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Affine and Projective Geometry AFFINE AND PROJECTIVE GEOMETRY Μ. K. Bennett Department of Mathematics and Statistics University of Massachusetts Amherst, Massachusetts A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York • Chichester • Brisbane • Toronto • Singapore This page intentionally left blank AFFINE AND PROJECTIVE GEOMETRY This page intentionally left blank AFFINE AND PROJECTIVE GEOMETRY Μ. K. Bennett Department of Mathematics and Statistics University of Massachusetts Amherst, Massachusetts A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York • Chichester • Brisbane • Toronto • Singapore This text is printed on acid-free paper. Copyright © 1995 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012. Library of Congress Cataloging in Publication Data: Bennett, Μ. K. (Mary Katherine) 1940- Affine and projective geometry / Μ. K. Bennett, p. cm. "A Wiley-Interscience publication." Includes index. ISBN 0-471-11315-8 1. Geometry, Affine. 2. Geometry, Projective. I. Title. QA477.B46 1995 516'.4—dc20 94-44365 10 987654321 To Tom who is always therefor me This page intentionally left blank Contents List of Examples xi Special Symbols xiii Preface xv 1. Introduction 1 1.1. Methods of Proof, 1 1.2. Some Greek Geometers, 2 1.3. Cartesian Geometry, 6 1.4. Hubert's Axioms, 11 1.5. Finite Coordinate Planes, 12 1.6. The Theorems of Pappus and Desargues, 14 Suggested Reading, 17 2. Affine Planes 18 2.1. Definitions and Examples, 18 2.2. Some Combinatorial Results, 23 2.3. Finite Planes, 28 2.4. Orthogonal Latin Squares, 33 2.5. Affine Planes and Latin Squares, 36 2.6. Projective Planes, 41 Suggested Reading, 46 3. Desarguesian Affine Planes 47 3.1. The Fundamental Theorem, 47 3.2. Addition on Lines, 48 3.3. Desargues' Theorem, 50 3.4. Properties of Addition in Affine Planes, 55 3.5. The Converse of Desargues' Theorem, 58 vii viii Contents 3.6. Multiplication on Lines of an Affine Plane, 62 3.7. Pappus' Theorem and Further Properties, 66 Suggested Reading, 70 4. Introducing Coordinates 71 4.1. Division Rings, 71 4.2. Isomorphism, 74 4.3. Coordinate Affine Planes, 76 4.4. Coordinatizing Points, 82 4.5. Linear Equations, 86 4.6 The Theorem of Pappus, 90 Suggested Reading, 92 5. Coordinate Projective Planes 93 5.1. Projective Points and Homogeneous Equations in D3, 93 5.2. Coordinate Projective Planes, 96 5.3. Coordinatization of Desarguesian Projective Planes, 99 5.4. Projective Conies, 102 5.5. Pascal's Theorem, 106 5.6. Non-Desarguesian Coordinate Planes, 110 5.7. Some Examples of Veblen-Wedderburn Systems, 114 5.8. A Projective Plane of Order 9, 117 Suggested Reading, 122 6. Affine Space 123 6.1. Synthetic Affine Space, 123 6.2. Flats in Affine Space, 128 6.3. Desargues' Theorem, 131 6.4. Coordinatization of Affine Space, 134 Suggested Reading, 143 7. Projective Space 144 7.1 Synthetic Projective Space, 144 7.2. Planes in Projective Space, 147 7.3. Dimension, 150 7.4. Consequences of Desargues' Theorem, 154 7.5. Coordinates in Projective Space, 159 Suggested Reading, 164 8. Lattices of Flats 165 8.1. Closure Spaces, 165 8.2. Some Properties of Closure Spaces, 167 Contents ix 8.3. Projective Closure Spaces, 170 8.4. Introduction to Lattices, 171 8.5. Bounded Lattices; Duality, 174 8.6. Distributive, Modular, and Atomic Lattices, 177 8.7. Complete Lattices and Closure Spaces, 181 Suggested Reading, 185 9. CoIIineations 186 9.1. General CoIIineations, 186 9.2. Automorphisms of Planes, 190 9.3. Perspectivities of Projective Spaces, 195 9.4. The Fundamental Theorem of Projective Geometry, 199 9.5. Projectivities and Linear Transformations, 205 9.6. CoIIineations and Commutativity, 209 Suggested Reading, 211 Appendix A. Algebraic Background 212 A.l. Fields, 212 A.2. The Integers Mod n, 214 A.3. Finite Fields, 217 Suggested Reading, 220 Appendix B. Hubert's Example of a Noncommutative Division Ring 221 Suggested Reading, 223 Index 225 This page intentionally left blank List of Examples Chapter 1 Example 1. Four point plane, 12 Example 2. Nine point plane, 13 Chapter 2 Example 1. Real coordinate plane, 19 Example 2. Rational affine plane, 19 Example 3. Four point plane, 19 Example 4. Independence of Al, 20 Example 5. Fano plane, 21 Example 6. Independence of A3, 21 Example 7. Nine point plane, 23 Example 8. Affine plane of order 4, 37 Example 9. Euclidean hemisphere, 42 Example 10. Real projective plane, 42 Chapter 3 Example 1. Affine plane of order 5, 69 Chapter 4 Example 1. Quaternions, 72 Example 2. Affine plane over GF(4), 80 Example 3. Affine plane over Z5, 81 Chapter 5 Example 1. Projective plane over Z2, 97 Example 2. Projective plane over Z3, 97 Example 3. Thirteen point projective plane, 101 Example 4. Twenty-one point projective plane, 102 xii List of Examples Chapter 6 Example 1. Affine planes as affine spaces, 124 Example 2. Coordinate affine spaces, 124 Example 3. Eight point affine space, 125 Example 4. Independence of AS1, 126 Example 5. Independence of AS2, 126 Example 6. Fano plane, 126 Example 7. Sasaki's example, 126 Chapter 7 Example 1. Projective planes as projective spaces, 145 Example 2. Fano space, 145 Example 3. Coordinate projective spaces, 145 Example 4. Independence of PS I, 146 Example 5. Independence of PS2, 146 Example 6. Independence of PS3, 146 Example 7. Fano plane, 163 Example 8. Fifteen point projective space, 164 Chapter 8 Example 1 Vector space as closure space, 165 Example 2. Centralizer closure space, 166 Example 3. Subfield closure space, 166 Example 4. Function closure space, 166 Example 5. Topological closure space, 166 Example 6. Convex closure space, 167 Example 7. Power set lattice, 171 Example 8. Subgroup lattice, 171 Example 9. Lattice of vector subspaces, 172 Example 10. Divisibility lattice, 172 Example 11. Closure space lattice, 172 Appendix A Example 1. Complex numbers, 213 Example 2. Integers mod n, 215 Example 3. Integers mod 3, 215 Example 4. Four element field, 217 Example 5. GF(9), 218 Example 6. GF(27), 218 Special Symbols The following is a list of special symbols used in this book, together with the page on which each first appears. R 7 U2 7 Q 7 Q2 7 Z„ 12 &) 18 /(P,Q) 18 /IU 18 ΤΓ(Γ) 28 R°S 29 Δ 29 m=„k 30 //(«) 36 {&>'t2") 41 {p,q,r) 42 ΧΦ 43 /„ 43 Η 73 D" 76 GF(n) 80 (/,Ο,Ι) 82 /X 84 [a,b,c] 97 PG2(D) 97 9) 123 124 P(iT) 129 ^ 130 S> 130 xiii xiv Special Symbols •5*w 131 AG(^) 134 I. 135 U.O.IJ 135 φ, 135 ΦΑ„ 135 <*,,...,*„> 146 PN 146 PG(F") 146 (M,/) 147 (M, a) 150 XR 158 PG(^) 159 ^xD 160 <P,1> 160 2r 165 (S,<) 171 V 171 Λ 171 & + W 172 Ν 172 a\b 172 W, j) 172 α,ν,Λ) 172 L* 176 23 177 M3 178 Ns 178 < 179 <· 179 L Χ Μ 184 VT 201 Q(v/2) 214 A' 2Π F[u] 217 F[K]/(?(II)) 218 Preface The purpose of this text is to tie together three different approaches to affine and projective geometry: the algebraic (or analytic) approach of coordinate geometry, the axiomatic approach of synthetic geometry, and the lattice theoretic approach through the flats of a geometry. It provides, in one volume, a complete, detailed, and self-contained description of the coordinatization of (Desarguesian) affine and projective space and a thorough discussion of the lattices of flats of these spaces, ending with the Fundamental Theory of Projective Geometry, relating synthetic collineations, lattice isomorphisms, and vector space isomorphisms. The text first aims to offer (for upper-level mathematics and mathe- matics education students) an opportunity to see, at a fairly early stage in their mathematical development, the basic correlation between two seem- ingly disparate but familiar branches of mathematics (synthetic geometry and linear algebra). The first five chapters are suitable for a one-semester course. They attempt to provide background and insight into this correla- tion, so important for prospective secondary teachers, and to develop an appreciation of the geometric nature of linear algebra. These chapters will also strengthen the student's understanding of abstract algebra, since they provide a different perspective on fields than is usually presented in an algebra course. They also will give the students who will not take an algebra course a modicum of familiarity with the most basic notions in abstract algebra. The nine chapters are suitable for a two-semester course. The last four chapters concentrate on nonplanar spaces and vector spaces of dimension 3 or more. They use geometry to introduce the students to the rudiments of lattice theory, with the final chapter providing insights into the geometric nature of linear and semilinear bijections on vector spaces. Throughout the text, geometry is used as a vehicle for introducing such topics as finite fields, some basic combinatorics, closure spaces, lattices, and algebraic systems such as division rings and those of Veblen and Wedderburn. Some of these topics may not appear in other under- graduate courses, and they may inspire some students to further investiga- tion. There are very interesting open questions in affine and projective xv xvi Preface geometry that can easily be understood by students at this level. Dis- cussing such questions always seems to interest students and whet their appetites for more mathematics.
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