On the general theory of hyperbolic functions based on the hyperbolic Fibonacci and Lucas functions and on Hilbert s Fourth Problem

Transcription

1 O the geeral theory of hyperbolic fuctios based o the hyperbolic Fiboacci ad Lucas fuctios ad o Hilbert s Fourth Problem Alexey Stakhov Doctor of Scieces i Computer Sciece, Professor The Iteratioal Club of the Golde Sectio 6 McCreary Trail, Bolto, ON, L7E C8, Caada goldemuseum@rogers.com Abstract. The article is devoted to descriptio of the ew classes of hyperbolic fuctios based o the golde ratio ad metallic proportios, what leads to the geeral theory of hyperbolic fuctios. This theory resulted i the origial solutio of Hilbert s Fourth Problem ad puts i frot to theoretical atural scieces a challege to search ew hyperbolic worlds of Nature.. Itroductio A iterest i the hyperbolic fuctios, itroduced i 77 by the Italia mathematicia Vicezo Riccati (707-77) sigificatly icreased i the 9 th cetury, whe the Russia geometer Nikolai Lobachevsky (79-86) used them to describe mathematical relatioships for the o-euclidia geometry. Because of this, Lobachevsky s geometry is also called hyperbolic geometry. Recetly, ew classes of hyperbolic fuctios have bee itroduced ito moder mathematics: ) hyperbolic Fiboacci ad Lucas fuctios [ - 4], based o the golde ratio ad ) hyperbolic Fiboacci ad Lucas lambda-fuctios [4, ], based o the metallic [6] or silver proprtios[7]. The scietific results, obtaied i [4, ], ca be cosidered as a geeral theory of hyperbolic fuctios. This theory puts forth a challege to search ew hyperbolic worlds of Nature. For the first time, the hyperbolic Fiboacci ad Lucas fuctios have bee described i 988 i the preprit of the Ukraiia mathematicias Alexey Stakhov ad Iva Tkacheko. I 993, the article of these authors "Fiboacci hyperbolic trigoometry" [] was published i the academic joural "Proceedigs of the Ukraiia Academy of Scieces." I further, the theory of the hyperbolic Fiboacci ad Lucas fuctios has bee developed i [ - 4]. The iterest i the hyperbolic Fiboacci ad Lucas fuctios icreased essetially after their use for the simulatio of the botaic pheomeo of phyllotaxis (Bodar s geometry) [8]. After Bodar s researches, it became clear that the hyperbolic Fiboacci ad Lucas fuctios have deep iterdiscipliary ature ad represet a iterest for all theoretical atural scieces [3, 4]. Hyperbolic Fiboacci ad Lucas fuctios are very commo i the wild (pie coes, cacti, pieapples, palm trees, suflower heads ad cauliflower, baskets of flowers), ad this raises their importace for the study of "hyperbolic worlds of Nature, which are studied i theoretical atural scieces (physics, chemistry, botay, biology, geetics, ad so o).

2 I the late 0 th ad early th ceturies, several researchers from differet coutries the Argetiea mathematicia Vera W. de Spiadel [6], the Frech mathematicia Midhat Ghazal [9], the America mathematicia Jay Kappraff [0], the Russia egieer Alexader Tatareko [], the Armeia philosopher ad physicist Hrat Arakelya [], the Russia researcher Victor Sheyagi [3], the Ukraiia physicist Nikolai Kosiov [4], Ukraiia-Caadia mathematicia Alexey Stakhov [4, ], the Spaish mathematicias Falco Sergio ad Plaza Agel [] ad others idepedetly bega to study a ew class of recurret umerical sequeces called Fiboacci -umbers [4, 6], which are a geeralizatio of the classical Fiboacci umbers. This study led to the itroductio of ew mathematical costats metallic meas [6] or silver meas [7]. These mathematical costats led to the itroductio of ew class of hyperbolic fuctios [4, ] called i [4] hyperbolic Fiboacci ad Lucas -fuctios. They are a wide geeralizatio of the classical hyperbolic fuctios ad hyperbolic Fiboacci ad Lucas fuctios itroduced i [, 3]. The mai goal of this article is to state a geeral theory of hyperbolic fuctios, which follows from the hyperbolic Fiboacci ad Lucas lambda-fuctios.. Biet formulas Biet formulas have expressed explicitly the Fiboacci ad Lucas umbers + through the golde ratio, as a fuctio of a discrete variable ± ± ±. I the book [7] Biet formulas are represeted as follows: ( 0,,, 3,... ) L + for k; for k + () + for k + ; F for k 3. Symmetric hyperbolic Fiboacci ad Lucas fuctios () 3.. Defiitio By usig () ad (), the followig hyperbolic fuctios have bee itroduced i [ - 4]: Symmetric hyperbolic Fiboacci sie

3 sfs x x (3) Symmetric hyperbolic Fiboacci cosie cfs x x + (4) Symmetric hyperbolic Lucas sie x x sls () Symmetric hyperbolic Lucas cosie x x cls + (6) where x is a cotiuous variable with values i the rage of {- + } Comparig Biet formulas (), () with (3) - (6), it is easy to see that i the discrete poits of the variable x (x0,±,±,±3, ) the fuctios (3) - (6) coicide with Biet formulas (), (), that is, sfs ( ) for k F cfs ( ) for k + L cls ( ) for k sls ( ) for k + where k takes the values from the set k0,±,±,±3, 3.. Graphs of the hyperbolic Fiboacci ad Lucas fuctios It follows from (7) that for all the eve values k the Fiboacci hyperbolic sie sfs()sfs(k) coicides with the Fiboacci umbers with the eve idexes F F k ad for the odd values k+ the hyperbolic Fiboacci cosie cfs()cfs(k+) coicides with the Fiboacci umbers with the odd idexes F F k+. At the same time, it follows from (8) that for all the eve values k the hyperbolic Lucas cosie cls() cls(k) coicides with the Lucas umbers with the odd idexes L L k. But for all the odd values k+ the hyperbolic Lucas sie sls()sls(k+) coicides with the Lucas umbers with the odd idexes L L k+. That is, the Fiboacci ad Lucas umbers iscribe ito the hyperbolic Fiboacci ad Lucas fuctios i the "discrete" poits of a cotiuous variable x0,±,±,±3,. This is clearly demostrated by the graphs of the hyperbolic Fiboacci ad Lucas fuctios preseted i Figures ad. (7) (8)

4 ycfs(x) Y O ysfs(x) Figure. The symmetric hyperbolic Fiboacci fuctios ycls(x) Y 0 ysls(x) Figure. The symmetric hyperbolic Lucas fuctios A detailed aalysis of the mathematical properties of a ew class of hyperbolic fuctios. is give i [ - 4]. It is show that the hyperbolic Fiboacci ad Lucas fuctios, o the oe had, have recursive properties, similar to Fiboacci ad Lucas umbers, ad, o the other had, hyperbolic properties, similar to the classical hyperbolic fuctios. 4. Recursive properties of the hyperbolic Fiboacci ad Lucas fuctios The simplest recursive properties of the hyperbolic Fiboacci fuctios, which are the cotiuous aalog of the discrete recurret relatio F + F + +F, are the followig: ( ) ( ) ( ) ( ) ( ) ( ) sfs x+ cfs x+ + sfs x ; cfs x+ sfs x+ + cfs x. (9) It is kow the followig idetity, which coects the three adjacet Fiboacci umbers: + ( ) F F F. (0) + This formula is called Cassii formula i hoour of the famous Frech astroomer Giovai Domeico Cassii (6-7)

5 It is proved [] that this discrete formula correspods to the two cotiuous idetities for the symmetric hyperbolic Fiboacci fuctios: ( ) ( ) ( ) sfs x cfs x+ cf x ; ( ) ( ) ( ) cfs x sfs x+ sf x. Below i Table, for compariso, there are give the "discrete" idetities for the Fiboacci ad Lucas umbers ad correspodig to them "cotiuous" idetities for the hyperbolic Fiboacci ad Lucas fuctios. Table. The recursive properties of the hyperbolic Fiboacci ad Lucas fuctios The idetities for the Fiboacci ad Lucas umbers The idetities for the hyperbolic Fiboacci ad Lucas fuctios F + F + + F sfs( x+ ) cfs( x+ ) + sfs ; cfs( x+ ) sfs( x+ ) + cfs L + L + + L sls( x+ ) cls( x+ ) + sls ; cls( x+ ) sls( x+ ) + cls + F ( ) F sfs sfs( x) ; cfs cfs( x) L ( ) L sls sls( x) ; cls cls( x) F+ 3+ F F+ sfs( x+ 3) + cfs cfs( x+ ); cfs( x+ 3) + sfs sfs( x+ ) F+ 3 F F + sfs( x+ 3) cfs sfs( x+ ); cfs( x+ 3) sfs cfs( x+ ) F F 4F sfs( x+ 6) cfs 4cFs( x+ 3 ); cfs( x+ 6) sfs 4cFs( x+ 3) F F + F ( ) sfs cfs( x ) cfs( x ) ; cfs + sfs( x+ ) sfs( x ) F + F+ + F cfs( x+ ) cfs( x ) sfs ; sfs( x ) sfs( x ) cfs L L s ( ) Ls + cls( x) ; cls cls + ( ) + ( + 3) ( + ); ( ) + ( + 3) ( + ) L L + 3 L + sls x cls x sls x cls x sls x cls x L L L sls x+ sls x cls x cls x+ cls x sls x + ( ) ( ) ( ) ( ) ; ( ) ( ) ( ) ( + 3) ( ) ( ); ( + 3) Fs cls ( ) ( ) ( ); ( ) ( ) ( ) + ( ) ( ) ( ); ( ) ( ) ( ) ( ) ( ) ( ); ( ) ( ) ( ) F F L sfs x cfs x sls x cfs x s + 3 L + L F sls x + cls x+ sfs x cls x + sls x+ cfs x + L + F L sls x + cfs x cls x+ cls x + sfs x sls x+ L + + L F + sls x+ + cls x cfs x+ cls x+ + sls x sfs x+ (). Hyperbolic properties of the hyperbolic Fiboacci ad Lucas fuctios I additio to the recursive properties, the hyperbolic Fiboacci ad Lucas fuctios preserve all the kow properties, iheret to the classical hyperbolic fuctios. The mai advatage of the symmetric hyperbolic Fiboacci ad Lucas fuctios () - (8), itroduced i [], is a preservatio of the parity property. It is proved i [] the followig: sfs( x) sfs; cfs( x) cfs () ( ) ( ); ( ) ( ) sls x sls x cls x cls x. (3)

6 But there are more profoud mathematical relatioships betwee the classical hyperbolic fuctios ad the hyperbolic Fiboacci ad Lucas fuctios. For example, there is the followig idetity for the classical hyperbolic fuctios: ( ) ( ) ch x sh x. (4) The idetity (4) takes the followig forms for the hyperbolic Fiboacci ad Lucas fuctios: 4 ( ) sfs cfs x () cls sls 4. (6) It is proved [, 3] that for each idetity for classical hyperbolic fuctios there is a aalog i the form of the correspodig idetity for the hyperbolic Fiboacci ad Lucas fuctios. I Table some formulas for the classical hyperbolic fuctios ad the correspodig formulas for the hyperbolic Fiboacci fuctios are represeted. Table. Hyperbolic properties for the hyperbolic Fiboacci fuctios Formulas for the classical hyperbolic fuctios ( + ) ( ) ( ) + ( ) ( ) ( y) sh ch ch sh ( + ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ) Formulas for the hyperbolic Fiboacci fuctios x x x x x x x x e e e + e + sh ; ch sfs ; cfs 4 ch sh cfs sfs sh x y sh x ch x ch x sh x sfs ( x + y) sfs cfs + cfs sfs sh x ch x y ch x ch x sh x sh x ( ) ( ) ( ) ( ) ( ) sfs x y sfs x cfs x cfs x sfs x ( + ) ( ) ( ) + ( ) ( ) cfs x y cfs x cfs x sfs x sfs x ch x y ch x ch x sh x sh x cfs x y cfs x cfs x sfs x sfs x ch sh ch cfs sfs cfs ( ) ( ) ( ) ( ) ( ) ch ± sh ch ± sh cfs ± sfs cfs ± sfs 6. Theory of Fiboacci umbers as a degeerate case of the theory of the hyperbolic Fiboacci ad Lucas fuctios It is show above, the two "cotiuous" idetities for the hyperbolic Fiboacci fuctios always correspod to every "discrete" idetity for the Fiboacci ad Lucas umbers. Coversely, oe may obtai a "discrete" idetity for the Fiboacci ad Lucas umbers by usig two correspodig cotiuous idetities for the hyperbolic Fiboacci ad Lucas fuctios. Sice the Fiboacci ad Lucas umbers, accordig to (7) ad (8), are "discrete" cases of the hyperbolic Fiboacci fuctios for the discrete values of the cotiuous variable, this meas that with the itroductio of the hyperbolic Fiboacci ad Lucas fuctios the classical theory of Fiboacci umbers" [8, 9] as if "degeerates," because this theory is special ("discrete") case of the more geeral ("cotiuous) theory of the hyperbolic Fiboacci ad Lucas fuctios. This coclusio is the first uexpected

7 result, which follows from the theory of the hyperbolic Fiboacci ad Lucas fuctios [, 3]. But oe more importat result, which cofirms a fudametal ature of the hyperbolic Fiboacci ad Lucas fuctios, is a ew geometric theory of phyllotaxis, described i [8]. 7. Commets 4.. Thus, the hyperbolic Fiboacci ad Lucas fuctios [ - 3], based o the socalled "Biet formulas," are a geeralizatio of Biet formulas for cotiuous domai. I cotrast to the classical hyperbolic fuctios, a ew class of hyperbolic fuctios has recursive properties, similar to the Fiboacci ad Lucas umbers. A theory of the hyperbolic Fiboacci ad Lucas fuctios is a extesio of the "Fiboacci umbers theory" [8, 9] for a cotiuous domai. With the itroductio of the hyperbolic Fiboacci ad Lucas fuctios, the classical "theory of Fiboacci umbers" [8, 9] as if "degeerates", because all mathematical idetities for the Fiboacci ad Lucas umbers ca be easily obtaied from the correspodig idetities for the hyperbolic Fiboacci ad Lucas fuctios by usig the elemetary formulas (7), (8) which lik these fuctios with Fiboacci ad Lucas umbers. 4.. There is show i [8], the hyperbolic Fiboacci ad Lucas fuctios are a basis of botaical phyllotaxis pheomeo kow sice Johaes Kepler s time. By usig the hyperbolic Fiboacci ad Lucas fuctios, the Ukraiia researcher Bodar revealed the "puzzle of phyllotaxis" ad gave aswer to the two importat questios: ) How the phyllotaxis objects (pie coe, pieapple, cactus, head of suflower, etc.) are growig? ) Why Fiboacci spirals appear o their surfaces? These facts are emphasizig fudametal character of the hyperbolic Fiboacci ad Lucas fuctios. 8. Fiboacci ad Lucas -umbers ad metallic meas I 999 the Argetiea mathematicia Vera W. de Spiadel has itroduced the so-called metallic meas, [6], which follows from the followig cosideratios. Let us cosider the followig recurret relatios: F ( ) F ( ) + F ( ); F ( 0) 0, F ( ) (7) where > 0 is a give positive real umber. The recurret relatio (7) gives a ew class of the recurret umerical sequeces called Fiboacci -umbers [4]. The followig characteristic algebraic equatio follows from (7): x x 0. (8) A positive root of Eq. (8) geerates a ifiite umber of ew mathematical costats metallic mas [6], which are expressed with the followig geeral formula: (9) Note that for the case the formula (9) gives the classical golde mea +. The metallic meas (9) possess the followig uique mathematical properties:

8 ; +, (0) which are geeralizatios of similar properties for the classical golde mea ( ): ; +. () Gazale formulas By studyig the recurret relatio (7), the Egyptia mathematicia Midchat Gazale [9] deduced the followig aalytical formula for the Fiboacci -umbers: ( ) F ( ), () 4 + where > 0 is a give positive real umber, is the metallic mea give by (9), 0, ±, ±, ±3,... By developig formula (), Alexey Stakhov deduced i [4, ] the Gazale formula for the Lucas -umbers: ( ) ( ) L +. (3) Gazale formulas () ad (3) are a wide geeralizatio of Biet formulas () ad () for the classical Fiboacci ad Lucas umbers ( ). 0. Hyperbolic Fiboacci ad Lucas -fuctios The most importat result is that the Gazale formulas () ad (3), resulted i a geeral theory of hyperbolic fuctios [4, ], which is give with the followig formulas: Hyperbolic Fiboacci -sie x x x x sf (4) Hyperbolic Fiboacci -cosie x x x x cf + () Hyperbolic Lucas -sie x x x sl Hyperbolic Lucas -cosie x (6)

9 x x x cl + +. (7) Note that the hyperbolic Fiboacci ad Lucas -fuctios coicide with the Fiboacci ad Lucas -umbers for the discrete values of the variable x0,±,±,±3,, that is, ( ) ( ) ( ) ( ) sf, k cl, k F( ) ; L( ). (8) cf, k + sl, k+. The mai properties ad idetities of the hyperbolic Fiboacci ad Lucas - fuctios The formulas (4) - (7) provide a ifiite umber of hyperbolic models of Nature because every real umber >0 origiates its ow class of hyperbolic fuctios of the kid (4) - (7). As is proved i [4, ], these fuctios have, o the oe had, the hyperbolic properties similar to the properties of classical hyperbolic fuctios, ad o the other had, recursive properties similar to the properties of the Fiboacci ad Lucas -umbers () ad (3). I particular, the classical hyperbolic fuctios are a partial case of the hyperbolic Lucas -fuctios (6) ad (7). For the case e e , the classical hyperbolic fuctios are coected with e hyperbolic Lucas -fuctios by the followig simple relatios: sl cl sh ad ch. (9) Note that for the case, the hyperbolic Fiboacci ad Lucas -fuctios (4) - (7) coicide with the hyperbolic Fiboacci ad Lucas fuctios (3) (6). It is appropriate to give the followig comparative Table 3, which gives a relatioship betwee the golde mea ad metallic meas as ew mathematical costats of Nature. Table 3. Golde Mea ad Metallic Meas x

10 The Golde Mea ( ) The Metallic Meas ( > 0) ( ) ( ) F F ( ) 4 + L + ( ) L ( ) + ( ) x x x x + + sfs ; cfs sf ; cf sls x ; cls x + sl ; cl + ( ) ( ) x x x x x x x x x x x x A beauty of these formulas is charmig. This gives a right to suppose that Dirac s Priciple of Mathematical Beauty is applicable fully to the metallic meas ad hyperbolic Fiboacci ad Lucas -fuctios. Ad this, i its tur, gives hope that these mathematical results ca become a base of theoretical atural scieces. sf x Table 4 gives the basic formulas for the hyperbolic Fiboacci -fuctios ( ) ad cf fuctios sh( x ) ad ch( x ). i compariso with correspodig formulas for the classical hyperbolic Table 4. Compariso of the classical hyperbolic fuctios to the hyperbolic Fiboacci -fuctios

11 Formulas for the classical hyperbolic fuctios ( ) ( ) ( ) ( ) ( + ) ( ) ( + ) + ( ) x x x x + x x x x e e e + e sh ; ch sf ; cf sh x + sh ch x + + sh x sf x + cf x + + sf x ch x sh sh x ch x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Formulas for the hyperbolic Fiboacci - fuctios ( ) ( ) ( ) ( + ) ( + ) + ( ) cf x sf x cf x ( ) ( ) ( ) sh x ch x + ch x ch sf x cf x + cf x ( ) ( ) ( ) ch x sh x + sh x ch cf x sf x + sf x 4 ch sh cf sf 4 + sf ( x + y) sf cf + cf sf sh( x + y) sh ch + ch sh 4 + sh( x y) sh ch ch sh sf ( ) ( ) ( ) ( ) ( ) x y sf x cf x cf x sf x 4 + cf ( x + y) cf cf + sf sf ch( x + y) ch ch + sh sh 4 + ch x y ch x ch x sh x sh x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) cf x y cf x cf x sf x sf x 4 + ch sh ch cf sf cf 4 + ch ± sh ch ± sh cf sf cf sf ± ± 4 + Remark. For the hyperbolic Lucas -fuctios sl ad cl the correspodig formulas ca be got by multiplicatio of the hyperbolic Fiboacci - fuctios sf ad cf by costat factor 4 +. Table 4 for the hyperbolic Fiboacci -fuctios sf ad cf to the above remark for the hyperbolic Lucas -fuctios sl ad cl, with regard, makes up a base of the geeral theory of hyperbolic fuctios [4]. This table is very covicig cofirmatio of the fact that we are talkig about a ew class of hyperbolic fuctios, sh x ad which keep all well-kow properties of the classical hyperbolic fuctios ( ) ch( x ), but, i additio, they posses additioal ( recursive ) properties, which uite them with two remarkable umerical sequeces Fiboacci ad Lucas -umbers F ( ) ad L ( ).. Hilbert s Fourth Problem I the lecture Mathematical Problems preseted at the Secod Iteratioal Cogress of Mathematicias (Paris, 900), David Hilbert (86-943) had formulated his famous 3 mathematical problems. These problems determied cosiderably the developmet of mathematics of 0th cetury. I particular, Hilbert s Fourth Problem is formulated i as follows:

12 Whether is possible from the other fruitful poit of view to costruct geometries, which with the same right ca be cosidered the earest geometries to the traditioal Euclidea geometry Note that Hilbert cosidered that Lobachevski s geometry ad Riemaia geometry are earest to the Euclidea geometry. I mathematical literature Hilbert s Fourth Problem is sometimes cosidered as formulated very vague what makes difficult its fial solutio. As it is oted i Wikipedia [0], the origial statemet of Hilbert, however, has also bee judged too vague to admit a defiitive aswer. I [] the America geometer Herbert Busema aalyzed the whole rage of issues related to Hilbert s Fourth Problem ad also cocluded that the questio related to this issue, uecessarily broad. Note also the book [] by Alexei Pogorelov (99-00) is devoted to a partial solutio to Hilbert s Fourth Problem. The book idetifies all, up to isomorphism, implemetatios of the axioms of classical geometries (Euclid, Lobachevski ad elliptical), if we delete the axiom of cogruece ad refill these systems with the axiom of "triagle iequality." I spite of critical attitude of mathematicias to Hilbert's Fourth Problem, we should emphasize great importace of this problem for mathematics, particularly for geometry. Without doubts, Hilbert's ituitio led him to the coclusio that Lobachevski's geometry ad Riemaia geometry do ot exhaust all possible variats of o-euclidea geometries. Hilbert s Fourth Problem pays attetio of researchers at fidig ew o-euclidea geometries, which are the earest geometries to the traditioal Euclidea geometry. I the articles [3 - ] there is give a ew mathematical result, which touches a ew approach to Hilbert s Fourth Problem based o the hyperbolic Fiboacci - fuctios (4) ad (). The mai mathematical result of this study is a creatio of ifiite set of the isometric -models of Lobachevski s plae that is directly relevat to Hilbert s Fourth Problem. As is kow [6], the classical model of Lobachevski s plae i pseudo-spherical u, v, 0 < u < +, < v < + with the Gaussia curvature K coordiates ( ) (Beltrami s iterpretatio of hyperbolic geometry o pseudo-sphere) has the followig form: ( ) ( ) ds du + sh ( u)( dv), (30) where ds is a elemet of legth ad sh(u) is the hyperbolic sie. Based o the hyperbolic Fiboacci -fuctios (4) ad (), Alexey Stakhov ad Samuil Araso deduced i [6] the metric -forms of Lobachevski s plae give by the followig formula: where ( ds) l ( )( du) + sf ( u) ( dv) is the metallic mea ad sf ( u) 4, (3) is hyperbolic Fiboacci -sie (4). Let us study partial cases of the metric -forms of Lobachevski s plae correspodig to the differet values of :

13 . The golde metric form of Lobachevski s plae. For the case we have the golde mea, ad hece the form (3) is reduced to the followig: ( ds) l ( )( du) + sfs( u) ( dv) 4 (3) u u + where l ( ) l 0.36 ad sfs( u) is symmetric hyperbolic Fiboacci sie (3).. The silver metric form of Lobachevski s plae. For the case we have the silver mea, ad hece the form (3) is reduced to the followig: ( ds) l ( )( du) + sf ( u) ( dv) u, (33) where l ( ) ad sf ( u). 3. The broze metric form of Lobachevski s plae. For the case 3 we have the broze mea, ad hece the form (3) is reduced to the followig: 3 ( ds) l ( 3 )( du) + sf3 ( u) ( dv) 4 (34) u u 3 3 where l ( 3 ).4746 ad sf3 ( u) The cooper metric form of Lobachevski s plae. For the case 4 we have the cooper mea, ad hece the form (3) is reduced to the followig: 4 4 where l ( ) ad sf ( u) 4 u ( ds) l ( )( du) + sf ( u) ( dv) u, (3). The classical metric form of Lobachevski s plae. For the case sh.3040 we have e.78 - Napier umber, ad hece the form e ( ) e (3) is reduced to the classical metric forms of Lobachevski s plae give by (30). Thus, the formula (3) sets a ifiite umber of metric forms of Lobachevski s plae. The formula (30), give the classical metric form of Lobachevski s plae is a partial case of the formula (3). This meas that there are ifiite umber of Lobachevski s golde geometries, which ca be cosidered the earest geometries to the traditioal Euclidea geometry (David Hilbert). Thus, the formula (3) ca be cosidered as a origial solutio to Hilbert s Fourth Problem. u

14 . A ew challege for the theoretical atural scieces Thus, the mai result of the research, described i [3, 3 - ], is a proof of the existece of a ifiite umber of hyperbolic fuctios (4) - (7) based o the "metallic proportios" (9). I additio, each class of hyperbolic fuctios, correspodig to (4) - (7), "geerates" for the give >0 its ow "hyperbolic geometry," which leads to the appearace i the "physical world" of specific properties, which deped o the "metallic proportios" (9 ). A strikig example is a ew geometric theory of phyllotaxis, created by Oleg Bodar [8]. Bodar proved that "the world of phyllotaxis" is a specific "hyperbolic world," i which a "hyperbolicity" maifests itself i the "gold" ad "Fiboacci spirals" o the surface of "phyllotaxis objects." However, the hyperbolic Fiboacci ad Lucas fuctios uderlyig the "hyperbolic phyllotaxis world " are a special case of the hyperbolic Fiboacci ad Lucas -fuctios (4) - (7). I this regard, there is every reaso to suppose that other types of hyperbolic fuctios (4) - (7) ca be the basis for modelig of ew "hyperbolic worlds" that ca really exist i Nature. Moder sciece caot fid these special hyperbolic worlds, because hyperbolic fuctios (4) - (7) were ukow for moder sciece. Basig o the success of Bodar s geometry " [8], oe ca put forward i frot to theoretical physics, chemistry, crystallography, botay, biology, ad other braches of theoretical atural scieces the challege to fid ew "hyperbolic Nature s worlds," based o other classes of hyperbolic fuctios (4) - (7). I this case, perhaps, the first cadidate for the ew "hyperbolic world" i Nature may be, for example, "silver ratio" + ad based o it "silver" hyperbolic fuctios [7]. 3. Refereces. Stakhov AP Tkacheko IS. Fiboacci hyperbolic trigoometry. Proceedigs of the Ukraiia Academy of Scieces, Vol. 08, 7, 993., P.9-4 (Russia).. Stakhov A, Rozi B. O a ew class of hyperbolic fuctio. Chaos, Solitos & Fractals 004, 3(): Stakhov A., Rozi B. The Golde Sectio, Fiboacci series, ad ew hyperbolic models of Nature. Visual Mathematics, Volume 8, No.3, Stakhov A.P. The Mathematics of Harmoy. From Euclid to Cotemporary Mathematics ad Computer Sciece. New Jersey, Lodo, Sigapore, Beijig, Shaghai, Hog Kog, Taipei, Cheai: World Scietific, р. Stakhov AP. Gazale formulas, a ew class of hyperbolic Fiboacci ad Lucas fuctios ad the improved method of the "golde" cryptography. // Academy of Triitarism., Мoscow: , Electroic publicatio 4098, Vera W. de Spiadel. The family of Metallic Meas. Visual Mathematics, Vol. I, No. 3, 999, 7. Silver ratio. From Wikipedia, the free ecyclopedia

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