Table Of Content

BenjaminFine,AnjaMoldenhauer,GerhardRosenberger,AnnikaSchürenberg, LeonardWienke GeometryandDiscreteMathematics Also of Interest AlgebraandNumberTheory.ASelectionofHighlights BenjaminFine,AnthonyGaglione,AnjaMoldenhauer,Gerhard Rosenberger,DennisSpellman,2017 ISBN978-3-11-051584-8,e-ISBN(PDF)978-3-11-051614-2, e-ISBN(EPUB)978-3-11-051626-5 2ndeditionisplanned! GeneralTopology.AnIntroduction TomRichmond,2020 ISBN978-3-11-068656-2,e-ISBN(PDF)978-3-11-068657-9, e-ISBN(EPUB)978-3-11-068672-2 APrimerinCombinatorics AlexanderKheyfits,2021 ISBN978-3-11-075117-8,e-ISBN(PDF)978-3-11-075118-5, e-ISBN(EPUB)978-3-11-075124-6 Discrete-TimeApproximationsandLimitTheorems.InApplicationsto FinancialMarkets YuliyaMishura,KostiantynRalchenko,2021 ISBN978-3-11-065279-6,e-ISBN(PDF)978-3-11-065424-0, e-ISBN(EPUB)978-3-11-065299-4 DiscreteAlgebraicMethods.Arithmetic,Cryptography,Automataand Groups VolkerDiekert,ManfredKufleitner,GerhardRosenberger,Ulrich Hertrampf,2016 ISBN978-3-11-041332-8,e-ISBN(PDF)978-3-11-041333-5, e-ISBN(EPUB)978-3-11-041632-9 Benjamin Fine, Anja Moldenhauer, Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke Geometry and Discrete Mathematics | A Selection of Highlights 2nd edition MathematicsSubjectClassification2010 Primary:00-01,05-01,06-01,51-01,60-01;Secondary:03G05,05A10,05A15,06A06,20H10,20H15, 51F15,53A04,55U05,60A10,94C15 Authors Prof.Dr.BenjaminFine AnnikaSchürenberg FairfieldUniversity GrundschuleHoheluft DepartmentofMathematics Wrangelstr.80 1073NorthBensonRoad 20253Hamburg Fairfield,CT06430 Germany USA LeonardWienke Dr.AnjaMoldenhauer UniversityofBremen 20535Hamburg DepartmentofMathematics Germany Bibliothekstr.5 28359Bremen Prof.Dr.GerhardRosenberger Germany UniversityofHamburg DepartmentofMathematics Bundesstr.55 20146Hamburg Germany ISBN978-3-11-074077-6 e-ISBN(PDF)978-3-11-074078-3 e-ISBN(EPUB)978-3-11-074093-6 LibraryofCongressControlNumber:2022934378 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2022WalterdeGruyterGmbH,Berlin/Boston Coverimage:GerhardRosenberger Typesetting:VTeXUAB,Lithuania Printingandbinding:CPIbooksGmbH,Leck www.degruyter.com Preface Tomanystudents,aswellastomanyteachers,mathematicsmayseemlikeamun- dane discipline, filled with rules and algorithms and devoid of beauty, culture and art.However,theworldofmathematicsispopulatedwithtruegemsandresultsthat astound.InourseriesHighlightsinMathematicsweintroduceandexaminemanyof thesemathematicalhighlights,thoroughlydevelopingwhatevermathematicalresults andtechniquesweneedalongtheway. WeregardourHighlightsasbooksforgraduatestudiesandplannedittobeused incoursesforteachersandforgeneralmathematicallyinterested,soitissomewhat betweenatextbookandacollectionofresults.Weassumethatthereaderisfamiliar withbasicknowledgeinalgebra,geometryandcalculus,aswellassomeknowledge ofmatricesandlinearequations.Beyondthesethebookisself-contained.Thechap- tersofthebookarelargelyindependent,andweinvitethereadertochooseareasto concentrateon. Westructureourbookin11chaptersthatarearrangedinthreeparts:Inthefirst seven chapters we examine results which are related to geometry. In Chapter 8 we giveaconnectionofgeometricideasandcombinatoricallydefinedobjects.Inthelast threechapterswefurtherinvestigatetopicsincombinatorics,discussaglimpseoffi- niteprobabilitytheoryandendourbookwithBooleanalgebrasandBooleanlattices. In Chapter 1 we look at general geometric ideas and techniques. In the second editionweaddedaprimeroncurvesintherealspaceℝ3tothischaptertogivealittle insightintotherichnessofdifferentialgeometry. InChapter2wediscusstheisometriesinEuclideanvectorspacesandtheirclassi- ficationinℝn.WerealizethatthestudyofplanarEuclideangeometrydependsupon theknowledgeofthegroupofallisometriesoftheEuclideanplaneandhencedevote asectiontothem.Thestudyofgeometryusingisometriesandgroupsofisometries wasdevelopedbyF.Klein,andthisapproachisfundamentalinthemodernapplica- tiontogeometry.AfirstapplicationisinChapter3wherewegiveaclassificationand ageometricdescriptionoftheconicsections. InChapter4wedescribecertainspecialgroupsofplanarisometries,morepre- cisely,wedescribethefixedpointgroupsandclassifythefriezegroupsandthepla- narcrystallographicgroups.Thisespeciallyleadstoaclassificationoftheregulartes- sellationsoftheplane.Inthissecondeditionweincludedabeautifulnon-periodic tessellationoftherealplaneℝ2,thePenrosetilingwhichgetsalongwithonlytwo prototiles. InChapter5wepresentgraphtheoryandgraphtheoreticalproblems.Inpartic- ular,wediscusscolorings,matchings,EulerlinesandHamiltonianlinesalongwith theirrichandcurrentapplicationssuchasthemarriageproblemandthetravelling salesmanproblem.Incontrasttothefirsteditionofthisbookthechapterongraph theoryisnowanextendedstand-alonechapterandthediscussionofsphericalgeom- etryandthePlatonicsolidstakesplaceinanewChapter6. https://doi.org/10.1515/9783110740783-201 VI | Preface There,wediscussthePlatonicsolidswhichhistoricallyhaveplayedanoutsider’s roleinourviewoftheuniverse.ForthedescriptionandtheclassificationofthePla- tonicsolidsweuseEuler’sformulaforplanar,connectedgraphsandthesphericalge- ometryofthesphereS2.InChapter7wecompletethediscussionofplanargeometries withtheintroductionofamodelforahyperbolicplaneandalookatthedevelopment andpropertiesofhyperbolicgeometry. In the second edition we added a new Chapter 8 on simplicial complexes and topologicaldataanalysis–twoimportantconceptsfromtheemergingfieldofapplied topology. Chapter9givesadetailedpaththroughcombinatorics,combinatorialproblems andgeneratingfunctions.Finiteprobabilitytheoryisheavilydependentoncombina- toricsandcombinatorialtechniques.HenceinChapter10weexaminefiniteprobabil- itytheorywithaspecialfocusontheBayesiananalysis. Finally,inChapter11weconsiderBooleanalgebrasandBooleanlatticesandgive aproofofthecelebratedtheoremofM.StonewhichsaysthataBooleanlatticeislattice isomorphictoaBooleansetlattice.Hence,BooleanalgebrasandBooleanlatticesare crucialinbothpuremathematics,especiallydiscretemathematics,anddigitalcom- puting. Wewouldliketothankthemanypeoplewhowereinvolvedinthepreparationof themanuscriptaswellasthosewhohaveusedthefirsteditioninclassesandseminars fortheirhelpfulsuggestions.Inparticular,wehavetomentionAnjaRosenbergerfor herdedicatedparticipationintranslatingandproofreading.WethankYannickLilie forprovidinguswithexcellentdiagramsandpictures.Thosemathematical,stylistic, andorthographicerrorsthatundoubtedlyremainshallbechargedtotheauthors.Last butnotleast,wethankDeGruyterforpublishingourbook.Wehopethatourreaders, oldandnew,willfindpleasureinthisreviewedandextendededition. BenjaminFine AnjaMoldenhauer GerhardRosenberger AnnikaSchürenberg LeonardWienke Contents Preface|V 1 GeometryandGeometricIdeas|1 1.1 GeometricNotions,ModelsandGeometricSpaces|1 1.1.1 GeometricNotions|3 1.2 OverviewofEuclid’sMethodandApproachestoGeometry|3 1.2.1 IncidenceGeometries–AffineGeometries,FiniteGeometries,Projective Geometries|6 1.3 EuclideanGeometry|7 1.3.1 Birkhoff’sAxiomsforEuclideanGeometry|9 1.4 NeutralorAbsoluteGeometry|9 1.5 EuclideanandHyperbolicGeometry|12 1.5.1 ConsistencyofHyperbolicGeometry|13 1.6 EllipticGeometry|14 1.7 DifferentialGeometry|14 1.7.1 SomeSpecialCurves|22 1.7.2 TheFundamentalExistenceandUniquenessTheorem|24 1.7.3 ComputingFormulasfortheCurvature,theTorsionandtheComponents ofAcceleration|25 1.7.4 IntegrationofPlanarCurves|27 2 IsometriesinEuclideanVectorSpacesandtheirClassificationinℝn|31 2.1 IsometriesandKlein’sErlangenProgram|31 2.2 TheIsometriesoftheEuclideanPlaneℝ2|42 2.3 TheIsometriesoftheEuclideanSpaceℝ3|54 2.4 TheGeneralCaseℝnwithn≥2|61 3 TheConicSectionsintheEuclideanPlane|69 3.1 TheConicSections|69 3.2 Ellipse|77 3.3 Hyperbola|79 3.4 Parabola|81 3.5 ThePrincipalAxisTransformation|82 4 SpecialGroupsofPlanarIsometries|87 4.1 RegularPolygons|91 4.2 RegularTessellationsofthePlane|93 4.3 GroupsofTranslationsinthePlaneℝ2|98 4.4 GroupsofIsometriesofthePlanewithTrivialTranslation Subgroup|100 VIII | Contents 4.5 FriezeGroups|101 4.6 PlanarCrystallographicGroups|103 4.7 ANon-PeriodicTessellationofthePlaneℝ2|117 5 GraphTheoryandGraphTheoreticalProblems|127 5.1 GraphTheory|127 5.2 ColoringofPlanarGraphs|139 5.3 TheMarriageTheorem|140 5.4 StableMarriageProblem|142 5.5 EulerLine|144 5.6 HamiltonianLine|148 5.7 TheTravelingSalesmanProblem|151 6 SphericalGeometryandPlatonicSolids|159 6.1 StereographicProjection|159 6.2 PlatonicSolids|161 6.2.1 Cube(C)|163 6.2.2 Tetrahedron(T)|163 6.2.3 Octahedron(O)|164 6.2.4 Icosahedron(I)|165 6.2.5 Dodecahedron(D)|166 6.3 TheSphericalGeometryoftheSphereS2|167 6.4 ClassificationofthePlatonicSolids|170 7 LinearFractionalTransformationandPlanarHyperbolicGeometry|177 7.1 LinearFractionalTransformations|177 7.2 AModelforaPlanarHyperbolicGeometry|185 ℍ 7.3 The(Planar)HyperbolicTheoremofPythagorasin |189 7.4 TheHyperbolicAreaofaHyperbolicPolygon|190 8 SimplicialComplexesandTopologicalDataAnalysis|205 8.1 SimplicialComplexes|205 8.2 Sperner’sLemma|209 8.3 SimplicialHomology|211 8.4 PersistentHomology|217 9 CombinatoricsandCombinatorialProblems|223 9.1 Combinatorics|223 9.2 BasicTechniquesandtheMultiplicationPrinciple|223 9.3 SizesofFiniteSetsandtheSamplingProblem|228 9.3.1 TheBinomialCoefficients|233 9.3.2 TheOccupancyProblem|236 Contents | IX 9.3.3 SomeFurtherComments|237 9.4 MultinomialCoefficients|239 9.5 SizesofFiniteSetsandtheInclusion–ExclusionPrinciple|241 9.6 PartitionsandRecurrenceRelations|248 9.7 DecompositionsofNaturalsNumbers,PartitionFunction|255 9.8 CatalanNumbers|257 9.9 GeneratingFunctions|259 9.9.1 OrdinaryGeneratingFunctions|259 9.9.2 ExponentialGeneratingFunctions|269 10 FiniteProbabilityTheoryandBayesianAnalysis|277 10.1 ProbabilitiesandProbabilitySpaces|277 10.2 SomeExamplesofFiniteProbabilities|279 10.3 RandomVariables,DistributionFunctionsandExpectation|281 10.4 TheLawofLargeNumbers|285 10.5 ConditionalProbabilities|288 10.6 TheGoatorMontyHallProblem|295 10.7 BayesNets|296 11 BooleanLattices,BooleanAlgebrasandStone’sTheorem|311 11.1 BooleanAlgebrasandtheAlgebraofSets|311 11.2 TheAlgebraofSetsandPartialOrders|311 11.3 Lattices|318 11.4 DistributiveandModularLattices|324 11.5 BooleanLatticesandStone’sTheorem|328 11.6 ConstructionofBooleanLatticesvia0–1Sequences|335 11.7 BooleanRings|338 11.8 TheGeneralTheoremofStone|341 Bibliography|347 Index|349