Applied Mathematics, 2011, 2, 808815 doi:10.4236/am.2011.27108 Published Online July 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Multi Parameters Golden Ratio and Some Applications Seyed Moghtada Hashemiparast1, Omid Hashemiparast2 1Department of Mathematics, Faculty of science, K. N. Toosi University of Technology, Tehran, Iran 2Department of Engineering, Tehran University, Kish Pardis, Iran Email: hashemiparast@kn tu.ac.ir Received January 9, 2011; revised April 30, 2011; accepted May 3, 2011 Abstract The present paper is devoted to the generalized multi parameters golden ratio. Variety of features like twodimensional continued fractions, and conjectures on geometrical properties concerning to this subject are also presented. Wider generalization of Binet, Pell and Gazale formulas and wider generalizations of symmetric hyperbolic Fibonacci and Lucas functions presented by Stakhov and Rozin are also achieved. Geometrical applications such as applications in angle trisection and easy drawing of every regular polygons are developed. As a special case, some famous identities like Cassini’s, Askey’s are derived and presented, and also a new class of multi parameters hyperbolic functions and their properties are introduced, finally a generalized Qmatrix called Gnmatrix of order n being a generating matrix for the generalized Fibonacci numbers of order n and its inverse are created. The corresponding code matrix will prevent the attack to the data based on previous matrix. Keywords: Generalized Golden Ratio, Trisection, Qmatrix, Fibonacci, Lucas, Gazale, Casseni 1. Introduction During the last centuries a great deal of scientist’s at tempts has been made on generalizing the mathematical aspects and their applications. One of the famous aspects in mathematics is called golden ratio or golden propor tion has been studied by many authors for generalization from different points of view. In recent years there were a huge interest of modern science in the application of the Golden section and Fibonacci numbers in different area of engineering and science. An extensive bibliogra phy of activities can be found in a paper by Stakhov [1,2] who has investigated the generalized principle of Golden section and its vast area of applications. The main goal of his attention is to state the fundamental elements of this subject i.e. description of its basic concepts and theories and discussion on its applications in modern science. In this connection he explains the idea of the creation of a new mathematical direction called mathematics of Har mony that moved to intend for the mathematical simula tion of those phenomena and processes of objective world for which Fibonacci numbers and Golden Section are their objective essence which can influence on the other areas of human culture. Stakhov considers the harmony mathematics from sacral geometry point of view and its applications in this field [3]. Some authors consider the extension of the Fibonacci numbers to cre ate a generalized matrix with various properties suitable for the coding theory [4,5]. Even they established gener alized relations among the code matrix elements for all values of Fibonacci pnumbers with a high ability. Or derm generalized of Fibonacci pnumbers for a matrix representation gives some identities [6] useful for further application in theoretical physics and Metaphysics [710]. Some authors determine certain matrices whose parame ters generate the Lucas pnumbers and their sums creates the continues functions for Fibonacci and Lucas pnumbers which is considered in [1114] and their gen eralized polynomials and properties are also introduced in [15], based on Golden section the attempt of build up the fundamental of a new kind of mathematical direction is addressed in [3] which is the requirements of modern natural science and art and engineering, these mathe matical theories are the source of many new ideas in mathematics, philosophy, botanic and biology, electrical and computer science and engineering, communication system, mathematical education, theoretical physics and particle physics [16]. Other area of applications returns to the connection between the Fibonacci sequence and Kalman filter, by exploiting the duality principle control [17], finally nu merical results of generalized random Fibonacci se
S. M. HASHEMIPARAST ET AL. 809 quences which are stochastic versions of classical Fibo nacci sequences are also obtained [18]. Recently Edu ardo Soroko developed very original approach to struc tural harmony of system [19], using generalized Golden sections. He claims the Generalized Golden sections are invariant which allow natural systems in process of their selforganization to find the harmonic structure, station ary regime of their existence, structural and functional stability. Following the introduction, in Section 2 we establish multi parameters Golden Ratio and geometric applica tions. In Section 3 we generalized the Gazale formula. Sections 4 and 5 investigate the generalized hyperbolic functions and double parameters matrices and applica tions. 2. Preliminaries In this paper we start a geometrical discussion which leads to a generalized form of golden ratio with multi parameters that for the case of single parameter it changes to the ordinary generalized form which has been consid ered by authors in the recent years. Suppose a line AB is to divided in two parts AC and CB (Figure 1) such that n BAC aAC CB (1) where and a are real positive numbers. Let AC CB , then 1 ABAC CBCB aaa aa CACAC x Thus we obtain n a ax x or 1n ax b (2) where ab . In a special case for we intend to divide 1n B such that BA aC CCB then we have 20xaxb (3) or 2 1 4 2 aa b x (4) and C B 2 a 2 4 ab 2 a1 b Figure 1. Dividing the line AB in generalized Golden sec tion. 2 2 4 2 aa b x (5) We call the positive root of this equation the general ized two parameters golden ratio ,ab 2 4 1 1 ,22 b a ab a for the case b = 1 it gives a uniparameter a 24 ,1 22 a aab a which is called generalized golden ratio and has been considered in recent years by some authors, for a = 1 gives the famous historical golden ratio 1 15 1,1 2 where the ratio of the adjacent numbers in Fibonacci series and Lucas series tends towards this irrational number. Let us consider the properties of this golden ratio, for a = b we have 4 1 1 22 a a which is one solution of equation . Now other properties of the ratio can be considered. We have 20xaxa b a the generalized golden ratio when a = 1 reduces to the wellknown historical golden ratio 15 2 which has many properties and application in art, engineering, physics and mathematics. Similar properties can be es Copyright © 2011 SciRes. AM
S. M. HASHEMIPARAST ET AL. 810 tablished in the generalized case. For instance the gener alized can be expressed in terms of itself, like 1 a that can be expanded into this fraction or nested roots equations that goes on for ever and called continued fraction or nested roots 111 111 , aaaa ab bababa Using the relation in term of itself we get the fol lowing continued fractions for , a , respec tively ,ab 1 1 11 11 1 1 1 , aa a aa b ab ab ab aa Stakhov and Rozin [20,21] give some interesting re sults for and , similar results are veri fied for different values of a and b, two dimension peri odic continued fractions are considered by some authors [22,23] using above formulas may generalize the appli cation in art and architecture. The above line AB can be divided in n sections and by the successive ratios a sys tem of equations will be created such that for a given value n, (1) be extended to multi parameters ratios to create a multi parameters Golden ratio. In this paper we concentrate on a generalized double parameters and ap plications of this ratio. 2n1ab If in (3), thenab 4 1 1 ,22 a aa a , the generalized single parameter ratio different from ,1a studied by Stakhov, actually (2) can be considered as a characteristic function of a difference equation of order n with parameters and b a 21 ,, nnn PabaPabbPab which can be generalized to a multi parameters cases. , 12 12 1 , ,,, ,...,, 2,3, p nppi npip i PaaaaPaa a p Theorem 2.1 Let DB be an isosceles triangle with the vertex angle , then the necessary and sufficient conditions for the angle equals to BAC (see Figure 2) 0π2 , is dividing the by in general ized Golden ratio, such that and BD DC C a1BC and 1 BC CDa . Proof Without losing the generality put 1b , train gle DB is isosceles and ˆˆπ2 AB , the condi tion is sufficient, because if and DC a1CB then DB CBDC or 1DB aa . For the trian gles DC and DB we have respectively 222 222 2. cos 2. cos ACDADCDA DC ABDADBDADB or, 2222 22222 2. cos 2.cos ACaaa a ABaaa a If BAC then the BC will be an isosceles tri angle and π 22 ABC ACB and also 1 cos 2 , so we will have ππ π 22 22 BAC . On the other hand the condition is necessary, because if BAC then π π 22 22 ACB π , hence CAB and triangle BC a will be isosceles and we conclude easily DC , 1 CB . D C B A ba α α π 22 a Figure 2. Dividing BD by point C in the generalized golden ratio. Copyright © 2011 SciRes. AM
S. M. HASHEMIPARAST ET AL. 811 The impossibility proofs are so advanced, that many people flatly refuse to accept the problem are impossible [24]. The proofs of impossibilities of certain geometric constructions like doubling the cube and squaring the circle, trisecting the angle is impossible and details can be found in some references [25], trisecting the angle is impossible that is there exist an angle that cannot be tri sected with a straightedge and compass. On the other hand there are some angles which may be trisected, some author give valuable discussion for ideal and classical trisection of an arbitrary angle [26]. Using generalized golden ratio one may establish the following criteria for trisection of a given angle. Now, we construct an isosceles triangle DB whose vertex angle is ADB (Figure 3), hence the base angles are π 22 BD . By taking the line and mark ing on the line in the generalized golden ratio, the angle C BAC , now we draw a circle with the center and radius D 1DA DB aa , then the line C will contacts with this circle at a point say , the arc 2BEEA . By a similar way we divide in generalized golden ratio and find such that DE DCCa and 1CE , now the line will contact with the circle in a point say, joining we have BC DEE DB BDE EDE The procedure can be carried on to pro duce a desired polygon by a proper selection of the angle . Thus we will have the following proposition for tri section of an arbitrary angle. Proposition For trisection of a given angle first con struct an isosceles triangle DB with vertex angle 3 and side DBDA a (Figure 4), then divide DB D E B A α α α α α α E C 1 1 C Figure 3. Trisecting the angle by using generalized golden section. DB C a C 1 1 M N A Figure 4. Algorithm for trisecting the angle by using gener alized golden section. into two parts DC a and 1CB , and find the middle point of , draw a line per pendicula to from CB CB , which contact the circle at point say , connect and by segment , then divide into two parts N DN DC DN DN and on golden ratio such that CN DC a and 1CN again find the point the middle point of CN draw a line perpendicular to CN at point , extend to contact the circle at point CN N , then draw , angle 3 DN will be tri sected by and DN DN . When is divided into two segment and DB a1 , then 1 cos 2 we can easily calculate a with re spect to cos , and we get 2 12cos1 2cos 1 b a having , a can easily be calculated, for given when the segments are and ab respectively, and are given in the Table 1 for and some nominal values of 1b . 3. Generalized Gazale Formula Let and b be any real positive number, and define a Generalized Gazale sequence n a ,,Pab P0,1, 2,n if and only if 0,Pab0 and and for all 1,Pab1 0n 21 ,, nnn PabaPabbPab , (6) To attain the solution, using characteristics method gives the following quadratic equation 20ab where Copyright © 2011 SciRes. AM
S. M. HASHEMIPARAST ET AL. 812 Table 1. Values of Cos (α), a and for the nominal values of α. 1 cos 2 60 a 2 cos 2 45 2a 21 51 cos 4 36 1a 51 2 3 cos 2 30 33 2 a 31 2 cos20 0.940 20 0.256a 1.136 cos20 0.985 10 0.061a 1.031 cos0 1 0 0a 1 2 1 4 2 aa b r and 2 2 4 2 aa b r are the roots of characteristics function and the positive root can be written as 1 r 2 1 4 1 1, 22 b a ra ab where is the Generalized Golden Ratio and ,ab 2 2 4 1 1 22 , b b a ra ab Hence the general solution of (1) can be written as 12 ,, , n n n b Pab CabCab Looking at the two initial values 0 and 1, the co efficients and can be determined, thus P P 1 C2 C 2 1 ,,( , 4 nn n b Pabab ab ab ) (7) For , , 1b ,1 nn Pa Pa 2 14 ,1 22 a a a 2 1 4 n n na a Pa a which is the Gazale formula for the Generalized Fibo nacci sequence, and if n is the ordinary Fibonacci sequence, the general solution to 21nnn FF will be 1 1 11 15 511 5 22 5 n n nn nn FP for the case 1a , and for the case , 2a 2,1 n P gives the John Pell (16101685) formula, and this for mula generates an infinite number of generalized Fibo nacci number of different orders for a and b. The following Theorem is a generalized aspect of theorems that have been established for Fibonacci and Lucas numbers. Theorem 3.1 If and are real positive numbers, then 1n,ab 1 2 11 ,,, n nn n PabPabPab b (9) and is independent of . a Proof We have 1 1 12 1 1 12 2 1 ,, , 4 1 ,, , 4 1 ,, , 4 n n n n n n n n n b Pabab ab ab b Pabab ab ab b Pabab ab ab Then 2 2 112 1 12 2 1 ,, ,, 4 ,, n n nn n n b Pab Pababab ab b bab ab and 2 2 2 2 , 1,2 ,4 n n n n Pab b ab b abab 1 (8) Thus Copyright © 2011 SciRes. AM
S. M. HASHEMIPARAST ET AL. 813 2 11 12 2 22 2 1 2 1 1 2 2 ,,, ,2 4, ,,4 4 4 nn n n n n n PabPabPab bb ab b ab ab bb ab abab bab b ab Similar identities like the Askey’s and Catelan’s and d’Ocagne’s can be generalized. For instance in Osler and Hilburn [2] Askcy’s hyperbolic sine identity for ordinary Fibonacci sequence is given as 2sinh log 5 nn ni i (10) and generalized form in [27,28] for real positive is 0a 2 2sinh log 4 na n G ia ni (11) If we generalize (11) for , n Pab we obtain 1 2 2 2 2 ,sinhlog 4 n nn b Pabn ibab ia b , (12) where and ,1 nn Pa G 1,1 nn PF. Because we have 1 1 2 log(,) 2 1 2 2 2 log(, ) 2 2 2 2 22 2 2 2 ,sinhlog 4 2 2 4 1 4 ,, ,, 4 1,, 4 n niba b nn n ibab n n n n nn nn nn n n n n n Pabn ibab ib ab ee ib ab ib ab ibabibab ab bab ab b ab ab ab , Thus 1 2 2 2 2 ,sinhlog 4 n nn b Pabn ibab ia b 4. Golden Hyperbolic Functions Similar to the Stakhov and Rozin hyperbolic functions [20], we can use the same approach and introduce the generalized hyperbolic functions of order (a,b) in the following form 2 2 2 2 2 1 ,, , 4 1 ,, , 4 111 ,, 4 ,, 4 11 4 n n n n n n n n n n n n n n b Pabab ab ab b Pababab ab ab bab ab bb ab ab ab bab , ,, n b ab ab Then we will have 1 ,, n nn PabPab b We define 22 22 ,, ,, nn nn nn nn nn nn SPSP abbabbab CPCP abbabbab , , Then, nn SP SP and . nn CP CP 2 2 21 , 2 n n nn n bSPnk Pab bCPn k where 2 4 1 1 ,22 b a ab a Theorem 4.1 Following recursive relation can be es tablished 21 ,, nnn PabaPabbPab , (13) Proof 2 1 ,, , 4 n n n b Pabab ab ab ,. Copyright © 2011 SciRes. AM
S. M. HASHEMIPARAST ET AL. 814 1 1 12 2 2 22 1 ,, , 4 1 ,, , 4 n n n n n n b Pabab ab ab b Pabab ab ab 2 12 2 2 2 1,, 4 ,, 1,, 4 , n nn n n n n aPbPa baabb ab bab b ab ab b ab ab ab Pab Similarly we define 2 2 , ,, ,, ,, , , x x x x xx x xx x x x x CPa b COTPa bSPa b ab bab ab bab ab b ab b The symmetric property is also satisfied, as we have for Fibonacci and Lucas hyperbolics sine and cosine. For example 2 2 , ,, , x x xx x x ab b TANP abTANPab ab b Similar to the De’Moivre formulas for the double pa rameter generalized hyperbolic functions following rela tion is satisfied 1 ,, 2 n xx n CP abSPab 2,, 4nx nx C P abSP ab ab 5. Generalized Double Parameter Matrices of n cci numbers by the fol lowing relation Hoggatt [3] introduced the following Qmatrix and The ory of Fibonacci Qmatrix 11 10 Q where n Qconnects to the Fibona 1 nFn Fn Q 1Fn Fn and 2 21 21 2 221 221 21 22 n n FnFn QFnFn Fn Fn QFnFn The matrix of order is then for any ,ab G ,ab , 1 0 ab a Gb integer 0, 2,n1, the th power of mae twparameter gol number andonaumbers such that n o cci n ,ab G den trix sets a connection with th two parameter Fib 1 , 1 ,, ,, nn n ab nn Pab Pab GPabP ab 1 , 1 ,, 1 ,, nn n ab n nn Pa b Pab GPab P ab b where , n n ab Det Gb the procedure can be rameter extended to the case of multi pa s, and we guess and also it maybe generalized to the multi dimension case which allows developing the application to the com munication engineering specially to the coding theory[4,5] The inverse can also be calculated similar to the by induction. 6. Conclusions he generalized golden ratio is extended to ,0, ,0, 00100 ab G 10 00 000 01 00 00 a b 1 1 2 ,, 10 00 01 00 aa a a G 1000 1 00 00 k k k a a ,ab G In this paper t Copyright © 2011 SciRes. AM
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