Geometries

A. B. Sossinsky

STUDENT MATHEMAT ICAL L IBRARYVolume 64

Geometries

http://dx.doi.org/10.1090/stml/064

Geometries

A. B. Sossinsky

STUDENT MATHEMAT ICAL L IBRARYVolume 64

American Mathematical SocietyProvidence, Rhode Island

Editorial Board

Gerald B. FollandRobin Forman

Brad G. Osgood (Chair)John Stillwell

2010 Mathematics Subject Classification. Primary 51-01;Secondary 51-02, 01A20, 01A55, 18-01.

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Sosinskii, A. B. (Aleksei Bronislavovich)Geometries / A. B. Sossinsky.

p. cm. – (Student mathematical library ; v. 64)Includes bibliographical references and index.ISBN 978-0-8218-7571-1 (alk. paper)1. Geometry–Textbooks. I. Title.

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10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12

Contents

Preface xiii

Chapter 0. About Euclidean Geometry 1

§0.1. The axioms of Euclidean plane geometry 2

§0.2. Commentary 5

§0.3. Rotations 7

§0.4. Parallel translations and vectors 11

§0.5. Triangles: congruence, properties 13

§0.6. Homothety and similitude 15

§0.7. Angle measure and trigonometry 18

§0.8. Properties of the circle 20

§0.9. Isometries of the plane 24

§0.10. Space geometry 28

Chapter 1. Toy Geometries and Main Definitions 33

§1.1. Isometries of the Euclidean plane and space 33

§1.2. Symmetries of some figures 35

§1.3. Transformation groups 41

§1.4. The category of geometries 46

§1.5. Some philosophical remarks 49

v

vi Contents

§1.6. Problems 50

Chapter 2. Abstract Groups and Group Presentations 53

§2.1. Abstract groups 53

§2.2. Morphisms of Groups 57

§2.3. Subgroups 58

§2.4. The Lagrange theorem 59

§2.5. Quotient groups 60

§2.6. Free groups and permutations 61

§2.7. Group presentations 62

§2.8. Cayley’s theorem 64

§2.9. Problems 65

Chapter 3. Finite Subgroups of SO(3) and the Platonic Bodies 67

§3.1. The Platonic bodies in art, philosophy, and science 68

§3.2. Finite subgroups of SO(3) 70

§3.3. The five regular polyhedra 77

§3.4. The five Kepler cubes 78

§3.5. Regular polyhedra in higher dimensions 79

§3.6. Problems 81

Chapter 4. Discrete Subgroups of the Isometry Group of the

Plane and Tilings 85

§4.1. Tilings in architecture, art, and science 85

§4.2. Tilings and crystallography 87

§4.3. Isometries of the plane 89

§4.4. Discrete groups and discrete geometries 90

§4.5. The seventeen regular tilings 90

§4.6. The 230 crystallographic groups 95

§4.7. Problems 95

Chapter 5. Reflection Groups and Coxeter Geometries 99

§5.1. An example: the kaleidoscope 99

Contents vii

§5.2. Coxeter polygons and polyhedra 100

§5.3. Coxeter geometries on the plane 101

§5.4. Coxeter geometries in Euclidean space R3 103

§5.5. Coxeter schemes and the classification theorem 105

§5.6. Problems 107

Chapter 6. Spherical Geometry 109

§6.1. A list of classical continuous geometries 109

§6.2. Some basic facts from Euclidean plane geometry 113

§6.3. Lines, distances, angles, polars, and perpendiculars 114

§6.4. Biangles and triangles in S2 116

§6.5. Other theorems about triangles 120

§6.6. Coxeter triangles on the sphere S2 121

§6.7. Two-dimensional elliptic geometry 121

§6.8. Problems 123

Chapter 7. The Poincare Disk Model of Hyperbolic Geometry 125

§7.1. Inversion and orthogonal circles 126

§7.2. Definition of the disk model 131

§7.3. Points and lines in the hyperbolic plane 133

§7.4. Perpendiculars 134

§7.5. Parallels and nonintersecting lines 134

§7.6. Sum of the angles of a triangle 135

§7.7. Rotations and circles in the hyperbolic plane 136

§7.8. Hyperbolic geometry and the physical world 138

§7.9. Problems 139

Chapter 8. The Poincare Half-Plane Model 143

§8.1. Affine and linear-fractional transformations of C 144

§8.2. The Poincare half-plane model 147

§8.3. Perpendiculars and parallels 148

§8.4. Isometries w.r.t. Mobius distance 150

§8.5. Problems 151

viii Contents

Chapter 9. The Cayley–Klein Model 153

§9.1. Isometry and the Cayley–Klein model 153

§9.2. Parallels in the Cayley–Klein model 156

§9.3. Perpendiculars in the Cayley–Klein model 158

§9.4. The hyperbolic line and relativity 159

§9.5. Problems 160

Chapter 10. Hyperbolic Trigonometry and Absolute Constants 163

§10.1. Isomorphism between the two disk models 163

§10.2. Isomorphism between the two Poincare models 168

§10.3. Hyperbolic functions 169

§10.4. Trigonometry on the hyperbolic plane 170

§10.5. Angle of parallelism and Schweikart constant 170

§10.6. Problems 173

Chapter 11. History of Non-Euclidean Geometry 177

§11.1. Euclid’s Fifth Postulate 177

§11.2. Statements equivalent to the Fifth Postulate 178

§11.3. Gauss 179

§11.4. Lobachevsky 180

§11.5. Bolyai 182

§11.6. Beltrami, Helmholtz, Lie, Cayley, Klein, Poincare 183

§11.7. Hilbert 184

Chapter 12. Projective Geometry 185

§12.1. The projective plane as a geometry 185

§12.2. Homogeneous coordinates 186

§12.3. Projective transformations 188

§12.4. Cross ratio of collinear points 191

§12.5. Projective duality 192

§12.6. Conics in RP 2 194

§12.7. The Desargues, Pappus, and Pascal theorems 194

§12.8. Projective space RP 3 199

Contents ix

§12.9. Problems 200

Chapter 13. “Projective Geometry Is All Geometry” 203

§13.1. Subgeometries 203

§13.2. The Euclidean plane as a subgeometry of the

projective plane RP 2 204

§13.3. The hyperbolic plane as a subgeometry of the

projective plane RP 2 205

§13.4. The elliptic plane as a subgeometry of RP 2 207

§13.5. Problems 209

Chapter 14. Finite Geometries 211

§14.1. Small finite geometries 212

§14.2. Finite fields 212

§14.3. Example: the finite affine plane over F(5) 213

§14.4. Example: the finite affine plane over F(22) 215

§14.5. Example of a finite projective plane 216

§14.6. Axioms for finite affine planes 217

§14.7. Axioms for finite projective planes 218

§14.8. Constructing projective planes

over finite fields 220

§14.9. The Desargues theorem 221

§14.10. Algebraic structures in finite projective planes 223

§14.11. Open problems and conjectures 226

§14.12. Problems 227

Chapter 15. The Hierarchy of Geometries 229

§15.1. Dimension one: lines 230

§15.2. Dimension two: planes 232

§15.3. From metric to affine to projective 234

§15.4. Three-dimensional space geometries 235

§15.5. Finite and discrete geometries 236

§15.6. The hierarchy of geometries 236

x Contents

§15.7. Problems 238

Chapter 16. Morphisms of Geometries 241

§16.1. Examples of geometric covering spaces 242

§16.2. Examples of geometric G-bundles 245

§16.3. Lie groups 247

§16.4. Examples of geometric vector bundles 248

§16.5. Geometric G-bundles 250

§16.6. The Milnor construction 251

§16.7. Problems 252

Appendix A. Excerpts from Euclid’s “Elements” 255

Postulates of Book I 256

The Common Notions 257

The Definitions of Book I 258

The Propositions of Book I 262

Conclusion 269

Appendix B. Hilbert’s Axioms for Plane Geometry 271

I. Axioms of connection 272

II. Axioms of order 274

III. Axiom of parallels 275

IV. Axioms of congruence 276

V. Axiom of continuity 279

Consistency of Hilbert’s axioms 280

Conclusion 281

Answers & Hints 283

Chapter 1 283

Chapter 2 285

Chapter 3 286

Chapter 4 288

Chapter 5 290

Contents xi

Chapter 6 290

Chapter 7 291

Chapter 8 294

Chapter 9 294

Chapter 10 295

Chapter 12 295

Chapter 13 296

Chapter 14 296

Chapter 15 296

Chapter 16 296

Bibliography 297

Index 299

Preface

This book is dedicated to the proposition that all geometries are cre-

ated equal. This was first pointed out by Felix Klein, who declared

that each individual geometry is a set with a transformation group

acting on it. Here we shall study geometries from this point of view

not only as individual objects, but also in their social life, i.e., in their

relationships (called morphisms or equivariant maps) within their so-

cium: the category of geometries.

Of course, some geometries are more equal than others. Accord-

ingly, we will ignore the most common ones (affine and Euclidean ge-

ometries, vector spaces), assuming that they are known to the reader,

and concentrate on the most distinguished and beautiful ones. (We

assume that our readers are familiar with elementary Euclidean ge-

ometry; those who aren’t may refer to Chapter 0, which is a precis of

the subject, whenever the need arises.)

The reader should not be deceived by the words groups, mor-

phisms, categories into thinking that this is a formal algebraic or

(heaven forbid) an analytic (coordinate) treatment of geometric top-

ics; it suffices to glance at the numerous figures in the book to realize

that we constantly privilege the visual aspect.

Category theory is not used in this book, but we do use some basic

category language, which, as the reader will see, is extremely natural

xiii

xiv Preface

in the geometric context. Thus, Cayley’s famous phrase: “projec-

tive geometry is all geometry” can be given a precise mathematical

meaning by using the term subgeometry (which means “image by an

injective equivariant map”). In the context of this book, it may be

rephrased as follows: “The geometries studied in this book (includ-

ing the three classical ones – hyperbolic, elliptic, and Euclidean) are

(almost all) subgeometries of projective geometry.”

There is very little in the main body of this book about the ax-

iomatic approach to geometry. This is one of the author’s biases: I

believe that the classical axiom systems for, say, Euclidean and hy-

perbolic geometry are hopelessly outdated and no longer belong in

contemporary mathematics. Their place is in the history of mathe-

matics and in the philosophy of science. Accordingly, here they only

appear in one chapter, devoted to the fascinating history of the cre-

ation of non-Euclidean geometry, while a detailed treatment of the

axiom systems of Euclid and Hilbert is relegated to Appendices A

and B.

The use of the plural (Geometries) in the title of the book indi-

cates that, to my mind, there is no such subject as “geometry”, but

there are some concrete mathematical objects called geometries. In

the singular, the word “geometry” should be understood as a way

of thinking about mathematics, in fact the original one: in Ancient

Greece, the word “geometry” was used as a synonym for “mathemat-

ics”. One can and should think geometrically not only when working

with circles and triangles, but also when using commutative diagrams,

morphisms, or groups. The famous phrase written above the entrance

to Plato’s Academy

Let no one enter who is not a geometer

should also be displayed on the gates leading to the world of mathe-

matics.

* * *

I will not give a systematic summary of the contents of this course

in this Preface, referring the reader to the Table of Contents. Look-

ing at it, the well-prepared reader may wonder why some of her/his

Preface xv

favorite geometric topics do not appear in this book among the “dis-

tinguished and beautiful” ones promised above. Let me comment on

some missing topics, explaining why they are not treated here.

First, there is no algebraic geometry in this book. This is because

the author believes that this beautiful field of mathematics belongs to

algebra, not geometry. Indeed, the mathematicians doing algebraic

geometry are typically algebraists, and this is not only true of the

great French school (following Grothendieck and his schemes), but

also of the more classical Russian school.

Neither is there any differential geometry: in its classical low-

dimensional aspect it is usually developed in calculus books (where it

indeed belongs); in its higher-dimensional modern aspect it is a part

of analysis (under the title “Calculus on Manifolds”) and topology

(under the title “Differential Topology”).

Other missing topics include convex geometry (part of analysis

and more specifically optimization theory as “convex analysis”), sym-

plectic geometry (part of classical mechanics and dynamical systems),

contact geometry (part of differential equations), etc.

Of course, contact geometry (say) is formally a geometry in the

sense of Klein. In fact, the ideology of transformation groups comes

from Sophus Lie as much as (if not more than) from Felix Klein, but

the context of Lie’s beautiful contact geometry is definitely differential

equations.

* * *

This book is based on lectures given in the framework of semester

courses taught in Russian at the Independent University of Moscow to

first-year students in 2003 and 2006, for which I prepared handouts,

written in “simple English” and posted on the IUM web site. These

lecture notes were published as a 100-page booklet by the Moscow

Center of Continuous Mathematical Education in 2006 and used in

geometry courses taught to Math in Moscow students of the Indepen-

dent University.

xvi Preface

The brevity of a short semester course (13 lectures) made me

restrict the study of such classical geometries as hyperbolic and pro-

jective to dimension two, regretfully shelving the three-dimensional

case. But then there are many occasions in the course for developing

one’s intuition of space, and indeed the general case is easier to treat

from the linear algebra coordinate point of view than from the rather

visual synthetic approach characterizing this course. For the reader

who wants to go further, I strongly recommend the book by Marcel

Berger [2]. I should add that, although my approach to the subject is

very different from Berger’s, I am heavily indebted to that remarkable

book in several specific parts of the exposition. For those who would

like to learn more about the axiomatic approach to the classical ge-

ometries, there is no better book, to my mind, than N.V. Efimov’s

Higher Geometry [6].

An important, if not the most important, aspect of this book

are the problems, which appear at the end of each chapter. It is by

solving these problems, much more than by learning the theory, that

the reader will become capable of thinking and working geometri-

cally. The sources of the problems are varied. Many were “stolen”

from books written by my friend and favorite co-author Victor Pra-

solov. In many cases, I simply don’t know where they originally

come from. As handouts for the exercise classes, they were grouped

together by Irina Paramonova, who contributed several, as did the

other instructors conducting the exercise classes (Vladimir Ivanov and

Oleg Karpenkov). I am grateful to all of the people mentioned above,

and also to Mikhail Panov, Anton Ponkrashov, and Victor Shuvalov,

who produced the computer versions of most of the illustrations, to

M.I. Bykova, who corrected many errors in the original handouts,

to Victor Prasolov, who found many more in the first draft of this

book, and to the anonymous referees, whose constructive criticism

was very helpful. Finally, I am indebted to Sergei Gelfand, without

whose encouragement this book would never have been written.

Bibliography

[1] A.F. Berdon, The Geometry of Discrete Groups, Springer, Berlin, 1983.

[2] M. Berger, Geometry, Springer, Berlin, 1987.

[3] A. Cayley, Collected Papers, Cambridge, 1889.

[4] H.S.M. Coxeter, Projective Geometry, Toronto, 1942.

[5] B.A. Dubrovin, S.P. Novikov, A.T. Fomenko, Modern Geometry.Methods and Applications, Springer, Berlin, 1985.

[6] N.V. Efimov, Higher Geometry, Moscow, Mir, 1980.

[7] B. Grunbaum, What symmetry groups are present in the Alhambra,Notices Amer. Math. Soc., 53, no. 6, 2006.

[8] T.L. Heath, The Thirteen Books of Euclid’s Elements, Cam-bridge, 1926.

[9] D. Hilbert, Grundlagen der Geometrie, 7th Edition, Leipzig & Berlin,1930; The Foundations of Geometry (Authorized translation by E.J.Townsend), Chicago, 1902.

[10] F. Klein, A comparative review of recent researches in geometry, Bull.New York Math. Soc., 2, 215–249, 1892–1893.

[11] N. Lobachevsky, Geometrical Researches on the Theory of Parallels,Austin, Texas, 1891.

[12] H. Poincare, Science and Hypothesis, London, 1905.

[13] B. Riemann, Gesammelte Mathematische Werke, Leipzig, 1892.

[14] D.E. Smith, A Source Book in Mathematics, New York, 1929.

[15] P. Stackel, Wolfgang und Johann Bolyai, Geometrische Unter-suchugen, Leipzig & Berlin, 1913.

[16] H.E. Wolfe, Non-Euclidean Geometry, Dryden Press, N.Y., 1945.

[17] J. Bolyai, Appendix, Edited by F. Karteszi, North-Holland Mathemat-ical Studies, Amsterdam, 1987.

297

Index

Abelian group, 54

absolute, 132, 147

adjacent angles, 4

affine space, 110

affine transformation, 144

angle, 4

angle of parallelism, 170

Artin relation, 63

axial symmetry, 30

axiom of Archimedes, 279

axiom of completeness, 280

axiom of continuity, 279

axioms of congruence, 276

axioms of connection, 272

axioms of order, 274

betweenness relation, 274

biangle on the sphere, 116

canonical Grassmann bundle, 249

Cayley–Klein model, 155

central symmetry, 8, 31

circle, 20

class formula, 43

classifying space, 251

common notions, 257

conic section, 194

consistency, 280

convex polyhedron, 30

convex set, 30

coset, 58

Coxeter geometry, 101

Coxeter polygon, 101

Coxeter polyhedron, 101

Coxeter scheme, 105

cross ratio of collinear points, 191

cross ratio of four complexnumbers, 144

crystallographic group, 95

cyclic group, 55

defining relations in a grouppresentation, 64

Desargues’ theorem, 194

discrete group action, 90

distance function, 2

distance in the Cayley–Kleinmodel, 154

dodecahedron, 78

duality in projective geometry, 193

duality principle in projectivegeometry, 193

elliptic geometry, 121

embedding of a geometry, 48

epimorphism, 44

equivariant map, 47

Erlangen program, 46

Euclid, xiv, 10

Euclidean geometry, 1

299

300 Index

Euclidean plane, 2Euclidean space, 111

Fano plane, 218Fedorov group, 87

Fifth Postulate, 177, 257finite affine plane, 217

over a field, 215

finite field, 212flat cylinder, 245flat torus, 244

free group, 56, 61fundamental domain, 43fundamental tile, 91

general linear group GL(n), 56, 110general position, 189generators of a group, 57

geometric G-bundle, 250geometric sum, 4geometry in the sense of Klein, 46

glide symmetry, 24Grassmann manifold, 248

Hilbert, xiv

homomorphism, 57of transformation groups, 43

honeycomb lattice, 91

Hopf bundle, 245hyperbolic circle, 137hyperbolic plane, 132

incidence, 192inverse element, 54inversion, 126

involution, 127isometry, 3, 34isometry group of the Riemannian

elliptic plane, 41

isomorphic geometries, 48isomorphic transformation groups,

44isomorphism of transformation

groups, 44

join, 251

Klein, Felix, xiii, xv, 46

Lagrange’s theorem, 59

Lie group, 247Lie, Sophus, xv

line at infinity, 188Lobachevsky distance, 168

Mobius distance, 150Mobius group, 145

monomorphism, 44

morphism of geometries, 47morphism of transformation

groups, 43

motion of the plane, 34

neutral element, 54

nonintersecting lines in theCaley–Klein model, 157

nonintersecting lines in thehyperbolic plane, 134

normal subgroup, 60

orbit, 42order of a transformation group, 45

order of an element g of atransformation group, 45

order of an element of a group , 57

orientation-reversing isometry ofthe plane, 34

oriented angle, 7orthogonal group, 111

orthonormal vector space, 110

Pappus’ theorem, 198

parallel lines, 2

in space, 28parallel planes, 28

parallel translation, 11parallelogram, 11

parallels in the Cayley–Kleinmodel, 157

parallels in the hyperbolic plane,134

Pascal’s theorem, 199

Penrose tiling, 87

permutation group, 42, 56, 62perpendicular line to a plane, 29

perpendiculars, 4Poincare disk model of the

hyperbolic plane, 131

point at infinity, 126

Index 301

polar of a point on the sphere, 115poles of a line on the sphere, 115presentation of a group, 63principal G-bundle, 250projective plane, 185projective space, 199

of arbitrary dimension, 187pullback, 248

quotient group, 61

reflection group, 99reflection in a line, 5reflection in a plane, 29

regular tiling, 91residues modulo m, 55Riemann sphere, 126rotation, 7rotation about an axis, 30

Schweikart constant, 171stabilizer, 42Stiefel manifold, 249Stiefel-over-Grassmann bundle, 249subgeometry, 48subgroup, 58

of a transformation group, 45symmetry group of the equilateral

triangle, 36symmetry group of the square, 38

tessellation, 90tiling, 90tiling geometry, 91tiling group, 91transitive action, 91triangle, 10

undefined notions, 272universal geometric G-bundle, 251

vector space, 110

vector, attached, 11vector, free, 12vertical angles, 4Vorderberg tiling, 86

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The book is an innovative modern exposition of geometry, or rather, of geometries; it is the fi rst textbook in which Felix Klein’s Erlangen Program (the action of transformation groups) is systematically used as the basis for defi ning various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal—although some, of course, remain more equal than others. The author concentrates on several of the more distinguished and beautiful ones, which include what he terms “toy geometries”, the geometries of Platonic bodies, discrete geometries, and classical continuous geometries.

The text is based on fi rst-year semester course lectures delivered at the Independent University of Moscow in 2003 and 2006. It is by no means a formal algebraic or analytic treatment of geometric topics, but rather, a highly visual exposition containing upwards of 200 illustrations. The reader is expected to possess a familiarity with elementary Euclidean geom-etry, albeit those lacking this knowledge may refer to a compendium in Chapter 0. Per the author’s predilection, the book contains very little regarding the axiomatic approach to geometry (save for a single chapter on the history of non-Euclidean geometry), but two Appendices provide a detailed treatment of Euclid’s and Hilbert’s axiomatics. Perhaps the most important aspect of this course is the problems, which appear at the end of each chapter and are supplemented with answers at the conclusion of the text. By analyzing and solving these problems, the reader will become capable of thinking and working geometrically, much more so than by simply learning the theory.

Ultimately, the author makes the distinction between concrete mathemat-ical objects called “geometries” and the singular “geometry”, which he understands as a way of thinking about mathematics. Although the book does not address branches of mathematics and mathematical physics such as Riemannian and Kähler manifolds or, say, differentiable manifolds and conformal fi eld theories, the ideology of category language and transfor-mation groups on which the book is based prepares the reader for the study of, and eventually, research in these important and rapidly developing areas of contemporary mathematics.