Applied Mathematics, 2011, 2, 808-815 doi:10.4236/am.2011.27108 Published Online July 2011 (http://www.SciRP.org/journal/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 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 cre- ate 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- Copyright © 2011 SciRes. AM S. M. HASHEMIPARAST ET AL. 809 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 struc- tural 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 applica- tion. 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- aa4 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 ACACACx 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 n1 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 aa4 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 Paaanp12,,, p aPaaa inpi 12, ,..., p , a i1 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 a1 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 a2.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 ab1, 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 ap- plications of this ratio. b π 4 B 22 1 C 1 If ab in (3), then aa, aa , 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 a characteristic function of a difference equation of order n with parameters a and b Figure 2.