9780387255309

The Four Pillars of Geometry

by
  • ISBN13:

    9780387255309

  • ISBN10:

    0387255303

  • Format: Hardcover
  • Copyright: 10/11/2005
  • Publisher: Textstream
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Summary

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.

Author Biography

John Stillwell is Professor of Mathematics at the University of San Francisco

Table of Contents

Preface vii
1 Straightedge and compass 1(19)
1.1 Euclid's construction axioms
2(2)
1.2 Euclid's construction of the equilateral triangle
4(2)
1.3 Some basic constructions
6(4)
1.4 Multiplication and division
10(3)
1.5 Similar triangles
13(4)
1.6 Discussion
17(3)
2 Euclid's approach to geometry 20(26)
2.1 The parallel axiom
21(3)
2.2 Congruence axioms
24(2)
2.3 Area and equality
26(3)
2.4 Area of parallelograms and triangles
29(3)
2.5 The Pythagorean theorem
32(2)
2.6 Proof of the Thales theorem
34(2)
2.7 Angles in a circle
36(2)
2.8 The Pythagorean theorem revisited
38(4)
2.9 Discussion
42(4)
3 Coordinates 46(19)
3.1 The number line and the number plane
47(1)
3.2 Lines and their equations
48(3)
3.3 Distance
51(2)
3.4 Intersections of lines and circles
53(2)
3.5 Angle and slope
55(2)
3.6 Isometries
57(4)
3.7 The three reflections theorem
61(2)
3.8 Discussion
63(2)
4 Vectors and Euclidean spaces 65(23)
4.1 Vectors
66(3)
4.2 Direction and linear independence
69(2)
4.3 Midpoints and centroids
71(3)
4.4 The inner product
74(3)
4.5 Inner product and cosine
77(3)
4.6 The triangle inequality
80(3)
4.7 Rotations, matrices, and complex numbers
83(3)
4.8 Discussion
86(2)
5 Perspective 88(29)
5.1 Perspective drawing
89(3)
5.2 Drawing with straightedge alone
92(2)
5.3 Projective plane axioms and their models
94(4)
5.4 Homogeneous coordinates
98(2)
5.5 Projection
100(4)
5.6 Linear fractional functions
104(4)
5.7 The cross-ratio
108(2)
5.8 What is special about the cross-ratio?
110(3)
5.9 Discussion
113(4)
6 Projective planes 117(26)
6.1 Pappus and Desargues revisited
118(3)
6.2 Coincidences
121(4)
6.3 Variations on the Desargues theorem
125(3)
6.4 Projective arithmetic
128(5)
6.5 The field axioms
133(3)
6.6 The associative laws
136(2)
6.7 The distributive law
138(2)
6.8 Discussion
140(3)
7 Transformations 143(31)
7.1 The group of isometries of the plane
144(2)
7.2 Vector transformations
146(5)
7.3 Transformations of the projective line
151(3)
7.4 Spherical geometry
154(3)
7.5 The rotation group of the sphere
157(2)
7.6 Representing space rotations by quaternions
159(4)
7.7 A finite group of space rotations
163(4)
7.8 The groups S³ and RP³
167(3)
7.9 Discussion
170(4)
8 Non-Euclidean geometry 174(39)
8.1 Extending the projective line to a plane
175(3)
8.2 Complex conjugation
178(4)
8.3 Reflections and Möbius transformations
182(2)
8.4 Preserving non-Euclidean lines
184(2)
8.5 Preserving angle
186(5)
8.6 Non-Euclidean distance
191(5)
8.7 Non-Euclidean translations and rotations
196(3)
8.8 Three reflections or two involutions
199(4)
8.9 Discussion
203(10)
References 213(2)
Index 215

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