Applied Mathematics, 2011, 2, 283-293
doi:10.4236/am.2011.23033 Published Online March 2011 (http://www.scirp.org/journal/am)
Hyperbolic Fibonacci and Lucas Functions, “Golden”
Fibonacci Goniometry, Bodnar’s Geometry, an d
Hilbert’s Fourth Problem
—Part III. An Original Solution of Hilbert’s Fourth Problem
Alexey Stakhov1,2, Samuil Aranson3
1International Higher Education Academy of Sciences, Moscow, Russia
2Institute of the Golden Section, Academy of Trinitarism, Moscow, Russia
3Russian Academy of Natural History, Moscow, Russia
Received June 25, 201 0; revised November 15, 2010; accepted November 18, 2010
Abstract
This article refers to the “Mathematics of Harmony” by Alexey Stakhov [1], a new interdisciplinary direction
of modern science. The main goal of the article is to describe two modern scientific discoveries—New Geo-
metric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic
Fibonacci and Lucas Functions and “Golden” Fibonacci

-Goniometry (

 is a given positive real
number). Although these discoveries refer to different areas of science (mathematics and theoretical botany),
however they are based on one and the same scientific ideas—The “golden mean,” which had been intro-
duced by Euclid in his Elements, and its generalization—The “metallic means,” which have been studied
recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary cha-
racter of the “Mathematics of Harmony”, which originates from Euclid’s Elements.
Keywords: Euclid’s Fifth Postulate, Lobachevski’s Geometry, Hyperbolic Geometry, Phyllotaxis, Bodnar’s
Geometry, Hilbert’s Fourth Problem, the “Golden” and “Metallic” Means, Binet Formulas,
Hyperbolic Fibonacci and Lucas Functions, Gazale Formulas, “Golden” Fibonacci

-Goniometry
1. Introduction
In Part III we study Hilbert’s Fourth Problem, concern-
ing to hyperbolic geometry, from new point of view-the
“Golden” Fibonacci

-Goniometry (

 is given
positive real number). This goniometry is based on a new
class of hyperbolic functions-hyperbolic Fibonacci and
Lucas

-functions [2,3], which are connected with the
“metallic” means and Gazale formulas. The main result
of this study is a creation of infinite set of the golden
isometric

-models of Lobachevski’s plane that is di-
rectly relevant to Hilbert’s Fourth Problem. Also we
discuss a connection between Poincare’s model of Lo-
bachevski’s plane on the unit disc and the golden

-models of Lobachevski’s plane. This study can be
considered as an unexpected variant of Hilbert’s Fourth
Problem solution based on the “metallic means” [4],
which are a generalization of the “golden mean” (Theo-
rem II.11 of Euclid’s Elements).
2. Euclid’s Fifth Postulate and
Lobachevski’s Geometry
On February 23, 1826 on the meeting of the Mathematics
and Physics Faculty of Kazan University the Russian
mathematician Nikolai Lobachevski had proclaimed on
the creation of new geometry named imaginary geometry.
This geometry was based on the traditional Euclid’s
postulates, excepting Euclid’s Fifth Postulate about par-
allels. New Fifth Postulate about parallels was formu-
lated by Lobachevski as follows: “At the plane through a
point outside a given straight line, we can conduct two
and only two straight lines parallel to this line, as well as
an endless set of straight lines, which do not overlap with
this line and are not parallel to this line, and the endless
set of straight lines, intersecting the given straight line.”
For the first time, a new geometry was published by Lo-
bachevski in 1829 in the article About the Foundations
of Geometry in the magazine Kazan Bulletin.

A. STAKHOV ET AL. 284
Independently on Lobachevski, the Hungarian mathe-
matician Janos Bolyai came to such ideas. He published
his work Appendix in 1832, that is, three years later Lo-
bachevski. Also the prominent German mathematician
Carl Friedrich Gauss came to the same ideas. After his
death some unpublished sketches on the non-Euclidean
geometry were found.
Lobachevski’s geometry got a full recognition and wide
distribution 12 years after his death, when it is became
clear that scientific theory, built on the basis of a system
of axioms, is considered to be fully co mpleted only when
the system of axioms meets three conditions: indepen-
dence, consistency and completeness. Lobachevski’s ge-
ometry satisfies these conditions. Finally this became
clear in 1868 when the Italian mathematician Eugenio
Beltrami in his memoirs The Experience of the Non-
Euclidean Geometry Interpretation showed that in Euc-
lidean space R3 at pseudospherical surfaces the geometry
of Lobachevski’s plane arises, if we take geodesic lines
as straight lines.
Later the German mathematician Felix Christian Klein
and the French mathematician Henri Poincare proved a
consistency of Non-Euclidean geometry, by means of the
construction of corresponding models of Lobachevski’s
plane. The interpretation of Lobachevski’s geometry on
the surfaces of Euclidean space contributed to general
recognition of Lobachevski’s ideas.
The creation of Riemannian geometry by Georg Frie-
drich Riemann became the main outcome of such Non-
Euclidean approach. The Riemannian geometry devel-
oped a mathematical doctrine about geometric space, a
notion of differential of a distance between elements of
diversity and a doctrine about curvature. The introduction
of the generalized Riemannian spaces, whose particular
cases are Euclidean space and Lobachevski’s space, and
the so-called Riemannian geometry, opened new ways in
the development of geometry. They found their applica-
tions in physics (theory of relativity) and other branches
of theoretical natural sciences.
Lobachevski’s geometry also is called hyperbolic geo-
metry because it is based on the hyperbolic functions (1.8)
(see Part I) introduced in 18th century by the Italian ma-
thematician Vincenzo Riccati.
The most famous classical isometric interpretations of
Lobachevski’s plane with the Gaussian curvature K = –1
are the following:
 Beltrami’s interpretation on a disk;
 Beltrami’s interpretation of hyperbolic geometry
on pseudo-sphere;
 Euclidean model by Keli-Klein;
 Projective model by Keli-Klein;
 Poincare’s interpretation at a half-plane;
 Poincare’s interpretation inside a circle;
 Poincare’s interpretation on a hyperboloid.
In particular, the classical model of Lobachevski’s
plane in pseudo-spherical coordinates

,,0 ,uv u


v

 with the Gaussian curvature 1K


(Beltrami’s interpretation of hyperbolic geometry on
pseudo-sphere) has the following form:
 
22 2
2
dsdush u dv , (3.1)
where ds is an element of length and sh(u) is hyperbolic
sine.
Lobachevski’s geometry has remarkable applications
in many fields of modern natural sciences. This concerns
not only applied aspects (cosmology, electrodynamics,
plasma theory), but, first of all, it concerns the most fun-
damental sciences and their foundation—Mathematics
(number theory, theory of automorphic functions created
by A. Poincare, the geometry of surfaces and so on).
Since on the closed surfaces of negative Gaussian
curvature, Lobachevski’s geometry is fulfilled and Lo-
bachevski’s plane is universal covering for these surfaces,
it is very fruitful to study v arious objects (dynamical sys-
tems with continuous and discrete time, layers, fabrics
and so on), defined on these surfaces. By developing this
idea, we can raise these objects to the level of universal
covering, which is replenished by the absolute (“infini-
ty”), and further we can study smooth topological prop-
erties of these objects with the help of the absolute.
Samuil Aranson studied this problem about four dec-
ades. The works [5-10] written by Samuil Aranson with
co-authors give presentation about these results and re-
search methods. Aranson’s DrSci dissertation “Global
problems of qualitative theory of dynamic systems on
surfaces” (1990) is devoted to this themes.
3. Hilbert’s Fourth P ro b l em
In the lecture “Mathematical Problems” presented at the
Second International Congress of Mathematicians (Paris,
1900), David Hilbert had formulated his famous 23 ma-
thematical problems. These problems determined consi-
derably the development of the 20th century mathematics.
This lecture is a unique phenomenon in the mathematics
history and in mathematical l iterature. Th e Russian trans-
lation of Hilbert’s lecture and its comments are given in
the work [11]. In particular, Hilbert’s Fourth Problem
asserts:
“Whether is possible from the other fruitful point of
view to construct geometries, which with the same right
can be considered the nearest geometries to the tradi-
tional Euclidean geometry”.
Note, Hilbert considered that Lobachevski’s Geome-
tryand Riemannian geometry are nearest to the Euclidean
geometry. In [12] the history of the Hilbert’s Fourth

A. STAKHOV ET AL. 285
Problem solution and some approaches to its solution are
described. Also Hilbert’s understanding of the Fourth
Problem is discussed. It is clear that in mathematics, Hil-
bert’s Fourth Problem was a fundamental problem in ge-
ometry. In the citation, taken from Hilbert’s original, we
found the following description of Hilbert’ Fourth Pro-
blem: “the problem is to find geometries whose axioms
are closest to those of Euclidean geometry if the ordering
and incidence axioms are retained, the congruence axi-
oms is weakened, and the equivalent of the parallel
postulate is omitted.”
In mathematical literature Hilbert’s Fourth Problem is
sometimes considered as formulated very vague what
makes difficult its final solution [12]. In [13] American
geometer Herbert Busemann analyzed the whole range of
issues related to Hilbert’s Fourth Problem and also con-
cluded that the question related to this issue, unnecessa-
rily broad. Note also the book [14] by Alexei Pogorelov
devoted to partial solution to Hilbert’s Fourth Problem.
The book identifies all, up to isomorphism, implementa-
tions of the axioms of classical geometries (Euclid, Lo-
bachevski and elliptical), if we delete the axiom of con-
gruence and refill these systems with the axiom of “tri-
angle inequality.”
In spite of critical attitude of mathematicians to Hil-
bert’s Fourth Problem, we should emphasize great im-
portance of this problem for mathematics, particularly
for geometry. Without doubts, Hilbert’s intuition led h im
to the conclusion that Lobachevski’s geometry and Rie-
mannian geometry do not exhaust all possible variants of
non-Euclidean geometries. Hilbert’s Fourth Problem di-
rects attention of researchers at finding new non-Eucli-
dean geometries, which are the nearest geometries to the
traditional Euclidean geometry.
In this connection, a discovery of new class of hyper-
bolic functions based on the “golden mean” and “metal-
lic means” [2,3,15] and following from them new geo-
metric theory of phyllotaxis (Bodnar’s geometry) [16]
have a principal importance for the development of geo-
metry because it shows an existence of new non-Eucli-
dean geometries in surrounding us world. Recently Ale-
xey Stakhov gave a wide generalization of the symmetric
hyperbolic Fibonacci and Lucas functions (1.9) and (1.10)
(see Part I) and developed the so-called hyperbolic Fi-
bonacci and Lucas

-functions [3]. It is proved in [3]
an existence of infinite variants of new hyperbolic func-
tions, which can be a base for new non-Euclidean geo-
metries.
The main purpose of Part III of the article is to devel-
op this idea, that is, to create new non-Euclidean geome-
tries based on the hyperbolic Fibonacci and Lucas

-
functions introduced in [3]. This study can be considered
as unexpected and original solution of Hilbert’s Fourth
Problem based on the the “metallic means” [4]. The au-
thors of this article announced this idea in [17]. In Part
III of the article we give a detailed proof of this idea.
4. The “Golden” Fibonacci λ-Goniometry
and Hilbert’s Fourth Problem
4.1. “Golden” Metric λ-Forms of
Lobachevski’s Plane
In connection with Hilbert’s Fourth Problem the au-
thors of the present article Alexey Stakhov and Samuil
Aranson suggested in [17] infinite set of metric forms of
Lobachevski’s plane in dependence on real parameter

. These metric forms are given in the coordinates


,,0 ,uv uv

 ; they have the Gaussian
curvature K= –1 and can be represented in the form:
    
22
22 2
24
ln 4 dsdusF udv

 
 
, (3.2)
where 2
4
2
λ



 is the “metallic mean” and


s
Fu

is hyperbolic Fibonacci

-sine. Let us name
the forms (3.2) metric

-forms of Lobachevski’s plane.
In [17] we asserted (without proof) that for any real
parameter

 the metric forms (3.2) are isometric on
the base of diffeomorphisms
 
 
2
2
4
,2
4
,,
2
uuu vArcchcFu
A
rcshsFuvvu vv

















(3.3)
to the classical metric forms of Lobachevski’s plane (3.1)
in semi-geodesic coordinates (u,v),0u,
v

 .
Since the forms of the kind (3.1) are isometric to all
previously known classical metric forms of Lobach-evs-
ki’s plane what is noted, for instance, in [18], th en it fol-
lows from here that the forms (3.2) are isometric to all
these classical forms.
Here we give direct proof of isometrics of the form
(3.2) and (3.1).
Next we describe the basic geometric objects so, for
instance, as geodesic lines and intersection angles, which
are induced by the form (3.2).
In the concluding part for completeness of presenta-
tion it is shown also isometrics between Poincare’s
model of Lobachevski’s plane on unit disc and the form
(3.2). From here it is easy to deduce for the form (3.2)
the formula for distance and also the formula for metrics
movement.

A. STAKHOV ET AL. 286
The proof of isometrics of the forms (3.2) and (3.1) is
fulfilled in tree steps.
Step 1. Let us prove that for the metric form (3.2)
Gaussian curvature 1K . For this purpose let us in-
troduce the following designations:
 
2
4
ln ,2
A
BsFu

 
  (3.4)
Here, according to (1.3 5) (see Part I),

2
4
uu
sF u





, therefore for the second correlation
in (3.4) we have:

2
4
22
uu
BsFu




 

(3.5)
where, by virtue (1.24) (see Part I),
2
4
2




 . Then, the expressions (3.4) can be
written in the form:


2
2
4
ln ln,
2
4.
22
uu
A
BsFu








 


 

(3.6)
Therefore, the metric form (3.2) can be written in the
form:
 
222
22
dsA duBdv. (3.7)
Taking into consideration the expression (3.6) and the
obvious conditions:
0

, 0u, v  ,
We can write: 0A, 0B. (3.8)
It is known [18] that for the metric form of the kind
(3.1) Gaussian curvature K is determined from the cor-
relation:
1
(,)
vu
vu
KKuvAB
AB
BA


 





. (3.9)
Here the symbols



and
vu
mean partial deriva-
tives on v and u.
By using definition (3.6), we can get the following
expressions:
 
2
0;ln ;ln;
22
uu uu
vu uu
AB B
 


 
 

ln ;0;
2
uu
v
v
A
AB B
 

 



 
ln ;
ln 2
uu
uuu
u
BB
A
 




 
 

1vu
vu
AB
KAB BA



 






 
1
ln ln
22
uu uu
 




 
 
 

 

 1


Step 2. Let us prove that the transformations (3.3) can
be written in the form:

ln ,uuvv




 (3.10)
Since


2
2
4
22
4
and ,
22
uu
uu
cF u
sF u








 
 
(3.11)
then, by virtue (3.11), from the transformations (3.3) for
the cases u>0, 0uuwe get:
 
and .
22
uu uu
ch ush u
 

 
 (3.12)
Take the differential d of the first correlation in (3.12):





2
ln2
uu
uu
dchu shudu
d
du





 






 

(3.13)
Since

2
uu
sh u





, then after substituting this
expression into (3.13) we get:






ln .
s
hu dushu du

 
 Since 0u, then after
the reduction by


s
hu we come to differential equa-
tion:

ln
du
du


. (3.14)
Hence we have:

lnuuC





, (3.15)
where C = const. Since in (3.3) we have u= 0 at u = 0,
then in (3.15) we can assume C = 0, and therefore we get
from (3.15) that


ln .uu


 

 Then at all
0,u

 v，
0,u

 v  (3.16)
instead (3.3) we can consider the transformations (3.10).
Step 3. Let us prove that the metric forms (3.1) and
(3.2) are isometric. For the proof we use the transforma-

A. STAKHOV ET AL. 287
tions (3.10), which is analytical diffeomorphism at the
values of variables given by (3.16).
With this purpose let us consider more general situa-
tion, when two isometric metric forms are given:
   
 
22 2
22 2
,2, ,
,2, ,
dsEuvduFuvdudvGuvdv
dsEuvduFu vdudvGuvdv
 

 


(3.17)
where


22
0, 0,0, 0, 0,0EGEGF EGEGF   ,
here isometrics is realized by using the diffeomorphism:
 
:,, ,.hu uuvv vuv (3.18)
Let us consider differential of u:


,
and ,
uv
uv
duduuvuduu dv
dvdvuvvduv dv
 

 

. (3.19)
Substitute (3.19) into (3.17) and note that according to
our assumption the first and second metric forms in (3.17)
are isometric. Taking into consideration that they have
common linear element ds, we get the following identity:



 
2
2
2
22 2
2
2
uv uv
uv uv
dsEuduudvFudu udv
v duvdvGv duvdv
dsEduF dudv Gdv
  


(3.20)
By equating in the left-hand and right-hand parts of
the identity (3.20) the equal coefficients at (du )2, du dv
and (dv)2, we get the following correlations:
 
 
22
22
2
2
uuuu
uvuv vuuv
v
vvv
Euuv vE
F
uuuu uuvvF
GG
uuvv





 


 

 

(3.21)
Also the back statement is correct, that is, if we have
two metric forms
   
 
22 2
22 2
,2, ,
,2, ,
dsEu vduFu vdudvGu vdv
dsEuv duFuvdudvGuv dv
 

 


(3.22)
and there is the diffeomorphism (3.18) so that the corre-
lations (3.21) are fulfilled, then
 
22
ds ds, that is,
the metric forms (3.22) are isometric.
In our situation we have two metric forms (3.1) and
(3.2); here, as is shown in correlation (3.6), the metric
form (3.2) can be rewritten in the form (3.7), whose
coefficients 2
A
and 2 Bare equal:

2
22 2
ln ,2
uu
AB





 


.
Let us prove isometrics of the forms (3.1) and (3.7):






   
22 2
2
2
22 2
2
ln 2
uu
dsdushu dv
ds dudv








 



(3.23)
This isometrics is proved by using the analytical dif-
feomorphism (3.10):





,ln ,,,uuuvuvvuv v





which, as is shown (step 2 ), can be rewritten in the form
(3.3).
Note that the area of parameters and variables change
has the following range:
0,(0 ,),
(0 ,)
uv
uv


    (3.24)
where, in virtue of (1.24) (Part I), we have:
2
4
2




 .
Then, in terms of the correlations (3.17) for the metric
forms (3.23) we get the following expressions for the
coefficients of these forms:
   
   
2
2
2
,1,,0,, ,
,ln ,,0,,2
uu
EuvFuvGuvshu
EuvFuv Guv








 



(3.25)
From the transformations (3.10) we get the following
derivatives:


ln,0,0, 1
uvuv
uuvv


 (3.26)
Then, with regard to (3.25) and (3.26), the transforma-
tion (3.21) can be written in the form:
 

22
2
2
lnln001
00000
001
2
uu
s
hu







 
  

 

 
 
 





(3.27)
From here we get the following identities:
 

22
2
2
ln ln
00
2
uu
s
hu






 



 








(3.28)

A. STAKHOV ET AL. 288
The first two identities from (3.28) are obvious. The
last identity from (3.28) follows from the second correla-
tion in (3.12), which is proved in step 2 at

ln ,uu




where>0 and >0λu .
Thus, by using the transformations (3.10), we have
proved that the metric forms (3.1) and (3.2 ) a re isom etric.
Hence, at all 0,(0 ,)uv

 , the
golden metric

-forms of Lobachevski’s plane of the
kind (3.2) are isometric to all previous known isometric
between themselves metric forms of Lobachevski’s plane.
4.2. Partial Cases of the Golden Metric
λ-Forms of Lobachevski’s Plane.
1) The golden metric form of Lobachevski’s plane
For the case 1

 we have115 1.61803
2

 —
the golden mean, and hence the form (3.2) is reduced to
the following:
  
2
22 2
25
ln 4 dsdusFs udv

  
 (3.29)
where

22
115
ln ln0.231565
2


 


 and

11
5
uu
sFs u


 is symmetric hyperbolic Fibonacci
sine (see Part I).
Let us name the metric form (3.29) the golden metric
form of Lobachevski’s plane.
2) The silver metric form of Lobachevski’s plane
For the case 2

 we have 2122.1421 —
The silver mean, and hence the form (3.2) is reduced to
the following:
 
2
22 2
22
ln 2dsdusF udv

  
, (3.30)
where

22
ln 0.776819 and

22
222
uu
sF u


.
Let us name the metric form (3.30) the silver metric
form of Lobachevski’s plane.
3) The bronze metric form of Lobachevski’s plane
For the case 3

 we have
3313
3.30278
2

  —the bronze mean, and hence
the form (3.2) is reduced to the following:
    
2
222
23
13
ln 4 dsdusFudv

  
 (3.31)
where

23
ln 1.42746 and

33
313
uu
sF u


.
Let us name the metric form (3.31) the bronze metric
form of Lobachevski’s plane.
4) The cooper metric form of Lobachevski’s plane
For the case 4


we have
425 4.23607 —The cooper mean, and hence
the form (3.2) is reduced to the following:
 
2
22 2
24
ln 5dsdusFudv

  
, (3.32)
where


24
ln 2.08408 and

44
4.
25
uu
sF u



Let us name the metric form (3.32) the cooper metric
form of Lobachevski’s plane.
5) The classical metric form of Loba chev sk i’s p lan e in
semi-geodesic coordinates. For the case


21
e
s
h  
2.350402

. we have 2.7182
ee


 -Napier number,
and hence the form (3.2) is reduced to the expression
(3.1), which gives classical metric form of Lobachevski’s
plane in semi-geodesic coordinates

,uv , where
0u

, v

 .
4.3. Geodesic Lines of the Golden Metric
λ-Forms of Lobachevski’s Plane and Other
Geometric Objects.
Geodesic lines and angles between these geodesic lines
are one of basic geometric concepts of inner geometry. If
metric form is given, then geodesic lines are determined
as extremums of functional of curve length.
We proved above (see step 3) that the golden metric

-forms of Lobachevski’s plane of the kind (3.2) coin-
cide with the metric forms of (3.7). For convenience, we
introduce new designations for these forms:
 
2
22 2
2
dsAduBudv 
 , (3.33)
where
 
2
ln 0,0,
2
4,0,0 ,
2
uu
ABu
uv







 


   
(3.34)
It is easy to prove (see also the formula (3.12)), that
for the conditions (3.34) we have:
 
 
0,
2
0
2
uu
uu
shAuBu
chAuC u










(3.35)
Let us consider three-dimensional Minkowski’s space


3,,LXYZ with Minkowski’s metrics

A. STAKHOV ET AL. 289

2222
dldXdY dZ, (3.36)
where dl is linear element of the space 3 L. Now let us
consider the upper half 2
M
of the two-sheet hype rboloid:
22 2
1, 0XYZ X . (3.37)
The surface 2
M
is given in implicit form. We can
give the surface 2
M
, if we fulfill the following special
parameterization:
 
 
 
,
,coscos
,sinsin
XXuv chAu Cu
YYuvshAu vBu v
Z
Zuv shAuv Buv


 

 

(3.38)
By direct testing, we can verify that
 
222
,,,1,,0.XuvY uvZ uvX uv  

(3.39)
Let us substitute (3.38) into the correlation (3.36); then
on the upper half 2
M
of two-sheet hyperboloid we get
the metric form:
 

2
22 2
2
dlAduBudv
 , (3.40)
where A and B(u) has a form (3.34). From here we get
the golden metric

-form of Lobachevski’s plane of the
kind (3.3 3):
 
2
22 22
2
dsdlAduBudv 

Let us consider in the space

3,,LXYZ the fol-
lowing planes

222
0, 0aX bY cZabc , (3.41)
which pass through the coordinate origin

0,0,0O and
intersect the upper half of the two-sheet hyperboloid
(3.37), if the coefficients of the equation (3.41) satisfy to
the following restriction:
222
0abc. (3.42)
Then the intersection lines of the planes (3.41) with
the surface (3.37) are geodesic lines on the surface (3.37)
in the metrics (3.36) (see [17]). This is analogous to the
case, when at the unit sphere 222 2
:1SX YZ the
intersection lines of the planes 0aXbY cZ
(where 222
0abc ) with this sphere are geodesic
lines in the metrics of constant Gaussian curvature
1
K
.
If we substitute (3.38) into (3.41), then in the golden

-metrics (3.33) in the coordinates


,uv we get the
following equation of geodesic lines in the following
implicit form:






cossin 0,ach Aubsh Auvcsh Auv

 (3.43)
where 0,(0 ,)uv

,

2
4
ln 0,1
2
A

 

 .
Note that coefficients a, b, c in (3.43) satisfy to the re-
strictions:
222
0abc

 , 222 0abc . (3.44)
Let us rewrite (3.43) in the form:
  
,cossin0.
uu
uu
Fuvabvcv







 (3.45)
Let


00
,uv be coordinates of the intersection point
of two geodesic lines given the equations:
 
  
111 1
222 2
,cossin0
,cossin0
uu
uu
uu
uu
Fuvabv cv
Fuvabv cv



















The angle

of intersection of two geodesic lines
(counted counter-clockwise), according to the formulas
of differential geometry, can be found from the correla-
tion:

21212
1112 1222
FF FF
EG Fdudvdv du
tg
F
FFFFFFF
EF G
du dvdudvdvdududv

 











(3.46)
where the right-hand parts in (3.46) are taken in the point


00
,uv and


,EEuv,

,
F
Fuv,


,GGuv
are coefficients of the metric form

22 2
2dsE dvFdudvGdv.
Note that the formulas for

sin

and


cos

are
written by analogy.
In our situation, acco rd ing to (3.2 ), (3. 4), (3 .33 ), (3. 34 ),
we have:




2
2
222
ln, 0,
4ln
22
uu
EF
GsFushu

 


 


 


  






.
(3.47)
Partial derivatives on u and v of the function


,
F
uv
of the kind (3.45) have the fo llowing fo rms:



 
2
ln ,sincos
ln
FF
abvcv
uv
ch u





 

 (3.48)

A. STAKHOV ET AL. 290
Further by analogy on Lobachevski’s plane, provided
by the golden metric

-forms (3.2), at any real number
0

 we can find corresponding formulas for distances
between two points, transformations of movement, and
all other mathematical objects, inherent in this remarka-
ble geometry.
The authors of the present article do not pursue a goal
to write out all corresponding formulas and geometric
constructions, which are connected with the golden me-
tric

-forms of Lobachevski’s plane because this prob-
lem is a subject of separate study.
In this conn ection, it would be very fruitful to unite in
further the developed by the authors golden metric

-forms of Lobachevski’s plane of the kind (3.2), rea-
lized at any real number

on the half-plane

,0 ,uvuv

  and having Gau-
ssian curvature 1K , with the well-known studied
and convenient for applications classical model of Lo-
bachevski’s plane, suggested in 1882 by Great French
mathematician, physicist, and astronomer Henry Poin-
care on a disc
22 2
:1Dx y
, (3.49)
completed by the absolute 22 :1Ex y, which plays a
role of a carrier of the infinitely distant points of Loba-
chevski’s plane.
4.4. Poincare’s Model of Lobachevski’s Plane on
the Unit Disc
Let us remind the basic facts of Lobachevski’s geometry
for Poincare’s realization on a disc (3.49). Information is
taken from [18].
Poincare’s metric form of Gaussian curvature
1
K
 h as th e following form:
  

22
2
2
22
4
1
dx dy
ds
xy



 . (3.50)
Geodesic planes for Poincare’s model are or circle
arcs, which are orthogonal to absolute (if these geodesic
lines do not contain the coordinate origin


0.0O) or
segments of right lines (if these geodesic lines pass
through the coordinate origin).
In general case the geodesic lines equation in Poin-
care’s model has the following form:


22 22
,1220, 1,Fxy axybxcyx y 
(3.51)
where 222 222 0, 0.abc abc 
The angle

of intersection of two geodesic lines,
which is counted counter-clockwise, is determined from
the correlations:

12 12
112 2
F
FFF
x
yyx
tg
F
FFF
x
yxy












(3.52)
where the right-hand parts in (3.52) is taken in the point


00
,
x
y being a common point of the geodesic lines
intersection:





22
11 11
22
22 22
,1 220
,1 220
Fxyaxybx cy
Fxyaxybx cy


 



 


(3.53)
Thus, in the metrics (3.50) the angles are measured in
Euclidean sense.
Let




1112 22
,and ,
A
xyAxy be arbitrary points of
Lobachevski’s plane, which is realized in the form of a
circle (3.49) with the metrics (3.50).
We use complex numbers further. We designate the
point


,
A
xy by zxiy

, where 1i is im-
aginary unit. A module of the complex number z is equal
to 2
2
z
y
x

. Let zxiy

 be a complex number
conjugate to the complex number zxiy.
For this case the points



111222
,and,
A
xyAx yin
complex notation can be represented as follows:
1112 22
,zxiy zxiy

. It is well-known that a dis-
tance


12
,
A
A between two points

111
,
A
xy and


222
,
A
xy in complex notation has the following form:

12
2
1
12
12
2
1
1
,ln
1
zz
zz
AA
zz
zz














(3.54)
In complex notation the metrics (3 .50) has the follow-
ing form :


2
2
2
4,1.
1
dsdzdz z
z



(3.55)
The movement of the metrics (3.55) of Lobachevski’s
plane is written as follows:

22
,1,
Az B
zfzA B
Bz A


 
 (3.56)
where andzxiyz xiy

. Note that at the move-
ments (3.56) the distances between points and angles
between geodesic lines are kept.
4.5. Connection Between Poincare’s Model of
Lobachevski’s Plane in the Unit Disc and the
Golden λ-Models of Lobachevski’s Plane
It is proved in [18] that the upper half 2
M
of the two-

A. STAKHOV ET AL. 291
sheet hyperboloid (3.37) assumes a parameterization of
the following kind:

22
2
,1,
1
XXxy xy  



22
22
2
,,
1
2
,,
1
x
YYxy xy
y
ZZxy
x
y




(3.57)
where 22 1.xy
Direct calculation shows that that if we substitute
(3.57) into the correlation (3.36), then we get a kind of
the metrics (3.50) on 2
M
in coordinates


,
x
y, that is,
the metrics of Poincare’s model of Lobachevski’s plane
in the unit disc 22 1xy of the kind
  

22
22
2
22
4,
1
dx dy
ds dl
xy





 
where dl is a linear element of the kind (3.36).
Thus, the transformations (3.57) result in Poincare’s
model on the unit disc; we have described its basic prop-
erties in Section 3.4 of this Part III.
In order to pass from Poincare’s model of Lobachevs-
ki’s plane in the unit disc 22
1xy
to the golden

-models of Lobachevski’s plane in the half-plane
0,uv , we introduce another para-
meterization of the upper half 2
M
of the two-sheet
hyperboloid (3.37), connected with the previous parame-
terization (3.57) by the following correlations:
 
  
 
22
22
22
2
,1
1
2
,cos
1
2
,sin
1
X
XxychAu
xy
x
YYxyshAu v
xy
x
Z
ZxyshAuv
xy
 


 


 


, (3.58)
where

2
4
ln0,, 0
2
A




  (3.59)
We have proved in Section 3.3 that if we consider the
upper half of 2
M
of the two-sheet hyperboloid (3.37) in
the form (3.38):
 




,cos,sin,
X
chAu YshAuvZshAuv
then we directly came to the golden

-forms of Loba-
chevski’s plane of the kind (3.2) or, in another notation,
of the kind (3.33 ):
 
2
22 2
2,dsAduBudv


where

0, 0,.
2
uu
Buu v




 (3.60)
According to the formulas (3.35), at the condition
0u

 the following correlations are valid:

0, 0.
22
uu uu
sh Auch Au
 

 

(3.61)
Then, from the correlations (3.58) and (3.61) we get
directly the following connection between parameters




,and,:
x
yuv



  
 
 
,cos cos
12
,sin sin
12
uu
uu
uu
uu
sh Au
x
xuv vv
ch Au
sh Au
yyuv vv
ch Au









 



 
 

(3.62)
Note that at any 0, 0u

 the transforma-
tions (3.62) are diffeomorphisms, because their jacobian
is not equal to 0, and they establish connection between
the golden

-models of Lobachevski’s plane in the co-
ordinates 0,uv

  and the classical
Poincare’s model of Lobach evski’s plane in the unit disc
22
1xy

.
Most in all, the transformations (3.62) establish an
isometry between Poincare’s metric form (3.50) and the
golden metric

-forms of the kind (3.2) or, in another
notation, of the kind (3.33).
By using the transformations (3.62) and the formulas
(3.54) and (3.56), for the golden

-models of Lobache-
vski’s plane in the coordinates0,u v ,
for every real 0

 we can got the formula for the dis-
tance between two arbitrary points



112 2
,and,uvu v
and the formula of metrics movement (3.2), that is, for
the metrics (3.33).
Thus, the main result of Part III of the article is an un-
expected variant of Hilbert’s Fourth Problem solution
based on the the “metallic mean s” [4], which is a genera-
lization of the “golden mean” (Theorem II.11 of Euclid’s
Elements).
5. General Conclusion to the Article
5.1. Euclid’s Fifth Postulate, Hyperbolic
Geometry, and Hilbert’s Fourth Problem
A study of Euclid’s Fifth Postulate led in 19th cen tury to
Lobachevski’s geometry, which can refer to one of the
greatest mathematical discoveries of 19th century. Lo-
bachevski’s hyperbolic geometry can be considered as a
break-through of hyperbolic ideas into mathematics and
theoretical physics. Interest in hyperbolic geometry,

A. STAKHOV ET AL. 292
which was studied during 19th cen tury by many outstan-
ding mathematicians Bolyai, Gauss, Beltrami, Klein,
Poincare, Riemann, increased in the end of 19th century
so much that Great mathematician David Hilbert in his
famous lecture Mathematical Problems (Second Interna-
tional Congress of Mathematicians, Paris, 1900) [11] was
forced to include problems of hyperbolic geometry to the
list of the most important 23 Mathematical Problems.
We are talking about Hilbert's Fourth Problem, which
sounds: “Whether is possible from the other fruitful point
of view to construct geometries, which with the same
right can be considered the nearest geometries to the
traditional Euclidean geometry.”
During 20th century many attempts to solv e this prob-
lem were undertaken. Finally, mathematicians came to
conclusion that Hilbert's Fourth Problem was formulated
very vague what makes difficult its final solution [12,
13].
5.2. Euclid’s Golden Section and the
Mathematics of Harmony
A problem of the Golden Section was formulated by
Euclid as a problem of division of line segments in ex-
treme and mean ratio (Theorem II.11). This problem was
introduced by Euclid with the purpose to create a full
geometric theory of Platonic Solids (the Book XIII), ex-
pressed in Plato’s cosmology the harmony of Universe.
During two last centuries the interest in the Golden
Section and Platonic Solids increased rapidly what led to
many scientific discoveries (quasi-crystals, fullerenes,
“golden” genomatrices and so on). Besides, the devel-
opment of this direction led to the creation of the Ma-
thematics of Harmony-a new interdisciplinary theory and
the “gol den” para digm of mo dern sci ence [ 1].
5.3. Hyperbolic Fibonacci and Lucas Functions
and Bodnar’s Geometry
Hyperbolic Fibonacci and Lucas functions based on Euc-
lid’s Golden Section united together two great problems
formulated by Euclid—Euclid’s Fifth Postulate, which
led to Lobachevski’s hyperbolic geometry, and Euclid’s
Golden Section problem (Teorem II.11), which led to the
“Mathematics of Harmony” [1]. The hyperbolic Fibo-
nacci and Lucas functions [2,15] led to Bodnar’s geome-
try [16], which discovered for us a new “hyperbolic
world”-the world of phyllotaxis.
5.4. The “Golden” Fibonacci Goniometry and
Hilbert’s Fourth Problem
The “golden” Fibonacci goniometry based on Spinadel’s
metallic means, which are a generalization of the classi-
cal Euclid’s Golden Section, led to obtaining an original
solution of Hilbert’s Fourth Problem, which considered
until now as formulated very vague, what makes difficult
its final solution [12,13].
The “golden” Fibonacci goniometry [3] generates a
theoretically infinite number of new hyperbolic functions,
in particular, hyperbolic Fibonacci and Lucas functions
[2,15], which underlie Bodnar’s geometry [16]. The
“golden” Fibonacci goniometry extends considerably the
sphere of hyperbolic researches and attracts an attention
of theoretical natural sciences to the question of a search
of new hyperbolic worlds of Nature, based on the hyper-
bolic Fibonacci and Lucas  -functions (  > 0 is a given
real number) [3].
6. References
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