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elseviercomlocatecamwa Acomputerapplicationinmathematics MSivasubramanian SKalimuthu DepartmentofMathematicsDrMahalingamCollegeofEngineeringandTechnologyPollachiTamilnadu642003India 2124KanjampattiPOPollachiviaTamilnadu642003India a r t i c l e i n f ID: 23201 Download Pdf

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ComputersandMathematicswithApplications59(2010)296297 Contentslistsavailableat ScienceDirect ComputersandMathematicswithApplicationsjournalhomepage: www.elsevier.com/locate/camwa AcomputerapplicationinmathematicsM.Sivasubramanian a ,  ,S.Kalimuthu b aDepartmentofMathematics,Dr.MahalingamCollegeofEngineeringandTechnology,Pollachi,Tamilnadu-642003,Indiab212/4,KanjampattiP.O.,Pollachivia,Tamilnadu-642003,India articleinfo Articlehistory:Received15April2009Accepted20July2009 Keywords:NumbertheoryAlgebraGeometryEuclideanpostulatesNon-Euclideangeometriesandphysicalapplicationstogeometryabstract Inthisstudy,acomputerapplicationwasusedtosolveamathematicalproblem.'2009ElsevierLtd.Allrightsreserved. 1.Introduction Geometryisthesecondfieldofmathematics.Itistheextensionofnumbertheory.Thereisnoexactperiodforthe originofclassicalgeometry.EuclidwasthefirstmathematicianwhocompiledElementswhichcontainspropositionsand constructions.InElements,Euclidassumedfivepostulates.Euclidcouldnotprovetheparallelpostulate.AfterEuclidalmost allmathematiciansattemptedtodeducethefifthpostulatefromthefirstfourpostulates.Butunfortunatelyallofthemfailed. Thestudiesonthisfamoushistoricalproblemgavebirthtotwoconsistentmodelsofnon-Euclideangeometries.Theseaffine geometriesarewidelyusedinquantumphysicsandrelativisticmechanics.Also,thesurveysandresearchledtoanumber ofpropositionsequivalenttothefifthpostulate.Saccheri'ssimilartrianglepropositioniswellknownequivalentaxiomto theparallelpostulate.Inthisworktheauthorsderivethepreliminaryresultandsincerelyproposetheopenproblemby usingaphysicalphenomena. 2.Preliminaryresult InclassicalandRiemanniangeometrieswecanconstructsimilartriangles.Butitisimpossibletodrawatrianglesimilar tothegiventriangleinLobachevskiangeometry.LetABCbethegivenLobachevskiantriangle.Magnifythistriangle.Andlet A0B0C0bethemagnifiedtriangleofthegivenLobachevskiantriangleABC.Itiswellknownthatinmagnificationtheangles arepreserved.So,theLobachevskiantrianglesABCandA0B0C0aresimilar.WithoutassumingEuclid'sfifthpostulate,we havederivedthispreliminaryresult.ThisestablishesSaccheri'stheorem[ 14 ].Butithasbeenshownonceandforallthat thefifthpostulateisaspecialcase.Theauthorshaveprovedthisimpossibilityintheirpaper[ 5 , 6 ]. Correspondingauthor.E-mailaddresses: profpk49@yahoo.com (M.Sivasubramanian), ohm@budweiser.com (S.Kalimuthu). 0898-1221/$seefrontmatter'2009ElsevierLtd.Allrightsreserved. doi:10.1016/j.camwa.2009.07.048 M.Sivasubramanian,S.Kalimuthu/ComputersandMathematicswithApplications59(2010)296297297 3.Conclusion ComputermagnificationisaUniversalcomputerphenomenon.Thistechniqueisappliedinphysics,astronomy,biology, medicine,architecture,particlephysics,genetics,microbiologyandinchemistry.Withoutmagnification,deepstudiesand researchareimpossible.Forthefirsttimeinthehistoryofmathematics,theauthorsappliedmagnificationtechnologyand obtainedasolutionforanearly4300yearoldparallelpostulateproblem.Inbriefanimpossiblepropositionwasprovedas possible.Thisisaproblematicproblem.Furtherstudieswillgivebirthtoanewbranchofmathematicalscience. Acknowledgements TheauthorsthanktheChairmanEmeritusDr.N.Mahalingam,ChairmanShri.M.Manickam,theCorrespondentShri. ShankarVanavarayar,theSecretaryProf.C.Ramasamy,theDirectorDr.S.Vijayarangan,thePrincipalDr.V.V.Sreenarayanan andtheHeadoftheDepartmentofMathematicsDr.M.Palanivelfortheirencouragementforthepreparationofthispaper. References [1] www.groups.dcs.standac.uk/~history/HisTopics/Non.Euclidean_geometry . [2] www.cut-the-knot.org . [3] http://www.softsurfer.com/history.html . [4] www.beva.org/math323/asgn6/nov19.htm . [5] M.Sivasubramanian,S.Kalimuthu,Onthenewbranchofmathematicalscience,JournalofMathematicsandStatistics04(2)(2008)122123. [6] M.Sivasubramanian,S.Kalimuthu,OnthenewbranchofmathematicalsciencePart2,JournalofMathematicsandStatistics04(3)(2008)146147.