Exploring Geometry

477 Pages·2012·5.36 MB·English

Exploring Geometry
Michael Hvidsten
Gustavus Adolphus College
DRAFT: August 7, 2012
ii
Copyright @ 2004 by Michael Hvidsten
Final draft, August 2004.
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 Go¨del’s Incompleteness 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 . . . . . . . . . . . . . . . . . . . 97
2.7.1 Orthogonal Circles Redux . . . . . . . . . . . . . . . . 102
3 Analytic Geometry 107
3.1 The Cartesian Coordinate System . . . . . . . . . . . . . . . 109
3.2 Vector Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.3 Project 5 - B´ezier Curves . . . . . . . . . . . . . . . . . . . . 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´e Model . . . . . . . . . . . . . . . . . . . . . . 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 Mo¨bius Transformations . . . . . . . . . . . . . . . . . . . . . 317
8.1.1 Fixed Points and the Cross Ratio . . . . . . . . . . . . 317
8.1.2 Geometric Properties of M¨obius Transformations . . . 319
8.2 Isometries in the Poincar´e Model . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . 413
B.3 Active vs. Inactive Tools . . . . . . . . . . . . . . . . . . . . . 416
B.4 Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
B.5 Object Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . 417
B.6 Online Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
B.7 Undo/Redo of Actions . . . . . . . . . . . . . . . . . . . . . . 418
B.8 Clearing and Resizing the Canvas . . . . . . . . . . . . . . . . 419
B.9 Saving Files as Images . . . . . . . . . . . . . . . . . . . . . . 420
B.10 Main Window Button Panels . . . . . . . . . . . . . . . . . . 421
CONTENTS vii
B.10.1 Create Panel . . . . . . . . . . . . . . . . . . . . . . . 421
B.10.2 Construct Panel . . . . . . . . . . . . . . . . . . . . . 421
B.10.3 Transform Panel . . . . . . . . . . . . . . . . . . . . . 425
B.11 Measurement in Geometry Explorer . . . . . . . . . . . . . . . 429
B.11.1 Neutral Measurements . . . . . . . . . . . . . . . . . . 430
B.11.2 Euclidean-only Measurements . . . . . . . . . . . . . . 431
B.11.3 Hyperbolic-only Measurements . . . . . . . . . . . . . 432
B.11.4 User Input Measurements . . . . . . . . . . . . . . . . 432
B.12 Using Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
B.13 Using the Calculator . . . . . . . . . . . . . . . . . . . . . . . 433
B.14 Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . 434
B.15 Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 436
B.16 Turtle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 436
C Birkhoff’s Axioms 441
D Hilbert’s Axioms 443
E The 17 Wallpaper Groups 445
Bibliography 451
Index 455

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,sofruitfulindelightfulsurprises,asPureMathematics.
—Lewis Carroll (Charles Dodgson), 1832–1898
Anexplorerisonewhoseeksoutnewworldsandideas. AsLewisCarroll
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 computer explorations in the text. The software used
in these explorations is Geometry Explorer, a virtual geometry laboratory
where one can create geometric objects (like points, circles, polygons, areas,
etc.), carry out transformations on these objects (dilations, reflections, ro-
tations, and translations), and measure aspects of these objects (like length,
area, radius, etc.). As such, it is much like doing geometry on paper (or
sand) with a ruler and compass. However, on paper such constructions are
static—points placed on the paper can never be moved again. In Geometry
Explorer, all constructions are dynamic. One can draw a segment and then
grab one of the endpoints and move it around the canvas, with the segment
movingaccordingly. Thus, onecanconstructageometricfigureandtestout
ix
x PREFACE
hypotheses by experimentation with the construction.
The development of intuitive notions of geometric concepts is a critical
first step in understanding such concepts. However, intuition alone cannot
providethebasisforprecisecalculationandanalysisofgeometricquantities.
For example, one may know experimentally that the sides of a right triangle
followthePythagoreanTheorem,butdataalonedonotshowwhythisresult
is true. Only a logical proof of a result will give us confidence in using it in
any given situation.
Throughout this text there is a dual focus on intuition/experimentation
on the one hand and on explanation/proofs on the other. This integration
of exploration and explanation can be seen most clearly in the use of major
projects to tie together concepts in each chapter. For example, the first
project explores the golden ratio and its amazing and ubiquitous properties.
Studentsnotonlyexperimentallydiscoverthepropertiesofthegoldenratio,
but are asked to dig deeper and analyze why these properties are true.
The goal of the projects is to have students actively explore geometry
through a three-fold approach. Students will first see a topic introduced in
the text. Then, they will explore that topic using Geometry Explorer or by
means of in-class group projects. Finally, they will review and report on
their exploration, discussing what was discovered, conjectured, and proved
during the course of the project.
The beginning of each project is designated by a special heading—the
projecttitlesetbetweentwohorizontallines. Theconclusionofeachproject
is designated by an ending horizontal line. Projects are illustrated with
screen shots from the Geometry Explorer program, which comes bundled
with the text.
Using Geometry Explorer
EachprojectincludesaseriesofspecificgeometricactivitiesusingGeometry
Explorer. The following conventions will be used for directing computer
explorations:
• Menu References All menu references will be in bold face type
and will reference the menu option to click on. Parent menus will be
listed in parentheses to assist in navigating to the correct menu. For
example, the phrase “Click on Hide (View menu)” means to go to
the Hide menu under the View menu and select that menu.