and editors Stan Selzer and Zaven Karien provided the technical support and positive encouragement needed to make this book a reality. Finally I thank my wife, Jennifer Galovich, for her unswerving support and love throughout the years of the writing and now rewriting. If you have comments or suggestions for improvement, please contact me by e-mail at [email protected].

Thomas Q. Sibley Mathematics Department St. John’s University Collegeville, MN 56321-3000

Answers to Selected Exercises

Section 1.1

1.1.1. 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 of 4, 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 reasoning about infinite processes based on finite steps. 8 2 − π 2 ≈ . 3 37.8 ≈ . 1.1.3. Difference is ( 9 20) (20) (10) (20) 37 8 (units) ,erroris π(10)2(20) 0 6%, 8 2 ≈ π 2 π ≈ . ( 9 2r) r reduces to 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 the wizard’s flight. Then x + z = 300 and z2 = 2002 + (x + 100)2.Sox = 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. n = + / (b) Divide oblong into two triangles, one upside down. Thus i=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 itself to get a · b. In geometric terms, the square with side ab has the same area as a rectangle with sides a and b. √ = + 2 − − 2 = = 1+ 5 ≈ . (b) b a(a b)orb ab a 0. Then b a( 2 ) 1 618a. 1.1.15. (a) 72◦,54◦, 108◦,36◦, 108◦,72◦,72◦,36◦. (b) Use angles. ABH ∼DAB ∼AFG.

(c) If AB = 1, BC = 1 and GC = 1, and AC = AG + GC = 1 + x. √ ∼ 1+x = 1 2 + − = = −1+ 5 + = (d) Since√ ABD BGC, 1 x and x x 1 0. So x 2 or 1 x (1 + 5)/2. c = a , c = b . = 2 1.1.17. Angles show similar triangles ADC and AC B.So a y b x So cy a and cx = b2.Alsox + y = c,givingc2 = c(x + y) = a2 + b2.

515 516 Answers to Selected Exercises

Section 1.2 1.2.1. From the construction use SSS (I-8) to show BAF =∼ BEF and so ∠ABF =∼ ∠EBF. 1.2.2. By construction and SSS (I-8) ABD =∼ GEH =∼ GEI and so ∠ABD =∼ ∠GEH =∼ ∠GEI. 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 congruence theorem—but shows similar triangles. Neither ASS nor SSA are congruence theorems. Consider PQR and PQT with R between P and T and PQ = 52, QT = 25, PT = 31.5 and PR = 16.5. Let V be between R and T with PV = 24 and use the converse of the Pythagorean theorem. 1.2.6. Suppose in ABC that AB =∼ AC.Now∠BAC =∼ ∠CAB. Then by SAS (I-4), ABC =∼ AC B. Hence ∠ABC =∼ ∠AC B. 1.2.8. Suppose that AC ⊥ BE, AE ⊥ CF and AD bisects ∠EAC. (a) In ADF and ADB, ∠AFD =∼ ∠ABD and ∠FAD =∼ ∠BAD. By Theorem 1.1.1 (I-32) we also have ∠ADF =∼ ∠ABF. Since AD =∼ AD, by ASA (I-26), we have ADF =∼ ADB. (b) From part (a) we have AF =∼ AB. By ASA (I-26), AFC =∼ ABE. 1.2.11. (a) For a + b on a line construct adjacent segments of lengths a and b.Fora − b,make the segments overlapping with a common endpoint. PQ = PS . (b) By similar triangles, PR PT (c) If PQ = 1, PR = a, and PS = b, then PT = a · b.IfPQ = 1, PT = a, and PR = b, then PS = a/b. 1.2.13. (a) Pick any A on a circle with center O and construct B and C on the circle so that AB = AC = AO. Construct D and E on the circle so that BD = CE = AB. Then ADE is equilateral. (b) Continue from (a) to construct F on the circle so that DF = BD. Then ABDFEC is a regular hexagon. (c) Pick any point A on a circle with center O and construct the diameter AC through O. Construct diameter BD perpendicular to AC at O. Then ABCD is a square. 1.2.15. (a) Let ABC be any triangle and G be the intersection of the perpendicular bisector GD of AB and the perpendicular bisector GE of BC. Then ADG =∼ BDG and BEG =∼ CEG by SAS (I-4). Then AG =∼ BG and BG =∼ CG. (b) (Continuing from part (a).) Let F be the midpoint of AC. Then AFG =∼ CFG by SSS (I-8). Thus ∠AFG =∼ ∠CFG. Together they make a straight angle, so they are each right angles, FG ⊥ AC and FG is the perpendicular bisector of AC. Answers to Selected Exercises 517

1.2.18. Let m∠ABC = p and m∠BCA = q. Use isosceles triangles, Theorem 1.1.1 and algebra to show that 3p = q and p = (180/7)◦ ≈ 25.7◦. 1.2.20. m∠QTS = 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 BD = x + a and c2 = a2 + b2 + 2ax. Section 1.3 1.3.1. (a) 55◦. Reasoning: By I-29, m(∠FGD) = 35◦. From Theorem 1.1.1, m(∠EFG) = 180◦ − m(∠FDG) − m(∠FGD) = 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 parallelogram with at least one right angle and at least two congruent adjacent sides. A rhombus is a parallelogram with at least two congruent adjacent sides. 1.3.3. (a) By I-29 ∠RPQ =∼ ∠PRS. Since PR =∼ PR and PQ =∼ RS,bySAS,PQR =∼ RSQ. (b) From part (a), ∠S =∼ ∠Q and ∠RPQ =∼ ∠PRS. From Theorem 1.1.1, m(∠S) + m(∠SRQ) = 180◦. Then from I-28, PS QR and PQRS is a parallelogram. 1.3.5. (a) Proof. Suppose that ABCD is a parallelogram with diagonal AC.ByI-29 ∠CAB =∼ ∠AC D and ∠AC B =∼ ∠CAD. Since AC =∼ AC, by ASA (I-26) ABC =∼ CDA. The two triangles have the same area and together they have the area of ABCD. So each have half of the area of ABCD. 1.3.6. (a) Proof. Assume the given relationships. Use I-23 to construct an angle ∠PEH ∠ congruent to ABE with P onthesamesideofBE as D.ByI-28PE AB.By←→ Playfair’s axiom, there is only one parallel to AB through E.SoP must be on DE and so ∠ABE =∼ ∠DEH. 1.3.7. Proof. Let ABCD be a parallelogram with (∠ABC) = 90◦. Since AB CD and by I-29 m∠ABC + m∠BCD = 180◦, m∠BCD = 90◦. The other angles are similar. 1.3.9. (b) Proof. Let ABCD be a kite with AB =∼ BC and CD =∼ AD.DrawBD. By SSS =∼ ∠ =∼ ∠ BAD BCD and so←→BAD←→ BCD. (c) Let O be intersection of BD and AC. Show CBO =∼ ABO.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 PR and S and T are on the same side of PR with PS =∼ QT and QS =∼ RT. By SSS PQS =∼ QRT. Then ∠SPQ =∼ ∠TQRand these are corresponding angles, showing PS QT. 518 Answers to Selected Exercises

(b) (continuing the proof) By PQS =∼ TSQ, ST =∼ PQ. Further, ∠PQS =∼ ∠TSQ.SobyI-27ST PQ.

Section 1.4

1.4.1. 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 AG with 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 ABEF with base AB and height AF equal to the height CDof ABC. Then ABEF contains ABC. Construct D on AB so that CD ⊥ AB. Then ADC =∼ CFA and BDC =∼ CEB. Hence the area of ABC is half of the area of ABEF. 1.4.4. (a) 2 : 1 because AFB ∼CFE and AB = 2 · CE. (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 1 3 so 4 of area of ABCD. Similarly for BCE. Hence ADEB has 4 oftheareaof ADEB. (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 from semicircles. 1.4.8. First construct DPQ congruent to ABC based on Exercise 1.2.2.−−→ Draw the parallel to PQ through E.LetF be the intersection of this parallel with DQ. Then DEF ∼ ABC. 1.4.10. (d) is similar since it halves both the x and y-values and in general, y = sin(x) is similar = 1 to y k sin(kx). 1.4.12. Use Theorem 1.4.4 to show PSU ∼PQR ∼SQT ∼UTR. Use Theorem 1.4.3 to show TUS ∼PQR. 1.4.13. (a) Proof. By Exercise 1.2.17, the measure of an inscribed angle of a given arc is half the measure of the corresponding central angle. Thus inscribed angles of the same arc of a circle are congruent. So ∠SPR =∼ ∠SQR and ∠PSQ =∼ ∠PRQ.By Theorem 1.4.2 PST ∼QRT.SoPT/ST = QT/RT. Cross multiply to get PT · RT = QT · ST. 1.4.14. Construct EF AB with F on BC. Because BFEDis a parallelogram, Exercise 1.3.8 implies BF = DE. As with the argument in the proof, BF, FC, and BC areinthe same 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 T1, T2 T2,...,Tn Tn. These triangles also have the ratio k. Then = n = n 2 = 2 n = 2 the Area(P) i=1Area(Ti ) i=1 k Area(Ti ) k i=1Area(Ti ) k Area(P ). Answers to Selected Exercises 519

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 the x-direction. = x (b) Use integration by substitution where u k . (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 as sides and so all angles formed by diagonals and sides as well as angles bounded by adjacent sides.

(b) Let A1 A2 A3 A4 and A1 A2 A3 A4 be two√ squares with√ sides of length s and ks, respectively. 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 1.5 1.5.2. Replace B by πr 2. 1.5.3. Use any interior point of the polyhedron as the vertex of pyramids whose bases are the faces 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)r 2. 1.5.10. Similar triangles have the same angle sum and so the same area. w 1 w 1.5.11. (a) The volume is l h. Each pyramid has volume 3 l h. (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 ABCA , A B C C and A BB 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 √1 l3 = 4 l3 2. √ 6 2 3 , / , 2 (d) l l 2 l 3 . 520 Answers to Selected Exercises

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, E P = E D, 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, πr 2/12. 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 ∠ABN and ∠ABP. 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 and two for each original edge.

Section 2.1 ←→ 2.1.1. PQ is the set of points R so that R is on the line PQ 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 definitions are 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23. (We don’t define some of these terms this way.) 2.1.3. (a) For a contradiction, suppose B(A)C. Then by axiom (i), C(A)B, which we showed was 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 and from 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 .So P, 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) x likes some y. Now use axioms (ii), (iii) and (v). (b) 4 (c) 8 Answers to Selected Exercises 521

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 PQcontains all points R on line PQso that in postulates 3 and 4 R corresponds with a non-negative number, provided P corresponds with 0 and Q corresponds with 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 2.2.5 there is exactly one line k in this plane perpendicular to OX through O. By postulate 3 there is a point Y on k so that the distance from O to Y equals the distance from O to X. By postulate 4 we can choose this common distance to be 1. For←→ any point A of this plane, construct the parallel m to k and call its intersection with OX the point Ax . Deﬁne Ay similarly on k.GiveA the coordinates (a, b), where a is the number corresponding 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, ∠BCA and ∠CAB. (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 there are P−1 and P0.5 different from P0 and P1 with P0 between P−1 and P1 and with P0.5 between P0 and P1. By II-3 P2, P−1 and P0.5 must also differ from each other.

including P0. Repeat with A = P2. 2.2.14. (e) Note that we have ∠ABD =∼ ∠A B C . Use SAS.

Section 2.3 2.3.1. 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. 522 Answers to Selected Exercises

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 empty set 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 first model, a1 and a2 both beat a3. However, in the second model no two teams beat the same 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 of generality let {A, B, C} be a circle. Now D must be on a circle with any 2 of the others, but we can’t have {A, B, C, D} as a circle because of axiom (ii). Hence each of 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.LetR be the third point on the line with P and Q. Then h has parallels through P and R.For exactly 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 3.1 3.1.1. 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.$ % 2 + 2 = 1 2 (c) x y 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. Answers to Selected Exercises 523

3.1.7. (a) {(x, y):(x − a)2 + (y − b)2 = r 2}. (c) {(x, y):(x − a)2 + (y − b)2 < r 2}. 2 2 2 2 3.1.9. Distance formula: k (x1 − x2) + j (y1 − y2) , Circle: ((x − a)/k)2 + ((y − b)/j)2 = r 2. 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 3.2 3.2.4. (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.

2 3.2.11. (c) 2xy − 1 = 0, ac − b =−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 3.3 3.3.1. 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 adjacent circles. 524 Answers to Selected Exercises

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 (−2sint, 2 cos t). The second is twice as long as the first because a 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), where the curve crosses itself. (c) (cos√ t − t sin t, sin t + t√cos t). Directions: (1, 2π) and (1, 4π), lengths: 1 + 4π 2 ≈ 6.36 and 1 + 16π 2 ≈ 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-ory-coordinate. Taxicab distance is greater by at most a factor of 2. (b) Squares at a 45◦ angle to the axes.

Section 3.4 3.4.3. 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. Answers to Selected Exercises 525

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 3.4.9. (a) T3(x) = x − x /3! 3 5 (b) T5(x) = x − x /3! + x /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 3.5 −→ −→ 3.5.1. 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,weareat t ,atα = 0, β = 1weareat 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 common coordinates. 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◦. 526 Answers to Selected Exercises

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

Dimension 12 3 4 e/v 12 3 4 f/v 01 3 6 c/v 00 1 4 f/e 01 2 3 c/e 00 1 3 c/f 00 1 2 v 24 8 16 e 141232 f 01 6 24 c 00 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. Dimension 1 2 3 4 e/v 123 4 f/v 013 6 c/v 001 4 f/e 012 4 (a) c/e 001 4 c/f 001 2 v 234 5 e 13610 f 01410 c0015

(b) For dimension n,ifn 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 4.1 4.1.1. 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. Answers to Selected Exercises 527

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 4.2 4.2.3. (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 sensed parallel 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 ultraparallel or sensed parallel. Since the angle of parallelism is acute, lines with a common perpendicular are ultraparallel. 4.2.7. Euclid’s parallel lines are ultraparallel in hyperbolic geometry. 4.2.9. ∠CB is bigger than ∠CA. 4.2.10. It is less than 360◦.

Section 4.3 4.3.1. 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 BA.← Then→ ABDX is a Saccheri quadrilateral.←→ Theorem 4.3.1 and Corollary 4.2.3 show←→ AX is ultraparallel to BD. 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 sensed parallel AC to another BD 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 Theorem 3.3.3. 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 triangle and a k-gon and apply the induction hypothesis and Theorem 3.3.3. This proof holds for nonconvex polygons, although one must show one can always divide the k + 1-gon into a triangle and an k-gon. 528 Answers to Selected Exercises

Section 4.4 4.4.1. Consider three congruent triangles ABC, BAD and EFG, where EFG is dis- joint from the other two and these other two overlap only on the side AB. By postulate 19 R =ABC ∪EFG has the same area as the quadrilateral AC BD.However,R has one more side EF then AC BD has since we don’t include AB twice in the quadrilateral. So the side EF must have area 0. Also, EF is made of the union of the point E and the rest 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◦,soK = 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 as i→∞.

4.4.8. Kn = (n − 2)K .

Section 4.5 4.5.1. 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 a given 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, 2πr 2.

Section 5.1 5.1.1. 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 are the 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) and y = β(x)forx 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. α β 3 . (d)√ For ,1;for , 0 5. 3 . + . (e) √x,05x 0 5. √ (g) 3 0.5x + 0.5, 0.5 3 x + 0.5. Answers to Selected Exercises 529

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 5.2 5.2.1. 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 point on m goes to another point on m,som 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 aspect of a glide reflection means only lines parallel to the line of reflection have a chance of being stable. But other lines parallel to this line switch sides from the reflection part. 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). μ , = , − μ , = − , μ , = − + , − + 5.2.9. (a) x (x y) (x y), y(x y) ( x y), 3(x y) ( y 1 x 1). μ ◦ μ , = − , − ◦ , (b) y x (x y) ( x y), rotation of 180 around (0 0). μ ◦ μ , = + , − + ◦ , μ ◦ μ , = (c) 3 x (x y) (y 1 x 1), a rotation of 270 around (1 0). x 3(x y) (−y + 1, x − 1), a rotation of 90◦ around (1, 0). μ ◦ μ ◦ μ , = + , + = (d) 3 y x (x y) (y 1 x 1), glide reflection along y x. μ ∼ μ ◦ μ μ ◦ 5.2.11. (a) m fixes points on m, namely A and B. AC D = k m (A)CD so A k μ = = m (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 that isometries preserve distance. 530 Answers to Selected Exercises

5.2.15. τ 1 ◦ τ 2 = τ 2 ◦ τ 1.LetP 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 SAS Pτ 1 ◦ τ 2(P) = Qτ 1 ◦ τ 2(Q). Thus PQτ 1 ◦ τ 2(Q)τ 1 ◦ τ 2(P) is a parallelogram and τ 1 ◦ τ 2 is a translation. 5.2.19. (a) V consists of translations, 180◦ rotations, vertical and horizontal mirror and glide reﬂections. Proof. Let l be a vertical line. ι(l) = l is vertical, so ι ∈ V.Ifα, β ∈ V, β(l) and so α(β(l)) are vertical, so α ◦ β ∈ V. Suppose α ∈ V and k the vertical linesothatα(k) = l. Then α−1(l) = k,soα−1 ∈ V.

Section 5.3 ⎡ ⎤ ab0 ab 5.3.3. The matrix becomes ⎣cd0⎦. cd 001 ⎡ ⎤ 10b 5.3.8. (a) ⎣010⎦. ⎡001⎤ 100 (b) ⎣0 −10⎦. ⎡001⎤ 10b (c) ⎣0 −10⎦. ⎡001⎤ 010 (d) ⎣100⎦. ⎡001⎤ 01b (e) ⎣10b⎦. 001 ⎡ ⎤ cos(30) − sin(30) 0 5.3.9. (a) ⎣sin(30) cos(30) 0⎦. ⎡ 001√ ⎤ cos(30) − sin(30) 3.5 −√ 3 (b) ⎣sin(30) cos(30) 2 − 3 3/2⎦. 00 1 5.3.10. A is a rotation, B is a translation, C is a mirror reﬂection, D is a glide reﬂection, E is a rotation. 5.3.12. (b) c = 2 f . ⎡ ⎤ −100 5.3.16. (a) ⎣ 0 −10⎦ . 001 Answers to Selected Exercises 531

⎡ ⎤ −102u (b) ⎣ 0 −12v⎦ . 001 (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 5.4 5.4.2. 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. ⎡ ⎤ ⎡ ⎤ 0 −4 −11 04 13 5.4.4. direct: ⎣40 −7 ⎦ and indirect: ⎣40−7⎦. 00 1 00 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. √ θ θ/ (c) M is a rotation by and a scaling by r, S is a rotation√ by 2 and a scaling by r. θ/ 3 θ/ (d) For C use a rotation√ by 3 and a scaling by r and for N use a rotation by n and a scaling by n r. ⎡ ⎤ ⎡ ⎤ 22−2 2 −2 −2 5.4.7. (a) ⎣−22 4⎦ and ⎣−2 −24⎦. ⎡ 001⎤ ⎡001⎤ −34 2 −3 −410 (b) ⎣−4 −311⎦ and ⎣−43 5⎦. 001 001 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 abc in that square. For ⎣de f⎦, the following must be between 0 and 1: c, a + c, b + c, 001 a + b + c, f , d + f , e + f , and d + e + f . 532 Answers to Selected Exercises

Section 5.5 5.5.3. 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 with 1s on the main diagonal, any numbers in the far right column (except for a 1 in the lower right corner) and 0s elsewhere. 5.5.7. (a) cos(180◦) =−1 gives the upper two diagonal entries. Note that (0, 0, z, 1) is mapped ⎡to itself. ⎤ ⎡ ⎤ 10 00 −1000 ⎢0 −100⎥ ⎢ 0100⎥ (b) ⎢ ⎥, ⎢ ⎥, rotation of 180◦ around the z-axis. ⎣00−10⎦ ⎣ 00−10⎦ 00 01 0001 The other of these two matrices. ◦ (d) Rotations of 120⎡ around opposite⎤ vertices of the cube. 1000 ⎢00−10⎥ (e) For example, ⎢ ⎥ is a 90◦ rotation around the x-axis and ⎣0100⎦ 0001 ⎡ ⎤ 0100 ⎢0010⎥ ⎢ ⎥ is a 120◦ rotation around the axis through (1, 1, 1, 1) and ⎣1000⎦ 0001 − , − , − , (⎡ 1 1 1 1). ⎤ 10 0 0 ⎢0 cos θ − sin θ 0⎥ (f) ⎢ ⎥. ⎣0sinθ cos θ 0⎦ 00 0 1 ⎡ ⎤ cos θ 0 − sin θ 0 ⎢ 01 0 k⎥ 5.5.9. (a) ⎢ ⎥ (or with the − on the other sin θ, depending on the ⎣sin θ 0 cos θ 0⎦ 00 0 1 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 = 0or[1, 0, 1, −2], the fixed 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], rotation of 120◦. 5.5.13. (a) A 4 × 4 affine matrix where the upper left 3 × 3 submatrix is r times an orthogonal matrix. (b) The dilation affects only the upper left 3 × 3 submatrix, multiplying each of its entries by r. Answers to Selected Exercises 533

Section 5.6 5.6.1. Circle x2 + y2 = r 2 inverts to x2 + y2 = 1/r 2.

5.6.2. d(O, P) · d(O,νc(P)) = d(O,νc(P)) · d(O, P). 5.6.3. Case 1. r = 0givesq = ps/r and f (z) = p.Case2.r = 0. Either p = 0ors = 0. For p = 0, f (z) = q/s.Fors = 0, f is undeﬁned. As z → s/r, the denominator of f (z) goes to 0. As z →∞the limit makes the constant terms irrelevant. To convert the formulas of Theorem 5.6.4, ﬁrst ignore the complex conjugate to rewrite r 2/(z − w) + w as (r 2 + wz − w2)/(z − w) = (wz + r 2 − w2)/(z − w), which is a Mobius¨ transformation. 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) ν D(1, 0) = (1, 0), νC (ν D(1, 0)) = (4, 0), νC (1, 0) = (4, 0) and ν D(νC (1, 0)) = (0.25, 0). No. (b) ν D(0, 3) = (0, 1/3), νC (ν D(0, 3)) = (0, 12), νC (0, 3) = (0, 4/3) and ν D(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;ifr < 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 6.1 6.1.1. 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 symmetry may be advantageous. 6.1.2. Rotations of multiples of 60◦ and mirror reﬂections over 6 lines through the center. 534 Answers to Selected Exercises

6.1.3. (a) No symmetry: F, G, J, L, P,Q, R; Vertical symmetry: A, M, T, U, V,W,Y; Horizontal symmetry: 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 order of 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: rotations of 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 reflections, including vertical; color switching include rotations of 60◦, 180◦ and 300◦ and 3 mirror reflections, including horizontal. 6.1.9. (a) Mexican: translations, 180◦ rotations, vertical mirror reflections, and horizontal glide reflections. Chinese: translations, 180◦ rotations, vertical and horizontal mirror reflections, and horizontal glide reflections. 6.1.11. (a) 25%, 25%. Use rotations. (b) 33.3%, 33.3%.

Section 6.2

6.2.1. 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 6.3 6.3.2. If τ is the smallest translation to the right, all other translations are of the form τ n,for some n.Ifn > 0, τ n represents a translation of n to the right. If n < 0, it is a translation of n to the left. τ 0 is the identity. Answers to Selected Exercises 535

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. Then the 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). μ , = − , (d) d (x y) (2d x y). Section 6.4 6.4.1. 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, the identity and the central symmetry.

6.4.3. 4n. Dnh. All symmetries of these prisms are symmetries of a prism with a regular 2n-gon as a base. ⎡ ⎤ −10 0 6.4.5. ζ = ⎣ 0 −10⎦. 00−1 6.4.6. (a) Identity, 9 rotations of 90◦, 180◦, and 270◦ around the centers of opposite faces, 8 rotations of 120◦ and 240◦ around opposite vertices, and 6 rotations of 180◦ around centers of opposite edges; mirror reﬂections over 6 planes through opposite edges and 3 planes between opposite faces. Switch roles of faces and vertices to match cube 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 and P each have 5 of them. Section 6.5 6.5.1. p6m. 6.5.2. 180◦ rotation through a vertical or horizontal axis going midway between adjacent atoms. 536 Answers to Selected Exercises

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 6.6 6.6.1. 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 of various lengths can easily vary, signiﬁcantly altering the estimates of d.)

Section 7.1 7.1.2. Larger. ←→ ←→ 7.1.5. (a) Let T be the intersection of PP and QQ . 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 replace quadrangle with quadrilateral. (b) m⊥l, m is the other bisector of j and k. = , = a , a 7.1.11. (a) T1 (0 a), T2 ( 2 2 ). (d) y = (1 − 2a)x/(2 − a) + a/(2 − a). Answers to Selected Exercises 537

7.1.14. (a) V , E, and D. 7.1.15. (a) y =−w. (b) midpoint. at inﬁnity

Section 7.2

7.2.1. Deﬁne X−n to be the point such that H(XX0, Xn X−n). 7.2.3. Two distinct lines have exactly one point on them. There are at least four lines with no three on the same point. Every two distinct points have at least one line on both points. 7.2.4. Dual of 7.2.1: Two distinct points have exactly one line on both points. Every point has at least four distinct lines on it. If H(pq, rs), then H(pq, sr). 7.2.2: If pq//rs, then qp//rs, qp//sr, rs//qp, sr//pq, and sr//qp.Ifa, b, c, and d are distinct concurrent lines, then exactly one of the following holds: ab//cd, ac//bd,orad//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 the lines vi on another point so that vi is the image of ui if and only if ui , vi and k are concurrent. 7.2.6: A perspectivity preserves harmonic sets of lines and the relation of separation. That is, if a perspectivity from k maps the concurrent lines p, q, r, and s to 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 completely determined 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. , a 7.2.12. (a) Given the construction for H(0 1 a 2a−1 ), draw parallel lines to these through ak the corresponding points 0, k, ak and 2a−1 . By Theorem 1.4.1 the corresponding ak triangles are similar. Hence the sixth new line must go through 2a−1 . = 1 = n+1 (b) In part (a) let k n+1 and a n . 7.2.15. (a) s < 0or1< 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, concurrent lines, pq//sr and rs//pq. 7.2.18. (b) In a complete quadrilateral with lines t , t , t , and t ,thediagonal lines are the three ←−−−−−−1 2→3 ←−−−−−−−−4 → ←−−−−−−−→ lines on the “opposite” points: (t1 · t2)(t3 · t4), (t1 · t3) · (t2 · t4), and (t1 · t4)(t2 · t3). 538 Answers to Selected Exercises

Section 7.3 7.3.1. (x, 2x, 0), [0, 0, c]. 7.3.2. Two points are two Euclidean lines through the origin O, which determine a unique Euclidean 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 O intersect in a Euclidean line. 7.3.3. Note that three of the four terms in R(a, b, c, d)arenegative. 7.3.6. y = x2 and y = 1/x are functions. [1, 0 − b] intersects x2 − yz = 0in(b, b2, 1) and (0, 1, 0). It intersects xy − z2 = 0in(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. , 2, = − 2 , − , − 2 (b) (x0 x0 1), y 2x0x x0 ,[2x0 1 x0 ]. Section 7.4 ab a 0 ab 7.4.2. (a) , where a = 0; , where d = 0; and , where a + b = c + d. ⎡0 d ⎤ cd⎡ ⎤ cd⎡ ⎤ abc a 0 c ab0 (b) ⎣0 ef⎦, where a = 0; ⎣de f⎦, where e = 0; ⎣de0⎦, where i = 0; 0 ⎡hi ⎤ g 0 i ghi abc and ⎣de f⎦, where a + b + c = d + e + f = g + h + i. gh i Answers to Selected Exercises 539

30 7.4.3. (a) , 2 goes to 3 and H(0 1, 3 3 ). 12 3 4 2 4 ⎡ ⎤ 3 −10 7.4.4. (a) ⎣442⎦. 112 ⎡ ⎤ a 00 7.4.7. (a) ⎣0 a 0⎦. Dilation ﬁxing the origin. 00i (b) The corresponding sides of ABC and A B C are parallel. ⎡ ⎤ 20 0 7.4.10. (a) ⎣0 −0.51⎦,4x2 − 2yz = 0. 0 −0.5 −1 (c) y = 2x2, a parabola. ⎡ ⎤ ⎡ ⎤ 100 10−w 7.4.12. (a) ⎣ 010⎦, ⎣01 0⎦, and (1 + 2w)x2 + y2 − 2xz = 0. −w 01 00 1 (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 7.5

2 7.5.1. Note that Xx · X−x = (1 − x )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◦. ⎡√ ⎤ 1 − b2 00 ⎣ ⎦ 7.5.9. Yb = 01b . Transposes of each other. 0 b 1 540 Answers to Selected Exercises

Section 7.6

7.6.1. 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.Theyareh-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]. , , , , , , , , , , , , 3 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 P are aeim bfjn coplanar if and only if the determinant = 0. cgko dhl p ⎡ ⎤ 2000 ⎢4 −100⎥ 7.6.4. (a) ⎢ ⎥. ⎣0 −220⎦ 0041 7.6.5. (a) By a quadric surface we mean a symmetric invertible 3 × 3 matrix. Two such matrices represent the same conic if and only if one is the multiple of the other by some real number λ = 0. A point P is on a quadric surface S if and only if P T SP = 0. ⎡ ⎤ 100 0 ⎢010 0⎥ (b) x2 + y2 + z2 − t2 = 0, ⎢ ⎥. ⎣001 0⎦ 000−1 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 the apparent position of the viewer closer to the cube. Vx = (2, 1.73), Vy = (−3.46, 1), and Vz = (0, −3.46). ⎡ ⎤ 10 0 7.6.12. (a) ⎣01 0⎦. 22−1 (b) The image of (x, 0, 1) is (x, 0, 2x − 1) = (x/(2x − 1), 0, 1), which is on the x-axis on either side of [0, 1]. Note that for 0 < x < 1/2, 2x − 1 < 0. The diagonal is pointwise ﬁxed.

Section 8.1 A1 B2 C3 8.1.1. (a) C2 A3 B1. B3 C1 A2 8.1.2. (a) Shift two to the right and one down. Shift one right and two down. Answers to Selected Exercises 541

8.1.3.123,456,789; 147,258,369; 168,249,357; and159,267,348.Fourdays. 8.1.5. (b) 4k + 1. 8.1.8. (a) A pentagon, 5 points.

Section 8.2 8.2.3. 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 geometry 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 parallel lines with points on one of those lines representing girls in the same row for that day. 8.2.8. (b) Suppose k l and m intersects k.Ifm didn’t intersect l, there would be two parallels to 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 than part (v). 8.2.17. (a) A triangle.

Section 8.3 8.3.1. 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 is odd. 8.3.7. Rotate {1, 2, 5, 7} to obtain the lines. 542 Answers to Selected Exercises

8.3.13. (a)

10111000100 01011100010 00101110001 10010111000 01001011100 00100101110 00010010111 10001001011 11000100101 11100010010 01110001001

. 5 and 2.

Section 8.4 8.4.1. 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. Z2 8.4.2. There are only 4 points on a line in P 3. 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 stable lines 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). Answers to Selected Exercises 543

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 9.1 9.1.1. (a) So the bending can match, x = 0, x2 + (y − b)2 = b2, or x2 + y2 − 2by = 0for 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 9.2 , , 9.2.1. 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.

2x 3/2 x 9.2.4. (a) r(x) = (1 + e ) /e √. (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 = 0.5ln(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. 544 Answers to Selected Exercises

−→ 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 9.3

9.3.4. 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. 2 2 2 2 2 2 9.3.10. (a) sx = (1, 0, ∓x/ R − x − y ), sy = (0, 1, ∓y/ R − x − y ), which are not orthogonal. 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)). √−a sin(u)cos(v) √ a cos(u)cos(v) √ a (d) Nu = , , 0 = su. a2 cos2(v)+sin2(v) a2 cos2(v)+sin2(v) a2 cos2(v)+sin2(v)

9.3.16. (a) su = (−(r cos(v) + R)sin(u), (r cos(v) + R) cos(u), 0) and sv = (−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 9.4 9.4.2. 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 10.1 10.1.1. 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 circle with center B going through A intersect in two points C and D. However, the distances AB and CD are unequal. 10.1.2. Yes. Answers to Selected Exercises 545

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. Place guards at A, B, and E. Extend CD, FE, GH, and JI to where they intersect AB at K , L, M, and N, respectively. There must be a guard in each of BCK, LFGM, and AJN 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 BAC . Then AC BC is a parallelogram and we can tile the plane as a slanted checkerboard with copies of AC 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 the perpendicular bisectors of A with each of these regions shrink the regions B, C, D and 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 cover it 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 10.2 10.2.1. 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 in d dimensions. Then {v1,...,vn} has at least Dd (n) distances, forcing V to have at 546 Answers to Selected Exercises

least that many. Hence Dd (n) ≤ Dd (n + 1). Now suppose W ={w1,...,wn} is a set of n points in d dimensions with exactly Dd (n) distances. We can embed W in d + 1 dimensions and still have the same number of distances. So Dd+1(n) ≤ Dd (n). 10.2.4. (a) n (b) Suppose we have n numbers1,2,...,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 we can 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 so every 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 ABH to see B and one must be in EFG 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 one design. 10.2.19. (a) n/2 . Position the guards to cover the outside. Any one of them also covers the inside.

Section 10.3 10.3.1. 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, height h, perpendicular to the plane of the tiling. Stack layers of these prisms on top of one another to ﬁll space. 10.3.4. No. Each angle is 120◦ so every vertex must have 3 polygons meeting at it. If a short edge of one polygon is matched with a long edge, there will be a 60◦ angle. So matching edges must be the same length. However at each vertex each polygon has a long and a short 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 three consecutive angles is 360. Answers to Selected Exercises 547

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 by polygons 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 10.4 10.4.1. The boundaries are two parallel lines. Three parallel lines.

10.4.2. All but the ones with P−1,1 and P1,−1.AllbutP1,1 and P−1,−1. 10.4.3. Suppose part of the perpendicular bisector of the sites A and B forms part of the boundary of the region surrounding B. Then the point C so that B is the midpoint of A and C is another site and the perpendicular bisector of the sites C and B forms part of the boundary of the region surrounding B. Thus these sides are centrally symmetric with 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 ABC and 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 equilateral triangle 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 center of 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 by regular hexagons and form part of a lattice. , a+b < , b−a > 10.4.15. (a) The ray of points (x 2 )forx 0, the ray (x 2 )forx a and the segment , a+b , b−a between (0 2 ) and (a 2 ). 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, New York, 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 Reinhold Co., New York, reprinted with permission. Figure 5.3, from Thompson, On Growth and Form, copyright 1942 by Cambridge University Press, 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,from Wade, 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 art of San Ildefonso,” Algebra, Groups and Geometries, copyright 1985 by Hadronic Press, Palm Harbor, 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 MIT Press, 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. All rights 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

Terms art gallery theorem, 463 AAS, 11, 493 (I-26) ASA, 11, 493 (I-26) absolute conic, 355, 358 arithmetic mean absolute quadric surface, 367 asymptote, 108, 341 acceleration vector, 509 atoms, 294 Achilles and the Tortoise, 4 axiom, 69 affine geometry, 358 axiom of linear completeness, 501 affine matrix, 215 axiomatic system, 68 plane, 215 axioms, projective geometry, 327ff three-dimensional, 235 axis of rotation, 237 n-dimensional, 241 affine plane, 379ff, 396 Babylonian mathematics, 2 affine space, 397, 400 balanced incomplete block design, affine transformation, 215, 243, 358 386 alternate interior angles, 27, 494 barycentric coordinates, 118, 366 alternate exterior angles, 31, 494 base of a pyramid, 42 altitude of a triangle, 41 base of a Saccheri quadrilateral, 169 analysis, 11 b.c.e. (before the common era), xviii analytic geometry, 98 between, 366 analytic projective geometry, 337 Bezier´ curve, 126ff analytic model, 98 BIBD, 386 angle, 3, 49, 70, 500 bilateral symmetry, 262 angle bisector, 13 binormal, 433 angle defect for polyhedra, 46 bipyramid, 53 angle of parallelism, 162 bisector, 12 angle of two great circles, 49 block, 386 angle-angle-side, 11, 493 (I-26) bonds, 294 angle-side-angle, 11, 493 (I-26) Brianchon’s theorem, 336 angle sum, 3 Bruck-Ryser theorem, 381 antipodal points, 49 antiprism, 51, 291 CAD, 126ff, 241, 362 apex, 42 cases, 514 arc length, 438 Catalan number, 461 Archimedean axiom, 501 Cavalieri’s principle, 56, 498 Archimedean solid, 63, 289 cell, 144 area, 36, 176 center of inversion, 246 arithmetic mean, 7 central angle, 21 art gallery problem, 449, 462 central symmetry, 221, 243

551 552 Index centroid of a triangle, 41 cross polytope, 139 chaos, 489 cross product, 503, 509 characteristic axiom, hyperbolic geometry, 153 cross ratio, 339 characteristic equation, 241 cross section, 510 chemical structure, 294 crystallographic group, 290 circle, 79 crystallographic restriction, 277 circular points at infinity, 358 crystals, 290, 294 circumcenter of a triangle, 20 cube, 51 circumscribed circle, 20 cuboctahedron, 52 closure, 200 curvature of a curve, 415 code, 390 curvature of a surface, 154, 427 codeword, 390 curve, 509 collinear, 29, 328 cyclic group, 267 collineation, 347, 362, 398 cycloid, 115 color group, 281 coloring, 473 decomposition of a figure, 17 color preserving group, 281 defect of a hyperbolic triangle, 178 color preserving symmetry, 271, 282 definition, 69 color switching symmetry, 271, 282 degenerate conic, 108, 341 color symmetry, 271, 282 derivative vector, 509 combinations, 448 Desargues’ theorem, 322, 370 combinatorics, 375 Descartes’ formula, 46 commensurable, 3 descriptive geometry, 136 Common Core State Standards (CCSS), xv, xvi, design theory, 385ff xx, 196, 208 determinant, 505 complete (axiomatic system), 86 diagonal points, 335 complete quadrangle, 320 diagrams in proofs, 70, 71 complete quadrilateral, 321 diameter of a circle, 79 complex conjugate, 103, 249 diameter of an ellipse, 107 complex numbers, 103–104, 248ff differential geometry, 409ff composition of functions, 198 dihedral angle, 140 computer-aided design, 126ff, 241, 362 dihedral group, 267 conclusion, 511 dilation, 225 concurrent, 20, 328 direct isometry, 206 conformal, 252 direct proof, 511 congruent, 3, 11, 76, 166 directrix, 106 conic, 104ff, 341, 370 discrete geometry, 448ff conic surface, 135 discrete pattern, 273 consistent, 83 distance, 99, 134, 214, 355 constructible angle, 25 dodecahedron, 55 constructible number, 24 dot product, 503, 509 constructible polygon, 19 double elliptic geometry, 185 constructions, 12 doubling the cube, 16 continuity axiom (projective), 330 dual, 92, 144, 331, 362, 382 contraction mapping, 230 dual of a polyhedron, 53 contradiction, 512 duality, 331ff, 362, 382 converse, 27 dynamical systems, 201, 489 convex, 7, 366 convex hull, 464 ear of a polygon, 462 convex set of points, 458 edges, 19, 42 correlation, 371 Egyptian mathematics, 2 corresponding angles, 31, 494 eigenvalue, 219, 506 counterexample, 513 eigenvector, 219, 506 Index 553

Elements, The, 10, 491 group, 200 ellipse, 105 guard point, 62 ellipsoid, 135 equal (functions), 197 h-inner product, 357 equal (in measure), 15 h-length, 357 equivalence relation, 259, 380 h-orthogonal, 357 equivalent polygons, 177 half plane, 76 Erlanger programm, 209, 212 half plane model, hyperbolic geometry, 157, 253 error-correcting code, 389 Hamming distance, 390 Euclidean geometry, 1ff, 358 harmonic (music), 326 Euclidean isometry, 202, 241 harmonic set of lines, 325 Euler’s formula, 44 harmonic set of points, 320 example, 513 Hausdorff dimension, 305 excess of a spherical triangle, 51 helix, 116 exterior of a conic, 353 Hilbert’s axioms, 77ff, 499 externally tangent circles, 40 homogeneous coordinates, 337 homogeneous second degree equation, 340 faces, 42 horocycle, 191, 258 ff (“following” for page references), 51 horolation, 258 field, 394 hyperbola, 105 15-schoolgirl problem, 375 hyperbolic distance, 182, 355 fifth postulate, 28, 492 hyperbolic geometry, 152ff, 247, 355 finite geometry, 373ff hyperbolic glide reflection, 258 first fundamental form, 437 hyperbolic isometry, 250, 355 first stellation, 55 hyperbolic translation, 258, 356 fixed point, 197, 347 hyperboloid of one sheet, 136 flow chart, wallpaper groups, 279 hyperboloid of two sheets, 135 foci, focus, 106 hypercube, 137 fortress problem, 454 hypercycle, 258 fortress theorem, 464 hyperplane, 134, 241 fractal, 228, 230, 304ff hypothesis, 511 fractal curve, 309 fractal dimension, 307 icosahedral group, 289 fractal surface, 309 icosahedron, 54 frequency, 47 ideal line, 318 frieze pattern, 273 ideal point, 318 fundamental theorem, projective geometry, 333 identity function, 199 identity matrix, 505 Gauss-Bonnet theorem, 439 if and only if, 514 Gaussian curvature of a surface, 427 IFS, 228, 230 general theory of relativity, 409, 440 image, 197 generate, 274 incenter, 21 generic, 451 incidence matrix, 390 geodesic, 155, 409, 433 incommensurable, 3 geodesic dome, 47 independent, 84 geometric mean, 7 indirect isometry, 206 geometry (definition), 209 induction proof, 512 glide reflection, 206 inscribed angle, 21 global positioning system, 440 inscribed circle, 21 golden ratio, 8 inscribed polygon, 19 GPS, 440 interior angles on the same side, 31, 494 great circle, 48 interior of a conic, 353, 355 Greek mathematics, 2ff inverse function, 199 554 Index inverse matrix, 505 minor of a matrix, 505 inversion, 246 mirror reflection, 203, 237 inversive line, 246 Mobius¨ transformation, 250 inversive plane, 246 model, 82 invertible matrix, 505 mod (n), modulo, 395 irrational, 3 monohedral tile, 450 isomer, 299 motif, 264 isometry, 202 Euclidean plane, 202 Napoleon’s theorem, 145 Euclidean n-dimensional, 241 National Council of Teachers of Mathematics spherical, 235, 241 (NCTM), xv, xx isometric, 212 n-gon, 19 isomorphic, 87 non-orientable surface, 188, 365 isosceles, 17 norm of a vector, 509 iterated function system, 228, 230 normal, 414 iteration, 304 normal line, 415

Kirkman’s 15 schoolgirl problem, 375 oblate, 430 kite, 32 oblong number, 7 Klein model, hyperbolic geometry, 157 octahedral group, 289 octahedron, 53 latitude, 49, 424 omega point, 165, 355 law of cosines, 23 omega triangle, 165 law of the lever, 26 on, 99, 134, 216, 337, 341 lemniscate, 117 one-point perspective, 323 length, 3, 237, 503 operational definition, 69 length of a vector, 418, 509 opposite interior angles, 29 lightlike (relativity), 303, 367 orbit of a point, 269 limiting curve, 191 orbit-stabilizer theorem, 269 line, 3, 99, 133, 216, 337, 362 order, affine plane, 379 line conic, 345 order, field, 395 line segment, 3, 366 order, projective plane, 383 linear algebra, 503 orientable, 188, 488 linear combination, 504 orientation, 196 linear transformation, 504 oriented point, 365 locus problem, 105 oriented projective geometry, 365 logically equivalent, 27 origin, 133 longitude, 49, 424 orthocenter, 41 Lorentz transformation, 297, 368 orthogonal, 237, 503, 509 lunes, 8, 49 orthogonal circles, 154, 247 orthogonal matrix, 241 manifold, 439 orthogonal polygon, 454 matrices, matrix, 504 orthonormal basis, 237, 504 matrix multiplication, 504 osculating circle, 414 mean curvature, 445 oval, 399 median of a triangle, 41, 100 metamathematics, 83 Pappus’ theorem, 336 Michelson-Morley experiment, 297 parabola, 106 midline, 273 paraboloid, 135 midpoint, 18 parallel, 26, 134, 379, 492 minimal surface, 445 parallel postulate, 26, 492, 498 Minkowski geometry, 367 parallelogram, 29 Index 555 parametric equations, 114 radians, 50 partial derivative, 510 radius, 79 Pascal’s theorem, 336 radius of curvature, 415 Pasch’s axiom, 77, 499 ratio of proportionality, 33 Penrose tiling, 296 rational line, 120 pentamino, 483 rational numbers, 120 perpendicular, 12, 143 rational point, 120 perpendicular bisector, 12 ray, 70, 500 perspective (art), 318 real numbers, 82 perspective (computers), 362ff rectangle, 30 perspective from a line, 321 rectangular box, 51 perspective from a point, 321 recursive, 461 perspectivity, 320 regular polygon, 19 Pick’s theorem, 145 regular polyhedron, 44 plane, 134, 238, 362 regular polytope, 137 plane curve, 509 regular star figure, 62 plane tiling, 450 regular star polygon, 61 platonic solid, 44 regular tiling, 484 Playfair’s axiom, 28, 498, 501 relatively consistent, 83 Poincare´ conjecture, 489 relativity theory Poincare´ model, hyperbolic geometry, 157, special, 263, 297 247 Galilean, 297 point, 99, 133, 214, 235, 337, 362 general, 409, 440 polar, 252, 326 reproducing an angle, 13 polar coordinates, 116 rep-tile, 476 pole, 252, 326 rhombus, 22 polygon, 19 Richardson’s equation (dimension), 308 polyhedra, polyhedron, 42 right angle, 12 polytope, 137 rotary reflection, 237 postulate, 12, 28, 76 rotation, 203, 237 power of a point, 39 principle of mathematical induction, 512 Saccheri quadrilateral, 169 prism, 43 SAS, 11, 492 (I-4), 498, 500 product of matrices, 505 scalar, 503 projection mapping, 242 scalar multiple, 133 projective geometry, 318ff scaling ratio, 223 projective plane, 382ff, 397 Schonflies’¨ notation, 289 projective space, 362ff, 399 screw motion, 240 projective transformation, 346ff segment, 3, 499 projectivity, 333, 347 self-dual, 53, 362 prolate, 430 self-similarity, 304 proof, 70 semi-regular tiling, 484 proof by contradiction, 512 sensed parallel lines, 162 properties of matrices, 506 separation (projective geometry), 329 proportional, 33 separation axiom (or postulate), 76, 77, 498, pseudosphere, 156, 431 500 pyramid, 42 separation axiom (projective geometry), 329 Pythagorean theorem, 3, 495 (I-47) set, 401 shape operator, 427 quadric surface, 368 shear, 227 quadrilateral, 29 side-angle-side, 11, 492 (I-4) quasicrystal, 296 side-side-side, 11, 492 (I-8) 556 Index signed curvature, 427 taxicab geometry, 84, 125 similar, 34, 41 tetrahedral group, 289 similarity, 223 tetrahedron, 42 simplex, simplices, 139 tetromino, 455 simply connected, 488 theorem, 70 single elliptic geometry, 185ff, 358 36-officer problem, 374 site, 453 three body problem, 489 smooth plane curve, 415 three-point perspective, 323 smooth surface, 426 tile, 450 SMSG postulates, 76ff, 497 tilings, 450ff spacelike (relativity), 303, 368 timelike (relativity), 303, 368 sphere, 48, 424 topology, 488 spherical cap, 58 torsion, 410, 420, 445 spherical excess, 51 torus, 425 spherical geometry, 185ff total angle defect, 46 spherical isometry, 235 tractrix, 431 spherical triangle, 50 transcendental, 17 spheroid, 430 transformation, 197 spiral of Archimedes, 418 transformation group, 200 spline, 130 translation, 203 square, 30 transpose, 237, 504 square matrix, 504 transversals, 26 squaring the circle, 16 trapezoid, 39 SSS, 11, 493 (I-8) triangle, 3 stabilizer of a point, 268 triangle inequality, 493 (I-20) stable, 197, 238, 347 triangular number, 7 standard basis, 504 triangulation, 449, 459 statistical design theory, 374 trilinear plot, 118 statistical self-similarity, 305 trisecting an angle, 16 Steiner quadruple system, 392 two-point perspective, 323 Steiner triple system, 388 stereographic projection, 254 ultraparallel lines, 162 straight angle, 12 undefined term, 69 straightedge and compass constructions, 12 unit normal, 426, 510 subgeometry, 354ff, 367 unit tangent vector, 419, 509 subgroup, 274 subspace, 504 vanishing point, 318 sum, 133 variety, 386 summit of Saccheri quadrilateral, 169 vector, 133, 503 supplementary angles, 79 vector function, 509 surface, 426ff, 509 velocity vector, 509 surface of revolution, 428 vertex, vertices, 42, 203 symmetric design, 391 vertical angles, 79, 493 (I-15) symmetric group, 291 volume, 43 symmetric matrix, 504 Voronoi diagram, 453 symmetries of a prism, 289 Voronoi edge, 478 symmetry, 264 Voronoi region, 453 symmetry group, 264 Voronoi vertex, 478 synthetic, 11 wallpaper group flow chart, 279 tangent, 15, 341, 399 wallpaper pattern, 273 tangent plane, 510 tangram, 61 Zeno’s paradoxes, 4 Index 557

Godel,¨ Kurt, 83, 94 People Grunbaum,¨ Branko, 477 (biography pages in bold) Archimedes, 25 Hamilton, William Rowan, 361 Aristotle, 5, 474 Hausdorff, Felix, 305 Helmholtz, Hermann, 235 Barnsley, Michael, 228, 230 Hessel, J. H. C., 263, 288, 290 Beltrami, Eugenio, 156, 161, 424 Hilbert, David, 77, 81 Bernoulli, Jakob, 116, 410 Bernoulli, Johann, 377 Jordan, Camille, 263 Bezier,´ Pierre Etiene,´ 126 Bolyai, Janos,´ 153, 169 Kant, Immanuel, 152 Bolyai, W., 17 Kelvin, Lord, 378 Bonnet, Pierre Ossian, 439 Kepler, Johannes, 267, 283, 318, 423 Bravais, Auguste, 263, 288 Khayyam, Omar, 169, 174 Brianchion, Charles Julien, 336 Kirkman, Rev. Thomas, 375, 378, 388 Klein, Felix, 185, 196, 209, 212, 263, 278, 318, Cayley, Arthur, 318, 354, 361 346, 354 Cavalieri, Bonaventura, 56 Koch, Helge von, 304 Chvatal,´ Vaclav,´ 450 Clairaut, Alexis-Claude, 410 Lagrange, Joseph Louis, 263 Coxeter, H. S. M., 180, 263, 288, 293 Leibniz, Gottlieb Wilhelm, 409, 423 Crowe, Don, 280 Lie, Sophus, 196, 234, 263 Lindemann, Ferdinand, 16 da Vinci, Leonardo, 267, 318 Lobachevsky, Nikolai, 153, 169 Dehn, Max, 17 Lorentz, Hendrik, 263, 297, 489 Desargues, Girard, 318, 322, 336 Descartes, Rene,´ 46, 98, 104, 423, 477, 487 Mandelbrot, Benoit, 304, 312, 488 Dirichlet, Peter Gustave Lejeune, 477 Minkowski, Hermann, 299 Durer,¨ Albrecht, 263, 318, 319 Mobius,¨ Augustus, 118, 137, 195, 250, 254, 263, 318, 487 Eddington, Sir Arthur, 409, 440 Monge, Gaspard, 136, 145, 318, 336 Einstein, Albert, 155, 297, 411, 444 Eratosthenes, 8 Nasir Eddin, 60 Erdos, ˝ Paul, 449, 457, 459, 466 Newton, Sir Isaac, 297, 409, 423, 430 Escher, M. C., 180, 288, 293 Euclid, 10ff, 318, 487 Oresme, Nicole, 98 Eudoxus, 4 Euler, Leonhard, 46, 98, 114, 374, 377, 410 Pappus, 336 Pascal, Blaise, 114, 318 Fano, Gino, 374, 382 Penrose, Sir Roger, 296 Fedorov, Vyatseglav, 278 Perelman, Grigori, 489 Fermat, Pierre de, 98, 114, 487 Plato, 4 Fisher, Sir Ronald A., 374, 385, 393 Plucker,¨ Julius, 149, 254, 318, 346, 388 Fisk, Steve, 462 Poincare,´ Henri, 212, 247, 263, 297, 488 Francesca, Piero della, 318 Poncelet, Jean Victor, 318, 336 Fuller, Buckminster, 1, 46, 59, 293 Pythagoras, 2 Pythagoreans, 2 Galileo Galilei, 36, 297, 423 Galois, Evariste,´ 263 Richardson, Lewis, 308 Gauss, Carl Friedrich, 19, 153, 160, 410, 424, Riemann, Georg, 154, 185 189, 411, 439 437, 439 Gerwien, P., 17 Saccheri, Giovanni, 152, 175 558 Index

Schlaﬂi,¨ Ludwig, 137 g1, g2,..., gn, the subgroup generated by Senechal, Margorie, 304 g1, g2,...,gn, 274 Snow, John, 447, 477 Steiner, Jacob, 346, 388 ·h , h-inner product, 357 H(PQ, RS), harmonic set, 320, 339 Tait, Peter Guthrie, 378 H( jk, lm), harmonic set, lines, 325 Thales, 33, 511 Theaetetus, 4 I , identity matrix, 505

Van Staudt, Karl, 318 k · l, intersection of lines, 328 Viete,` Franc¸ois, 98 κ(p, q), curvature of a surface, 427 Voronoi, Georgy, 477 κ(t), curvature of a curve, 416

Wantzel, Pierre, 16 M, shape operator, 427 Washburn, Dorothy, 280 M−1,inverseofM, 216, 505 Wiles, Andrew, 114 M T , transpose of M, 504 m∠ABC, measure of ∠ABC, 21 Zeno, 4 N, the natural numbers, 512 !n", the greatest integer less than or equal to n, 459 Notation n , the least integer greater than or equal to n, AB, length of AB, 3 ←→ 467 AB, line on A and B, 3 $n!,% n factorial, 290 −→ n AB, ray from A through B, 70 k , combinations of n, k at a time, 448 AB, segment between A and B, 3 N(u,v), unit normal vector, 426, 510 ∠ABC, angle with vertex at B, 3 ABC, triangle with vertices A, B,andC, 3 P, icosahedral group, 289 −−−−→ −−−−−→ (a, b, c) · (d, e, f ), dot product, 338, 503, 509 P, oriented point, 365 a + bi, complex conjugate, 103, 249 p111, etc. frieze pattern groups, 276 2 α−1,inverseofα, 199 PF , projective plane over F, 397 d AF2, afﬁne plane over F, 396 PF , projective space over F, 399 AFd , afﬁne space over F, 400 pg, etc. wallpaper groups, 279 PQ//RS, separation, 329, 339

B(t0), binormal, 433 Q, the rational numbers, 120 cm, etc. wallpaper groups, 279 R C, complex numbers, 249 , the real numbers, 82 , , , C#, extended complex numbers, 248 R(P S U W), cross ratio, 339 C , cyclic group, 267 r(x), radius of curvature, 416 −→n c(t), curve, 415, 509 −→ −→ Sn, symmetric group, 290 c(t) , norm (length) of c(t), 418, 503, 509 −−→ −→ su , partial derivative of surface, 426, 510 c (t), derivative (tangent) of c(t), 418, 509 ,v −−→ s(u ), surface, 426 c (t), acceleration vector, 509 T, tetrahedral group, 289 Dd (n), minimum number of distances, 449 T (t), unit tangent vector, 419, 509 det(M), determinant of M, 505 T (n), triangulations of a convex n-gon, Dn, dihedral group, 267, 289 461 Dnh, symmetry group of a prism, 289 ∂x v, ,λ ∂u , partial derivative, 510 ( k ), parameters of a BIBD, 385 Vor(S), Voronoi diagram of S, 453 −→ −→ F,ﬁeld,395 v × w , cross product, 503, 509 Index 559

W, octahedral group, 289 * exercise answered in back of book, xvii ∼ X x , hyperbolic translation, 356 =, congruent, 3 ⊥, perpendicular, 12

Zn, integers modulo n, 395 , parallel, 26, 379 ∼, similar, 34 end of a proof, xvii ⊕, velocity addition in relativity, 298 ♦ end of an example, xvii ∞, added point for inversive plane, 246

AMS / MAA TEXTBOOKS

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 mathematical 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.