Exploring geometry

545 Pages·2017·4.735 MB·English

by  Hvidsten, Michael

Exploring
Geometry
Second Edition
Michael Hvidsten
Gustavus Adolphus College
St. Peter, Minnesota, USA
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2017 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
Version Date: 20161019
International Standard Book Number-13: 978-1-4987-6080-5 (Hardback)
Library of Congress Cataloging‑in‑Publication Data
Names: Hvidsten, Michael. | Hvidsten, Michael. Geometry with Geometry
Explorer.
Title: Exploring geometry / Michael Hvidsten.
Other titles: Geometry with Geometry Explorer
Description: Second edition. | Boca Raton : CRC Press, 2017. | Series:
Textbooks in mathematics series | Previous edtion: Geometry with Geometry
Explorer / Michael Hvidsten (Dubuque, Iowa : McGraw-Hill, 2005).
Identifiers: LCCN 2016027443 | ISBN 9781498760805 (hardback : alk. paper)
Subjects: LCSH: Geometry--Textbooks.
Classification: LCC QA445 .H84 2017 | DDC 516--dc23
LC record available at https://lccn.loc.gov/2016027443
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface to the Second Edition xiii
Chapter 1(cid:4) 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 17
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 Gödel’s Incompleteness Theorem 31
1.6 EUCLID’S AXIOMATIC GEOMETRY 35
1.6.1 Euclid’s Postulates 36
1.7 PROJECT 2- A CONCRETE AXIOMATIC SYSTEM 43
Chapter 2(cid:4) Euclidean Geometry 49
2.1 ANGLES, LINES, AND PARALLELS 51
2.2 CONGRUENT TRIANGLES AND PASCH’S AXIOM 62
2.3 PROJECT 3- SPECIAL POINTS OF A TRIANGLE 70
vi (cid:4) Contents
2.3.1 Circumcenter 70
2.3.2 Orthocenter 72
2.3.3 Incenter 75
2.4 MEASUREMENT AND AREA 76
2.4.1 Mini-Project - Area in Euclidean Geometry 77
2.4.2 Cevians and Areas 82
2.5 SIMILAR TRIANGLES 85
2.5.1 Mini-Project - Finding Heights 92
2.6 CIRCLE GEOMETRY 94
2.6.1 Chords and Arcs 95
2.6.2 Inscribed Angles and Figures 96
2.6.3 Tangent Lines 99
2.6.4 General Intercepted Arcs 101
2.7 PROJECT 4- CIRCLE INVERSION AND ORTHOGONAL-
ITY 108
2.7.1 Orthogonal Circles Redux 112
Chapter 3(cid:4) Analytic Geometry 117
3.1 THE CARTESIAN COORDINATE SYSTEM 119
3.2 VECTOR GEOMETRY 123
3.3 PROJECT 5- BÉZIER CURVES 128
3.4 ANGLES IN COORDINATE GEOMETRY 135
3.5 THE COMPLEX PLANE 141
3.5.1 Polar Form 143
3.6 BIRKHOFF’S AXIOMATIC SYSTEM 144
Chapter 4(cid:4) Constructions 151
4.1 EUCLIDEAN CONSTRUCTIONS 151
4.2 PROJECT 6- EUCLIDEAN EGGS 163
4.3 CONSTRUCTIBILITY 167
4.3.1 Mini-Project - Origami Construction 177
Contents (cid:4) vii
Chapter 5(cid:4) Transformational Geometry 185
5.1 EUCLIDEAN ISOMETRIES 186
5.2 REFLECTIONS 192
5.2.1 Mini-Project - Isometries through Reflection 194
5.2.2 Reflection and Symmetry 196
5.3 TRANSLATIONS 200
5.3.1 Translational Symmetry 203
5.4 ROTATIONS 205
5.4.1 Rotational Symmetry 210
5.5 PROJECT 7- QUILTS AND TRANSFORMATIONS 213
5.6 GLIDE REFLECTIONS 218
5.6.1 Glide Reflection Symmetry 221
5.7 STRUCTURE AND REPRESENTATION OF ISOMETRIES 224
5.7.1 Matrix Form of Isometries 225
5.7.2 Compositions of Rotations and Translations 227
5.7.3 Compositions of Reflections and Glide Reflections 230
5.7.4 Isometries in Computer Graphics 231
5.7.5 Summary of Isometry Compositions 232
5.8 PROJECT 8- CONSTRUCTING COMPOSITIONS 234
Chapter 6(cid:4) Symmetry 239
6.1 FINITE PLANE SYMMETRY GROUPS 241
6.2 FRIEZE GROUPS 246
6.3 WALLPAPER GROUPS 253
6.4 TILING THE PLANE 263
6.4.1 Escher 263
6.4.2 Regular Tessellations of the Plane 265
6.5 PROJECT 9- CONSTRUCTING TESSELLATIONS 267
Chapter 7(cid:4) Hyperbolic Geometry 271
7.1 BACKGROUND AND HISTORY 271
7.2 MODELS OF HYPERBOLIC GEOMETRY 274
viii (cid:4) Contents
7.2.1 Poincaré Model 274
7.2.2 Mini-Project - The Klein Model 279
7.3 BASIC RESULTS IN HYPERBOLIC GEOMETRY 284
7.3.1 Parallels in Hyperbolic Geometry 284
7.3.2 Omega Points and Triangles 290
7.4 PROJECT 10- THE SACCHERI QUADRILATERAL 296
7.5 LAMBERT QUADRILATERALS AND TRIANGLES 300
7.5.1 Lambert Quadrilaterals 300
7.5.2 Triangles in Hyperbolic Geometry 303
7.6 AREA IN HYPERBOLIC GEOMETRY 308
7.7 PROJECT 11- TILING THE HYPERBOLIC PLANE 312
Chapter 8(cid:4) Elliptic Geometry 317
8.1 BACKGROUND AND HISTORY 317
8.2 PERPENDICULARS AND POLES IN ELLIPTIC GEOME-
TRY 319
8.3 PROJECT 12- MODELS OF ELLIPTIC GEOMETRY 325
8.3.1 Double Elliptic Model 326
8.3.2 Spherical Lunes 327
8.3.3 Single Elliptic Geometry 331
8.4 BASIC RESULTS IN ELLIPTIC GEOMETRY 334
8.4.1 Stereographic Projection Model 335
8.4.2 Segments and Triangle Congruence in Elliptic
Geometry 339
8.5 TRIANGLES AND AREA IN ELLIPTIC GEOMETRY 342
8.5.1 Triangle Excess and AAA 345
8.5.2 Area in Elliptic Geometry 346
8.6 PROJECT 13- ELLIPTIC TILING 351
Chapter 9(cid:4) Projective Geometry 359
9.1 UNIVERSAL THEMES 359
9.1.1 Central Projection Model of Euclidean Geometry 361
9.1.2 Ideal Points at Infinity 362
Contents (cid:4) ix
9.2 PROJECT 14- PERSPECTIVE AND PROJECTION 363
9.3 FOUNDATIONS OF PROJECTIVE GEOMETRY 368
9.3.1 Affine Geometry 369
9.3.2 Axioms of Projective Geometry 370
9.3.3 Duality 370
9.3.4 Triangles and Quadrangles 372
9.3.5 Desargues’ Theorem 375
9.4 TRANSFORMATIONS AND PAPPUS’S THEOREM 380
9.4.1 Perspectivities and Projectivities 381
9.4.2 Projectivity Constructions and Pappus’s Theorem 384
9.5 MODELS OF PROJECTIVE GEOMETRY 391
9.5.1 The Real Projective Plane 391
9.5.2 Transformations in the Real Projective Plane 395
9.5.3 Collineations 400
9.5.4 Homogeneous Coordinates and Perspectivities 404
9.5.5 Elliptic Model 405
9.6 PROJECT 15- RATIOS AND HARMONICS 406
9.6.1 Ratios in Affine Geometry 407
9.6.2 Cross-Ratio 409
9.6.3 Harmonious Ratios 413
9.7 HARMONIC SETS 416
9.7.1 Harmonic Sets of Points 416
9.7.2 Harmonic Sets of Lines 418
9.7.3 Harmonic Sets and the Cross-Ratio 421
9.8 CONICS AND COORDINATES 426
9.8.1 Conic Sections Generated by Euclidean Trans-
formations 426
9.8.2 Point Conics in Projective Geometry 429
9.8.3 Non-singular Conics and Pascal’s Theorem 432
9.8.4 Line Conics 435
9.8.5 Tangents 436
9.8.6 Conics in Real Projective Plane 439
x (cid:4) Contents
Chapter 10(cid:4) Fractal Geometry 445
10.1 THE SEARCH FOR A “NATURAL” GEOMETRY 445
10.2 SELF-SIMILARITY 447
10.2.1 Sierpinski’s Triangle 447
10.2.2 Cantor Set 450
10.3 SIMILARITY DIMENSION 451
10.4 PROJECT16-ANENDLESSLYBEAUTIFULSNOWFLAKE
454
10.5 CONTRACTION MAPPINGS 458
10.6 FRACTAL DIMENSION 469
10.7 PROJECT 17- IFS FERNS 472
10.8 ALGORITHMIC GEOMETRY 479
10.8.1 Turtle Geometry 480
10.9 GRAMMARS AND PRODUCTIONS 483
10.9.1 Space-Filling Curves 484
10.10 PROJECT 18- WORDS INTO PLANTS 489
Appendix A(cid:4) A Primer on Proofs 495
A.1 AXIOMATIC SYSTEMS AND PROOFS 495
A.2 DIRECT PROOFS 497
A.3 PROOFS BY CONTRADICTION 500
A.4 CONVERSE STATEMENTS 502
A.5 IF AND ONLY IF STATEMENTS 503
A.6 PROOFS AND WRITING 504
Appendix B(cid:4) Book I of Euclid’s Elements 505
B.1 DEFINITIONS 505
B.2 THE POSTULATES (AXIOMS) 507
B.3 COMMON NOTIONS 507
B.4 PROPOSITIONS (THEOREMS) 507
Appendix C(cid:4) Birkhoff’s Axioms 515
C.1 AXIOMS 515
Contents (cid:4) xi
Appendix D(cid:4) Hilbert’s Axioms 517
D.1 INCIDENCE AXIOMS 517
D.2 BETWEENESS AXIOMS 517
D.3 CONGRUENCE AXIOMS 518
D.4 PARALLELISM AXIOM 518
D.5 CONTINUITY AXIOM 518
Appendix E(cid:4) Wallpaper Groups 521
E.1 PATTERNS 521
References 527
Index 531
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.–LewisCarroll(CharlesDodgson),1832–1898
Geometry is one of the most ancient subjects within mathematics. It
is also one of the most vibrant and modern. One can simply walk down
any street and see geometry come to life. The shapes that one sees in
buildings, traffic signs, billboards, and paving stones are all based on
some fundamental geometric design.
Geometry is also key to the development of computational “virtu-
alities” that are now ubiquitous. These include the three-dimensional
computer graphics animations used in television and film, virtual-reality
displays, robotics, and the layout and design of architectural structures.
Geometryisarichandrewardingareaformathematicalexploration.
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.
The interplay between the practical and the abstract is one of the
main features of this text. For students new to geometry, there are nu-
merous concrete exploratory activities to aid in the understanding of ge-
ometric concepts. For students who have prior experience in geometry,
the text provides detailed proofs and explanations and offers advanced
topics such as elliptic and projective geometry. In general, the pedagog-
ical approach used in this text is to give equal weight to the intuitive
andimaginativeexplorationofgeometryaswellastoabstractreasoning
and proofs.