Applied Mathematics, 2011, 2, 808-815 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

E-mail: hashemiparast@kntu.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 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

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 p-numbers with a high ability. Or-der-m generalized of Fibonacci p-numbers for a matrix representation gives some identities [6] useful for further application in theoretical physics and Metaphysics [7-10]. Some authors determine certain matrices whose parame-ters generate the Lucas p-numbers and their sums creates the continues functions for Fibonacci and Lucas p-numbers which is considered in [11-14] 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 self-organization 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

nAB AC

aAC CB

(1)

where and a are real positive numbers. Let

ACx

CB

, then

1AB AC CB CB

a a a a aAC AC AC x

Thus we obtain

naa x

x

or 1nx ax b (2)

where a b . In a special case for we intend to divide

1n AB such that

AB Aa

C

AC CB

then we have 2 0x ax b (3)

or

2

1

4

2

a a bx

(4)

and

A C B2

a

2

4

ab

E

2

a 1

b

Figure 1. Dividing the line AB in generalized Golden sec-tion.

2

2

4

2

a a bx

(5)

We call the positive root of this equation the general-ized two parameters golden ratio ,a b

2

41

1,

2 2

b

aa b a

for the case b = 1 it gives a uniparameter a

2 4

,12 2a

a a ba

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

1 51,1

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

41

1

2 2aa

which is one solution of equation . Now other properties of the ratio can be considered. We have

2 0x ax a

ba

the generalized golden ratio when a = 1 reduces to

the well-known historical golden ratio 1 5

2 which

has many properties and application in art, engineering, physics and mathematics. Similar properties can be es-

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S. M. HASHEMIPARAST ET AL. 810

tablished in the generalized case. For instance the gener-alized can be expressed in terms of itself, like

1a

that can be expanded into this fraction or nested roots equations that goes on for ever and called continued fraction or nested roots

1 1 1

1 1 1

,

a a a a

a b b a b a b a

Using the relation in term of itself we get the fol-lowing continued fractions for , a , respec-tively

,a b

11

11

11

11

1

,

a aa

aa

ba b a

ba

ba

a

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.

2n 1a b

If in (3), thena b 4

11

,2 2

aa a 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

2 1, ,n n nP a b aP a b bP a b

which can be generalized to a multi parameters cases.

,

1 2 1 21

, , , , ,..., ,

2,3,

p

n p p i n p i pi

P a a a a P a a a

p

Theorem 2.1 Let ADB be an isosceles triangle with the vertex angle , then the necessary and sufficient conditions for the angle equals to BAC (see Figure 2) 0 π 2 , is dividing the by in general-ized Golden ratio, such that and

BDDC

Ca 1BC

and 1

BC CD a

.

Proof Without losing the generality put 1b , train- gle ADB is isosceles and ˆ ˆ π 2A B , the condi-tion is sufficient, because if and DC a 1CB then DB CBDC or 1DB a a . For the trian-gles ADC and ADB we have respectively

2 2 2

2 2 2

2 . cos

2 . cos

AC DA DC DA DC

AB DA DB DA DB

or,

2 2 2 2

2 2 2 2 2

2 . cos

2 . cos

AC a a a a

AB a a a a

If AB AC then the ABC will be an isosceles tri-

angle and π

2 2ABC ACB

and also

1cos

2

, so we will have

π ππ

2 2 2 2BAC

. On the other hand

the condition is necessary, because if BAC then

ππ

2 2 2 2ACB

π , hence AC AB

and triangle ABC

a

will be isosceles and we conclude

easily DC , 1

CB

.

D

C B

A

b

ba

α

α

π

2 2

a

Figure 2. Dividing BD by point C in the generalized golden ratio.

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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 ADB whose vertex angle is ADB (Figure 3), hence the base

angles are π

2 2

BD . By taking the line and mark-

ing on the line in the generalized golden ratio, the angle

CBAC , now we draw a circle with the center

and radiusD 1DA DB a a , then the line AC will contacts with this circle at a point say E , the

arc 2BEEA . By a similar way we divide in generalized golden ratio and find such that

DEDCC a

and 1C E , now the line will contact with the circle in a point say , joining we have

BCDEE ADB

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 ADB with vertex angle 3 and side DB DA 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.

D B C

a

C

1

1

M

M N

N A

Figure 4. Algorithm for trisecting the angle by using gener-alized golden section. into two parts DC a and 1CB , and find M the middle point of , draw a line per pendicula to

from CB

CB M , which contact the circle at point say , connect and by segment , then divide

into two parts

ND N

DCDN

DN and on golden ratio such that

C NDC a and 1C N again find the point

M the middle point of C N draw a line perpendicular to C N at point M , extend to contact the circle at point

C NN , then draw , angle 3DN will be tri-

sected by and DN DN . When is divided into two segment and DB a 1 ,

then 1cos

2

we can easily calculate a with re-

spect to cos , and we get

21 2cos 1

2cos 1

ba

having , a can easily be calculated, for given when the segments are and a b 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 , ,P a b P 0,1, 2,n

if and only if 0 ,P a b 0 and and for all

1 ,P a b 10n

2 1, ,n n nP a b aP a b bP a b , (6)

To attain the solution, using characteristics method gives the following quadratic equation

2 0a b

where

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S. M. HASHEMIPARAST ET AL. 812

Table 1. Values of Cos (α), a and for the nominal values of α.

1cos

2 60 a

2cos

2 45 2a 2 1

5 1cos

4 36 1a 5 1

2

3cos

2 30 3 3

2a

3 1

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

a a br

and

2

2

4

2

a a br

are the roots of characteristics function and the positive root can be written as 1r

2

1

41

1,

2 2

b

ar a a b

where is the Generalized Golden Ratio and ,a b

2

2

41

1

2 2 ,

bbar aa b

Hence the general solution of (1) can be written as

1 2, ,,

nn

n

bP a b C a b C

a b

Looking at the two initial values 0 and 1 , the co-efficients and can be determined, thus

P P

1C 2C

2

1, , (

,4

n nn

bP a b a b

a ba b

) (7)

For , , 1b ,1n nP a P a 21 4

,12 2a

aa

2

1

4

n

nn a

a

P aa

which is the Gazale formula for the Generalized Fibo-nacci sequence, and if nF is the ordinary Fibonacci sequence, the general solution to

2 1n n nF F F

will be

11

1 11

5

5 1 1 52 2

5

n

nn n

n n

F P

for the case 1a , and for the case , 2a 2,1nP gives the John Pell (1610-1685) 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 ,a b

121 1, , ,

n

n n nP a b P a b P a b b

(9)

and is independent of . aProof We have

1

11 2

1

11 2

2

1, ,

,4

1, ,

,4

1, ,

,4

n

nn

n

nn

n

nn

bP a b a b

a ba b

bP a b a b

a ba b

bP a b a b

a ba b

Then

2

21 1 2

11 2

2

1, , ,

,4

,,

n

nn n

nn

bP a b P a b a b

a ba b

bb a b

a b

and

2

2

22

,

1, 2

,4

n

n

nn

P a b

ba b b

a ba b

1 (8)

Thus

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S. M. HASHEMIPARAST ET AL. 813

21 1

1 22

2 2

21

2

112

2

, , ,

, 24 ,

,,4

44

n n n

n

n

nn

P a b P a b P a b

b ba b b

a b a b

b ba b

a ba b

ba b b

a b

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

5n n

F n ii

(10)

and generalized form in [27,28] for real positive is

0a

2

2sinh log

4n an

Gi a

n i

(11)

If we generalize (11) for ,nP a b we obtain

122

2

2, sinh log

4

n

n n

bP a b n ib a b

i a b

, (12)

where and ,1n nP a G 1,1n nP F . Because we have

112log( , )2

1

2

22

log( , )

22

22

2 2

2

2

2, sinh log

4

2

24

1

4

, ,

, ,

4

1,

,4

nn ib a b

n nn

ib a b

nn

nn

n nn n n n

nnn

n

n

P a b n ib a b

i b a b

e e

i b a b

i b a b

i b a b i b a b

a b b a b

a b

ba b

a ba b

,

Thus

122

2

2, sinh log

4

n

n n

bP a b n ib a b

i a 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

1 1 1,

,4

,,4

1 1

4

n

nn

n

nn

nn

n

nn

n

nn

bP a b a b

a ba b

bP a b a b

a ba b

a bb a ba b

b ba b

a ba b

b a b

,

,

,

nb

a ba b

Then we will have

1, ,

n

n nP a b P a bb

We define

2 2

2 2

, ,

, ,

n nn n

n n

n nn n

n n

SP SP a b b a b b a b

CP CP a b b a b b a b

,

,

Then, n nSP SP and . n nCP CP

2

2

2 1,

2

n

nn n

n

b SP n kP a b

b CP n k

where

2

41

1,

2 2

b

aa b a

Theorem 4.1 Following recursive relation can be es-

tablished

2 1, ,n n nP a b aP a b bP a b , (13)

Proof

2

1, ,

,4

n

nn

bP a b a b

a ba b

, .

Copyright © 2011 SciRes. AM

S. M. HASHEMIPARAST ET AL. 814

1

11 2

2

22 2

1, ,

,4

1, ,

,4

n

nn

n

nn

bP a b a b

a ba b

bP a b a b

a ba b

21 2 2

2

2

1, ,

4

, ,

1,

,4

,

nn n

n

n

n

n

aP bP a b a a b ba b

b abb

a b a b

ba b

a ba b

P a b

Similarly we define

2

2

,,

,

, ,

, ,

,

,

xx

x

xx x

xx x

xx

xx

CP a bCOTP a b

SP a b

a b b a b

a b b a b

a b b

a b b

The symmetric property is also satisfied, as we have for Fibonacci and Lucas hyperbolics sine and cosine. For example

2

2

,, ,

,

xx

x xxx

a b bTANP a b TANP a b

a b b

Similar to the De’Moivre formulas for the double pa-rameter generalized hyperbolic functions following rela-tion is satisfied

1

, ,

2

n

x x

n

CP a b SP a b

2, ,

4nx nxCP a b SP a b

a b

5. Generalized Double Parameter Matrices of n

cci numbers by the fol-

lowing relation

Hoggatt [3] introduced the following Q-matrix and The-ory of Fibonacci Q-matrix

1 1 1 0

Q

where nQ connects to the Fibona

1n F n F nQ

1F n F n and

2

2 1

2 1 2

2 2 1

2 2 1

2 1 2 2

n

n

F n F nQ

F n F n

F n F nQ

F n F n

The matrix of order is

then for any

,a bG ,a b

,

1

0a b

aG

b

integer 0, 2,n 1, the th power of ma e tw parameter gol number and ona umbers such that

no

cci n,a bG

den trix sets a connection with th

two parameter Fib

1,

1

, ,

, ,n nn

a bn n

P a b P a bG

P a b P a b

1

,1

, ,1

, ,n nn

a b nn n

P a b P a bG

P a b P a bb

where

,

nna bDet G b

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,

0 0 1 0 0

a bG

1 0 0 0

0 0 0 0 1

0 0 0 0

a

b

1

1

2

, ,

1 0 0 0

0 1 0 0

a a

a

a

G

1 0 0 0 1

0 0 0 0

k

k

k

a

a

,a bG

In this paper t

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S. M. HASHEMIPARAST ET AL.

Copyright © 2011 SciRes. AM

815

case and the properties are also gen-eralized to the related historical and recently developed

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