1 The Mathematics of Harmony: Clarifying the Origins and Development of Mathematics A.P. Stakhov International Club of the Golden Section 6 McCreary Trail, Bolton, ON, L7E 2C8, Canada goldenmuseum@rogers.com - www.goldenmuseum.com/ The article is published in the Journal Congressus Numerantium, 193 (2008), pp. 5 - 48 Abstract.Thisstudydevelopsanewapproachtotheoriginsandhistoryofmathematics.We analyzestrategicmistakesinthedevelopmentofmathematicsandmathematicaleducation (includingseveranceoftherelationshipbetweenmathematicsandthetheoreticalnaturalsciences, neglect of thegolden section, the one-sided interpretation of Euclids Elements, and the distorted approachtotheoriginsofmathematics).WedeveloptheMathematicsofHarmonyasanew interdisciplinary direction for modern science by applying to itDiracs Principle of Mathematical Beautyanddiscussingitsroleinovercomingthesestrategicmistakes.Themainconclusionis thatEuclidsElementsareasourceoftwomathematicaldoctrinestheClassicalMathematics basedonaxiomaticapproachandtheMathematicsofHarmonybasedontheGoldenSection (Theorem II.11 of Euclids Elements) and Platonic Solids (Book XIII of Euclids Elements). Keywords:goldenmeanFibonacciandLucasnumbersBinetformulas-Gazaleformulas hyperbolicFibonacciandLucasfunctionsFibonaccimatrices-Bergmansnumbersystem FibonaccicodesFibonaccicomputersternarymirror-symmetricalarithmeticacodingtheory based on Fibonacci matrices golden cryptography -mathematics of harmony

Geometryhastwogreattreasures:oneistheTheoremof Pythagoras; theother, thedivision of a lineinto extremeand meanratio.Thefirst,wemaycomparetoameasureofgold; the second, we may name a precious stone. Johannes Kepler 1.Introduction 1.1.Diracs Principle of Mathematical Beauty Recently the author studied the contents of a public lecture: The complexity of finitesequencesofzerosandunits,andthegeometryoffinitefunctional spaces[1]byeminentRussianmathematicianandacademicianVladimir Arnold,presented beforetheMoscow Mathematical Societyon May 13,2006. Let us consider some of its general ideas. Arnold notes: 1.Inmyopinion,mathematicsissimplyapartofphysics,thatis,itisan experimentalscience,whichdiscoversformankindthemostimportantand simple laws of nature. 2 2.Wemustbeginwithabeautifulmathematicaltheory.Diracstates:Ifthis theory is really beautiful, thenit necessarily will appear as a fine model of important physical phenomena. I t is necessary to search for these phenomena to develop applications of the beautiful mathematical theory and to interpret them as predictions of new laws of physics. Thus, according to Dirac, all new physics, including relativistic and quantum, develop in this way. AtMoscowUniversitythereisatraditionthatthedistinguished visiting-scientistsarerequestedtowriteonablackboardaself-chosen inscription. When Dirac visited Moscow in 1956, he wrote "A physical law must possessmathematicalbeauty."ThisinscriptionisthefamousPrincipleof MathematicalBeautythatDiracdevelopedduringhisscientificlife.Noother modernphysicisthasbeenpreoccupiedwiththeconceptofbeautymorethan Dirac.Thus,accordingtoDirac,thePrincipleofMathematicalBeautyisthe primarycriterionforamathematicaltheorytobeconsideredasamodelof physicalphenomena.Ofcourse,thereisanelementofsubjectivityinthe definitionofthebeauty"ofmathematics,butthemajorityofmathematicians agreesthat"beauty"inmathematicalobjectsandtheoriesneverthelessexist. Let'sexaminesomeofthem,whichhaveadirectrelationtothethemeofthis paper. 1.2. Platonic Solids. We can find the beautiful mathematical objects in Euclids Elements. As is well known, in Book XIII of his Elements Euclid stated a theory of 5 regular polyhedrons called Platonic Solids (Fig. 1). Figure 1. Platonic Solids: tetrahedron, octahedron,cube, icosahedron, dodecahedron 3 Andreallytheseremarkablegeometricalfiguresgotverywideapplicationsin theoretical natural sciences, in particular, in crystallography (Shechtmans quasi-crystals), chemistry (fullerenes), biology and so on what is brilliant confirmation of Diracs Principle of Mathematical Beauty. 1.3. Binomial coefficients, the binomial formula, and Pascals triangle. For the given non-negative integers n and k, there is the following beautiful formula that sets the binomial coefficients:!!( ) !nkCnk n k=, (1) where n!=123n is a factorial of n.Oneofthemostbeautifulmathematicalformulas,thebinomial formula, is based upon the binomial coefficients: ( )1 1 2 2 2 1 1... ... .nn n n k n k k n n na b a C a b C a b C a b C ab bn n n n + = + + + + + + + (2) Thereisaverysimplemethodforcalculationofthebinomial coefficients based on their following graceful properties called Pascals rule:1.1k n kC C Cn n n= ++(3) Usingtherecurrencerelation(3)andtakingintoconsiderationthat 01nC Cn n= = and k n kC Cn n= ,wecanconstructthefollowingbeautifultableof binomial coefficients called Pascals triangle (see Table 1).Table 1. Pascals triangle 11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 11 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1 Hereweattributebeautifultoallthemathematicalobjectsabove. They are widely used in both mathematics and physics. 1.4.Fibonacciand Lucasnumbers, the Golden Mean andBinetFormulas. Let us consider the simplest recurrence relation:1 2F F Fn n n= + , (4) wheren=0,1,2,3,.Thisrecurrencerelationwasintroducedforthefirst timebythefamousItalianmathematicianLeonardoofPisa(nicknamed Fibonacci).4 For the seeds0 10 and 1 F F = = , (5) therecurrencerelation(4)generatesanumericalsequencecalledFibonacci numbers (see Table 2). Inthe19thcenturytheFrenchmathematicianFrancoisEdouard Anatole Lucas (1842-1891) introduced the so-called Lucas numbers (see Table 2) given by the recursive relation 1 2L L Ln n n= + (6) with the seeds 0 12 and 1 L L = =(7) Table 2. Fibonacci and Lucas numbers 0 1 2 3 4 5 6 7 8 9 100 1 1 2 3 5 8 13 21 34 550 1 1 2 3 5 8 13 21 34 552 1 3 4 7 11 18 29 47 76 1232 1 3 4 7 11 18 29 47 76 123nnnnnFFLL It follows from Table 2 that the Fibonacci and Lucas numbers build up twoinfinitenumericalsequences,eachpossessinggracefulmathematical properties.AscanbeseenfromTable2,fortheoddindices2 1 n k = + the elementsFnandF n oftheFibonaccisequencecoincide,thatis, 2 1 2 1F Fk k=+ , and for the even indices2 n k =they are opposite in sign, that is, 2 2F Fk k= .FortheLucasnumbersLnallisviceversa,thatis, ;2 2 2 1 2 1L L L Lk k k k= = + . Inthe17thcenturythefamousastronomerGiovanniDomenico Cassini(1625-1712)deducedthefollowingbeautifulformula,whichconnects three adjacent Fibonacci numbers in the Fibonacci sequence: 2 1( 1)1 1nF F Fn n n+ = +. (8) Thiswonderful formulaevokes areverent thrill,if onerecognizes that it is valid for any value of n (n can be any integer within the limits of - to +). Thealternationof+1and-1intheexpression(8)withinthesuccessionofall Fibonaccinumbersresultsintheexperienceofgenuineaestheticenjoymentof its rhythm and beauty.IfwetaketheratiooftwoadjacentFibonaccinumbers/1F Fn n and direct this ratio towards infinity, we arrive at the following unexpected result: 1 5l i m21FnnFn+= t =,(9) 5 wheretisthefamousirrationalnumber,whichisthepositiverootofthe algebraic equation: 21 x x = + . (10) Thenumberthasmanybeautifulnamesthegoldensection,goldennumber, goldenmean,goldenproportion,andthedivineproportion.SeeOlsenpage2 [2]. Notethatformula(9)issometimescalledKeplersformulaafter Johannes Kepler (1571-1630) who deduced it for the first time.Inthe19thcentury,FrenchmathematicianJacquesPhilippeMarie Binet (1786-1856) deduced the two magnificent Binet formulas: ( 1)5n n nFnt t=(11) ( 1)n n nLn= t + t .(12) Thegoldensectionordivisionofalinesegmentinextremeandmean ratiodescendedtousfromEuclidsElements.Overthemanycenturiesthe goldenmeanhasbeenthesubjectofenthusiasticworshipbyoutstanding scientists and thinkers including Pythagoras, Plato, Leonardo da Vinci,Luca Pacioli,JohannesKeplerandseveralothers.Inthisconnection,weshould recall Kepler's saying concerning the golden section. This saying was taken the epigraph of the present article.AlexeyLosev,theRussianphilosopherandresearcherintothe aestheticsofAncientGreeceandtheRenaissance,expressedhisdelightinthe golden section and Platos cosmology in the following words: FromPlatospointofview,andgenerallyfromthepointofviewof allantiquecosmology,theuniverseisacertainproportionalwholethatis subordinatedtothelawofharmoniousdivision,theGoldenSection...This systemofcosmicproportionsissometimesconsideredbyliterarycriticsasa curiousresultofunrestrainedandpreposterousfantasy.Totalanti-scientific weaknessresoundsintheexplanationsofthosewhodeclarethis.However,we canunderstandthishistoricalandaestheticphenomenononlyinconjunction withanintegralcomprehensionofhistory,thatis,byemployingadialectical andmaterialisticapproachtocultureandbysearchingfortheanswerinthe peculiarities of ancient social existence.Wecanaskthequestion:inwhatwayisthegoldenmeanreflectedin contemporarymathematics? Unfortunately, theanswer forced upon us is-only in the most impoverished manner. In mathematics, Pythagoras and Platos ideas 6 are considered to be a curious result of unrestrained and preposterous fantasy. Therefore,themajorityofmathematiciansconsiderstudyofthegoldensection asamerepastime,whichisunworthyoftheseriousmathematician. Unfortunately,wecanalsofindneglectofthegoldensectionincontemporary theoreticalphysics.In2006BINOMpublishinghouse(Moscow)published theinterestingscientificbookMetaphysics:CenturyXXI[3].InthePrefaceto thebook,itscompilerandeditorProfessorVladimirov(MoscowUniversity) wrote: Thethirdpartofthisbookisdevotedtoadiscussionofnumerous examplesofthemanifestationofthegoldensectioninart,biologyandour surroundingreality.However,paradoxically,thegoldenproportionisnot reflectedincontemporarytheoreticalphysics.Inordertobeconvincedofthis fact, it is enough to merely browse 10 volumes of Theoretical Physics by Landau and Lifshitz. The time has come to fill this gap in physics, all the more given that the golden proportion isclosely connected with metaphysics andtrinitarity [the triune nature of things]. During several decades, the authorhas developed a new mathematical directioncalledTheMathematicsofHarmony[4-39].Forthefirsttime,the name of The Harmony of Mathematicswas introduced by the author in 1996 in thelecture,TheGoldenSectionandModernHarmonyMathematics[14], presented atthesession of the7th InternationalconferenceFibonacci Numbers and Their Applications (Austria, Graz, July 1996). The present article pursues three goals: 1.Toanalyzethestrategicmistakesinthemathematicsdevelopmentandto showaroleoftheMathematicsofHarmonyinthegeneraldevelopmentofthe mathematics 2. To examine the basic theories of the Mathematics of Harmony from a point of view of Diracs Principle of Mathematical Beauty 3.TodemonstrateapplicationsoftheMathematicsofHarmonyinmodern science

2.The Strategic mistakes in the development of mathematics 2.1.Mathematics: The Loss of Certainty. ThebookMathematics:TheLossof Certainty[40]byMorrisKline(1908-1992)isdevotedtotheanalysisofthe crisis of the 20th century mathematics. Kline wrote: The history of mathematics is crowned with glorious achievements but alsoarecordofcalamities.Thelossoftruthiscertainlyatragedyofthefirst magnitude,fortruthsaremansdearestpossessionsandalossofevenoneis cause for grief.The realization that the splendid showcaseof human reasoning exhibitsabynomeansperfectstructurebutonemarredbyshortcomingsand vulnerabletothediscoveryofdisastrouscontradictionsatanytimeisanother blowtothestatureofmathematics.Butthesearenottheonlygroundsfor distress. Grave misgivings and cause for dissension among mathematicians stem 7 from the direction which research of the past one hundred years has taken. Most mathematicianshavewithdrawnfromtheworldtoconcentrateonproblems generatedwithinmathematics.Theyhaveabandonedscience.Thischangein directionisoftendescribedastheturntopureasopposedtoapplied mathematics. Further we read:Sciencehadbeenthelifebloodandsustenanceofmathematics. Mathematicianswerewillingpartnerswithphysicists,astronomers,chemists, andengineersinthescientificenterprise.Infact,duringthe17thand18th centuriesandmostofthe19th,thedistinctionbetween mathematicsand theoreticalsciencewasrarelynoted.Andmanyoftheleadingmathematicians didfargreaterworkinastronomy,mechanics,hydrodynamics,electricity, magnetism,andelasticitythantheydidinmathematicsproper.Mathematics was simultaneously the queen and the handmaiden of the sciences. Klinenotesthatourgreatpredecessorswerenotinterestedinthe problems of pure mathematics, which were put forward in the forefront of the 20th century mathematics. In this connection, Kline writes:However,puremathematicstotallyunrelatedtosciencewasnotthemain concern.Itwasahobby,adiversionfromthefarmorevitalandintriguing problems posed by the sciences. Though Fermat was the founder of the theory of numbers, hedevoted most of his efforts to thecreation of analyticgeometry, to problemsofthecalculus,andtooptics....HetriedtointerestPascaland Huygens in thetheoryof numbers but failed.Very few men of the17th century took any interest in that subject.FelixKlein(18491925),whowasarecognizedheadofthe mathematical world at the boundary of the 19th and 20th centuries, considered it necessary to protest against striving for abstract, pure mathematics:Wecannothelpfeelingthatintherapiddevelopmentsofmodern thought,ourscienceisindangerofbecomingmoreandmoreisolated.The intimatemutualrelationbetweenmathematicsandtheoreticalnaturalscience which, to the lasting benefit of both sides, existed ever since the rise of modern analysis, threatens to be disrupted.RichardCourant(1888-1972),whoheadedtheInstituteof Mathematical Sciences of New York University, also treated disapprovingly the passion for pure mathematics. He wrote in 1939:Aseriousthreattotheverylifeofscienceisimpliedintheassertion thatmathematicsisnothingbutasystemofconclusionsdrawnfromthe definitionandpostulatesthatmustbeconsistentbutotherwisemaybecreated bythefreewillofmathematicians.Ifthisdescriptionwereaccurate, mathematicscouldnotattractanyintelligentperson.Itwouldbeagamewith definitions, rules, and syllogisms without motivation or goal. The notion that the intellectcan createmeaningful postulational systemsat its whim is a deceptive half-truth. Only under the discipline of responsibility to the organic whole, only 8 guidedbyintrinsicnecessity,canthefreemindachieveresultsofscientific value. At present, mathematicians turned their attention to the solution of old mathematicalproblemsformulatedbythegreatmathematiciansofthepast. FermatsLastTheoremisoneofthem.Thistheoremcanbeformulatedvery simply.Letusprovethatforn>2anyintegersx,y,zdonotsatisfythe correlationxn+ yn= zn.Thetheoremwasformulated by Fermatin1637inthe margins of Diophantus of Alexandrias book Arithmetica along with a postscript thatthewittyproofhefoundwastoolongtobeplacedthere.Overtheyears manyoutstandingmathematicians(includingEuler,Dirichlet,Legandreand others)triedtosolvethisproblem.TheproofofFermat'sLastTheoremwas completedin1993byAndrewWiles,aBritishmathematicianworkinginthe UnitedStatesatPrincetonUniversity.Theproofrequired130pagesinthe Annals of Mathematics.JohannCarlFriedrichGauss(17771855)wasarecognized specialistinnumbertheory,confirmedbythepublicationofhisbook Arithmetical Researchers (1801). In this connection, it is curious tofind Gauss opinion about Fermats Last Theorem. Gauss explained in one of his letters why hedidnotstudyFermatsproblem.Fromhispointofview,Fermats hypothesisisanisolatedtheorem,connectedwithnothing,andtherefore this theoremholds nointerest[40].Weshould not forget that Gauss treated withgreatinterestall19thcenturymathematicalproblemsanddiscoveries.In particular,GausswasthefirstmathematicianwhosupportedLobachevskis researchers on Non-Euclidean geometry. Without a doubt, Gauss opinion about FermatsLastTheoremsomewhatdiminishesWilesproofofthistheorem.In this connection, we can ask the following questions: 1. What significance does Fermats Last Theoremhold for the development of modern science? 2. Can we compare the solution to Fermats problem with the discovery of non-Euclidean geometry in the first half of the 19th centuryand othermathematical discoveries? 3. Is Fermats Last Theorem an aimless play of intellect and its proof merely a demonstration of the imaginative power of human intellect - and nothing more? Thus,followingFelixKlein,RichardCourantandotherfamous mathematicians,MorrisKlineassertedthatthemainreasonforthe contemporarycrisisinmathematicswastheseveranceoftherelationship betweenmathematicsandtheoreticalnaturalsciencesthatisthegreatest strategic mistake of 20th century mathematics. 2.2. The neglect of the beginnings. Eminent Russian mathematician Andrey Kolmogorov(1903-1987)wroteaprefacetotheRussiantranslationof LebeguesbookAbouttheMeasurementofMagnitudes[41].Hestatedthat thereisatendencyamongmathematicianstobeashamedoftheoriginof 9 mathematics.Incomparisonwiththecrystalclarityofthetheoryofits development it seems unsavory and an unpleasant pastime to rummage through theoriginsofitsbasicnotionsandassumptions.Allbuildingupofschool algebraandallmathematicalanalysismightbeconstructedonthenotionof realnumberwithoutanymentionofthemeasurementofspecificmagnitudes (lengths, areas, time intervals, and so on). Therefore, one and the same tendency showsitselfatdifferentstagesofeducationandwithdifferentdegreesof inclination to introduce numbers possibly sooner, and furthermore to speak only aboutnumbersandrelationsbetweenthem.Lebegueprotestsagainstthis tendency! Inthisstatement,Kolmogorovrecognizedapeculiarityof mathematicians-thediffidentattitudetowardstheoriginsofmathematics. However,longbeforeKolmogorov,NikolayLobachevski(17921856)also recognized this tendency:AlgebraandGeometryhaveoneandthesamefate.Theirveryslow successesfollowedafterthefastonesatthebeginning.Theyleftscienceina stateveryfarfromperfect.Itprobablyhappened,becausemathematicians turnedalltheirattentiontowardstheadvancedaspectsofanalytics,andhave neglectedtheoriginsofmathematicsbybeingunwillingtodiginthefield already harvested by them and now left behind.However,justasLobachevskidemonstratedbyhisresearchthatthe originsofmathematicalsciences,inparticular,Euclid'sElementsarean inexhaustiblesourceofnewmathematicalideasanddiscoveries.Geometric Researches on Parallel Lines(1840) by Lobachevski openswith the following words: Ihavefoundsomedisadvantagesingeometry,reasonswhythis science did notuntil now step beyondtheboundsof Euclids Elements.Weare talkinghereaboutthefirstnotionssurroundinggeometricmagnitudes, measurementmethods,andfinally,theimportantgapinthetheoryofparallel lines .... Thankfully,Lobachevski,unlike other mathematicians did not neglect concernwithorigins.HisthoroughanalysisoftheFifthEuclideanPostulate (theimportantgapinthetheoryofparallellines)ledhimtothecreationof Non-Euclideangeometrythemostimportantmathematicaldiscoveryofthe 19th century. 2.3.TheneglectoftheGoldenSection.Pythagoreansadvancedforthefirst timethebrilliantideaabouttheharmonicstructureoftheUniverse,including not only nature and people, but also everything in the entire cosmos. According tothePythagoreans,harmonyisaninnerconnectionofthingswithoutwhich thecosmoscannotexist.Atlast,accordingtoPythagoras,harmonyhad numericalexpression,thatis,itisconnectedwiththeconceptofnumber. 10 Aristotle (384 BC 322 BC) noticed in his Metaphysics just this peculiarity of the Pythagorean doctrine:Theso-calledPythagoreans,whowerethefirsttotakeup mathematics, not only advanced this study, but also having been brought up in it theythoughtitsprinciplesweretheprinciplesofallthings...since,then,all otherthingsseemedintheirwholenaturetobemodeledonnumbers,and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole cosmos to be a harmony and a number. ThePythagoreans recognized thattheshapeof theUniverseshould be harmoniousandallitselementsconnectedwithharmoniousfigures. PythagorastaughtthattheEartharosefromcube,Firefrompyramid (tetrahedron),Airfromoctahedron,Waterfromicosahedron, thesphereof the Cosmos (the ether) from dodecahedron.ThefamousPythagoreandoctrineoftheharmonyofspheresisof course connected with the harmony concept. Pythagoras and his followers held thatthemovementofheavenlybodiesaroundthecentralworldfirecreatesa wonderfulmusic,whichisperceivednotbyear,butbyintellect.Thedoctrine abouttheharmonyofthespheres,theunityofthemicrocosmandmacrocosm, andthedoctrineaboutproportions-unifiedtogetherprovidethebasisofthe Pythagorean doctrine. Themainconclusion,followingfromPythagoreandoctrine,isthat harmonyisobjective;itexistsindependentlyfromourconsciousnessandis expressedintheharmoniousstructureoftheUniversefromthemacrocosm downtothemicrocosm.However,ifharmonyisinfactobjective,itshould become a central subject of mathematical research. ThePythagoreandoctrineofnumericalharmonyintheUniverse influencedthedevelopmentofallsubsequentdoctrinesaboutnatureandthe essence ofaesthetics.This brilliant doctrinewasreflectedanddevelopedin the works of great thinkers, in particular, in Platos cosmology. In his works, Plato (428/427BC348/347BC)developedPythagoreandoctrineandespecially emphasizedthecosmicsignificanceofharmony.Hewasfirmlyconvincedthat harmonycanbeexpressedbynumericalproportions.ThisPythagorean influencewastraced especially inhisTimaeus,wherePlato, afterPythagoras, developedadoctrineaboutproportionsandanalyzedtheroleoftheregular polyhedra (Platonic Solids), which, in his opinion, underlie the Universe itself.Thegoldensection,whichwascalledinthatperiodthedivisionin extremeandmeanratio,playedaspecialroleinancientscience,including Platos cosmology. Abovewe presented Keplers and Losevs statements about the role of the golden section in geometry and Greek culture. Keplers assertion raisesthesignificanceofthegoldensectionuptothelevelofthePythagorean Theorem-oneofthemostfamoustheoremsofgeometry.Asaresultofthe unilateralapproachtomathematicaleducationeachschool-childknowsthe 11 Pythagorean Theorem, but has a rather vague concept of the golden section - the secondtreasureofgeometry.Themajorityofschooltextbooksongeometry go back in their origin to Euclids Elements. But then we may ask the question: whyinthemajorityofthemistherenorealsignificantmentionofthegolden section,describedforthefirsttimeinEuclidsElements?Theimpression createdisthatthematerialisticpedagogyhasthrownoutthegoldensection frommathematicaleducationontothedumpheapof"doubtfulscientific conceptstogetherwithastrologyandotherso-calledesotericsciences(where the golden section is widely emphasized). We consider this sad fact to be one of the strategic mistakes of modern mathematical education.ManymathematiciansinterprettheaboveKeplerscomparisonofthe goldensectionwithPythagoreanTheoremasagreatoverstatementregarding thegoldensection.However,weshouldnotforgetthatKeplerwasnotonlya brilliantastronomer,butalsoagreatphysicistandgreatmathematician(in contrasttothemathematicianswhocriticizeKepler).Inhisfirstbook MysteriumCosmographicum(TheCosmographicMystery),Keplercreatedan original model of the Solar System based on the Platonic Solids. He was one of thefirstscientists,whostartedtostudytheHarmonyoftheUniverseinhis book Harmonices Mundi (Harmony of the World). In Harmony, he attempted to explain theproportions of thenaturalworldparticularly theastronomicaland astrological aspects in terms of music. The Musica Universalis or Music of the Spheres, studied by Ptolemy and many others before Kepler, was his main idea. Fromthere,heextendedhisharmonicanalysistomusic,meteorologyand astrology;harmonyresultedfromthetonesmadebythesoulsofheavenly bodiesandinthecaseofastrology,theinteractionbetweenthosetonesand humansouls.Inthefinalportionofthework(BookV),Keplerdealtwith planetarymotions,especiallyrelationshipsbetweenorbitalvelocityandorbital distancefromtheSun.Similarrelationshipshadbeenusedbyother astronomers, but Kepler with Tycho's data and his own astronomical theories treatedthemmuchmorepreciselyandattachednewphysicalsignificanceto them.Thus, the neglect of the golden section and its associated idea of harmonyisonemorestrategicmistakeinnotonlymathematicsand mathematical education, but also theoretical physics. This mistake resulted in anumberofotherstrategicmistakesinthedevelopmentofmathematicsand mathematical education. 2.4.Theone-sided interpretation ofEuclidsElements. EuclidsElementsis theprimaryworkofGreekmathematics.Itisdevotedtotheaxiomatic constructionofgeometry,andledtotheaxiomaticapproachwidelyusedin mathematics.ThisviewoftheElementsiswidespreadincontemporary mathematics.InhisElementsEuclidcollectedandlogicallyanalyzedall achievements of the previous period in the field of geometry. At the same time, 12 hepresentedthebasisofnumbertheory.Forthefirsttime,Euclidprovedthe infinity of prime numbers and constructed a full theory of divisibility. At last, in Books II, VI and X,wefind thedescription ofaso-calledgeometrical algebra thatallowedEuclidtonotonlysolvequadraticequations,butalsoperform complex transformations on quadratic irrationals. Euclid'sElementsfundamentallyinfluencedmathematicaleducation. Withoutexaggerationitisreasonabletosuggest,thatthecontentsof mathematicaleducationinmodernschoolsisonthewholebaseduponthe mathematical knowledge presented in Euclids Elements. However,thereisanotherpointofviewonEuclidsElements suggestedbyProclusDiadochus(412-485),thebestcommentatoronEuclids Elements.ThefinalbookofEuclidsElements,BookXIII,isdevotedtoa description of the theory of the five regular polyhedra that played a predominate roleinPlatoscosmology.Theyarewellknowninmodernscienceunderthe namePlatonicSolids.Proclusdidpayspecialattentiontothisfact.Asis generallythecase,themostimportantdataarepresentedinthefinalpartofa scientificbook.Basedonthisfact,ProclusassertsthatEuclidcreatedhis Elementsprimarilynot to presentan axiomaticapproach to geometry, but inordertogiveasystematictheoryoftheconstructionofthe5Platonic Solids,inpassingthrowinglightonsomeofthemostimportant achievements of Greek mathematics. Thus, Proclus hypothesis allows one to supposethatitwaswell-knowninancientsciencethatthePythagorean Doctrine about the Numerical Harmony of the Cosmos and Platos Cosmology, basedontheregularpolyhedra,wereembodiedinEuclidsElements,the greatest Greek work of mathematics. From this point of view, we can interpret EuclidsElements asthefirstattempttocreateaMathematical Theory of Harmony which was the primary idea in Greek science.ThishypothesisisconfirmedbythegeometrictheoremsinEuclids Elements.Theproblemofdivisioninextremeandmeanratiodescribedin Theorem II.11 is one of them. This division named later the golden section was usedbyEuclidforthegeometricconstructionoftheisoscelestrianglewiththe angles72 , 72 and36 , (the golden isosceles triangle) and then of the regular pentagonanddodecahedron.WeascertainwithgreatregretthatProclus hypothesiswasnotreallyrecognizedbymodernmathematicianswhocontinue toconsidertheaxiomaticstatementofgeometryasthemainachievementof Euclids Elements. However, as Euclids Elements are thebeginnings ofschool mathematical education, we should ask the question: whydo the golden section andPlatonicSolidsoccupysuchamodestplaceinmodernmathematical education?Thenarrowone-sidedinterpretationofEuclidsElementsisone morestrategicmistakeinthedevelopmentofmathematicsand mathematicaleducation.Thisstrategicmistakeresultedinadistorted picture of the history of mathematics.13 2.5.Theone-sidedapproachtotheoriginofmathematics.Thetraditional approachtotheoriginofmathematicsconsistsofthefollowing[42]. Historically, two practical problems stimulated the development of mathematics oninitsearlierstagesofdevelopment.Wearereferringtothecountproblem andmeasurementproblem.Thecountproblemresultedinthecreationofthe firstmethodsofnumberrepresentationandthefirstrulesforthefulfillmentof arithmeticaloperations(includingtheBabyloniansexagecimalnumbersystem andEgyptiandecimalarithmetic).Theformationoftheconceptofnatural numberwasthemainresultofthislongperiodinthemathematicshistory.On theotherhand,themeasurementproblemunderliesthecreationofgeometry (Measurement of the Earth). The discovery of incommensurable line segments isconsideredtobethemajormathematicaldiscoveryinthisfield.This discoveryresultedintheintroductionofirrationalnumbers,thenext fundamental notion of mathematics following natural numbers.Theconceptsofnaturalnumberandirrationalnumberarethemajor fundamentalmathematicalconcepts,withoutwhichitisimpossibletoimagine theexistenceofmathematics.TheseconceptsunderlietheClassical Mathematics. Neglectoftheharmonyproblem andgoldensectionby mathematicianshasanunfortunateinfluenceonthedevelopmentof mathematics and mathematical education. As a result, we have a one-sided view of the origin of mathematics, which is one more strategic mistake in the development of mathematics and mathematical education. 2.6.Thegreatestmathematicalmystificationofthe19thcentury.The strategic mistake influenced considerably on the development of mathematics and mathematical education, was made in the 19th century. We are talking about Cantors Theory of Infinite Sets. Remind that George Cantor (1845 1918) was a German mathematician, born in Russia. He is best known as the creator of set theory,which has become a fundamental theory in mathematics. Unfortunately, Cantorssettheorywasperceivedbythe19thcenturymathematicianswithout proper critical analysis. The end of the 19th century was a culmination point in recognizing of Cantorssettheory.Theofficialproclamationofthesettheoryasthe mathematicsfoundationwasheldin1897:thisstatementwascontainedin HadamardsspeechontheFirstInternationalCongressofMathematiciansin Zurich(1897).InhislecturetheGreatmathematicianJacquesHadamard (1865-1963)didemphasizethatthemainattractivereasonofCantor'sset theoryconsistsofthefactthatforthefirsttimeinmathematicshistorythe classification of the sets was made on the base of a new concept of "cardinality" andtheamazingmathematicaloutcomesinspiredmathematiciansfornewand surprising discoveries. 14 However, very soon the mathematical paradise" based on Cantor's set theorywasdestroyed.FindingparadoxesinCantorssettheoryresultedinthe crisis in mathematics foundations, what cooled enthusiasm of mathematicians to Cantorssettheory.TheRussianmathematicianAlexanderZenkinfinisheda critical analysis of Cantors set theory and a concept of actual infinity, which is the main philosophical idea of Cantors set theory. AfterthethoroughanalysisofCantorscontinuumtheorem,inwhich AlexanderZenkingavethe"logic"substantiationforlegitimacyoftheuseof the actual infinity in mathematics, he did the following unusual conclusion [43]: 1.CantorsproofofthistheoremisnotmathematicalproofinHilbertssense and in the sense of classical mathematics.2.Cantorsconclusionaboutnon-denumerabilityofcontinuumisa"jump through a potentially infinite stage, that is, Cantors reasoning contains the fatal logic error of unproved basis" (a jump to the wishful conclusion"). 3. Cantors theorem, actually, proves, strictly mathematically, the potential, that is, not finishedcharacter of the infinity of the set of all real numbers, that is, Cantorprovesstrictlymathematicallythefundamentalprincipleofclassical logic and mathematics: "Infinitum Actu Non Datur" (Aristotle). However, despite of so sharp critical attitude to Cantor's set theory, the theoretic-setideashadappearedrather"hardy"andwereappliedinmodern mathematicaleducation.Inanumberofcountries,inparticular,inRussia,the revisionoftheschoolmathematicaleducationonthebaseoftheoretic-set approachwasmade.Asiswellknown,thetheoretic-setapproachassumes certainmathematicalculture.Amajorityofpupilsandmanymathematics teachersdonotpossessandcannotpossessthisculture.Whatasaresulthad happened?InopinionoftheknownRussianmathematician,academicianLev Pontrjagin(1908-1988)[44],thisbroughttoartificialcomplicationofthe learningmaterialandunreasonableoverloadofpupils,totheintroductionof formalisminmathematicaltrainingandisolationofmathematicaleducation fromlife,frompractice.Manymajorconceptsofschoolmathematics(suchas conceptsoffunction,equation,vector,etc.)becamedifficultformasteringby pupils...Thetheoretic-setapproachisalanguageofscientificresearches convenientonlyformathematicians-professionals.Thevalidtendencyofthe mathematicsdevelopmentisinitsmovementtospecificproblems,topractice. Therefore,modernschoolmathematicstextbooksareastepbackin interpretationofthisscience,theyareunfoundedessentiallybecausethey emasculate an essence of mathematical method. Thus,Cantorstheoryofinfinitesetsbasedontheconceptof actualinfinitycontainsfatallogicerrorandcannotbeconsideredas mathematicsbase.Itsacceptanceasmathematicsfoundation,without proper critical analysis, is one more strategic mistake in the mathematics development;Cantorstheoryisoneofthemajorreasonsofthe contemporarycrisisinmathematicsfoundations.Auseoftheoretic-set 15 approachinschoolmathematicaleducationhasledtoartificial complication of the learning material, unreasonable overload of pupils and to the isolation of mathematical education from life, from practice.

2.6. The underestimation of Binet formulas. In the 19th century, a theory of the goldensectionandFibonaccinumberswassupplementedbyoneimportant result.Thiswaswiththeso-calledBinetformulasforFibonacciandLucas numbers given by (11) and (12).TheanalysisoftheBinetformulas(11)and(12)givesonethe opportunity to sense the beauty of mathematics and once again be convinced of thepowerofthehumanintellect!Actually,weknowthattheFibonacciand Lucasnumbersarealwaysintegers.Butanypowerofthegoldenmeanisan irrational number. As it follows from the Binet formulas, the integer numbers Fn and Ln can be represented as the difference or sum of irrational numbers, namely the powers of thegolden mean! We know it is not easy to explain to pupilsthe conceptofirrationals.Forlearningmathematics,theBinetformulas(10)and (11), which connect Fibonacci andLucasnumberswith thegolden meant, are very important because they demonstrate visually a connection between integers and irrational numbers.Unfortunately, in classical mathematics and mathematical education the Binetformulasdidnotgettheproperkindofrecognitionasdid,forexample, Eulerformulasandotherfamousmathematicalformulas.Apparently,this attitudetowardstheBinetformulasisconnectedwiththegoldenmean,which always provoked anallergic reaction in manymathematicians. Therefore, the Binet formulas are not generally found in school mathematics textbooks.However,themainstrategicmistakeintheunderestimationof the Binet formulas is the fact that mathematicians could not see in the Binet formulas a prototype for a new class of hyperbolic functions the hyperbolic Fibonacci and Lucas functions. Such functions were discovered roughly 100 yearslaterbyUkrainianresearchersBodnar[45],Stakhov,Tkachenko, andRozin[9,13,20,29,33].IfthehyperbolicfunctionsonFibonacciand Lucas had been discovered in the 19th century, hyperbolic geometry and its applicationstotheoreticalphysicswouldhavereceivedanewimpulsein their development. 2.7.TheunderestimationofFelixKlein'sideaconcerningtheRegular I cosahedron.ThenameoftheGermanmathematicianFelixKlein(18491925)iswellknowninmathematics.Inthe19thcenturyFelixKleintriedto unite all branches of mathematics on the base of the regular icosahedron dual to the dodecahedron [46].Klein interprets the regular icosahedron based on the golden section as ageometricobject,connectedwith5mathematicaltheories:geometry,Galois theory,grouptheory,invarianttheory,anddifferentialequations.Kleinsmain 16 idea is extremely simple:Each unique geometric objectis connectedoneway or another with the properties of the regular icosahedron. Unfortunately, this remarkable idea was not developed in contemporary mathematics, which is one more strategic mistake in the development of mathematics. 2.8.TheunderestimationofBergmansnumbersystem.Onestrange traditionexistsinmathematics.Itisusuallythecasethatmathematicians underestimatethemathematicalachievementsoftheircontemporaries.The epochalmathematicaldiscoveries,asarule,inthebeginninggounrecognized by mathematicians. Sometimes they are subjected to sharp criticism and even to gibes. Only after approximately 50 years, as a rule, after the death of the authors of theoutstandingmathematical discoveries,thenewmathematical theories are recognized and take their place of worth in mathematics. The dramatic destinies of Lobachevski, Abel, and Galois are very well-known.In1957theAmericanmathematicianGeorgeBergmanpublishedthe articleAnumbersystemwithanirrationalbase[47].InthisarticleBergman developedaveryunusualextensionofthenotionofthepositionalnumber system.Hesuggestedthatoneusethegoldenmean 1 52+t = asthe basisofa specialpositionalnumbersystem.Ifweusethesequencesti{i=0,1,2,3, }asdigitweightsofthebinarynumbersystem,wegetthebinary number system with irrational base t: iA aii = t ,(13) where is a real number, ai are binary numerals 0 or 1, i = 0, 1, 2, 3 , ti is the weight of the i-th digit, t is the base of the number system (13).Unfortunately,Bergmansarticle[47]wasnotnoticedby mathematiciansofthatperiod.Onlythejournalistsweresurprisedbythefact that George Bergman made his mathematical discovery at the age of 12! In this connection,TIMEMagazinepublishedanarticleaboutmathematicaltalentin America.In 50years, according to "mathematical tradition"thetimehad come toevaluatetheroleofBergmanssystemforthedevelopmentofcontemporary mathematics.ThestrategicimportanceofBergmanssystemisthefactthatit overturnsourideasaboutpositionalnumbersystems,moreover,ourideas about correlations between rational and irrational numbers.Asiswellknown,historicallynaturalnumberswerefirstintroduced, afterthemrationalnumbersasratiosofnaturalnumbers,andlaterafterthe discoveryoftheincommensurablelinesegments-irrationalnumbers,which cannotbeexpressedasratiosofnaturalnumbers.Byusingthetraditional positionalnumbersystems(binary,ternary,decimalandsoon),wecan 17 represent any natural, real or irrational number byusing number systems with a baseof(2,3,10andsoon).ThebaseinBergmanssystem[47]isthegolden mean.ByusingBergmanssystem(13),wecanrepresentallnatural,realand irrational numbers. As Bergmans system (13) is fundamentally a new positional numbersystem,itsstudyisveryimportantforschoolmathematicaleducation becauseitexpandsourideasaboutthepositionalprincipleofnumber representation.Thestrategicmistakeof20thcenturymathematiciansisthat theytooknonoticeofBergmansmathematicaldiscovery,whichcanbe consideredasthemajormathematicaldiscoveryinthefieldofnumber systems(followingtheBabyloniandiscoveryofthepositionalprincipleof number representation and also decimal and binary systems). 3.Threekeyproblemsofmathematicsandanewapproachtothe mathematics origins ThemainpurposeoftheHarmonyMathematicsistoovercomethestrategic mistakes, which arose in mathematics in process of its development.Wecanseethatthreekeyproblemsthecountproblem,the measurementproblem,andtheharmonyproblem-underlietheoriginsof mathematics(seeFig.2).Thefirsttwokeyproblemsresultedinthecreation oftwofundamentalnotionsofmathematicsnaturalnumberandirrational numberthatunderlietheClassicalMathematics.Theharmonyproblem connectedwiththedivisioninextremeandmeanratio(TheoremII.11of EuclidsElements) resulted in theorigin oftheHarmonyMathematicsanew interdisciplinarydirectionofcontemporaryscience,whichisrelatedto contemporary mathematics, theoretical physics, and computer science. Thisapproachleadstoaconclusion,whichisstartlingformany mathematicians. It proves to be, in parallel withthe Classical Mathematics, one more mathematical direction theHarmony Mathematics already developing inancientscience.SimilarlytotheClassicalMathematics,theHarmony MathematicshasitsorigininEuclidsElements.However,theClassical Mathematicsfocusesitsattentionontheaxiomaticapproach,whilethe HarmonyMathematicsisbasedonthegoldensection(TheoremII.11)and PlatonicSolidsdescribedinBookXIIIofEuclidsElements.Thus,Euclid's Elementsisthesoureoftwoindependentdirectionsinthedevelopmentof mathematics the Classical Mathematics and the Harmony Mathematics.Formanycenturies,themainfocusofmathematicianswasdirected towards the creation of the Classical Mathematics, which became the Czarina of NaturalSciences.However,theforcesofmanyprominentmathematicians- sincePythagoras,PlatoandEuclid,Pacioli,KepleruptoLucas,Binet, Vorobyov, Hoggatt and so forth - were directed towards the development of the basicconceptsandapplicationsoftheHarmonyMathematics.Unfortunately, theseimportantmathematicaldirectionsdevelopedseparatelyfromoneother. 18 ThetimehascometounitetheClassicalMathematicsandHarmony Mathematics.Thisunusualunioncanleadtonewscientificdiscoveriesin mathematicsandnaturalsciences.Someofthelatestdiscoveriesinthenatural sciences, in particular, Shechtmans quasi-crystals based on Platos icosahedron andfullerenes(NobelPrizeof1996)basedontheArchimedeantruncated icosahedrondo demandthisunion.Allmathematicaltheoriesshouldbeunited for one unique purpose: to discover and explain Nature's Laws. The "key" problems of the ancient mathematicsCountMeasurementproblemproblemPositional Incommensurableprinciple of linenumber segmentsrepresentation+ + ++ + ++HarmonyproblemDivision in extreme andmeanratioNumber theory Measurementand natural theory andnumbers irrationalrepresentation numbersClassical mathematicsTheoretical physicsComputer sc+ ++ + +Theory of FibonacciandLucas numbersand the Golden SectionienceHarmony mathematics"Golden" theoretical physics"Golden" computer science Figure 2. Three key problems of the ancient mathematics

Anewapproachtothemathematicsorigins(seeFig.2)isvery important for school mathematical education. This approach introduces in a very naturalmannertheideaofharmonyandthegoldensectionintoschool mathematical education. This provides pupils access to ancient science and to its mainachievementtheharmonyideaandtotellthemaboutthemost importantarchitecturalandsculpturalworksofancientartbaseduponthe goldensection(includingpyramidofKhufu(Cheops),Nefertiti,Parthenon, Doryphorus, Venus). 4.ThegeneralizedFibonaccinumbersandthegeneralizedgolden proportions 4.1. Pacals Triangle, the generalized Fibonacci p-numbers, the generalized p-proportions,thegeneralizedBinetformulas,andthegeneralizedLucasp-19 numbers. Pascals triangle (Table 1) is recognized as one of the most beautiful objectsofmathematics.Andwecanexpectfurtherbeautifulmathematical objectsstemmingfromPascalstriangle.Intherecentdecades,many mathematiciansfoundaconnectionbetweenPascalstriangleandFibonacci numbersindependentofeachother.ThegeneralizedFibonaccip-numbers, whichcanbeobtainedfromPascalstriangleasitsdiagonalsums[4]arethe most important of them.For a given integer=0, 1, 2, 3, ... , they aregiven by the recurrence relation:( ) ( 1) ( 1);(0) (1) ... ( ) 1F n F n F n pp p pF F F pp p p= + = = = = (14) It is easyto seethatfor thecase=1therecurrence relation(14)is reduced to the recurrence relation for the classical Fibonacci numbers:( ) ( 1) ( 1);1 1 1(0) 0, (1) 11 1Fn Fn FnF F= + = =(15) Itfollowsfrom(14)thattheFibonacci-numbersexpressmore complicatedharmoniesthantheclassicalFibonaccinumbersgivenby(15). Notethattherecurrenceformula(14)generatesaninfinitenumberofdifferent recurrencenumericalsequencesbecauseeverypgeneratesitsownrecurrence sequences,inparticular,thebinarynumbers1,2,4,8,16, forthecasep=0 and the classical Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, for the case p=1. It is important to notethat therecurrencerelation (14) expresses some deepmathematicalpropertiesofPascalstriangle(thediagonalsums).The Fibonacci p-numbers can be represented by the binomial coefficients as follows [4]:( )0 1 3 41 ...4 2...kF n C C C C Cp n n p n p n p n kp+ = + + + + + + , (16) where the binomial coefficient0kCn kp =for the case k>n-kp. Notethatforthecasep=0theformula(16)isreducedtothewell-known formula of combinatorial analysis: 0 12 ... .n nC C Cn n n= + + +(17) It is easy to prove[4] thatinthelimit(n)theratio of theadjacent Fibonaccip-numbers( ) ( ) / 1 F n F np p aimsforsomenumericalconstant,that is,( )( )l i m ,1F nppnF np= t (18) where tp is the positive root of the following algebraic equation:xp+1 = xp + 1, (19) which for =1 is reduced to the golden algebraic equation (10) generating the classical golden mean (9).20 Note that the result (16) is a generalization ofKeplers formula (9) for the classical Fibonacci numbers (p=1).The positive roots of Eq.(17) were named the golden -proportions [4]. It is easy to prove [4] that the powers of the golden -proportions are connected between themselves by the following identity:1 1 1 n n n p np p p p p t = t + t = t t , (20) thatis,eachpowerofthegolden-proportionisconnectedwiththepreceding powers by the additive relation 1 1 n n n pp p p t = t + tand by the multiplicative relation 1 n np p pt = t t (similar to the classical golden mean).It is proved in [23] that the Fibonacci p-numbers can be represented in the following analytical form:( ) ( ) ( ) ( )... ,1 1 2 2 1 1 nn nF n k x k x k xp p p= + + ++ +

(21) wheren=0,1,2,3,...,x1,x2,,xp+1aretherootsofEq.(19),andk1,k2,, kp+1 are constant coefficients that depend on the initial elements of the Fibonacci p-series and are solutions to the following system of algebraic equations:( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )0 ... 01 2 11 ... 11 1 2 2 122 22 ... 11 1 2 2 1 1............................................................................. 1.1 1 2 2 1 1F k k kp pF k x k x kp pF k x k x k xp p ppp pF p k x k x k xp p p= + + + =+= + + + =+= + + + =+ += + + + =+ + Note that for the case p=1, the formula (21) is reduced to the Binet formula (11) for the classical Fibonacci numbers.In[23]thegeneralizidLucasp-numbersareintroduced.Theyare represented in the following analytical form: ( ) ( ) ( ) ( )... .1 2 1 nn nL n x x xp p= + + ++(22) where n=0,1,2,3,..., x1, x2, , xp+1 are the roots of Eq. (19).Noticethatforthecasep=1,theformula(22)isreducedtotheBinet formula (12) for the classical Lucas numbers.Directly from (22) we can deduce the following recurrence relation( ) ( ) ( ) 1 1 L n L n L n pp p p= + , (23) which at the seeds ( ) ( ) ( ) ( ) 0 1and1 2 ... 1 L p L L L pp p p p= + = = = =(24) producesanewclassofnumericalsequencesLucasp-numbers.Theyarea generalization of the classical Lucas numbers for the case p=1. 21 Thus, a study of Pascals triangle produces the following beautiful mathematical results: 1. The generalized Fibonacci p-numbers, which are expressed through binomial coefficients by the graceful formula (16). 2. The golden p-proportions pt(p=0, 1, 2, 3, ), a new class of mathematical constants,whichexpresssomeimportantmathematicalpropertiesofPascals triangle and possess unique mathematical properties (20). 3.Anewclassofthegoldenalgebraicequations(19),whichareawide generalization of the classical golden equation (10). 4. A generalization of the Binet formulas for Fibonacci and Lucas p-numbers. DiscussingapplicationsoftheFibonaccip-numbersandgoldenp-proportions to contemporary theoretical natural sciences, we find two important applications: 1.Asymmetricdivisionofbiologicalells.Theauthorsofthearticle[48] provedthattheFibonaccip-numberscanmodelthegrowthofbiologicalcells. Theyconcludethatbinarycelldivisionisregularlyasymmetricinmost species.Growthbyasymmetricbinarydivisionmayberepresentedbythe generalizedFibonacciequation.Ourmodels,forthefirsttimeatthesingle celllevel,providearationalbasisfortheoccurrenceofFibonacciandother recursivephyllotaxisandpatterninginbiology,foundedontheoccurrenceof the regular asymmetry of binary division.2.Structuralharmonyofsystems.Studyingtheprocessofsystemself-organizationindifferentaspectsofnature,BelarusianphilosopherEduard Soroko formulated the Law of Structural Harmony of Systems [49] based on the goldenp-proportions:Thegeneralizedgoldenproportionsareinvariantsthat allownaturalsystemsintheprocessoftheirself-organizationtofinda harmonious structure, a stationary regime for their existence, and structural and functional stability. 4.2. The generalized Fibonacci -numbers, metallic means, Gazale formulas andageneraltheoryofhyperbolicfunctions.Anothergeneralizationof FibonaccinumberswasintroducedrecentlybyVeraW.Spinadel[50], MidchatGazale[51],JayKappraff[52]andotherscientists.Wearetalking aboutthegeneralizedFibonacci -numbersthatforagivenpositivereal number > 0are given by the recurrence relation:( ) ( ) ( )( ) ( )1 2 ;0 0, 1 1F n F n F nF F= + = = . (25) Firstofall,wenoticethattherecurrencerelation(25)isreducedtothe recurrence relation (4) for the case = 1 . For other values of , the recurrence relation(25)generatesaninfinitenumberofnewrecurrencenumerical sequences. The following characteristic algebraic equation follows from (25):22 21 0 x x = , (26) whichforthecase = 1 isreducedtoEq.(10).ApositiverootofEq.(26) generatesaninfinitenumberofnewharmonicproportionsMetallic MeansbyVeraSpinadel[50],whichareexpressedbythefollowinggeneral formula:242 + + u =. (27) Notethatforthecase = 1 theformula(27)givestheclassicalgoldenmean 1 512+t = u= . The metallic means possessthefollowinguniquemathematical properties:11 1 1 ...11... mu = + + + u = + + ++, (28) whicharegeneralizationsofsimilarpropertiesfortheclassicalgoldenmean ( ) 11t = u = : 11 1 1 ... 111111 ...t = + + + t = ++++. (29) Notethattheexpressions(27),(28)and(29),withoutdoubt,satisfyDiracs Principle of Mathematical Beauty and emphasize a fundamental characteristic of both the classical golden mean and the metallic means. Recently,bystudyingtherecurrencerelation(25),theEgyptian mathematician Midchat Gazale [51] deduced the following remarkable formula given by Fibonacci -numbers:( 1)( )24n n nF nu u =+ , (30) where > 0isagivenpositiverealnumber,uisthemetallicmeangivenby (27), n = 0, 1, 2, 3, .... The author of the present article named theformula (30)in[33]formulaGazalefortheFibonacci -numbersafterMidchat Gazale. The similar Gazale formula for the Lucas -numbers is deduced by the author in [33]:( ) ( ) 1 nn nL n= u+ u . (31) 23 Firstofall,wenotethattheGazaleformulas(30)and(31)areawide generalization of the Binet formulas (11) and (12) for the classical Fibonacci and Lucas numbers ( = 1 ).ThemostimportantresultisthattheGazaleformulas(30)and(31) resulted in a general theory of hyperbolic functions [33]. Hyperbolic Fibonacci -sine ( )24x xsF xu u =+ (32) Hyperbolic Fibonacci -cosine( )24x xcF xu + u =+ (33) Hyperbolic Lucas -sine( )x xsL x= u u (34) Hyperbolic Lucas -cosine( )x xcL x= u + u , (35) where 242 + + u =is the metallic means. NoticethatthehyperbolicFibonacciandLucas -functionscoincide with the Fibonacci and Lucas -numbers for the discrete values of the variable x=n=0,1,2,3, , that is, , 2, 2 1, 2, 2 1sF n n kF ncF n n kcL n n kL nsL n n k . (36) Theformulas(32)-(35)provideaninfinitenumberofhyperbolic modelsofnaturebecauseeveryrealnumber originatesitsownclassof hyperbolic functions of the kind (32)-(35). As is proved in [33], these functions have,ontheonehand,thehyperbolicpropertiessimilartothepropertiesof classicalhyperbolicfunctions,andontheotherhand,recursiveproperties similar to the properties of the Fibonacci and Lucas -numbers (30) and (31). In particular, the classicalhyperbolic functions area partial case of thehyperbolic 24 Lucas -functions(34)and(35).Forthecase 12.35040238... eee = ~ ,the classical hyperbolic functions are connected with hyperbolic Lucas -functions by the following simple relations:( ) ( )( ) and ( )2 2sL x cL xsh x ch x = = . (37) Notethatforthecase = 1 ,thehyperbolicFibonacciandLucas -functions (32)-(35) coincide with the symmetric hyperbolic Fibonacci and Lucas functions introduced by Alexey Stakhov and Boris Rozin in the article [20]:Symmetric hyperbolic Fibonacci sine and cosine ( ) ; ( )5 5x x x xsFs x cFs x t t t + t= =(38) Symmetrical hyperbolic Fibonacci sine and cosine ( ) ; ( )x x x xsLs x cLs x = = +t t t t(39) where 1 52+t =is the golden mean. In the book [45], theUkrainian researcherOleg Bodnarused Stakhov and Rozins symmetric hyperbolic Fibonacci and Lucas functions(38) and (39) forthecreationofagracefulgeometrictheoryofphyllotaxis.Thismeansthat thesymmetricalhyperbolicFibonacciandLucasfunctions(38)and(39) and their generalization the hyperbolic Fibonacci and Lucas -functions (32)-(35)canbeascribedtothefundamentalmathematicalresultsof modernsciencebecausetheyreflectNaturesphenomena,inparticular, phyllotaxis phenomena [45]. These functions set a general theory of hyperbolic functionsthatis offundamental importancefor contemporarymathematics and theoretical physics.WeproposethatthehyperbolicFibonacciandLucas -functions, whichcorrespondtothedifferentvaluesof ,canmodeldifferentphysical phenomena.Forexample,inthecaseof = 2 therecurrencerelation(25)is reduced to the recurrence relation( ) ( ) ( )( ) ( )2 1 22 2 20 0, 1 1,2 2F n F n F nF F= + = = (40) which gives the so-called Pell numbers: 0, 1, 2, 5, 12, 29, ... . In this connection, theformulasforthemetallicmeanandhyperbolicFibonacciandLucas -numbers take for the case = 2the following forms, respectively:25 1 22u= +(41) 2 2( )22 2x xsF xu u=(42) 2 2( )22 2x xcF xu + u=(43) ( )2 2 2x xsL x= u u (44) ( )2 2 2x xcL x= u + u .(45) It is appropriate to give the following comparative Table 3, which gives arelationshipbetweenthegoldenmeanandmetallicmeansasnew mathematical constants of Nature.Table 3 21 2 1 1 2 11 5 42 21 1 1 ... 1 1 1 ...1 111 111 111 ... ...( 1) ( 1)( )54n n n n n n n nn n n n n nnF Fn + + + t = u =t = + + + u = + + + t = + u = ++ ++ ++ +t = t + t = t t u = u + u = u uu u t t= =+The Golden Mean ( = 1) The Metallic Means ( 0) >( ) ( )22 2( 1) ( ) ( 1)( ) ; ( ) ( ) ; ( )5 54 4; ( ) ; ( )n n n n n nnx x x x x x x xx x x x x x x xL L nsFs x cFs x sF x cF xsLs x cLs x sL x cL x = t + t = u + uu u u + u t t t + t= = = =+ + = t t = t + t = u u = u + u Abeautyoftheseformulasischarming.Thisgivesarighttosuppose thatDiracsPrincipleofMathematicalBeautyisapplicablefullytothe metallicmeansandhyperbolicFibonacciandLucas -functions.Andthis,in itsturn,giveshopethatthesemathematicalresultscanbecomeabaseof theoretical natural sciences. 5. A new geometric definition of a number

5.1. Euclidean and Newtonian definition of a real number. The first definition ofanumberwasmadeinGreekmathematics.Wearetalkingaboutthe Euclidean definition of natural number: 26 1 1 ... 1 NN= ++ +. (46) InspiteoftheutmostsimplicityoftheEuclideandefinition(46),itplayeda decisiveroleinmathematics,inparticular,innumbertheory.Thisdefinition underliesmanyimportantmathematicalconcepts,forexample,theconceptsof prime and composite numbers, and also a concept of divisibility that is one of the major concepts of number theory. Over the centuries, mathematicians developed and defined more exactly the concept of a number. In the 17th century, that is, in theperiodofthecreationofnewscience,inparticular,newmathematics,a number of methods for the study of continuous processes were developed and theconceptofarealnumberagainmovesintotheforeground.Mostclearly,a new definition of this concept was given by Isaac Newton (1643 1727), one of the founders of mathematical analysis, in his Arithmetica Universalis (1707):We understand a number not as a set of units, but as the abstract ratio of one magnitude to another magnitude of the same kind taken for the unit. This formulation gives us a general definition of numbers, rational and irrational. For example, the binary system2iN ai+ =(47) is the example ofNewtons definition,when we choose the number of 2 for the unit and represent a number as the sum of the powers of number 2. 5.2.Numbersystemswithirrationalradicesasanewdefinitionofreal number. Let us consider the set of the powers of the golden p-proportions:{ }, 0,1,2,3,..; 0, 1, 2, 3,...iS p ip= t = = (48) Byusing(48),wecanconstructthefollowingmethodofpositional representation of real number A:,iA ap ii = t(49) where ai is the binary numeral of the i-th digit; iptis the weight of the i-th digit; pt istheradixofthenumeralsystem(47),0, 1, 2, 3,... i = .Thepositional representation (49) is called code of the golden p-proportion [6].Notethatforthecasep=0thesum(49)isreducedtotheclassical binary representation of real numbers given by (47).Forthecasep=1,thesum(49)isreducedtoBergmanssystem(13). For the case p, the sum (49) strives for the expression similar to (46).Intheauthorsarticle[17],anewapproachtogeometricdefinitionof real numbers based on (49) was developed. A new theory of real numbers based onthedefinition(49)containsanumberofunexpectedresultsconcerning numbertheory.LetusstudytheseresultsasappliedtoBergmanssystem(13). We shall represent a natural number N in Bergmans system (13) as follows: 27 iN aii = t . (50) The following theorems are proved in [17]:1. Every natural number N can be represented in the form (50) as a finite sum of the golden powers it( 0, 1, 2, 3,... i = ). Note that this theorem is not a trivial property of natural numbers.2.Z-propertyofnaturalnumbers.Ifwesubstitutein(50)theFibonacci number Fi for the golden power ti ( 0, 1, 2, 3,... i = ), then the sum that appears asaresultofsuchasubstitutionisequalto0independentoftheinitialnatural number N, that is,0. a Fiii =(51) 3. D-property of natural numbers. If we substitute in (50) the Lucas number Li forthegoldenpowerti( 0, 1, 2, 3,... i = ),thenthesumthatappearsasa result of suchasubstitutionis equaltothedoublesum (50)independentofthe initial natural number N, that is,2 . a L Niii =(52) 4. F-code of natural number N. If we substitute in (50) the Fibonacci number Fi+1 for thegolden powerti ( 0, 1, 2, 3,... i = ), then thesum that appearsas a resultofsuchasubstitutionisanewpositionalrepresentationofthesame natural number N called the F-code of natural number N, that is, ( ) 0, 1, 2, 3,... .1N a F iiii = = + (53) 5. L-code of natural number N. If we substitute in (50) the Lucas number Li+1 forthegoldenpowerofti( 0, 1, 2, 3,... i = ),thenthesumthatappearasa resultofsuchasubstitutionisanewpositionalrepresentationofthesame natural number N called L-code of natural number N, that is, ( ) 0, 1, 2, 3,... .1N a L iiii = = + (54) Notethatsimilar propertiesfor naturalnumbersareprovedin [17]for the code of the golden p-proportion given by (49).Thus, after 2.5 millennia, we have discovered new properties of natural numbers(Z-property,D-property,F-andL-codes)thatconfirmthe fruitfulnessofsuchanapproachtonumbertheory[17].Theseresultsareof greatimportanceforcomputerscienceandcouldbecomeasourcefornew computer projects.Asthestudyofthepositionalbinaryanddecimalsystemsarean importantpartofmathematicaleducation,thenumbersystemswithirrational radicesgivenby(13)and(49)areofgeneralinterestformathematical education. 28 6. Fibonacci and golden matrices 6.1.Fibonacci matrices. For thefirsttime,atheoryoftheFibonacciQ-matrix wasdevelopedinthebook[53]writtenbyeminentAmericanmathematician VernerHoggattfounderoftheFibonacciAssociationandTheFibonacci Quarterly.Thearticle[54]devotedtothememoryofVernerE.Hoggatt contained a history and extensive bibliography of theQ-matrix and emphasized Hoggattscontributiontoitsdevelopment.AlthoughthenameoftheQ-matrix was introduced beforeVerner E. Hoggatt, he was the first mathematician who appreciatedthemathematicalbeautyoftheQ-matrixandintroduceditinto Fibonacci numberstheory. Thanks toHoggattswork,theidea of theQ-matrix caughtonlikewildfireamongFibonaccienthusiasts.Numerouspapers appeared in The Fibonacci Quarterly authored by Hoggatt and/or his students and other collaborators wheretheQ-matrix method becamethecentral toolin the analysis of Fibonacci properties [54]. The Q-matrix1 11 0Q = | |\ . (55) isageneratingmatrixforFibonaccinumbersandhasthefollowingwonderful properties: 11F Fnn nQF Fn n+=+| | |\ . (56) ( )2det 11 1nnQ F F Fn n n= = + +. (57) NotethatthereisadirectrelationbetweentheCassiniformula(8)andthe formula (57) given for the determinant of the matrix (56).Inthearticle[15],Alexey Stakhovintroduced ageneratingmatrixfor the Fibonacci p-numbers called Qp-matrix (p=0, 1, 2, 3, ): 1 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 0 0 0Qp =| | | | | | | |\ . . (58) The following properties of the Qp-matrices (58) are proved in [15]: 29 ( 1) ( ) ( 2) ( 1)( 1) ( ) ( 2 2) ( 2 1)( 1) ( 2) ( ) ( 1)( ) ( 1) ( 1) ( )F n F n F n p F n pp p p pF n p F n p F n p F n pp p p pnQpF n F n F n p F n pp p p pF n F n F n p F n pp p p p+ + + + + += + | | | | | | |\ . (59) ( ) det 1 pnnQp = ,(60) where p=0, 1, 2, 3, ... ; and0, 1, 2, 3,... n = The generating matrixG for the Fibonacci-numbers( ) F n 11 0G=| |\ . (61) was introduced by Alexey Stakhov in [33]. The following properties of theG-matrix (61) are proved in [33]: ( 1) ( )( ) ( 1)F n F nnGF n F n+ = | | |\ . (62) ( ) det 1 nnG = . (63) ThegeneralpropertyoftheFibonacciQ-,Qp-,andG-matrices consistsofthefollowing.ThedeterminantsoftheFibonacciQ-,Qp-,andG-matricesandalltheirpowersareequalto+1or-1.Thisuniqueproperty emphasizesmathematical beautyintheFibonacci matricesandcombinesthem intoaspecialclassofmatrices,whichareoffundamentalinterestformatrix theory. 6.2. The golden matrices. Integer numbers the classical Fibonacci numbers, the Fibonacci p- and -numbers - are elements of the Fibonacci matrices (56), (59)and(62).In[28]aspecialclassofthesquarematricescalledgolden matriceswasintroduced.Theirpeculiarityisthefactthatthehyperbolic Fibonacci functions (38) or the hyperbolic Fibonacci -functions (32) and (33) are elements of these matrices. Let us consider the simplest of them [28]: ( 2 1) ( 2 ) 2( 2 ) ( 2 1)( 2 2) ( 2 1) 2 1( 2 1) ( 2 )cFs x sFs x xQsFs x cFs xsFs x cFs x xQcFs x sFs x+=+ + +=+| |\ .| |\ .(64) Ifwecalculatethedeterminantsofthematrices(64),weobtainthefollowing unusual identities:2 2 1det 1; det 1x xQ Q+= = . (65) ThegoldenmatricesbasedonhyperbolicFibonacci -functions (32) and (33) take the following form [33]: 30 ( 2 1) ( 2 )2( 2 ) ( 2 1)( 2 2) ( 2 1)2 1( 2 1) ( 2 )cF x sF xxGsF x cF xsF x cF xxGcF x sF x+ = + ++ =+ | | |\ .| | |\ .. (66) Itisproved[33]thatthegoldenG-matrices(66)possessthefollowing unusual properties:2 2 1det 1; det 1x xG Gm m += = . (67) Themathematicalbeautyofthegoldenmatrices(64)and(66)are confirmed by their unique mathematical properties (65) and (67). 7. Applications in computer science: the Golden Information Technology 7.1.Fibonaccicodes,FibonacciarithmeticandFibonaccicomputers.The conceptofFibonaccicomputerssuggestedbyAlexeyStakhovinthespeech AlgorithmicMeasurementTheoryandFoundationsofComputerArithmetic giventothejointmeetingofComputerandCyberneticsSocietiesofAustria (Vienna,March1976)anddescribedinthebook[4]isoneofthemore importantideasofmoderncomputerscience.Theessenceoftheconcept amounts to the following: modern computers are based on a binary system (47), which represents all numbers as sums of the binary numbers( ) 2 0, 1, 2, 3,...ii = withbinarynumerals,0and1.However,thebinarysystem(47)isnon-redundant and does not allow for detection of errors, which could appear inthe computerduringtheprocessofitsexploitation.Inordertoeliminatethis shortcoming, Alexey Stakhov suggested in [4] the use of Fibonacci p-codes ( ) ( ) ( ) ( ) 1 ... ... 11 1N a F n a F n a F i a Fn p p p p i n= + + + + + (68) whereNisanaturalnumber,aie{0,1}isabinarynumeralofthei-thdigitof the code (68); n is the digit number of the code (68); Fp(i) is the i-th digit weight calculated in accordance with the recurrence relation (14). Thus,Fibonacci p-codes(6) represent allnaturalnumbersas thesums of Fibonacci p-numbers with binary numerals, 0 and 1. In contrast to the binary numbersystem(47),theFibonaccip-codes(68)areredundantpositional methodsofnumberrepresentation.Thisredundancycanbeusedforchecking differenttransformationsofnumericalinformationinthecomputer,including arithmetical operations. A Fibonacci computer project was developed by Alexey Stakhovin the former Soviet Union from 1976 right up to the disintegration of the Soviet Union in 1991. Sixty-five foreign patents in the U.S., Japan, England, France,Germany,Canadaandothercountriesareofficialjuridicaldocuments, which confirm Soviet priority (and Stakhovs priority) in Fibonacci computers. 31 7.2.Ternary mirror-symmetrical arithmetic. Computerscanbeconstructedby usingdifferentnumbersystems.TheternarycomputerSetundesignedin Moscow University in 1958 was thefirst computer basednot on binarysystem butonternarysystem[55].Thegoldenternarymirror-symmetricalnumber system[16]isanoriginalsynthesisoftheclassicalternarysystem[55]and Bergmanssystem(13)[47].Itrepresentsintegersasthesumofgoldenmean squares with ternary numerals {-1, 0, 1}. Each ternary representation consists of twopartsthataredisposedsymmetricallywithrespecttothe0-thdigit. However, one part is mirror-symmetrical to another part with respect to the 0-th digit. At the increase of a number, its ternary mirror-symmetrical representation isexpandingsymmetricallytotheleftandtotherightwithrespecttothe0-th digit.Thisuniquemathematicalpropertyproducesaverysimplemethodfor checking numerical information in computers. It is proved [16] that the mirror-symmetricpropertyisinvariantwithrespecttoallarithmeticaloperations,that is, theresults of all arithmeticaloperations havemirror-symmetricalform.This meansthatthegoldenmirror-symmetricalarithmeticcanbeusedfor designing self-controlling and fault-tolerant processors and computers.ThearticleBrousentsovsTernaryPrinciple,BergmansNumber SystemandTernaryMirror-SymmetricalArithmetic[16]publishedinThe ComputerJournal(England)gotahighapprovalfromtwooutstanding computer specialists - Donald Knut, Professor-Emeritus of Stanford University andtheauthorofthefamousbookTheArtofComputerProgramming,and Nikolay Brousentsov,ProfessoratMoscow University, aprincipaldesigner of thefist ternary computer"Setun."And thisfactgivesahopethatthegolden ternarymirror-symmetricalarithmetic[16]canbecomeasourceofnew computer projects in the near future.7.3.A new theory of error-correcting codes based upon Fibonacci matrices. Theerror-correctingcodes[56,57]areusedwidelyinmoderncomputerand communication systems forthe protection of information from noise. Themain idea of error-correcting codes consists of the following [56]. Let us consider the initialcodecombinationthatconsistsofndatabits.Weaddtotheinitialcode combinationmerror-correctionbitsandbuildupthek-digitcodecombination oftheerror-correctingcode,or(k,n)-code,wherek n m = + .Theerror-correction bits are formed from the data bits as the sums by module 2 of certain groups of the data bits. There are two important coefficients, which characterize an ability of error-correcting codes to detect and correct errors [56].The potential detecting ability 112Sd m= (69) The potential correcting ability 12Sc n= , (70) 32 where m is the number of error-correction bits, n is the number of data bits.Theformula(70)showsthatthecoefficientofpotentialcorrecting abilitydiminishespotentiallyto0asthenumbernofdatabitsincreases.For example,theHamming(15,11)-codeallowsonetodetect ( )11 15 112 2 2 62,914,560 = erroneoustransitions;here,theHamming(15,11)-code can only correct 15 112 2 30,720 = erroneous transitions. Theirratioisequalto30,720 : 62,914,560 0.0004882 = ,thatis,the Hamming(15,11)-codecancorrectpotentiallyonly(0.04882%)erroneous transitions.Ifwetaken=20,thenaccordingto(70)thepotentialcorrecting abilityoftheerror-correcting(k,n)-codediminishesto0.00009%.Thus,the potential correcting ability of the classical error-correcting codes [56, 57] is very low.Thisconclusionisoffundamentalimportance!Onemorefundamental shortcomingofallknownerror-correctingcodesisthefactthattheverysmall informationelements,bitsandtheircombinations,areobjectsofdetectionand correction.Thenewtheoryoferror-correctingcodes[7,27]thatisbasedon Fibonaccimatriceshasthefollowingadvantagesincomparisontotheexisting algebraic error-correcting codes [56, 57]:1.TheFibonaccicoding/decodingmethodisreducedtomatrixmultiplication, thatis,tothewell-knownalgebraicoperationthatiscarriedoutsowellin modern computers.2. The main practical peculiarity of the Fibonacci encoding/decoding method is the fact that large information units, in particular, matrix elements, are objects of detection and correction of errors. 3.ThesimplestFibonaccicoding/decodingmethod(p=1)canguaranteethe restorationofallerroneous( ) 2 2 -codematriceshavingsingle,double and triple errors. 4.Thepotentialcorrectingabilityofthemethodforthesimplestcasep=1is between26.67%and93.33%thatexceedsthepotentialcorrectingabilityofall known algebraic error-correcting codes by 1,000,000 or more times. This means thatanewcodingtheorybaseduponthematrixapproachisofgreatpractical importance for modern computer science. 7.4.Thegoldencryptography.Allexistingcryptographicmethodsand algorithms[58]werecreatedforidealconditionswhenweassumethatthe coder,communicationchannel,andthedecoderoperateideally,thatis,the codercarriesouttheidealtransformationofplaintextintociphertext,the communicationchanneltransmitsideallyciphertextfromthesendertothe receiver and the decoder carries out the ideal transformation of ciphertext into plaintext.Itisclearthatthesmallestbreachoftheidealtransformationor transmissionisacatastropheforthecryptosystem.Allexistingcryptosystems baseduponbothsymmetricandpublic-keycryptographyhaveessential 33 shortcomingsbecausetheydonothaveintheirprinciplesandalgorithmsan inner checking relation that allows checking the informational processeswithin the cryptosystems.The golden cryptography developed in [28, 33] is based upon the use ofthegoldenmatrices.Thismethodofcryptographypossessesunique mathematical properties based on (65) and (67). It is proved in [28, 33] that the determinantsofthecodematrixanddatamatrixcoincidebyabsolutevalue. Thankstothisproperty,wecancheckallinformationalprocessesinthe cryptosystem,includingencryption,decryptionandtransmissionofthe ciphertext via the channel. Such an approach can result in designing simple and reliablecryptosystemsfortechnicalrealization.Thus,goldencryptography openswithanewstageinthedevelopmentofcryptographydesigning super-reliable cryptosystems. 8.Fundamentaldiscoveriesofmodernsciencebaseduponthegolden section and Platonic Solids 8.1.Shechtmansquasi-crystals. It is necessary to notethatright up tothelast quarterofthe20thcenturytheuseofthegoldenmeanandPlatonicSolidsin theoretical science, in particular, in theoretical physics, was very rare. In order to be convinced ofthis, it is enough to browse 10 volumes ofTheoretical Physics byLandauandLifshitz.Wecannotfindanymentionaboutthegoldenmean andPlatonicSolids.Thesituationintheoreticalsciencechangedfollowingthe discoveryofQuasi-crystalsbytheIsraelresearcherDanShechtmanin1982 [59].One type of quasi-crystal was based upon the regular icosahedron (Fig. 1) described in Euclids Elements!Quasi-crystalsareofrevolutionaryimportanceformoderntheoretical science.Firstofall,thisdiscoveryisthemomentofagreattriumphforthe icosahedron-dodecahedrondoctrine,whichproceedsthroughoutallthehistory ofthenaturalsciencesandisasourceofdeepandusefulscientificideas. Secondly,thequasi-crystalsshatteredtheconventionalideathattherewasan insuperablewatershedbetweenthemineralworldwherethe"pentagonal" symmetrywasprohibited,andthelivingworld,wherethe"pentagonal" symmetryisoneofmostwidespread.NotethatDanShechtmanpublishedhis firstarticleaboutthequasi-crystalsin1984,thatis,exactly100yearsafterthe publicationofFelixKleinsLecturesontheIcosahedron(1884)[46].This meansthatthisdiscoveryisaworthygifttothecentennialanniversaryof Kleinsbook[46],inwhichKleinpredictedtheoutstandingroleofthe icosahedron in the future development of science.8.2.Fullerenes.Thediscoveryoffullerenesisoneofthemoreoutstanding scientificdiscoveriesofmodernscience.Thisdiscoverywasmadein1985by RobertF.Curl,HaroldW.KrotoandRichardE.Smalley.Thetitle "fullerenes"referstothecarbonmoleculesofthetype60,70,76,84,in 34 whichallatomsareonasphericalorspheroidsurface.Inthesemoleculesthe atomsofcarbonarelocatedatthevertexesofregularhexagonsandpentagons that cover the surface of a sphere or spheroid. The molecule C60 (Fig. 3-a) plays a special role amongst fullerenes. This molecule is based upon the Archimedean truncated icosahedron (Fig. 3-b). (a)(b) Figure 3. Archimedean truncated icosahedron (a) and the molecule C60 (a)ThemoleculeC60ischaracterizedbythegreatestsymmetryandasa consequenceisofthegreateststability.In1996RobertF.Curl,HaroldW. KrotoandRichardE.SmalleywontheNobelPrizeinchemistryforthis discovery. 8.3.El-NaschiesE-infinitytheory.Prominenttheoreticalphysicistand engineering scientist Mohammed S. El Naschie is a world leader in the field of goldenmeanapplicationstotheoreticalphysics,inparticular,quantumphysics [6063].ElNaschiesdiscoveryofthegoldenmeaninthefamousphysical two-slitexperimentwhichunderliesquantumphysicsbecamethesource of many important discoveries in this area, in particular, of E-infinity theory. It is alsonecessarytonotethattheimportantcontributionofSlavicresearchersin thisarea.Thebook[64]writtenbyBelarusianphysicistVasylPertrunenkois devoted to applications of the golden mean in quantum physics and astronomy.8.4.Bodnarsgeometry. Accordingtothelawofphyllotaxis,thenumberson theleft-handandright-handspiralsonthesurfaceofphyllotaxisobjectsare always adjacent Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, .... Their ratios 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, ... are called asymmetry order of phyllotaxis objects.SinceJohannesKepler,thephyllotaxisphenomenaexcitedthebest minds of humanity during the centuries. The puzzle of phyllotaxis consists of the fact that a majority of bio-forms change their phyllotaxis orders during their growth.Itisknown,forexample,thatsunflowerdisksthatarelocatedon differentlevelsofthesamestalkhavedifferentphyllotaxisorders;moreover, thegreatertheageofthedisk, thehigherits phyllotaxis order. This means that 35 during thegrowth ofthe phyllotaxis object, a naturalmodification (increase)in symmetry happens and this modification of symmetry obeys the law: 2 3 5 8 13 21...1 2 3 5 8 13 . (71) The law (71) is called dynamic symmetry [45].RecentlyUkrainianresearcherOlegBodnardevelopedavery interestinggeometrictheoryofphyllotaxis[45].Heprovedthatphyllotaxis geometry is a special kind of non-Euclidean geometry based upon the golden hyperbolicfunctionssimilartothehyperbolicFibonacciandLucasfunctions (38)and(39).Suchapproachallowsonetoexplaingeometricallyhowthe Fibonacci spirals appear on the surface of phyllotaxis objects (for example, pine cones,ananas,andcacti)intheprocessoftheirgrowthandthusdynamic symmetry (71) appears. Bodnars geometry is of essential importance because it concernsfundamentalsofthetheoreticalnaturalsciences,inparticular,this discoverygivesastrictgeometricalexplanationofthephyllotaxislawand dynamic symmetry based upon Fibonacci numbers.8.5.Petoukhovsgoldengenomatrices.Theideaofthegeneticcodeis amazinglysimple.Therecordofthegeneticinformationinribonucleicacids (RNA) of any living organism, uses the "alphabet" that consists of four "letters" or thenitrogenous bases:adenine(A),cytosine(C),guanine(G),uracil(U) (in DNAinsteadoftheuracilitusestherelatedthymine(T)).Petoukhovsarticle [65]isdevotedtothedescriptionofanimportantscientificdiscoverythe goldengenomatrices,whichaffirmthedeepmathematicalconnectionbetween the golden mean and genetic code.8.6.Fibonacci-Lorenztransformationsandgoldeninterpretationofthe Universeevolution.Asisknown,Lorentzstransformationsusedinspecial relativitytheory(SRT)arethetransformationsofthecoordinatesoftheevents (x, y, z, t) at the transition from one inertial coordinate system (ICS) K to another ICSK' , which is moving relatively to ICS K with a constant velocity V. The transformations were named in honor of Dutchphysicist Hendrik AntoonLorentz(1853-1928),whointroducedtheminordertoeliminatethe contradictionsbetweenMaxwellselectrodynamicsandNewton'smechanics. Lorentzstransformationswerefirstpublishedin1904,butatthattimetheir form was not perfect. The French mathematician Jules Henri Poincar(1854-1912) brought them to modern form.In 1908, that is, three years after the promulgation of SRT, the German mathematicianHermannMinkowski(1864-1909)gavetheoriginal geometrical interpretation of Lorentzs transformations. In Minkowskis space, a geometricallinkbetweentwoICSKandK' areestablishedwiththehelpof hyperbolic rotation, a motion similar to a normal turn of the Cartesian system in 36 Euclideanspace.However,thecoordinatesofx' andt ' intheICSK' are connectedwiththecoordinatesofxandtoftheICSKbyusingclassical hyperbolic functions. Thus,LorentzstransformationsinMinkowskisgeometryarenothing astherelationsofhyperbolictrigonometryexpressedinphysicsterms.This meansthatMinkowskisgeometryishyperbolicinterpretationofSRTand thereforeitisarevolutionarybreakthroughingeometricrepresentationsof physics, a way out on a qualitatively new level of relations between physics and geometry. AlexeyStakhovandSamuilAransonputforwardin[39]the following hypotheses concerning the SRT :1.The first hypothesis concerns the light velocity in vacuum.As is well known, themaindisputeconcerningtheSRT,basically,isabouttheprincipleofthe constancy of the light velocity in vacuum. In recent years a lot of scientists in the field of cosmology put forward a hypothesis,which puts doubt the permanence of the light velocity in vacuum- a fundamental physical constant, on which the basic laws of modern physics are based [66]. Thus, the first hypothesis is that thelightvelocityinvacuumwaschangedinprocessoftheUniverse evolution.2.AnotherfundamentalideainvolveswiththefactoroftheUniverseself-organization in the process of its evolution [67, 68]. According to modern view [68],afewstagesofself-organizationanddegradationcanbeidentifiedin processoftheUniversedevelopment:initialvacuum,theemergenceof superstrings,thebirthofparticles,theseparationofmatterandradiation,the birth of theSun, stars, and galaxies,theemergence of civilization, thedeath of Sun, thedeath of theUniverse.Themain idea ofthearticle[39] is to unitethe fact of the light velocity change during the Universe evolution with the factor of itsself-organization,thatis,tointroduceadependenceofthelightvelocityin vacuumfromsomeself-organizationparameter ,whichdoesnothave dimensionandischangingwithin:( ) < < + .Thelightvelocityin vacuum c is depending on the self-organization parameter( ) < < +and this dependence has the following form: 0( ) ( ) c c c c = = . (72) Asfollowsfrom(72)thelightvelocityinvacuumisaproductofthetwo parameters:c0 and ( ) c . The parameter c0 = const, having dimension [m.sec-1], is called normalizing factor. It is assumed in [39] that constant parameter c0 is equaltoEinsteins light velocityin vacuum(2.998- 108 m.sec-1) divided by the goldenmean( )1 5 / 2 1,61803 t = + ~ ).Thedimensionlessparameter( ) c is called non-singular normalized Fibonacci velocity of light in vacuum. 37 3.ThegoldenFibonaccigoniometryisusedfortheintroductionofthe Fibonacci-Lorentztransformations,whichareageneralizationoftheclassical Lorentz transformations. We are talking aboutthe matrix( )( )( 1) ( 2)( ) ( 1)cFs sFssFs cFs O = ,(73) whoseelementsaresymmetrichyperbolicfunctionssFs,cFs,introducedby Alexey Stakhov and Boris Rozin in [20].The matrix( ) O of the kind (73) is callednon-singulartwo-dimensionalFibonacci-Lorentzmatrixandthe transformations( 1) ( 2)( ) ( 1)11cFs sFssFs cFsxx' = '| || | | | |\ . \ .\ . are called non-singular two-dimensional Fibonacci-Lorentz transformations.The above approach to the SRT led to the new (golden) cosmological interpretationoftheUniverseevolutionandtothechangeofthelightvelocity before, in the moment, and after the bifurcation, called Big Bang. 8.7. Hilberts Fourth Problem. In the lecture Mathematical Problems presented attheSecondInternationalCongressofMathematicians(Paris,1900),David Hilbert(1862-1943)hadformulatedhisfamous23mathematicalproblems. Theseproblemsdeterminedconsiderablythedevelopmentofmathematicsof 20thcentury.Thislectureisauniquephenomenoninthemathematicshistory andinmathematicalliterature.TheRussiantranslationofHilbertslectureand itscommentsaregivenintheworks[69-71].Inparticular,HilbertsFourth Problem is formulated in [69] as follows: Whetherispossiblefromtheotherfruitfulpointofviewtoconstruct geometries, which with the same right can be considered the nearest geometries to the traditional Euclidean geometry NotethatHilbertconsideredthatLobachevskisgeometryand Riemannian geometry are nearest to the Euclidean geometry. InmathematicalliteratureHilbertsFourthProblemissometimes considered as formulated very vague what makes difficult its final solution. As it is noted in Wikipedia [72], the original statement of Hilbert, however, has also been judged too vague to admit a definitive answer.In[70]AmericangeometerHerbertBusemannanalyzedthewhole range of issues related toHilberts Fourth Problemand also concluded that the questionrelatedtothisissue,unnecessarilybroad.Notealsothebook[71]by AlexeiPogorelov(1919-2002)isdevotedtoapartialsolutiontoHilberts Fourth Problem. The book identifies all, up to isomorphism, implementations of theaxiomsofclassicalgeometries(Euclid,Lobachevskiandelliptical),ifwe deletetheaxiomofcongruenceandrefillthesesystemswiththeaxiomof "triangle inequality." InspiteofcriticalattitudeofmathematicianstoHilbert'sFourth Problem,weshouldemphasizegreatimportanceofthisproblemfor 38 mathematics,particularlyforgeometry.Withoutdoubts,Hilbert'sintuitionled him to the conclusion that Lobachevski's geometry and Riemannian geometry do not exhaust all possiblevariants ofnon-Euclideangeometries.Hilberts Fourth Problemdirectsattentionofresearchersatfindingnewnon-Euclidean geometries,whicharethenearestgeometriestothetraditionalEuclidean geometry. Themostimportantmathematicalresultpresentedin[39]isanew approachtoHilbertsFourthProblembasedonthehyperbolicFibonacci -functions (32) and (33). The main mathematical result of this study is a creation of infinite set of theisometric -models of Lobachevskis plane that is directly relevant to Hilberts Fourth Problem.Asisknown[69],theclassicalmodelofLobachevskisplanein pseudo-sphericalcoordinates( ) , , 0 , < < + < < + u v u v withtheGaussian curvature1 = K(Beltramis interpretation of hyperbolic geometry on pseudo-sphere) has the following form:() ( ) ( )( )2 2 22= + ds du sh u dv ,(74) where ds is an element of length and sh(u) is the hyperbolic sine. The metric -forms of Lobachevskis plane are given by the following formula [39]: ( ) ( )( ) ( ) ( )22 4 2 2 22ln4ds du sF u dv+ = u + ( ,(75) where0 > is a given realnumber, 242+ +u=is themetallic meanand ( ) sF u is hyperbolic Fibonacci -sine (32).Note that the formula (75) gives an infinite number of different metric forms of Lobachevskis plane because every real number0 >generates its own metric form of Lobachevskis plane of the kind (75). Letusstudyparticularcasesofthemetric -formsofLobachevskis plane corresponding to the different values of : 1.ThegoldenmetricformofLobachevskisplane.Forthecase =1we have 1 51.6180312+u = ~ the golden mean, and hence the form (75) is reduced to the following: ( ) ( )( ) ( ) ( )2 5 2 2 22l n4ds du sFs u dv = u +1 ( (76) 39 where( )1 52 2l n l n 0.23156512+u = ~| | |\ .and( )1 15u usFs uu u= issymmetric hyperbolic Fibonacci sine (38). 2. The silver metric form of Lobachevskis plane. For the case = 2we have 1 2 2.14212u= + ~- the silver mean, and hence theform (75) is reduced to the following: ( ) ( )( ) ( ) ( )2 2 2 22ln 22ds du sF u dv = u +2 ( ,(77) where( )2ln 0.7768192u ~ and( )2 222 2u usF uu u= isthehyperbolicFibonacci 2-sine (42).3.Thebronzemetricform ofLobachevskisplane.Forthecase = 3 we have 3 133.3027832+u= ~ -thebronzemean,andhencetheform(75)is reduced to the following: ( ) ( )( ) ( ) ( )2 13 2 2 22l n34ds du sF u dv = u +3 ( (78) where ( )2ln 1.427463u ~and ( )3 3313u usF uu u=is the hyperbolic Fibonacci 3-sine of the kind (32).4.ThecoopermetricformofLobachevskisplane.Forthecase = 4 we have2 5 4.236074u= + ~ - the cooper mean, and hence the form (75) is reduced to the following: ( ) ( )( ) ( ) ( )2 2 2 22ln 54ds du sF u dv = u +4 ( ,(79) where( )2ln 2.084084u ~and ( )4 442 5u usF uu u=is the hyperbolic Fibonacci 4-sine of the kind (32). 5.TheclassicalmetricformofLobachevskisplane.Forthecase ( ) 2 1 2.350402 she = = ~ we have2.7182 eeu = ~ - Napier number, and hence theform(75)isreducedtotheclassicalmetricformsofLobachevskisplane given by (74).Thus,theformula(75)setsaninfinitenumberofmetricformsof Lobachevskisplane.Theformula(74)giventheclassicalmetricformof Lobachevskisplaneisaparticularcaseoftheformula(75).Thismeansthat there are infinite number of Lobachevskis golden geometries, which can be 40 consideredthenearestgeometriestothetraditionalEuclideangeometry (DavidHilbert).Thus,theformula(75)canbeconsideredasasolutionto Hilberts Fourth Problem.9.Conclusion The following conclusions follow from this study: 9.1. The first conclusion touches on a question of the origin of mathematics and itsdevelopment.Thisconclusioncanbemuchunexpectedformany mathematicians.WeaffirmthatsincetheGreekperiod,thetwo mathematicaldoctrinestheClassicalMathematicsandtheHarmony Mathematicsbeguntodevelopinparallelandindependentoneanother. TheybothoriginatedfromoneandthesamesourceEuclidsElements,the greatestmathematicalworkoftheGreekmathematics.Geometricaxioms,the beginningsofalgebra,theoryofnumbers,theoryofirrationalsandother achievements of the Greek mathematics were borrowed from Euclids Elements bytheClassicalMathematics.Ontheotherhand,aproblemofdivisionin extremeandmeanratio(TheoremII.11)calledlaterthegoldensectionanda geometrictheoryofregularpolyhedrons(BookXIII),expressedtheHarmony of the Cosmos in Platos Cosmology, were borrowed from Euclids Elements by the Mathematics of Harmony. We affirm that Euclids Elements were the first attempttoreflectinmathematicsthemajorscientificideaoftheGreek science,theideaofHarmony.AccordingtoProclus,thecreationof geometrictheoryofPlatonic Solids(BookXIIIofEuclidsElements)was the main purpose ofEuclids Elements. 9.2. The second conclusion touches on the development of number theory. We affirmthatnewconstructivedefinitionsofrealnumbersbasedon Bergmanssystem(13)andthecodesofthegoldenp-proportion(49) overturnourideasaboutrationalandirrationalnumbers[17].Aspecial classofirrationalnumbersthegoldenmeanandgoldenp-proportions- becomesabaseofnewnumbertheorybecauseallrestrealnumberscanbe reduced to them by using the definitions (13) and (49). New properties of natural numbers(Z-property(51),F-code(53)andL-code(54)),followingfromthis approach, confirm a fruitfulness of this approach to number theory.9.3. The third conclusion touches on the development ofhyperbolic geometry.WeaffirmthatanewclassofhyperbolicfunctionsthehyperbolicFibonacci and Lucas -functions (32)-(35) [33] can become inexhaustible source for the developmentofhyperbolicgeometry.Weaffirmthattheformulas(32)-(35) giveaninfinitenumberofhyperbolicfunctionssimilartotheclassical hyperbolicfunctions,whichunderlieLobachevskisgeometry.This affirmationcanbereferredtooneofthemainmathematicalresultsofthe Mathematics of Harmony. A solution to Hilberts Fourth Problem [39] confirms a fruitfulness of this approach to hyperbolic geometry.9.4.ThefourthconclusiontouchesontheapplicationsoftheMathematicsof Harmonyintheoreticalnaturalsciences.WeaffirmthattheMathematicsof 41 Harmonyisinexhaustiblesourceofthedevelopmentoftheoreticalnatural sciences. This confirms by the newest scientific discoveries based on the golden meanandPlatonicSolids(quasi-crystals,fullerenes,goldengenomatrices,E-infinitytheoryandsoon).TheMathematicsofHarmonysuggestsfor theoreticalnaturalsciencesatremendousamountofnewrecurrence relations,newmathematicalconstants,andnewhyperbolicfunctions, whichcanbeusedintheoreticalnaturalsciencesforthecreationofnew mathematical models of natural phenomena and processes.A new approach totherelativitytheoryandUniverseevolutionbasedonthehyperbolic Fibonacci and Lucas functions [39] confirms a fruitfulness of this study.9.5.ThefifthconclusiontouchesontheapplicationsoftheMathematicsof Harmony in computer science. We affirm that the Mathematics of Harmony is a source for the development of new information technology the Golden I nformationTechnologybasedontheFibonaccicodes(68),Bergmans system (13),codes of the golden p-proportions (49),golden ternarymirror-symmetricalrepresentation[16]andfollowingfromthemnewcomputer arithmetics:Fibonacciarithmetic,goldenarithmetic,and ternary mirror-symmetricalarithmetic;theyallcanbecomeasourceofnewcomputer projects.Alsothisconclusionisconfirmedbythenewtheoryoferror-correctingcodesbasedonFibonaccimatrices[27]andthegolden cryptography [28].9.6.ThesixthconclusiontouchesontheapplicationsoftheMathematicsof Harmony in modern mathematical education. We affirm that the Mathematics of Harmonyshould becomea baseforthe reform ofmodernmathematical educationonthebaseoftheancient idea of Harmonyandgolden section. Suchanapproachcanincreaseaninterestofpupilstostudyingmathematics because this approach brings together mathematics and natural sciences. A study of mathematics turns into fascinating search of new mathematical regularities of Nature.9.7. The seventh conclusiontouches on the general role of theMathematics of Harmonyintheprogressofcontemporarymathematics.Weaffirmthatthe MathematicsofHarmonycanovercomeacontemporarycrisisinthe developmentofthe20thcenturymathematicswhatresultedinthe severanceoftherelationshipbetweenmathematicsandtheoreticalnatural sciences [40].TheMathematics of Harmonyis atrueMathematics of Nature incarnatedinmanywonderfulstructuresofNature(pinecones,pineapples, cacti,headsofsunflowersandsoon)anditcangivebirthtonewscientific discoveries. Acknowledgements TheauthorwouldliketoexpressacknowledgementstoProfessorScottOlsen (U.S.) for the English edition of this article and valuable remarks. References:42 1.ArnoldV.Acomplexityofthefinitesequencesofzerosandunitsandgeometryofthe finite functional spaces. Lecture at the se