Multi Parameters Golden Ratio and Some Applications

Home , Angle trisection, Doubling the cube, Golden ratio

Applied Mathematics, 2011, 2, 808-815 doi:10.4236/am.2011.27108 Published Online July 2011 (http://www.SciRP.org/journal/am)

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 E-mail: [email protected] 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 two-dimensional 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 Q-matrix called Gn-matrix 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, Q-matrix, Fibonacci, Lucas, Gazale, Casseni

1. Introduction consider the extension of the Fibonacci numbers to create a generalized matrix with various properties suitable During the last centuries a great deal of scientist’s at- for the coding theory [4,5]. Even they established gener- tempts has been made on generalizing the mathematical alized relations among the code matrix elements for all aspects and their applications. One of the famous aspects values of Fibonacci p-numbers with a high ability. Or- in mathematics is called golden ratio or golden propor- der-m generalized of Fibonacci p-numbers for a matrix tion has been studied by many authors for generalization representation gives some identities [6] useful for further from different points of view. In recent years there were application in theoretical physics and Metaphysics [7-10]. a huge interest of modern science in the application of Some authors determine certain matrices whose parame- the Golden section and Fibonacci numbers in different ters generate the Lucas p-numbers and their sums creates area of engineering and science. An extensive bibliogra- the continues functions for Fibonacci and Lucas phy of activities can be found in a paper by Stakhov [1,2] p-numbers which is considered in [11-14] and their gen- who has investigated the generalized principle of Golden eralized polynomials and properties are also introduced section and its vast area of applications. The main goal of in [15], based on Golden section the attempt of build up his attention is to state the fundamental elements of this the fundamental of a new kind of mathematical direction subject i.e. description of its basic concepts and theories is addressed in [3] which is the requirements of modern and discussion on its applications in modern science. In natural science and art and engineering, these mathe- this connection he explains the idea of the creation of a matical theories are the source of many new ideas in new mathematical direction called mathematics of Har- mathematics, philosophy, botanic and biology, electrical mony that moved to intend for the mathematical simula- and computer science and engineering, communication tion of those phenomena and processes of objective system, mathematical education, theoretical physics and world for which Fibonacci numbers and Golden Section particle physics [16]. are their objective essence which can influence on the Other area of applications returns to the connection other areas of human culture. Stakhov considers the between the Fibonacci sequence and Kalman filter, by harmony mathematics from sacral geometry point of exploiting the duality principle control [17], finally nu- view and its applications in this field [3]. Some authors merical results of generalized random Fibonacci se-

1 quences which are stochastic versions of classical Fibo- a a nacci sequences are also obtained [18]. Recently Edu- A 2 2 C  B ardo Soroko developed very original approach to structural harmony of system [19], using generalized Golden a2  b sections. He claims the Generalized Golden sections are 4 invariant which allow natural systems in process of their b self-organization to find the harmonic structure, station- ary regime of their existence, structural and functional stability. E Following the introduction, in Section 2 we establish Figure 1. Dividing the line AB in generalized Golden sec- multi parameters Golden Ratio and geometric application. tions. In Section 3 we generalized the Gazale formula.

Sections 4 and 5 investigate the generalized hyperbolic 2 functions and double parameters matrices and applica- aa4 b x2  (5) tions. 2 We call the positive root of this equation the general- 2. Preliminaries ized two parameters golden ratio ab,  4b  In this paper we start a geometrical discussion which 1 1 2  leads to a generalized form of golden ratio with multi  ab,  a a  parameters that for the case of single parameter it changes 22   to the ordinary generalized form which has been consid-   ered by authors in the recent years. for the case b = 1 it gives a uniparameter  Suppose a line AB is to divided in two parts AC and a CB (Figure 1) such that aab2  4 a,1 n a  ABAC  22 a   (1) AC CB which is called generalized golden ratio and has been where a and  are real positive numbers. Let considered in recent years by some authors, for a = 1 gives the famous historical golden ratio  AC  x , then 15 CB  1,1  1 2 AB AC CB CB  aa a1 aa ACACACx where the ratio of the adjacent numbers in Fibonacci series and Lucas series tends towards this irrational Thus we obtain number.

a n Let us consider the properties of this golden ratio, for ax x a = b we have or  4   1  n1 1 x ax b (2)  a  a  22 where a  b. In a special case for n 1 we intend to   divide AB such that   ABA C which is one solution of equation xaxa2 0 . Now a  ACCB other properties of the ratio can be considered. We have b then we have a     xaxb2 0 (3) or the generalized golden ratio  when a = 1 reduces to 15 2 the well-known historical golden ratio   which aa4 b 2 x1  (4) 2 has many properties and application in art, engineering, and physics and mathematics. Similar properties can be es-

Copyright © 2011 SciRes. AM 810 S. M. HASHEMIPARAST ET AL. tablished in the generalized case. For instance the gener- which can be generalized to a multi parameters cases. alized  can be expressed in terms of itself, like p 1 Paaanp12,,, p  aPaaa inpi 12, ,..., p ,  a i1  p  2,3, that can be expanded into this fraction or nested roots Theorem 2.1 Let ADB be an isosceles triangle with equations that goes on for ever and called continued the vertex angle  , then the necessary and sufficient fraction or nested roots conditions for the angle BAC equals to  (see Figure 2) 0  π 2 , is dividing the BD by C in general-  111 ized Golden ratio, such that DC  a and BC 1  1 a 111aaa and BC CD a .   ab,  bababa   Proof Without losing the generality put b 1 , train- gle ADB is isosceles and ABˆ ˆ π 2  , the condi- Using the  relation in term of itself we get the fol- tion is sufficient, because if DC a and CB  1 then lowing continued fractions for  ,  ,  ab, respec- a  DB DC  CB or DB a1  a . For the trian- tively gles ADC and ADB we have respectively 1 222   AC DA DC2. DA DC cos 1 1 222 1 AB DA DB2. DA DB cos 1 1 or, 1  a AC2222 a a2. a a cos a 1  a  1 AB22222 a a2.cos a a a   a  If ABAC then the ABC will be an isosceles tri- b  ab,  a   π   b angle and ABC ACB  and also a  22 b a   1 a  cos  , so we will have 2 Stakhov and Rozin [20,21] give some interesting re-  ππ sults for n  2 and ab1, similar results are veri- BAC  π  . On the other hand fied for different values of a and b, two dimension peri- 22 22 odic continued fractions are considered by some authors the condition is necessary, because if BAC   then [22,23] using above formulas may generalize the appli-  π π cation in art and architecture. The above line AB can be ACB  π  , hence AC AB 22 22 divided in n sections and by the successive ratios a sys-  tem of equations will be created such that for a given and triangle ABC will be isosceles and we conclude 1 value n, (1) be extended to multi parameters ratios to easily DC  a , CB  . create a multi parameters Golden ratio. In this paper we  concentrate on a generalized double parameters and applications of this ratio. b π   4  B 22 1 C 1 If ab in (3), then aa,  aa , the 22 a   α generalized single parameter ratio different from  a,1 α D A studied by Stakhov, actually (2) can be considered as a b characteristic function of a difference equation of order n  a  with parameters a and b Figure 2. Dividing BD by point C in the generalized golden PabaPabbPabnnn21,,  , ratio.

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 1 be found in some references [25], trisecting the angle is C  impossible that is there exist an angle that cannot be tri- D a M B sected with a straightedge and compass. On the other hand there are some angles which may be trisected, some C M author give valuable discussion for ideal and classical 1 trisection of an arbitrary angle [26]. Using generalized N  golden ratio one may establish the following criteria for trisection of a given angle. Now, we construct an isosceles triangle ADB whose N vertex angle is ADB   (Figure 3), hence the base A π   Figure 4. Algorithm for trisecting the angle by using gener- angles are  . By taking the line BD and mark- 22 alized golden section. ing C on the line in the generalized golden ratio, the into two parts DC a and CB 1  , and find M angle BAC   , now we draw a circle with the center the middle point of CB , draw a line per pendicula to D and radius DA DB a1  a , then the line   CB from M , which contact the circle at point say N , AC will contacts with this circle at a point say E , the connect D and N by segment DN , then divide arc EABE  2 . By a similar way we divide DE in DN into two parts DC and CN on golden ratio generalized golden ratio and find C such that DC  a such that DC  a and CN 1  again find the point and CE 1  , now the line BC will contact with the M  the middle point of CN draw a line perpendicular circle in a point say E , joining DE we have ADB  to CN at point M  , extend CN to contact the circle BDE  EDE The procedure can be carried on to pro- at point N , then draw DN , angle 3 will be tri- duce a desired polygon by a proper selection of the angle sected by DN and DN .  . Thus we will have the following proposition for tri- When DB is divided into two segment a and 1  , section of an arbitrary angle.  1 Proposition For trisection of a given angle first con- then cos  we can easily calculate a with re- struct an isosceles triangle ADB with vertex angle 3 2 and side DB DA a (Figure 4), then divide DB spect to cos  , and we get 12cos1b 2 a  2cos  1 having  , a can easily be calculated, for given  E when the segments are a and b  respectively, and are given in the Table 1 for b 1 and some nominal values of  .

1 3. Generalized Gazale Formula D α C  α Let a and b be any real positive number, and define α E a Generalized Gazale sequence Pabn ,, P n  0,1,2,

if and only if Pab0  ,  0 and Pab1 ,1 and for all α C α n  0 1 PabaPabbPab,,, (6)  nnn21 

α B To attain the solution, using characteristics method gives the following quadratic equation A 2  ab0 Figure 3. Trisecting the angle by using generalized golden section. where

Table 1. Values of Cos (α), a and  for the nominal values of α. which is the Gazale formula for the Generalized Fibo-

nacci sequence, and if Fn is the ordinary Fibonacci 1 cos    60 a    sequence, the general solution to 2 Fnn21 FF n 2 cos    45 a  2  21 2 will be n 51 51  cos    36 a 1   11n  FPnn1 1  4 2  5 1 3 33 31 cos    30 a    nn 2 2 2 51 1 5  cos20 0.940   20 a  0.256  1.136 22    5 cos20 0.985  10 a  0.061  1.031

 for the case a 1 , and for the case a  2 , P 2,1 cos0 1   0 a  0  1 n gives the John Pell (1610-1685) formula, and this formula generates an infinite number of generalized Fibo- aa2 4 b nacci number of different orders for a and b. r  1 2 The following Theorem is a generalized aspect of theorems that have been established for Fibonacci and and Lucas numbers. aa2 4 b Theorem 3.1 If n 1 and ab, are real positive r2  numbers, then 2 are the roots of characteristics function and the positive 2 n1 PabPabPabbnn11,,,  n  (9) root r1 can be written as and is independent of a . 4b 1 Proof We have 1 2 raa  ab, 1   n1  22 1  n1 b   Pabn1 ,,  ab  2  ab, ab 4   where  ab,  is the Generalized Golden Ratio and  n1  1  n1 b  Pabn1 ,,  ab 4b 2  1 ab 4   ab,  1 2 b  ra  a  2 22 ab , n  1 n b  Pab,,  ab   n 2  ab, ab 4  Hence the general solution of (1) can be written as Then n n b  2n Pabn ,, C12  ab C 1  b  ab, P ab,, P ab 2n ab ,   nn11  2   ab 4  ab,   Looking at the two initial values P0 and P1 , the co- n1  efficients C1 and C2 can be determined, thus n12 b  bab , 2   ab, 1 nnb  Pabn ,,(  ab ) (7) ab2  4  ab, and 2 2 14a  Pabn  ,  For b  1,Pann,1  Pa , a,1 a 22  2n  1  2n b n   ab,2 b  n ab2  4  ab, 1n 1    Pana  (8) 2  a  4 a Thus

2 4. Golden Hyperbolic Functions PabPabPabnn11,,,  n 

n1 2 b 2 b Similar to the Stakhov and Rozin hyperbolic functions 22 ab,2 b ab 4,  ab [20], we can use the same approach and introduce the generalized hyperbolic functions of order (a,b) in the n1 2 b b following form  ab, ab2  4  ab,    n 1 n b n1 Pabn ,,  ab , b 2   2 n1 ab 4  ab, 2 ab4  b  ab 4 n 1 n b Similar identities like the Askey’s and Catelan’s and Pab,,  ab n 2  ab, d’Ocagne’s can be generalized. For instance in Osler and ab 4  Hilburn [2] Askcy’s hyperbolic sine identity for ordinary  n  Fibonacci sequence is given as 111 n     ab,  2 n 2 ab 4  bab ,  Fn  sinhni log   (10) in 5 n n b n b  ab,  and generalized form in [27,28] for real positive a  0 2  ab 4  ab, is  n n 2 11n b Gna sinhn logi  (11)    ab,  n 2 b 2  ab, ia 4 ab 4  If we generalize (11) for Pabn  ,  we obtain Then we will have n 1 n 2  1 2b 2 Pab,sinhlog  n ib  ab, (12) Pabnn,,  Pab n n 2  b ia 4 b   We define where Pann,1  G and PFnn1,1  . Because we have nn   nn SP SPab,, b22 ab b ab, 1  nn    2 2  nn  Pabn ,sinhlog n n ib  ab, nn  CP CPab,, b22 ab b ab, n 2  nn   ib2 a 4 b  1 1   n Then, SPnn  SP and CPnn CP . 2 n log(ib2 a , b ) 2 eelog(ib a , b )   n  n 2  2 bSPn n21 k n 2 2 Pab,  ib a 4 b n  n  2 1  bCPn n2 k  n  ibn 2 a2  4 b where nn   4b  ibnn22 ab,, i nn b ab 1   1 2    ab,  a a  22 n n n   ab,, b ab      ab2  4 Theorem 4.1 Following recursive relation can be es- n  1 n b tablished  ab,  2  ab 4  ab,  PabaPabbPabnnn21 ,,  , (13) Thus Proof

n n 1   2  1 b 2b 2  n  Pab,sinhlog  n ib  ab,. Pabn ,,  ab  n n 2  2  ab, ia 4 b  ab 4  

n1 lowing relation 1 n1 b Pabn1 ,,  ab 2  ab, n Fn 1 Fn ab 4  Q   Fn Fn1  n2  1 n2 b Pab,,  ab and n2 2  ab, ab 4  Fn21 Fn 2 Q2n    Fn221 Fn   1  n2 aP bP a,, b a  a b  b nn122  21n Fn221 Fn   ab 4  Q   Fn21 Fn 22  n2  bab The G matrix of order ab, is   b  ab,  ab,, ab  a 1 Gab,   n b 0 1 n b  ab,  2  ab, then for any integer n  0, 1, 2, the n th power ab 4  of Gab, matrix sets a connection with the two parameter  Pabn , golden number and two parameter Fibonacci n umbers Similarly we define such that PabPab ,, CPx  a, b n nn1 COTP a, b  Gab,   x  Pab,, P ab SP a, b nn1  x xxx  Pa,,b Pab ab,, b  ab n 1 nn1    Gab,  n  xxx  Pab,, P ab ab,, b  ab b nn1   2x ab,  b x where  n n 2x x Det G b  ab,  b  ab,   The symmetric property is also satisfied, as we have the procedure can be extended to the case of multi pa- for Fibonacci and Lucas hyperbolics sine and cosine. For rameters, and we guess example a 10 0 0 x   2x ab,  b 001 0 0 TANPab,, TANP ab  xx2x x    ab,  b Gab,0, ,0,     000 0 1 Similar to the De’Moivre formulas for the double pa-  rameter generalized hyperbolic functions following rela- b 00 0 0 tion is satisfied and also it maybe generalized to the multi dimension n case which allows developing the application to the com- CPab,, SPab  xx munication engineering specially to the coding theory[4,5] n1 2  a 10 0 0 CP ab,, SP  ab 1 2 nx nx  ab 4 a 01 0 0  2

Gaa,,   5. Generalized Double Parameter Matrices 1 k  ak1 00 0 1 of n  ak 00 0 0 Hoggatt [3] introduced the following Q-matrix and The- The inverse can also be calculated similar to the Gab, ory of Fibonacci Q-matrix by induction. 11 Q   6. Conclusions 10 where Qn connects to the Fibonacci numbers by the fol- In this paper the generalized golden ratio is extended to

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