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@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
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
n
A
BAC
aAC CB

(1)
where and
a
are real positive numbers. Let
AC
x
CB
, then
1
ABAC CBCB
aaa aa
A
CACAC x







Thus we obtain
n
a
ax
x




or
1n
x
ax b
 (2)
where ab
. In a special case for we intend to
divide
1n
A
B such that
BA
aC
A
CCB
then we have
20xaxb (3)
or
2
1
4
2
aa b
x
(4)
and
A
C B
2
a
2
4
ab
E
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 well-known 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
A
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
A
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
A
DC and
A
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
A
BAC
then the
A
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
A
CAB
and triangle
A
BC
a
will be isosceles and we conclude
easily DC
, 1
CB
.
D
C B
A
b
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
A
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
A
C 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
DE
DCCa
and 1CE
, now the line will contact with the
circle in a point say, joining we have
BC
DEE 
A
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
A
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
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
N
DN
DC
DN
DN
and on golden ratio
such that
CN
DC a
and 1CN
again find the point
M
the middle point of CN
draw a line perpendicular
to CN
at point
M
, 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
F
is the ordinary Fibonacci
sequence, the general solution to
21nnn
F
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 (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,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
F
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 Q-matrix and The-
ory of Fibonacci Q-matrix
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
S. M. HASHEMIPARAST ET AL.
Copyright © 2011 SciRes. AM
815
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