The Four Pillars of Geometry (Undergraduate Texts in Mathematics)
By: John Stillwell
ISBN: 0387255303
ISBN-13: 9780387255309
Publisher: Springer - 2005-08-09
Hardcover | 1 Edition | 229 Pages
Editorial Review:
This new textbook demonstrates that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. In this book, two chapters are devoted to each approach, the first being concrete and introductory, while the second is more abstract.
Geometry, of all subjects, should be about taking different viewpoints, and geometry is unique among mathematical disciplines in its ability to look different from different angles. Some students prefer to visualize, while others prefer to reason or to calculate. Geometry has something for everyone, and students will find themselves building on their strengths at times, and working to overcome weaknesses at other times. This book will be suitable for a second course in geometry and contains more than 100 figures and a large selection of exercises in each chapter.
Table of Contents
Preface vii 1 Straightedge and compass 1 1.1 Euclid's construction axioms 2 1.2 Euclid's construction of the equilateral triangle 4 1.3 Some basic constructions 6 1.4 Multiplication and division 10 1.5 Similar triangles 13 1.6 Discussion 17 2 Euclid's approach to geometry 20 2.1 The parallel axiom 21 2.2 Congruence axioms 24 2.3 Area and equality 26 2.4 Area of parallelograms and triangles 29 2.5 The Pythagorean theorem 32 2.6 Proof of the Thales theorem 34 2.7 Angles in a circle 36 2.8 The Pythagorean theorem revisited 38 2.9 Discussion 42 3 Coordinates 46 3.1 The number line and the number plane 47 3.2 Lines and their equations 48 3.3 Distance 51 3.4 Intersections of lines and circles 53 3.5 Angle and slope 55 3.6 Isometries 57 3.7 The three reflections theorem 61 3.8 Discussion 63 4 Vectors and Euclidean spaces 65 4.1 Vectors 66 4.2 Direction and linear independence 69 4.3 Midpoints and centroids 71 4.4 The inner product 74 4.5 Inner product and cosine 77 4.6 The triangle inequality 80 4.7 Rotations, matrices, and complex numbers 83 4.8 Discussion 86 5 Perspective 88 5.1 Perspective drawing 89 5.2 Drawing with straightedge alone 92 5.3 Projective plane axioms and their models 94 5.4 Homogeneous coordinates 98 5.5 Projection 100 5.6 Linear fractional functions 104 5.7 The cross-ratio 108 5.8 What is special about the cross-ratio? 110 5.9 Discussion 113 6 Projective planes 117 6.1 Pappus and Desargues revisited 118 6.2 Coincidences 121 6.3 Variations on the Desargues theorem 125 6.4 Projective arithmetic 128 6.5 The field axioms 133 6.6 The associative laws 136 6.7 The distributive law 138 6.8 Discussion 140 7 Transformations 143 7.1 The group of isometries of the plane 144 7.2 Vector transformations 146 7.3 Transformations of the projective line 151 7.4 Spherical geometry 154 7.5 The rotation group of the sphere 157 7.6 Representing space rotations by quaternions 159 7.7 A finite group of space rotations 163 7.8 The groups S[superscript 3] and RP[superscript 3] 167 7.9 Discussion 170 8 Non-Euclidean geometry 174 8.1 Extending the projective line to a plane 175 8.2 Complex conjugation 178 8.3 Reflections and Mobius transformations 182 8.4 Preserving non-Euclidean lines 184 8.5 Preserving angle 186 8.6 Non-Euclidean distance 191 8.7 Non-Euclidean translations and rotations 196 8.8 Three reflections or two involutions 199 8.9 Discussion 203 References 213 Index 215
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