Geometries and transformations

472 Pages·2018·4.42 MB·English

by  Johnson, Norman W

GeometriesandTransformations
Euclideanandothergeometriesaredistinguishedbythetransformations
that preserve their essential properties. Using linear algebra and trans-
formationgroups,thisbookprovidesareadableexpositionofhowthese
classical geometries are both differentiated and connected. Following
CayleyandKlein,thebookbuildsonprojectiveandinversivegeometry
to construct “linear” and “circular” geometries, including classical real
metric spaces like Euclidean, hyperbolic, elliptic, and spherical, as well
as their unitary counterparts. The first part of the book deals with the
foundations and general properties of the various kinds of geometries.
The latter part studies discrete-geometric structures and their symme-
tries in various spaces. Written for graduate students, the book includes
numerous exercises and covers both classical results and new research
in the field. An understanding of analytic geometry, linear algebra, and
elementarygrouptheoryisassumed.
NormanW.JohnsonwasProfessorEmeritusofMathematicsatWheaton
College, Massachusetts. Johnson authored and coauthored numerous
journal articles on geometry and algebra, and his 1966 paper ‘Convey
PolyhedrawithRegularFaces’enumeratedwhathavecometobecalled
the Johnson solids. He was a frequent participant in international con-
ferences and a member of the American Mathematical Society and the
MathematicalAssociationofAmerica.
NormanJohnson:AnAppreciation
byThomasBanchoff
Norman Johnson passed away on July 13, 2017, just a few months short
ofthescheduledappearanceofhismagnumopus,GeometriesandTrans-
formations. It was just over fifty years ago in 1966 that his most famous
publication appeared in the Canadian Journal of Mathematics: “Convex
PolyhedrawithRegularFaces,”presentingthecompletelistofninety-two
examples of what are now universally known as Johnson Solids. He not
only described all of the symmetry groups of the solids; he also devised
names for each of them, containing enough information to exhibit their
constituentelementsandthewaytheywerearranged.
Now,asaresultofthisnewvolume,hewillbefurtherappreciatedfor
his compendium of all the major geometric theories together with their
fullgroupsoftransformationsthatpreservetheiressentialcharacteristic
features.Inthisprocess,hefollowsthespiritofFelixKleinandhisErlan-
gen Program, first enunciated in 1870. Once again, a specific personal
contribution to this achievement is his facility for providing descriptive
namesforeachofthetypesoftransformationsthatcharacterizeagiven
geometry.Hisbookwillbeaninstantclassic,givinganoverviewthatwill
serveasanintroductiontonewsubjectsaswellasanauthoritativerefer-
enceforestablishedtheories.Inthelastfewchapters,heincludesasimilar
treatment of finite symmetry groups in the tradition of his advisor and
mentor,H.S.M.Coxeter.
Itisfittingthatthecoverimageandtheformalannouncementofthe
forthcoming book were first released by Cambridge University Press at
theannualsummerMathFestoftheMathematicalAssociationofAmer-
ica.NormanJohnsonwasaregularparticipantintheMathFestmeetings,
andin2016hewashonoredasthepersontherewhohadbeenamember
oftheMAAforthelongesttime,atsixty-threeyears.Hisfriendsandcol-
leaguesaresaddenedtolearnofhispassingevenaswelookforwardto
hisnewgeometricalmasterwork.
GEOMETRIES AND
TRANSFORMATIONS
Norman W. Johnson
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Informationonthistitle:www.cambridge.org/9781107103405
©NormanW.Johnson2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2018
PrintedintheUnitedStatesofAmericabySheridanBooks,Inc.
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Johnson,NormanW.,1930–2017author.
Title:Geometriesandtransformations/NormanW.Johnson,
WheatonCollege.
Description:Cambridge,UnitedKingdom:
CambridgeUniversityPress,2017.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2017009670|ISBN9781107103405(hardback)
Subjects:LCSH:Geometry–Textbooks.|BISAC:MATHEMATICS/Topology.
Classification:LCCQA445.J642017|DDC516–dc23
LCrecordavailableathttps://lccn.loc.gov/2017009670
ISBN978-1-107-10340-5Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyInternetWebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchWebsitesis,orwillremain,
accurateorappropriate.
ToEva

CONTENTS
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pagexi
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1 RealMetricSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 UnitarySpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Linear Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 ProjectivePlanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Projectiven-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 EllipticandEuclideanGeometry. . . . . . . . . . . . . . . . . . 46
2.4 HyperbolicGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Circular Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 InversiveandSphericalGeometry. . . . . . . . . . . . . . . . . 57
3.2 PseudosphericalGeometry . . . . . . . . . . . . . . . . . . . . . . 61
3.3 ConformalModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 TrianglesandTrigonometry. . . . . . . . . . . . . . . . . . . . . . 74
3.5 Non-EuclideanCircles . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6 SummaryofRealSpaces. . . . . . . . . . . . . . . . . . . . . . . . 86
4 Real Collineation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1 LinearTransformations. . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 AffineCollineations . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 HomogeneousCoordinates . . . . . . . . . . . . . . . . . . . . . . 96
4.4 ProjectiveCollineations . . . . . . . . . . . . . . . . . . . . . . . . 102
vii
viii Contents
4.5 ProjectiveCorrelations . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6 SubgroupsandQuotientGroups . . . . . . . . . . . . . . . . . . 110
5 Equiareal Collineations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1 TheRealAffinePlane. . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 OrtholinearTransformations. . . . . . . . . . . . . . . . . . . . . 117
5.3 ParalinearTransformations . . . . . . . . . . . . . . . . . . . . . . 121
5.4 MetalinearTransformations . . . . . . . . . . . . . . . . . . . . . 125
5.5 SummaryofEquiaffinities. . . . . . . . . . . . . . . . . . . . . . . 129
5.6 SymplecticGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Real Isometry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.1 SphericalandEllipticIsometries . . . . . . . . . . . . . . . . . . 138
6.2 EuclideanTransformations . . . . . . . . . . . . . . . . . . . . . . 143
6.3 HyperbolicIsometries. . . . . . . . . . . . . . . . . . . . . . . . . . 151
7 Complex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1 AntilinearGeometries . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 AnticircularGeometries . . . . . . . . . . . . . . . . . . . . . . . . 162
7.3 SummaryofComplexSpaces . . . . . . . . . . . . . . . . . . . . 166
8 Complex Collineation Groups. . . . . . . . . . . . . . . . . . . . . . . . 168
8.1 LinearandAffineTransformations . . . . . . . . . . . . . . . . 168
8.2 ProjectiveTransformations . . . . . . . . . . . . . . . . . . . . . . 175
8.3 AntiprojectiveTransformations. . . . . . . . . . . . . . . . . . . 178
8.4 SubgroupsandQuotientGroups . . . . . . . . . . . . . . . . . . 180
9 CircularitiesandConcatenations . . . . . . . . . . . . . . . . . . . . . . 183
9.1 TheParabolicn-Sphere. . . . . . . . . . . . . . . . . . . . . . . . . 183
9.2 TheRealInversiveSphere . . . . . . . . . . . . . . . . . . . . . . 186
9.3 TheComplexProjectiveLine . . . . . . . . . . . . . . . . . . . . 194
9.4 InversiveUnitaryGeometry . . . . . . . . . . . . . . . . . . . . . 199
10 Unitary Isometry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.1 UnitaryTransformations . . . . . . . . . . . . . . . . . . . . . . . . 203
10.2 TransunitaryTransformations . . . . . . . . . . . . . . . . . . . . 206
10.3 Pseudo-unitaryTransformations . . . . . . . . . . . . . . . . . . 209
10.4 QuaternionsandRelatedSystems . . . . . . . . . . . . . . . . . 211
11 Finite Symmetry Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
11.1 PolytopesandHoneycombs. . . . . . . . . . . . . . . . . . . . . . 223
11.2 PolygonalGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
11.3 Pyramids,Prisms,andAntiprisms . . . . . . . . . . . . . . . . . 231