The ''golden'' hyperbolic models of Universe - Student Oulu

Chaos, Solitons and Fractals 34 (2007) 159–171

www.elsevier.com/locate/chaos

The ‘‘golden’’ hyperbolic models of Universe

Alexey Stakhov, Boris Rozin *

International Club of the Golden Section, 6 McCreary Trail, Bolton, Ont., Canada L7E 2C8

Accepted 31 March 2006

Abstract

This article presents a review of new mathematical models of the hyperbolic space. These models are based on thegolden section. In this article, the authors discuss the hyperbolic Fibonacci and Lucas functions and the surface of thegolden shofar, which are the most important of these models. The authors also introduce, within this article, the goldenhyperbolic approach for modeling the universe.� 2006 Elsevier Ltd. All rights reserved.

0960-0doi:10.

* CoE-m

UR

I want to know how God created this world. I am not interested in this or that phenomenon, in the spectrum ofthis or that element. I want to know His thoughts; the rest are details.Albert Einstein

1. Introduction

Discovering that the world around us is hyperbolic is probably one of the major achievements of science. The pri-ority of creation of the non-Euclidean geometry belongs to the Russian mathematician Nikolay Lobachevsky. In 1827,Lobachevsky offered a new geometric system based on hyperbolic functions. The need for new geometrical ideas becameapparent in physics at the beginning of the 20th century and proceeded from Einstein’s special relativity theory, pre-sented in 1905. In 1908, three years after publication of special relativity theory, the German mathematician HermanMinkovsky gave the original geometrical interpretation of special relativity theory based on hyperbolic ideas.

In 1993, the Ukrainian mathematicians Stakhov and Tkachenko developed a new approach to the hyperbolic geom-etry [4]. Using Binet formulas, they developed a new class of hyperbolic functions called the Hyperbolic Fibonacci andLucas functions [4,5]. The authors of the present article further developed the ideas of the hyperbolic Fibonacci andLucas functions and introduced the symmetric hyperbolic Fibonacci and Lucas functions in paper [6]. In [7], theauthors developed a new surface of the second degree, called the golden shofar. The hyperbolic Fibonacci and Lucasfunctions and the surface of the golden shofar are the representatives of the golden mathematical models used for mod-eling hyperbolic space.

Further development of the models of the universe requires the concepts of the generalized Fibonacci numbers andthe generalized golden proportions [8], the golden algebraic equations [9], the generalized Binet formulas and the

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rresponding author.ail addresses: [email protected] (A. Stakhov), [email protected] (B. Rozin).

Ls: www.goldenmuseum.com (A. Stakhov), www.goldensection.net (B. Rozin).

mailto:[email protected]

mailto:[email protected]

www.goldenmuseum.com

www.goldensection.net

160 A. Stakhov, B. Rozin / Chaos, Solitons and Fractals 34 (2007) 159–171

generalized Lucas numbers [10], the continuous functions for the generalized Fibonacci and Lucas functions [11], theFibonacci matrices [12], and the golden matrices [13]. These mathematical results were used in the algorithmic measure-ment theory [8,14,15], a new theory of real numbers [16], the Harmony Mathematics [17–20], a new computer arithme-tic’s [8,21–23], a new coding theory [24,25], and in the formulation of the generalized principle of the golden section [13].

The present article is a review of all these models. The main purpose is to give a brief description of the golden math-ematical models for modeling the hyperbolic space that the authors developed in the works [4–7].

2. The golden section, Fibonacci and Lucas numbers, and Binet formulas

In Euclid’s The Elements we find a geometric problem called ‘‘the problem of division of a line segment in theextreme and middle ratio’’. Often this problem is called the golden section problem [1–3]. Solution of the golden sectionproblem reduces to the following algebraic equation:

TableExtend

n

Fn

F�n

Ln

L�n

x2 ¼ xþ 1; ð1Þ

Eq. (1) has two roots. We call the positive root, s ¼ 1þffiffi5p

2, the golden proportion, golden mean, or golden ratio.

The root follows from (2), the following identity for the golden ratio:

sn ¼ sn�1 þ sn�2 ¼ s� sn�1; ð2Þ

where n takes its values from the set: {0,±1,±2,±3, . . .}.The Fibonacci numbers

F n ¼ f1; 1; 2; 3; 5; 8; 13; 21; 34; 55; . . .g ð3Þ

are a numerical sequence given by the following recurrence relation:

F n ¼ F n�1 þ F n�2 ð4Þ

with the initial terms

F 1 ¼ F 2 ¼ 1: ð5Þ

The Lucas numbers

Ln ¼ f1; 3; 4; 7; 11; 18; 29; 47; 76; . . .g ð6Þ

are a numerical sequence given by the following recurrence relation:

Ln ¼ Ln�1 þ Ln�2 ð7Þ

with the initial terms

L1 ¼ 1; L2 ¼ 3: ð8Þ

The Fibonacci and Lucas numbers can be extended to encompass negative values of the index, n. We present theextended Fibonacci and Lucas numbers in Table 1.

The data from Table 1 demonstrates that the terms of the extended series Fn and Ln have a number of interestingmathematical properties. For example, for the odd n = 2k + 1 the terms of the sequences Fn and F�n coincide. That is,F2k+1 = F�2k�1. For the even indices, n = 2k they are of opposite signs. That is, F2k = �F�2k. Note that for the Lucasnumbers Ln, results are exactly opposite, i.e. L2k = L�2k and L2k+1 = �L�2k�1.

There are a number of fundamental results in modern Fibonacci number theory [2,3]. In the 19th century, thefamous French mathematician Binet found one of them. Studying the golden ratio, Fibonacci and Lucas numbers,he discovered two remarkable formulas, called Binet formulas. These formulas, shown below, connect the Fibonacciand Lucas numbers with the golden ratio

1ed Fibonacci and Lucas numbers

0 1 2 3 4 5 6 7 8 9 100 1 1 2 3 5 8 13 21 34 550 1 �1 2 �3 5 �8 13 �21 34 �552 1 3 4 7 11 18 29 47 76 1232 �1 3 �4 7 �11 18 �29 47 �76 123

A. Stakhov, B. Rozin / Chaos, Solitons and Fractals 34 (2007) 159–171 161

F n ¼

s2kþ1 þ s�ð2kþ1Þffiffiffi5p ;

s2k � s�2kffiffiffi5p ;

8>>><>>>:

ð9Þ

Ln ¼s2k þ s�2k ;

s2kþ1 � s�ð2kþ1Þ;

�ð10Þ

where the discrete variable k takes its values from the set: 0, ±1, ±2, ±3.

3. Hyperbolic Fibonacci and Lucas functions

3.1. Stakhov and Tkachenko’s approach

If we compare Binet formulas (9) and (10) to the classical hyperbolic functions

sinh x ¼ ex � e�x

2and ð11Þ

cosh x ¼ ex þ e�x

2; ð12Þ

we can see that Binet formulas (9) and (10) are similar to the hyperbolic functions (11) and (12).In [4], Stakhov and Tkachenko replaced the discrete variable k in formulas (9) and (10) with the continuous variable

x that takes its values from the set of the real numbers. Consequently, we have the following continuous functions,called the hyperbolic Fibonacci and Lucas functions:

The hyperbolic Fibonacci sine

sFðxÞ ¼ s2x � s�2xffiffiffi5p : ð13Þ

The hyperbolic Fibonacci cosine

cFðxÞ ¼ s2xþ1 þ s�ð2xþ1Þffiffiffi5p : ð14Þ

The hyperbolic Lucas sine

sLðxÞ ¼ s2xþ1 � s�ð2xþ1Þ: ð15Þ

The hyperbolic Lucas cosine

cLðxÞ ¼ s2x þ s�2x: ð16Þ

The following equations denote the relationship between the Fibonacci and Lucas numbers and the hyperbolicFibonacci and Lucas functions (14)–(17):

sFðkÞ ¼ F 2k ; cFðkÞ ¼ F 2kþ1; sLðkÞ ¼ L2kþ1; cLðkÞ ¼ L2k ; ð17Þ

where k = 0;±1;±2;±3, . . .In [5], Stakhov devotes his discussion to the theory of the functions (13)–(16) and their application to modern science

and mathematics. The most important application is a new continuous approach to the Fibonacci numbers theory,based on the equations at (17), which connects the Fibonacci and Lucas numbers with the functions (13)–(16).

3.2. A symmetrical representation of the hyperbolic Fibonacci and Lucas functions (Stakhov and Rozin’s approach)

Let us compare the hyperbolic Fibonacci and Lucas functions given in (13)–(16) to the classical hyperbolic functions.It is easy to see that, in contrast to the classical hyperbolic functions, the graph of the Fibonacci cosine is asymmetricrelative to the x-axis, while the graph of the Lucas sine (15) is asymmetric relative to the origin. This limits the area ofeffective application of the new class of hyperbolic functions given by Eqs. (13)–(16).

Based on the analogy between Binet formulas and the classical hyperbolic functions [6], we are able to give the fol-lowing different definitions of the hyperbolic Fibonacci and Lucas functions:


Symmetrical hyperbolic Fibonacci sine

sFsðxÞ ¼ sx � s�xffiffiffi5p : ð18Þ

Symmetrical hyperbolic Fibonacci cosine

cFsðxÞ ¼ sx þ s�xffiffiffi5p : ð19Þ

Symmetrical hyperbolic Lucas sine

sLsðxÞ ¼ sx � s�x: ð20Þ

Symmetrical Lucas cosine

cLsðxÞ ¼ sx þ s�x: ð21Þ

The Fibonacci and Lucas numbers are determined identically through the symmetrical Fibonacci and Lucas func-tions (18)–(21) as follows:

F n ¼sFsðnÞ for n ¼ 2k;

cFsðnÞ for n ¼ 2k þ 1;

�Ln ¼

cLsðnÞ for n ¼ 2k;

sLsðnÞ for n ¼ 2k þ 1:

�ð22Þ

We designate the representations of the hyperbolic Fibonacci and Lucas functions, given in Eqs. (18)–(21), the sym-

metrical representations.It is easy to construct the graphs for the symmetrical Fibonacci and Lucas functions (Figs. 1 and 2).Their graphs have a symmetrical form and are similar to the graphs of the classical hyperbolic functions. Note that

when x = 0, the symmetrical Fibonacci cosine, cFs(x), takes the value cFsð0Þ ¼ 2ffiffi5p and the symmetrical Lucas cosine,

cLs(x), takes the value cLs(0) = 2. We also now emphasize that the Fibonacci numbers, Fn, with even indicesn = {0,±2,±4,±6, . . .} are points on the symmetrical Fibonacci sine curve, sFs(x) when x = {0,±2,±4,±6, . . .}. Also,for odd values of n, the Fibonacci numbers are points on the symmetrical Fibonacci cosine curve, cFs(x) whenx = {±1,±3,±5, . . .}). Similarly, the Lucas numbers, with the even indices, are points on the symmetrical Lucas cosinecurve cLs(x) for x = {0,±2,±4,±6, . . .}. The Lucas numbers with the odd indices are points on the symmetrical Lucascosine curve sLs(x) whenever x = {±1,±3,±5, . . .}.

We show that the symmetrical hyperbolic Fibonacci and Lucas functions above are related to the classical hyper-bolic functions using the following simple equations:

Fig. 1. The symmetrical Fibonacci functions.

Fig. 2. The symmetrical Lucas functions.

TableThe re

The id

Fn+2 =F 2

n � FLn+2 =L2

n � 2ðFn+1 +Fn + L


sFsðxÞ ¼ 2ffiffiffi5p shðlnðaÞ � xÞ; cFsðxÞ ¼ 2ffiffiffi

5p chðlnðaÞ � xÞ;

sLsðxÞ ¼ 2shðlnðaÞ � xÞ; cLsðxÞ ¼ 2chðlnðaÞ � xÞ:

The symmetrical hyperbolic Fibonacci and Lucas functions are related to each other by the following simplecorrelations:

sFsðxÞ ¼ sLsðxÞffiffiffi5p ; cFsðxÞ ¼ cLsðxÞffiffiffi

5p :

3.3. The recurrent properties of the symmetrical hyperbolic Fibonacci and Lucas functions

The symmetrical hyperbolic Fibonacci and Lucas functions (18)–(21) are a generalization of the Fibonacci andLucas numbers. Therefore, they have recurrent properties. Also, they are similar to the classical hyperbolic functionsand, hence, have hyperbolic properties.

Table 2 gives some of the recurrent properties of the symmetrical hyperbolic Fibonacci and Lucas functions.Note that, as the table shows, we can write the famous Cassini formula [26]

F 2n � F nþ1F n�1 ¼ ð�1Þnþ1

; ð23Þ

an important mathematical identity that connects three adjacent Fibonacci numbers, in the form of two continuousidentities for the symmetrical hyperbolic Fibonacci functions (18) and (19)

2current properties of the symmetrical hyperbolic Fibonacci and Lucas functions

entities for Fibonacci and Lucas numbers The identities for the symmetrical hyperbolic Fibonacci and Lucas functions

Fn+1 + Fn sFs(x + 2) = cFs(x + 1) + sFs(x) cFs(x + 2) = sFs(x + 1) + cFs(x)

nþ1F n�1 ¼ ð�1Þnþ1 [sFs(x)]2 � cFs(x + 1)cFs(x � 1) = �1 [cFs(x)]2 � sFs(x + 1)sFs(x � 1) = 1Ln+1 + Ln sLs(x + 2) = cLs(x + 1) + sLs(x) cLs(x + 2) = sLs(x + 1) + cLs(x)�1Þn ¼ L2n [sLs(x)]2 + 2 = cLs(2x) [cLs(x)]2 � 2 = cLs(2x)Fn�1 = Ln cFs(x + 1) + cFs(x � 1) = cLs(x) sFs(x + 1) + sFs(x � 1) = sLs(x)

n = 2Fn+1 cFs(x) + sLs(x) = 2sFs(x + 1) sFs(x) + cLs(x) = 2cFs(x + 1)

Table 3The hyperbolic properties of the symmetrical hyperbolic Fibonacci and Lucas functions

Classical hyperbolic function Symmetrical hyperbolic Fibonacci function Symmetrical hyperbolic Lucas function

[cosh(x)]2 � [sinh(x)]2 = 1 ½cFsðxÞ�2 � ½sFsðxÞ�2 ¼ 45 [cLs(x)]2 � [sLs(x)]2 = 4

cosh(x ± y) = cosh(x)cosh(y) ±sinh(x)sinh(y)

2ffiffi5p cFsðx� yÞ ¼ cFsðxÞcFsðyÞ � sFsðxÞsFsðyÞ 2cLs(x ± y) = cLs(x)cLs(y) ± sLs(x)sLs(y)

sinh(x ± y) = sinh(x)cosh(y) ±cosh(x)sinh(y)

2ffiffi5p sFsðx� yÞ ¼ sFsðxÞcFsðyÞ � cFsðxÞsFsðyÞ 2sLs(x ± y) = sLs(x)cLs(y) ± cLs(x)sLs(y)

cosh(2x) = [cosh(x)]2 + [sinh(x)]2 2ffiffi5p cFsð2xÞ ¼ ½cFsðxÞ�2 þ ½sFsðxÞ�2 2cLs(2x) = [cLs(x)]2 + [sLs(x)]2

sinh(2x) = 2sinh(x)cosh(x) 1ffiffi5p sFsð2xÞ ¼ sFsðxÞcFsðxÞ sLs(2x) = sLs(x)cLs(x)

½coshðxÞ�ðnÞ ¼ sinhðxÞ for n ¼ 2k þ 1coshðxÞ for n ¼ 2k

�½cFsðxÞ�ðnÞ ¼ ðlnðsÞnsFsðxÞ for n¼ 2kþ1

ðlnðsÞÞncFsðxÞ for n¼ 2k

�½cLsðxÞ�ðnÞ ¼ ðlnðsÞÞnsLsðxÞ for n¼ 2kþ1

ðlnðsÞÞncLsðxÞ for n¼ 2k

�


½sFsðxÞ�2 � cFsðxþ 1ÞcFsðx� 1Þ ¼ �1; ð24Þ½cFsðxÞ�2 � sFsðxþ 1ÞsFsðx� 1Þ ¼ 1: ð25Þ

Clearly, we may consider identities (24) and (25) as generalizations of the Cassini formula to a continuous domain.

3.4. The hyperbolic properties of the symmetrical hyperbolic Fibonacci and Lucas functions

In Table 3, we present some of the hyperbolic properties of the symmetrical hyperbolic Fibonacci and Lucas func-tions. The table shows that the most important identity for the classical hyperbolic functions

½coshðxÞ�2 � ½sinhðxÞ�2 ¼ 1 ð26Þ

has the following counterparts in each of the hyperbolic Fibonacci and Lucas functions, respectively:

½cFsðxÞ�2 � ½sFsðxÞ�2 ¼ 4

5; ð27Þ

½cLsðxÞ�2 � ½sLsðxÞ�2 ¼ 4: ð28Þ

Therefore, Table 3 shows that the symmetric hyperbolic functions have all the properties of the classical hyperbolicfunctions. At the same time, they possess new recurrent properties that are similar to the properties of the Fibonacciand Lucas numbers (Table 2). We see in Eq. (22) that the new hyperbolic functions, in contrast to the classical hyper-bolic functions, have discrete analogs in the form of the Fibonacci and Lucas numbers whenever x has an integer value.Note that identities (24)–(28) and those given in Tables 2 and 3 demonstrate this characteristic.

3.5. Bodnar’s geometry

As is well known, the Fibonacci and Lucas numbers provide the basis for the Phyllotaxis Law [26]. According to thislaw, the ratios of the adjacent Fibonacci numbers represent the number of the left and right spirals on the surface ofphyllotaxis objects (pinecones, pineapples, cacti, the heads of sunflowers, etc.). That is:

F nþ1

F n:

2

1;3

2;5

3;8

5;13

8;21

13; . . .! s ¼ 1þ

ffiffiffi5p

2: ð29Þ

These ratios characterize the symmetry of the phyllotaxis objects. Thus, every phyllotaxis object has its own ratio ofadjacent Fibonacci numbers from figure (29), which is the characteristic of its structure. We call this ratio the order

of symmetry [26].When we observe phyllotaxis objects in the completed form and enjoy the ordered picture on their surface, we

always ask ourselves how Fibonacci’s lattice is formed the surface of these objects during their growth. This problemis the main mystery of phyllotaxis and one of the most intriguing phenomena of botany. The essence of the problem isthe fact that, during their growth period, most types of biologic forms change their orders of symmetry. We know, forexample, that the heads of a sunflower have different orders of symmetry at different heights of the same stalk, thatolder disks have higher orders of symmetry. This means that, during growth, there is a natural change (increase) inthe order of symmetry. This change of symmetry follows the pattern below:


2

1! 3

2! 5

3! 8

5! 13

8! 21

13! � � � ð30Þ

This change in the orders of symmetry of phyllotaxis objects is called dynamic symmetry [26]. Often scientists whoresearch this problem assume that the phyllotaxis phenomenon has fundamental interdisciplinary importance. Forexample, in Vernadsky’s opinion, the problem of biological symmetry is the key problem of the biology.

Thus, the phenomenon of dynamic symmetry (30) has a special role in the geometrical problem of phyllotaxis.Ukrainian researcher Bodnar [26] recently offered the fundamental solution to this problem. Bodnar developed an ori-ginal geometrical theory of phyllotaxis based on the assumption that the geometry of phyllotaxis objects is hyperbolic.He also assumed that the change in the orders of symmetry of phyllotaxis objects during their growth follows the hyper-

bolic turn, the basic transforming movement of hyperbolic geometry.Note that Bodnar primarily uses the golden hyperbolic functions to describe mathematical relations in the new

geometry. The golden hyperbolic functions differ from the symmetrical hyperbolic Fibonacci and Lucas functions onlyin constant coefficients. Thus, Bodnar’s geometry brilliantly and effectively applies the hyperbolic Fibonacci and Lucasfunctions to simulate the growth of phyllotaxis objects. Hence, the hyperbolic Fibonacci and Lucas functions are newand very effective mathematical models of that part of the biological world that considers the phyllotaxis phenomenon.

4. The golden shofar

4.1. The quasi-sine Fibonacci function

Let us consider the following widely used representation of the Binet formula for the Fibonacci numbers [7,8]:

F n ¼sn � ð�1Þns�nffiffiffi

5p ; ð31Þ

where n = {0,±1,±2,±3, . . .}.Comparing the Binet formula in (31) to the symmetric hyperbolic Fibonacci functions in (19) and (20), we see that

the continuous functions sx and s�x in (19) and (20) correspond to the discrete sequences sn and s�n in Eq. (24). Con-sequently, it is possible to replace the alternating sequence (�1)n in Eq. (31) with a continuous function that takes thevalues �1 and 1 whenever x = {0,±1,±2,±3, . . .}. The trigonometric function cos(px) is the simplest. Using this rea-soning, we may now introduce a set of new, continuous functions related to the Fibonacci numbers.

Y

O 1

1

y=sFs(x)

y=cFs(x)

y=FF(x)

Fig. 3. The graph of the quasi-sine Fibonacci function.

Table 4Some identities for the Fibonacci numbers and the quasi-sine Fibonacci function

The identities for Fibonacci numbers The identities for the quasi-sine Fibonacci function

Fn+2 = Fn+1 + Fn: FF(x + 2) = FF(x + 1) + FF(x)F 2

n � F nþ1F n�1 ¼ ð�1Þnþ1 [FF(x)]2 � FF(x + 1)FF(x � 1) = �cos(px)Fn+3 + Fn = 2Fn+2 FF(x + 3) + FF(x) = 2FF(x + 2)Fn+3 � Fn = 2Fn+1 FF(x + 3) � FF(x) = 2FF(x + 1)Fn+6 � Fn = 4Fn+3 FF(x + 6) + FF(x) = 4FF(x + 3)F 2nþ1 ¼ F 2

nþ1 þ F 2n FF(2x + 1) = [FF(n + 1)]2 + [FF(x)]2


Definition 1. The following continuous function is the quasi-sine Fibonacci function:

FFðxÞ ¼ ax � cosðpxÞa�xffiffiffi5p : ð32Þ

The following correlation exists between Fibonacci numbers (Fn), as given by (31), and the quasi-sine Fibonaccifunction given by (32)

F n ¼ FFðnÞ ¼ an � cosðpnÞa�nffiffiffi5p ; ð33Þ

where n = {0,±1,±2,±3, . . .}.The graph of the quasi-sine Fibonacci function is the quasi-sine curve that passes through all points corresponding

to the Fibonacci numbers given by (31) on the coordinate plane (Fig. 3). The symmetric hyperbolic Fibonacci functions(19) and (20) in Fig. 1 bound the quasi-sine Fibonacci function (33).

The quasi-sine Fibonacci function (32) has properties that are similar to the Fibonacci numbers given by (33) (seeTable 4).

Now we can represent the Cassini formula, Eq. (23), in the terms of the quasi-sinusoidal Fibonacci function with thefollowing ‘‘continuous’’ identity:

½FFðxÞ�2 � FFðxþ 1ÞFFðx� 1Þ ¼ � cosðpxÞ:

4.2. The three-dimensional Fibonacci spiral

It is well known that we can define the sine and cosine functions as projections of the translational movement of apoint on the surface of an infinite rotating cylinder with the radius 1 and a center of symmetry that coincides with the

Fig. 4. The three-dimensional Fibonacci spiral.


axis OX. Such a three-dimensional spiral is projected into the sine function on a plane and described by the complexfunction f(x)= cos(x) + i sin(x). If we assume that the quasi-sine Fibonacci function (32) is a projection of the three-dimensional spiral that is on some funnel-shaped surface and reason as we did for the sine curve, it is possible to con-struct a three-dimensional Fibonacci spiral.

Definition 2. The following function is the three-dimensional Fibonacci spiral:

CFFðxÞ ¼ ax � cosðpxÞa�xffiffiffi5p þ i

sinðpxÞa�xffiffiffi5p : ð34Þ

This function, by its shape and having a bent end, reminds one of a spiral drawn on a crater (Fig. 4).

The authors proved the following theorem in [7].

Theorem 1. For the three-dimensional Fibonacci spiral (34) the following equation is valid:

CFFðxþ 2Þ ¼ CFFðxþ 1Þ þ CFFðxÞ:
Note that this is similar to the recurrence relation for the Fibonacci numbers Fn+2 = Fn+1 + Fn.
4.3. The golden shofar

Let us consider the real and imaginary parts of the three-dimensional Fibonacci spiral (34)

Re½CFFðxÞ� ¼ ax � cosðpxÞa�xffiffiffi5p ; ð35Þ

Im½CFFðxÞ� ¼ sinðpxÞa�xffiffiffi5p : ð36Þ

If we consider the axis OY as a real axis and the axis OZ as an imaginary axis, then we get the following system ofequations from (34)–(36):

yðxÞ � axffiffiffi5p ¼ � cosðpxÞa�xffiffiffi

5p ;

zðxÞ ¼ sinðpxÞa�xffiffiffi5p :

8>><>>:

ð37Þ

Let us square both expressions of the equation system (37) and add them. Taking y and z as independent variables,we get the curvilinear surface of the second degree.

Definition 3. The following curved surface of the second degree is the golden shofar:

y � axffiffiffi5p

� �2

þ z2 ¼ a�xffiffiffi5p� �2

: ð38Þ

Fig. 5 is the three-dimensional representation of formula (38).

Fig. 5. The golden shofar.

Fig. 6. The projection of the golden shofar on the plane XOY.


Clearly, the shape of the golden shofar in Fig. 5 reminds one of a horn because of the narrow, bent end. Translatingfrom the Hebrew language, the word ‘‘Shofar’’ means a horn that is a symbol of power or might. The shofar is blown inThe Judgment Day (the Jewish New Year) and the Day of Atonement (the Yom Kippur).

The formula (38) for the golden shofar may take following form:

z2 ¼ ½cFsðxÞ � y�½sFsðxÞ þ y�; ð39Þ

where sFs(x) and cFs(x) are the symmetrical hyperbolic Fibonacci functions (20) and (21), respectively.Fig. 6 shows the projection of the golden shofar onto the plane XOY into the space between the graphs of the sym-

metric hyperbolic Fibonacci sine and cosine. This results in the projection of the three-dimensional Fibonacci spiral (34)onto the quasi-sine Fibonacci function (32). We see that function (32) lies on the golden shofar and ‘‘pierces’’ the planeXOY in the points that correspond to the terms of the Fibonacci sequence (Fig. 6).

Fig. 7. The projection of the golden shofar on the plane XOZ.

Fig. 8. The ‘‘shofarable’’ topology.


Fig. 7 shows the projection of the golden shofar on the plane XOZ and into the space between the graph of the twoexponent functions �s�xffiffi

5p and s�xffiffi

5p .

5. A general model of the hyperbolic space with a ‘‘shofarable’’ topology

In 2004, Aurich and his colleagues published a sensational cosmological article [27]. The article, based on the exper-imental data obtained in 2003 by the NASAs Wilkinson Microwave Anisotropy Probe (WMAP), developed a newhypothesis about the Universe structure. According to [28], the Universe geometry has hyperbolic character and theUniverse under its shape is similar to a horn or a pipe with an extending bell.

Based on the data obtained by the German astrophysicists in [28] authors make the following assumption:

Assumption 1. The Universe has a shofarable topology (Fig. 8).Fig. 8 gives the image of the hyperbolic space with the shofarable topology.The golden section and Fibonacci numbers display the harmony of the universe as a unification of parts in the

whole. In addition, according to works [4–7], the Fibonacci sequence generates a new class of hyperbolic functions thatpossesses all properties of the classical hyperbolic functions and recurrent properties. We can call this synthesis ofharmony, recursion and hyperbolic functions a golden hyperbolic approach. It is clear that the golden hyperbolic

approach that follows from Assumption 1 is of great interest to the modern physics, biology, and cosmology.

6. Conclusion

Consequently, the main result of authors’ works [4–7] is the development of the new mathematical models of thehyperbolic space based on the golden section. The most important golden mathematical models are the following:

1. The hyperbolic Fibonacci and Lucas functions. The first important consequence of their introduction is comprehen-sion of the fact that the classical hyperbolic functions, used in mathematics and theoretical physics, are not uniquemathematical models of the hyperbolic space. Corresponding to the hyperbolic space based on the classical hyperbolicfunctions (Lobachevsky’s hyperbolic geometry, Minkovsky’s geometry, etc.), there is the golden hyperbolic space basedon the hyperbolic Fibonacci and Lucas functions [4–7]. The golden hyperbolic space exists objectively and indepen-dently of our awareness and persistently shows itself in Living Nature. In particular, it appears in pinecones, sunflowerheads, pineapples, cacti, and the inflorescences of various flowers in the form of the Fibonacci and Lucas spirals foundon the surfaces of these biological objects (the phyllotaxis law). Note that the hyperbolic Fibonacci and Lucas functions[4–7], which underlie phyllotaxis phenomena, are not simply due to the ‘‘inventiveness’’ of Fibonacci-mathematiciansbecause they objectively reflect the mathematical laws of the geometry of Living Nature.

2. The golden shofar. The surface of the golden shofar is a new model of the golden hyperbolic space, as are thehyperbolic Fibonacci and Lucas functions. The article ‘‘Hyperbolic Universes with a Horned Topology and theCMB Anisotropy’’ [27] proved to be a great surprise for the authors because it shows that the geometry of the universeis possibly ‘‘shofarable’’ by its structure.


In summary, we might be surprised to find that, for many centuries, mathematicians and theoretical physicists didnot give proper attention to the development of mathematical apparatus for modeling the golden hyperbolic space.However, toward the end of 20th century, with honor due to certain physicists, the situation in theoretical physics beganto change sharply. Articles [28–36] demonstrate the substantial interest of modern theoretical physicists in the goldensection and the physical golden hyperbolic space. The works of Shechtman, Butusov, Mauldin and William, El Naschie,Vladimirov and other physicists show that it is impossible to imagine a future progress in physical and cosmologicalresearch without the golden section. Article [34] is a brilliant example. In this article, Prof. El Naschie shows thatthe golden proportion can express the majority of fundamental physical constants. The famous Russian physicist, Prof.Vladimirov from Moscow University, finishes his book Metaphysics [36] with the following words: ‘‘Thus, it is possibleto assert that in the theory of electroweak interactions there are relations that are approximately coincident with the‘‘Golden Section’’ that play an important role in the various areas of science and art’’.

Acknowledgements

The authors are most grateful to Dr. Pamela Ryan (Milligan College) for help in translating this article into English.

References

[1] Vorobyov NN. Fibonacci numbers. Moscow: Nauka; 1978 [in Russian].[2] Hoggat VE. Fibonacci and Lucas numbers. Palo Alto, CA: Houghton-Mifflin; 1969.[3] Vajda S. Fibonacci and Lucas numbers, and the golden section. Theory and applications. Ellis Horwood Limited; 1989.[4] Stakhov AP, Tkachenko IS. Hyperbolic Fibonacci trigonometry. J Rep Ukr Acad Sci 1993;208(7):9–14 [in Russian].[5] Stakhov AP. Hyperbolic Fibonacci and Lucas Functions: a new mathematics for the living nature. Vinnitsa: ITI; 2003. ISBN 966-

8432-07-X.[6] Stakhov A, Rozin B. On a new class of hyperbolic function. Chaos, Solitons & Fractals 2004;23(2):379–89.[7] Stakhov A, Rozin B. The golden shofar. Chaos, Solitons & Fractals 2005;26(3):677–84.[8] Stakhov AP. Introduction into algorithmic measurement theory. Moscow: Soviet Radio; 1977.[9] Stakhov A, Rozin B. The ‘‘golden’’ algebraic equations. Chaos, Solitons & Fractals 2005;27(5):1415–21.

[10] Stakhov A, Rozin B. Theory of Binet formulas for Fibonacci and Lucas p-numbers. Chaos, Solitons & Fractals2005;27(5):1162–77.

[11] Stakhov A, Rozin B. The continuous functions for the Fibonacci and Lucas p-numbers. Chaos, Solitons & Fractals2006;28(4):1014–25.

[12] Stakhov AP. A generalization of the Fibonacci Q-matrix. J Rep Ukr Acad Sci 1999;9:46–9.[13] Stakhov A. The generalized principle of the golden section and its applications in mathematics, science, and engineering. Chaos,

Solitons & Fractals 2005;26(2):263–89.[14] Stakhov AP. Algorithmic measurement theory. Moscow: Znanie; 1979.[15] Stakhov AP. The golden section in the measurement theory. Comput Math Appl 1989;17(4-6):613–38.[16] Stakhov AP. Generalized golden sections and a new approach to geometric definition of a number. Ukr Math J

2004;56(8):1143–50.[17] Stakhov AP. The golden section and modern harmony mathematics. Appl Fibonacci Numbers 1998;7:393–9.[18] Stakhov AP. Sacral geometry and harmony mathematics. Moscow: Academy of Trinitarism, Electronic publication, No. 77-6567,

published 11176, 26.04.2004.[19] Stakhov AP. Harmony Mathematics as a new interdisciplinary direction of modern science. Moscow: Academy of Trinitarism,

No. 77-6567, publication 12371, 19.08.2005.[20] Stakhov A. Fundamentals of a new kind of mathematics based on the golden section. Chaos, Solitons & Fractals

2005;27(5):1124–46.[21] Stakhov AP. Codes of the golden proportion. Moscow: Radio and Communications; 1984.[22] Stakhov AP. Brousentsov’s ternary principle, Bergman’s number system and ternary mirror-symmetrical arithmetic. The Comput

J 2002;45(2):222–36.[23] Stakhov AP. Brousentsov’s ternary principle, Bergman’s number system and the ‘‘golden’’ ternary mirror-symmetrical arithmetic.

Moscow: Academy of Trinitarism, No. 77-6567, publication 12355, 15.08.2005 [in Russian].[24] Stakhov A, Massingue V, Sluchenkova A. Introduction into Fibonacci coding and cryptography. Kharkov: Osnova; 1999, ISBN

5-7768-0638-0.[25] Stakhov AP. Data coding based on the Fibonacci matrices. In: Proceedings of the international conference problems of harmony,

symmetry and the golden section in nature, science, and art, Vinnitsa: Vinnitsa State Agricultural University 2003:311–25.[26] Bodnar OY. The golden section and non-Euclidean geometry in Nature and Art. Lvov: Svit 1994 [in Russian].


[27] Aurich R, Lusting S, Steiner F, Then H. Hyperbolic universes with a horned topology and the CMB anisotropy. Classical QuantGrav 2004;21:4901–26.

[28] Gratia D. Quasi-crystals. Usp physicheskich nauk 1988;56:347–63 [in Russian].[29] Butusov KP. The golden section in the solar system. Prob issledovania vselennoi 1978;7:475–500.[30] Mauldin RD, Willams SC. Random recursive construction. Trans Am Math Soc 1986;295:325–46.[31] El Naschie MS. On dimensions of cantor set related systems. Chaos, Solitons & Fractals 1993;3:675–85.[32] El Nashie MS. Quantum mechanics and the possibility of a cantorian space–time. Chaos, Solitons & Fractals 1992;1:485–7.[33] El Nashie MS. Is quantum space a random cantor set with a golden mean dimension at the core? Chaos, Solitons & Fractals

1994;4(2):177–9.[34] El Naschie MS. Elementary number theory in superstrings, loop quantum mechanics, twistors and E-infinity high energy physics.

Chaos, Solitons & Fractals 2006;27(2):297–330.[35] Vladimirov YS. Quark icosahedron, charges and Vainberg’s angle. In: Proceedings of the international conference problems of

harmony, symmetry and the golden section in nature, science, and art, Vinnitsa: Vinnitsa State Agricultural University 2003; p.69–79.

[36] Vladimirov YS. Metaphysics. Moscow: Binom; 2002.

The ''golden'' hyperbolic models of Universe - Student Oulu

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