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

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AFFINE AND PROJECTIVE GEOMETRY

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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 9 8 7 6 5 4 3 2 1

To Tom

who is always therefor me

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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 D 3 , 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

6.1. 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

6. Affine Space 123

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

Suggested Reading, 223

221

Index 225

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List of Examples

Chapter 1

Example 1. Example 2.

Chapter 2

Example 1. Example 2. Example 3. Example 4. Example 5. Example 6. Example 7. Example 8. Example 9. Example 10.

Chapter 3

Example 1.

Chapter 4

Example 1. Example 2. Example 3.

Chapter 5

Example 1. Example 2. Example 3. Example 4.

Four point plane, 12 Nine point plane, 13

Real coordinate plane, 19 Rational affine plane, 19 Four point plane, 19 Independence of Al, 20 Fano plane, 21 Independence of A3, 21 Nine point plane, 23 Affine plane of order 4, 37 Euclidean hemisphere, 42 Real projective plane, 42

Affine plane of order 5, 69

Quaternions, 72 Affine plane over GF(4), 80 Affine plane over Z 5 , 81

Projective plane over Z 2 , 97 Projective plane over Z 3 , 97 Thirteen point projective plane, 101 Twenty-one point projective plane, 102

xii List of Examples

Chapter 6

Example 1. Example 2. Example 3. Example 4. Example 5. Example 6. Example 7.

Chapter 7

Example 1. Example 2. Example 3. Example 4. Example 5. Example 6. Example 7. Example 8.

Chapter 8

Example 1 Example 2. Example 3. Example 4. Example 5. Example 6. Example 7. Example 8. Example 9. Example 10. Example 11.

Appendix A

Example 1. Example 2. Example 3. Example 4. Example 5. Example 6.

Affine planes as affine spaces, 124 Coordinate affine spaces, 124 Eight point affine space, 125 Independence of AS1, 126 Independence of AS2, 126 Fano plane, 126 Sasaki's example, 126

Projective planes as projective spaces, 145 Fano space, 145 Coordinate projective spaces, 145 Independence of PS I, 146 Independence of PS2, 146 Independence of PS3, 146 Fano plane, 163 Fifteen point projective space, 164

Vector space as closure space, 165 Centralizer closure space, 166 Subfield closure space, 166 Function closure space, 166 Topological closure space, 166 Convex closure space, 167 Power set lattice, 171 Subgroup lattice, 171 Lattice of vector subspaces, 172 Divisibility lattice, 172 Closure space lattice, 172

Complex numbers, 213 Integers mod n, 215 Integers mod 3, 215 Four element field, 217 GF(9), 218 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

Q 2 7 Z„ 1 2

& ) 1 8

/ ( P , Q ) 1 8

/ I U 1 8 ΤΓ(Γ) 2 8

R ° S 2 9

Δ 2 9

m=„k 3 0 / / ( « ) 3 6

{&>'t2") 4 1 {p,q,r) 4 2 Χ Φ 4 3

/ „ 4 3

Η 7 3 D" 7 6 GF(n) 8 0 ( / , Ο , Ι ) 8 2

/ X 8 4

[a,b,c] 9 7 P G 2 ( D ) 9 7

9) 123 124

P(iT) 129 ^ 1 3 0

S> 130

xiii

xiv Special Symbols

•5* w 131

A G ( ^ ) 134

I . 135

U . O . I J 135

φ, 135 ΦΑ„ 135

< * , , . . . , * „ > 146

P N 146

PG(F") 146 ( M , / ) 147

( M , a) 150 X R 158

P G ( ^ ) 159

^ x D 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

2 3 177 M 3 178

Ns 178 < 179

< · 179

L Χ Μ 184 V T 201

Q(v /2) 2 1 4

A' 2Π F[u] 2 1 7 F [ K ] / ( ? ( I I ) ) 2 1 8

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.

There are many legitimate and interesting possibilities for a one- or two-semester course in geometry for mathematics majors. In general, geometry courses tend to be either survey courses covering several types of geometry or courses concentrating on Euclidean or one particular sort of non-Euclidean geometry. This text is of the latter variety, and focuses on affine geometry.

The main mathematical distinction between this and other single-geometry texts is the emphasis on affine rather than projective geometry. Although projective geometry is, with its duality, perhaps easier for a mathematician to study, an argument can be made that affine geometry is intuitively easier for a student. The reason for this is twofold: the Euclidean plane, already familiar, is an affine plane, and in coordinatizing affine geometry by the methods described here, one sees the actual vectors, rather than one-dimensional subspaces, in a vector space.

The use of basic algebra and linear algebra should help the students to see how one branch of mathematics can sometimes be used to prove theorems in another. The text is designed so that it can be used by students who have little or no background in algebra, as well as by students who do.

I owe thanks to several people who read earlier drafts of this text and made helpful comments. Joseph Bonin of George Washington University, Richard Greechie of Louisiana Tech University, and Robert Piziak of Baylor University used earlier drafts with their students and made wel-come suggestions. I am particularly grateful to Garrett Birkhoff of Harvard University, who encouraged me in this project and who made many invaluable comments, sharing with me his wisdom and his many years of experience in mathematical writing.

For their patience and good cheer, and for the help they've given me, I would like to thank the staffs of the University of Massachusetts Physical Sciences Library and of John Wiley & Sons.

Finally, the appearance of a book gives its author the opportunity to express appreciation to those people who, while not directly involved with this project, nevertheless have had a profound influence on her mathemat-ical life. I wish to thank Sister Eileen Flanagan who taught me the challenge of mathematics in high school, Florence D. Jacobson of Albertus Magnus College, who first made me love abstract algebra, and David J. Foulis of the University of Massachusetts, who opened for me the doors to mathematical research in lattice theory as well as affine and projective geometry. These extraordinary people have made a real difference in my life.

Amherst, Massachusetts

May 1995

AFFINE AND PROJECTIVE GEOMETRY

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CHAPTER 1

Introduction

The formal study of an area of mathematics is based on definitions, axioms, and theorems. The basic terms needed are either defined or assumed to be familiar; basic facts, called axioms, are assumed to be true, and the theorems that can be proved using the axioms and previously proved theorems make up the mathematical content of the subject under investigation.

1.1 METHODS OF PROOF

There are two basic methods of proof: direct proof, and proof by con-tradiction. The following examples of mathematical proofs come from elementary number theory, first treated systematically in Euclid's Ele-ments. In these examples it is assumed that the terms "integer" and "divisor" are familiar. An even integer is defined to be one of the form 2k, and an odd integer is defined to be one of the form 2k + 1 (k an integer). First is a simple direct proof that the product of two odd integers is odd.

• THEOREM 1. If η and m are odd integers, their product η X m is also odd.

Proof: Let η = 2k + 1 and m = 2h + 1. Then η X m = (2k + 1) (2Λ + 1) = Akh + 2k + 2h + 1 = 2(2kh + k + h) + 1, which is odd. •

In a proof by contradiction, one assumes the opposite of what is to be proved and deduces an untrue statement (or contradiction) as in the following example:

• THEOREM 2. For any integer n, if n2 is even, then η is even.

1

2 Introduction

Proof: Suppose that η is odd. Then η = 2k + 1 for some k; thus n2

is the product of odd numbers and is therefore odd. This contradicts the hypothesis that n2 is even; hence the assumption that η is odd must be false. Therefore η is even. •

An integer is called prime if it has exactly two divisors, itself and 1. Two positive integers a and b are said to be relatively prime if their only common divisor is 1. (In particular, when a and b are relatively prime, the fraction a/b is in lowest terms.) Take as an axiom the fact that integers a and b are relatively prime if and only if there are integers m and η with 1 = ma + nb.

The following is a second example of a proof by contradiction:

• THEOREM 3. Let p, u, and ν be integers with ρ prime. Then, if ρ divides the product uv, ρ divides u or ρ divides v.

Proof: If ρ divides neither u nor v, then ρ and u are relatively prime, as are ρ and v. Thus 1 = mp + nu, and 1 = m'p + n'v. Multiplying these two equations gives 1 = (mm'ρ + mn'v + m'nu)p + nn'uv. Thus ρ is relatively prime to uv, a contradiction. •

EXERCISES

1. Assume that every rational number can be written as a fraction in lowest

terms. Prove that -Jl is irrational by assuming that ^2 = a/b, a fraction in lowest terms.

2. Prove that is irrational.

*3. Prove that, for any whole number «, V̂T is either a whole number or is irrational.

4. By "p divides a" (written p\a) is meant that a can be written as mp for some integer m. Suppose that p, a, and η are positive integers with ρ prime and η greater than or equal to two. Prove that ρ divides a if and only if ρ divides a".

5. Prove that yp is irrational if η s 2 and ρ is a prime number. 6. By trying some examples, discover which integers η have the property that

n|(n - 1).

1.2 SOME GREEK GEOMETERS

The first person whose name was actually associated to some proofs was one Thales of Miletus (supposedly a retired olive oil merchant) who lived around 600 BC. The following five theorems are now attributed to him:

1. Any diameter of a circle bisects the circle. 2. The base angles of an isosceles triangle are equal.

1.2. Some Greek Geometers 3

3. Vertical angles are equal. 4. An angle inscribed in a semicircle is a right angle.

5. If two triangles have two angles and a side of one equal respectively to the corresponding two angles and side of the other, then the triangles are congruent.

Thales' student, Pythagoras, who became even more famous than his teacher, founded a school at Crotona (in what is now southern Italy) around 540 BC. Since the members of his group (or cult) dispersed their results under the name of their founder, it is difficult to know exactly what was due to Pythagoras and what was due to his followers. The famous Pythagorean Theorem which bears his name states that the square of the hypoteneuse of any right triangle is equal to the sum of the squares of its legs. This theorem was known before the time of Pythagoras, but was probably first proved by his followers. They also maintained an interest in number lore, and hoped to explain all the mysteries (i.e., physics) of the universe in terms of the integers and fractions, but they fell upon hard

times when they discovered (and tried to suppress the knowledge) that is irrational.

A neighboring rival school had sprouted up—the Eleatics (whose members included Zeno and Eudoxus) and these rival groups were con-stantly challenging and trying to best each other mathematically. Doubt-less the Eleatics were delighted to draw attention to 4Ϊ at every possible opportunity. The Eleatics were the precursors of the discoverers of the idea of a limit which is the foundation on which calculus is based. One "proof that the area of a circle is -nr2 goes as follows: If a circle is divided into sectors (as shown in Fig. 1.1), then the area of the circle is the sum of the areas of the sectors. If the sectors are made smaller and smaller, they resemble miniscule triangles whose areas are each \bh. Adding all the 6's gives the circumference of the circle; each h is equal to the radius of the circle. Thus

1

A = — r X (Sum of the bases)

1

= — r x (Circumference of circle)

1 = -r X (2 irr) = 7ΓΓ 2 .

The so-called proof has two things to recommend it. It seems reasona-ble, and it gives the right answer. All it lacks is logical rigor in passing

4 Introduction

Figure 1.1.

from sectors to triangles. This was supplied centuries later when limits were discovered.

The person who contributed most to Greek geometry was not mainly an innovator at all, but an organizer and expositor. Euclid's name is probably known to more people than that of any other mathematician. His Elements was used as the geometry text for over two thousand years, and he gave his name to the "Euclidean geometry"studied in high school and "Euclidean space" studied in linear algebra to this day. Euclid was at the University of Alexandria (the first real university) around 300 BC, and undertook the enormous task of collecting and arranging in logical order all of the geometry (as well as number theory) known then. The result of his labors, Euclid's Elements, has been studied, copied, criticized, and generalized but certainly never ignored. Euclid started from five axioms (or self-evident properties of magnitudes) and five postulates (self-evident geometrical truths):

AXIOMS

1. Things equal to the same thing are equal to each other. 2. If equals are added to equals, their sums are equal. 3. If equals are subtracted from equals, their differences are equal. 4. Things that coincide with one another are equal to one another. 5. The whole is greater than any of its parts.

1.2. Some Greek Geometers 5

ΔΒ = Z.C (Thales' Theorem 2),

Z.ADB = /.ADC (Euclid's Postulate 4),

AD = AD (Euclid's Axiom 4),

Δ ADB = Δ ADC (Thales' Theorem 5). •

POSTULATES

1. A straight line can be drawn between any two points. 2. A line can be extended indefinitely in either direction. 3. A circle can be described with any point as center and any segment as

radius. 4. All right angles are equal. 5. Through a point not on a line there exists a unique line parallel to the

given line.

The fifth or parallel postulate (a later formulation due to John Playfair is given here) puzzled mathematicians of the eighteenth and nineteenth centuries. Until around 1830 mathematicians tried to prove it from the other axioms, and it is part of the genius of Euclid that he recognized that it had to be assumed. When this postulate is dropped or changed, different kinds of geometries result, but this was not recognized until 2000 years after Euclid's work.

Much has been written about the flaws in Euclid's logic; the essential points are these: Euclid did not have set theory for describing undefined terms such as "point" and "line," the algebra of real and rational numbers was not yet developed, and some of his postulates could have been more rigorously stated. For example, mathematicians feel that Euclid tacitly assumed the uniqueness of the line drawn in the first postulate. The visual appeal in Euclid's proofs was both a strength and a weakness. Its strength was as an aid to intuition; however, it is well known that Euclid's procedures, when applied to skillfully misdrawn figures, can produce false results, such as "every triangle is isosceles" (Kline 1972, pp. 1006-1007). Other examples may be found in (Northrup 1944, ch. 6).

This section concludes with a formal example of a geometric theorem and its direct proof based on Euclid's Axioms and Postulates, and Thales' Theorems 2 and 5 given above.

• THEOREM 4. Any altitude of an equilateral triangle divides it into two congruent triangles.

Proof: Let A, B, and C be the vertices of an equilateral triangle, and let D be the endpoint of its altitude AD. Then

6 Introduction

Observe that the first sentence of the proof names a triangle and its relevant parts. Generally, it is worthwhile to proceed in this way so that specific and rigorous statements can then be given. In the rest of the proof above, each statement was accompanied by a legitimate reason (one of Euclid's assumptions or one of Thales' theorems). The student without much experience in proving theorems is strongly encouraged to write proofs carefully, making sure that every statement is based on axioms or previously proved theorems. The best was (in fact the only way) to learn to prove theorems is by practice, using the examples given in the text as models.

EXERCISES

In the following exercises, assume Euclid's Axioms and Postulates, the theo-rems of Thales given above, and the Pythagorean Theorem.

1. Prove that if a line bisects an angle of a triangle and is perpendicular to the opposite side, then it bisects that side.

2. Prove that in an isosceles triangle, altitudes to the equal sides are equal. 3. Prove that the area of any equilateral triangle, each of whose sides is s units in

length, is sV3/A. 4. Derive the formula for the area of a trapezoid with (parallel) bases bl and b2

whose height is h. 5. Assuming that lines are parallel exactly when they have empty intersection,

prove that if a line intersects one of two parallel lines, it intersects the other.

1.3 CARTESIAN GEOMETRY

In the seventeenth century the French philosopher and mathematician Rene Descartes published La Geometrie in which he sketched what is today called analytic (or Cartesian) geometry. The idea is quite simple. One can associate to each point of the Euclidean plane, a pair of real numbers (or coordinates) in such a way that every straight line is repre-sented by a linear equation of the form ax + by = c, where a, b, and c are constants (a and b not both zero). Circles and conic sections are represented by quadratic equations of the form ax2 + bxy + cy2 + dx + ey =/.

Descartes began his book by asserting that "any problem in geometry can be reduced to such terms that one only needs to know the lengths of certain straight lines" to solve it. He discussed the operations of addition, subtraction, multiplication, division, and extraction of square roots, which can be performed on lengths of segments, and pointed out how easy it is to add and multiply lengths geometrically (as Euclid had done) by what is today called geometric algebra. Descartes' rectangular x- and y-axes