Exploring Geometry Michael Hvidsten Gustavus Adolphus College DRAFT: October 18, 2012 ii Copyright @ 2004 by Michael Hvidsten Revised draft, August 2012. All rights reserved. No part of this publication may be reproduced, stored, or transmitted without prior consent of the author. Contents Preface ix Acknowledgments xiii 1 Geometry and the Axiomatic Method 1 1.1 Early Origins of Geometry . 1 1.2 Thales and Pythagoras . 4 1.2.1 Thales . 6 1.2.2 Pythagoras . 7 1.3 Project 1 - The Ratio Made of Gold . 8 1.3.1 Golden Section . 9 1.3.2 Golden Rectangles . 14 1.4 The Rise of the Axiomatic Method . 18 1.5 Properties of Axiomatic Systems . 26 1.5.1 Consistency . 27 1.5.2 Independence . 28 1.5.3 Completeness . 29 1.5.4 G¨odel'sIncompleteness Theorem . 30 1.6 Euclid's Axiomatic Geometry . 33 1.6.1 Euclid's Postulates . 34 1.7 Project 2 - A Concrete Axiomatic System . 40 2 Euclidean Geometry 51 2.1 Angles, Lines, and Parallels . 52 2.2 Congruent Triangles and Pasch's Axiom . 63 2.3 Project 3 - Special Points of a Triangle . 68 2.3.1 Circumcenter . 68 2.3.2 Orthocenter . 70 2.3.3 Incenter . 73 iii iv CONTENTS 2.4 Measurement and Area . 74 2.4.1 Mini-Project - Area in Euclidean Geometry . 75 2.4.2 Cevians and Areas . 78 2.5 Similar Triangles . 80 2.5.1 Mini-Project - Finding Heights . 87 2.6 Circle Geometry . 89 2.7 Project 4 - Circle Inversion . 98 2.7.1 Orthogonal Circles Redux . 103 3 Analytic Geometry 107 3.1 The Cartesian Coordinate System . 109 3.2 Vector Geometry . 112 3.3 Project 5 - B´ezierCurves . 117 3.4 Angles in Coordinate Geometry . 124 3.5 The Complex Plane . 130 3.5.1 Polar Form . 131 3.5.2 Complex Functions . 133 3.5.3 Analytic Functions and Conformal Maps (Optional) . 136 3.6 Birkhoff's Axiomatic System . 140 4 Constructions 147 4.1 Euclidean Constructions . 147 4.2 Project 6 - Euclidean Eggs . 159 4.3 Constructibility . 163 4.4 Mini-Project - Origami Construction . 173 5 Transformational Geometry 181 5.1 Euclidean Isometries . 182 5.2 Reflections . 187 5.2.1 Mini-Project - Isometries through Reflection . 189 5.2.2 Reflection and Symmetry . 190 5.3 Translations . 193 5.3.1 Translational Symmetry . 196 5.4 Rotations . 199 5.4.1 Rotational Symmetry . 203 5.5 Project 7 - Quilts and Transformations . 206 5.6 Glide Reflections . 212 5.6.1 Glide Reflection Symmetry . 215 5.7 Structure and Representation of Isometries . 217 5.7.1 Matrix Form of Isometries . 218 CONTENTS v 5.7.2 Compositions of Rotations and Translations . 221 5.7.3 Compositions of Reflections and Glide Reflections . 223 5.7.4 Isometries in Computer Graphics . 224 5.7.5 Summary of Isometry Compositions . 225 5.8 Project 8 - Constructing Compositions . 227 6 Symmetry 233 6.1 Finite Plane Symmetry Groups . 235 6.2 Frieze Groups . 239 6.3 Wallpaper Groups . 244 6.4 Tiling the Plane . 254 6.4.1 Escher . 254 6.4.2 Regular Tessellations of the Plane . 256 6.5 Project 9 - Constructing Tessellations . 259 7 Non-Euclidean Geometry 263 7.1 Background and History . 263 7.2 Models of Hyperbolic Geometry . 266 7.2.1 Poincar´eModel . 266 7.2.2 Mini-Project - The Klein Model . 271 7.3 Basic Results in Hyperbolic Geometry . 275 7.3.1 Parallels in Hyperbolic Geometry . 276 7.3.2 Omega Points and Triangles . 281 7.4 Project 10 - The Saccheri Quadrilateral . 286 7.5 Lambert Quadrilaterals and Triangles . 290 7.5.1 Lambert Quadrilaterals . 290 7.5.2 Triangles in Hyperbolic Geometry . 293 7.6 Area in Hyperbolic Geometry . 297 7.7 Project 11 - Tiling the Hyperbolic Plane . 301 7.8 Models and Isomorphism . 306 8 Non-Euclidean Transformations 313 8.1 M¨obiusTransformations . 317 8.1.1 Fixed Points and the Cross Ratio . 317 8.1.2 Geometric Properties of M¨obiusTransformations . 319 8.2 Isometries in the Poincar´eModel . 322 8.3 Isometries in the Klein Model . 327 8.4 Mini-Project - The Upper Half-Plane Model . 330 8.5 Weierstrass Model . 333 8.6 Hyperbolic Calculation . 333 vi CONTENTS 8.6.1 Arclength of Parameterized Curves . 334 8.6.2 Geodesics . 336 8.6.3 The Angle of Parallelism . 337 8.6.4 Right Triangles . 338 8.6.5 Area . 340 8.7 Project 12 - Infinite Real Estate? . 342 9 Fractal Geometry 347 9.1 The Search for a \Natural" Geometry . 347 9.2 Self-Similarity . 349 9.2.1 Sierpinski's Triangle . 349 9.2.2 Cantor Set . 352 9.3 Similarity Dimension . 353 9.4 Project 13 - An Endlessly Beautiful Snowflake . 356 9.5 Contraction Mappings . 362 9.6 Fractal Dimension . 372 9.7 Project 14 - IFS Ferns . 375 9.8 Algorithmic Geometry . 385 9.8.1 Turtle Geometry . 385 9.9 Grammars and Productions . 388 9.9.1 Space-filling Curves . 389 9.10 Project 15 - Words into Plants . 394 A Book I of Euclid's Elements 401 A.1 Definitions . 401 A.2 The Postulates (Axioms) . 403 A.3 Common Notions . 403 A.4 Propositions (Theorems) . 403 B Brief Guide to Geometry Explorer 411 B.1 The Main Geometry Explorer Window . 412 B.2 Selecting Objects . 414 B.3 Active vs. Inactive Tools . 417 B.4 Labels . 417 B.5 Object Coloring . 418 B.6 Online Help . 419 B.7 Undo/Redo of Actions . 419 B.8 Clearing and Resizing the Canvas . 420 B.9 Saving Files as Images . 421 B.10 Main Window Button Panels . 422 CONTENTS vii B.10.1 Create Panel . 422 B.10.2 Construct Panel . 422 B.10.3 Transform Panel . 426 B.11 Measurement in Geometry Explorer . 430 B.11.1 Neutral Measurements . 431 B.11.2 Euclidean-only Measurements . 432 B.11.3 Hyperbolic-only Measurements . 433 B.11.4 User Input Measurements . 433 B.12 Using Tables . 433 B.13 Using the Calculator . 434 B.14 Hyperbolic Geometry . 435 B.15 Analytic Geometry . 437 B.16 Turtle Geometry . 437 C Birkhoff's Axioms 443 D Hilbert's Axioms 445 E The 17 Wallpaper Groups 447 Bibliography 453 Index 457 Preface It may well be doubted whether, in all the range of science, there is any field so fascinating to the explorer, so rich in hidden treasures, so fruitful in delightful surprises, as Pure Mathematics. |Lewis Carroll (Charles Dodgson), 1832{1898 An explorer is one who seeks out new worlds and ideas. As Lewis Carroll would probably agree, exploration is not always easy|the explorer can at times find the going tough. But, the treasures and surprises that active exploration of ideas brings is worth the effort. Geometry is one of the richest areas for mathematical exploration. The visual aspects of the subject make exploration and experimentation natural and intuitive. At the same time, the abstractions developed to explain geometric patterns and connections make the subject extremely powerful and applicable to a wide variety of physical situations. In this book we give equal weight to intuitive and imaginative exploration of geometry as well as to abstract reasoning and proofs. As any good school teacher knows, intuition is developed through play, the sometimes whimsical following of ideas and notions without clear goals in mind. To encourage a playful appreciation of geometric ideas, we have incorporated many.