Applied Mathematics, 2011, 2, 181-188
doi:10.4236/am.2011.22020 Published Online February 2011 (
Copyright © 2011 SciRes. AM
Hyperbolic Fibonacci and Lucas Functions, “Golden”
Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s
Fourth Problem
—Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)
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 20, 2010
This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary di-
rection of modern science. The main goal of the article is to describe two modern scientific discoveries–New
Geometric Theory of Phyllotaxis (Bodnars Geometry) and Hilberts Fourth Problem based on the Hyper-
bolic 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 introduced
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 character 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 Formukas,
Hyperbolic Fibonacci and Lucas Functions, Gazale Formulas, “Golden” Fibonacci -Goniometry
1. Omnipresent Phyllotaxis
1.1. Examples of Phyllotaxis Objects
Everything in Nature is subordinated to stringent mathe-
matical laws. Prove to be that leaf’s disposition on plant’s
stems also has stringent mathematical regularity and this
phenomenon is called phyllotaxis in botany. An essence
of phyllotaxis consists in a spiral d isposition of leaves on
plant’s stems of trees, petals in flower baskets, seeds in
pine cone and sunflower head etc. This phenomenon,
known already to Kepler, was a subject of discussion of
many scientists, including Leonardo da Vinci, Turi ng, Veil
and so on. In phyllotaxis phenomenon more complex
concepts of symmetry, in particular, a concept of helical
symmetry, are used.
The phyllotaxis phenomenon reveals itself especially
brightly in inflorescences and densely packed botanical
structures such, as pine cones, pineapples, cacti, heads of
sunflower and cauliflower and many other objects (Fig-
ure 1).
On the surfaces of such objects their bio-organs (seeds
on the disks of sunflower heads and pine cones etc.) are
placed in the form of the left-twisted and right-twisted
spirals. For such phyllotaxis objects, it is used usually the
number ratios of the left-hand and right-hand spirals ob-
served on the surface of the phyllotaxis objects. Botanists
proved that these ratios are equal to the ratios of the ad-
jacent Fibonacci numbers, that is,
1235813211 5
:,,,, ,,
1235813 2
 (2.1)
The ratios (2.1) are called phyllotaxis orders. They are
different for different phyllotaxis objects. For example, a
head of sunflower can have the phyllotaxis orders given
by Fibonacci’s ratios 89 144
5589 and even 233
144 .
Copyright © 2011 SciRes. AM
(a) (b) (c)
(d) (e) (f)
Figure 1. Phyllotaxis structures: (а) cactus; (b) head of sunflower; (c) coneflower; (d) romanescue cauhflower; (e) pineapple;
(f) pinecone.
Geometric models of phyllotaxis structures in Figure
2 give more clear representation about this unique bo-
tanical phenomenon.
1.2. Puzzle of Phyllotaxis
By observing the subj ects of ph yllotax is in th e co mpleted
form and by enjoying the well organized picture on its
surface, we always ask a question: how are Fibonacci’s
spirals forming on its surface during its growth? It is pro-
ved that a majority of bio-forms changes their phyllo-
taxis orders during their growth. It is known, for example,
that sunflower disks located on the different levels of the
same stalk have the different phyllotaxis orders; more-
over, the more an age of the disk, the more its phyllotaxis
order. This means that during the growth of the phyllo-
taxis subject, a natural modification (an increase) of sym-
metry happens and this modification of symmetry obeys
the law:
12358 13
  (2.2)
The modification of the phyllotaxis orders according
to (2.2) is named dynamic symmetry [1]. All the above
data are the essence of the well known “puzzle of phyllo-
taxis”. Many scientists, who investigated this problem,
did believe what the phenomenon of the dynamical sym-
metry (2.2) is of fundamental interdisciplinary impor-
tance. In opinion of Vladimir Vernadsky, the famous
Russian scientist-encyclopedist, a problem of biological
symmetry is the key problem of biology.
Thus, the phenomenon of the dynamic symmetry (2.2)
plays a special role in the geometric problem of phyllo-
taxis. One may assume that the numerical regularity (2.2)
reflects some general geometric laws, which hide a secret
of the dynamic mechanism of phyllotaxis, and their un-
covering would be of great importance for understanding
the phyllotaxis phenomenon in the whole.
A new geometric theory of phyllotaxis was developed
recently by Ukrainian architect Oleg Bodnar. This origi-
nal theory is stated in Bodnar’s book [1].
2. Bodnar’s Geometry
2.1. Structural-Numerical Analysis of
Phyllotaxis Lattices
Let’s consider the basic ideas and concepts of Bodnar’s
geometry [1]. We can see in Figure 3(a) a cedar cone as
characteristic example of phyllotaxis subject.
On the surface of the cedar cone its each seed is blocked
with the adjacent seeds in three directions. As the outcome
Copyright © 2011 SciRes. AM
Figure 2. Geometric models of phyllotaxis structures: (а) pineapple; (b) pine cone; (c) head of sunflower.
we can see the picture, which consists of three types of
spirals; their numbers are equal to the Fibonacci numbers:
3, 5, 8. With the purpose of the simplification of the
geometric model of the phyllotaxis object in a Figures
3(a) and (b), we will represent the phyllotaxis object in
the cylindrical form (Figure 3(c)). If we cut the surface
of the cylinder in Figure 3(c) by the vertical straight line
and then unroll th e cylinder on a p lane (Figure 3(d)), we
will get a fragment of the phyllotaxis lattice bounded by
the two parallel straight lines, which are traces of the
cutting line. We can see that the three groups of parallel
straight lines in Figure 3(d), namely, the three straight
lines 0-21, 1-16, 2-8 with the right-hand small declina-
tion; the five straight lines 3-8, 1-16, 4-19, 7-27, 0-30
Copyright © 2011 SciRes. AM
with the left-hand declin ation; and the eight straig ht lines
0-24, 3-27, 6-30, 1-25, 4-25, 7-28, 2-18, 5-21 with the
right-hand abrupt declination, correspond to three types
of spirals on the surface of the cylinder in Figure 3(c).
We will use the following method of numbering the
lattice nodes in Figure 3(d). We will introduce now the
following system of coordinates. We will use the direct
line OO as the abscissa axis and the vertical trace,
which passes through the point O, as the ordinate axis.
We will take now the ordinate of the point 1 as the leng th
unit, then the number, ascribed to some point of the lat-
tice, will be equal to its ordinate. The lattice, numbered
by the indicated method, has a few characteristic proper-
ties. Any pair of the points gives a certain direction in the
lattice system and, finally, the set of the three parallel
directions of the phyllotaxis lattice. We can see that the
lattice in Figure 3(d) consists of triangles. The vertices
of the triangles are numbered by the numbers a, b, c. It is
clear that the lattice in Figure 3(d) consists of the set
triangles of the kind {c, b, a}, for example, {0, 3, 8}, {3,
6, 11}, {3, 8, 11}, {6, 11, 14} an so on. It is important to
note that the sides of the triangle {c, b, a} are equal to
the remainders between the numbers a, b, c of the trian-
gle {a, b, c} and are the adjacent Fibonacci numbers: 3, 5,
8. For example, for the triangle {0, 3, 8} we have the
following remainders: 3 – 0 = 3, 8 – 3 = 5, 8 – 0 = 8.
This means that the sides of the triangle {0, 3, 8} are
equal respectively 3, 5, 8. For the triangle {3, 6, 11} we
have: 6 – 3 = 3, 11 – 6 = 5, 11 – 3 = 8. This means that
its sides are equal 3, 5, 8, respectively. Here each side of
the triangle defines one of three declinations o f the strai-
ght lines, which make th e lattice in Figure 3(d). In parti-
cular, the side of the length 3 defines the right-hand small
declination, the side of the length 5 defines the left-hand
declination and the side of the length 8 defines the right-
hand abrupt declination. Thus, Fibonacci numbers 3, 5, 8
determines a structure of the phyllotaxis lattice in Figure
The second property of th e lattice in Figure 3(d) is the
following. The line segment OO can be considered as
a diagonal of the parallelogram constructed on the basis
of the straight lines corresponding to the left-hand decli-
nation and the right-hand small declination. Thus, the
given parallelogram allows to evaluate symmetry of the
(a) (b)
(c) (d)
Figure 3. Analysis of structure-numerical properties of the phyllotaxis lattice.
Copyright © 2011 SciRes. AM
lattice without the use of digital numbering. We will
name this parallelogram by coordinate parallelogram.
Note that the coordinate parallelograms of different sizes
correspond to the lattices with different symmetry.
2.2. Dynamic Symmetry of the Phyllotaxis Object
We will start the analysis of the phenomenon of dynamic
symmetry. The idea of the analysis consists of the com-
parison of the series of the phyllotaxis lattices (th e unrol-
ling of the cylindrical lattice) with different symmetry
(Figure 4).
In Figure 4 the variant of Fibonacci’s phyllotaxis is
illustrated, when we observe the following modification
of the dynamic symmetry of the phyllotaxis object dur-
ing its growth:
1:2:1 2:3:1 2:5:3 5:8:3 5:13:8. 
Note that the lattices, represented in Figure 4, are con-
sidered as the sequential stages (5 stages) of the trans-
formation of one and the same phyllotaxis object during
its grows. There is a question: how are carrying out the
transformations of the lattices, that is, which geometric
movement can be used to provide the sequential passing
all the illustrated stages of the phyllotaxis lattice?
2.3. The Key Idea of Bodnar’s Geometry
We will not go deep into Bodnar’s original reasoning’s,
which resulted him in a new geometrical theory of phyl-
lotaxis, and we send the readers to the remarkable Bod-
nar’s book [1] for more detailed acquaintance with his
original geometry. We will turn our attention only to two
key ideas, which underlie this geometry.
Now we will begin from the analysis of the phenome-
non of dynamic symmetry. The idea of the analysis con-
sists of the comparison of the series of the phyllotaxis
lattices of different symmetry (Figure 4). We will start
from the comparison of the stages I and II. At these sta-
ges the lattice can be transformed by the compression of
the plane along the direction 0-3 up to the position, wh en
the line segment 0-3 attains the edge of the lattice. Si-
multaneously the expansion of the plane in the direction
1-2, perpendicular to the compression direction, should
happen. At the passing on from the stage II to the stage
III, the compression should be made along the direction
О-5 and the expansion along the perpendicular direction
2-3. The next passage is accompanied by the similar de-
formations of the plane in the direction О-8 (compres-
sion) and in the perpendicular direction 3-5 (expansion).
But we know that the compression of a plane to any
straight line with the coefficient k and the simultaneous
expansion of a plane in the perpendicular direction with
the same coefficient k are nothing as hyperbolic rotation
[2]. A scheme of hyperbolic transformation of the lattice
fragment is presented in Figure 5. The scheme corres-
ponds to the stage II of Figure 4. Note that the hyperbola
of the first quadrant has the equation xy = 1, and the hy-
perbola of the fourth quadrant has the equation xy = –1.
Figure 4. Analysis of the dynamic symmetry of phyllotaxis object.
Copyright © 2011 SciRes. AM
It follows from this consideration the first key idea of
Bodnar’s geometry: the transformation of the phyllotaxis
lattice in the process o f its growth is carried out b y means
of the hyperbolic rotation, the main geometric transfor-
mation of hyperbolic geometry.
This transformation is accompanied by a modification
of dynamic symmetry, which can be simulated by the se-
quential passage from the object with the smaller sym-
metry order to the object with the larger symmetry order.
However, this idea does not give the answer to the
question: why the phyllotaxis lattices in Figure 4 are
based on Fibonacci numbers?
2.4. The “Golden” Hyperbolic Functions
For more detail study of the metric properties of the lat-
tice in Figure 5 we will consider its fragment repre-
sented in Figure 6. Here the disposition of the points is
similar to Figure 5.
Let us note the ba sic peculiarities of the dispo sition of
the points in Figure 6:
1) the points М1 and М2 are symmetrical regarding to
the bisector of the right angle YOX;
2) the geometric figures OM1M2N1, OM2N2N1, OM2M3N2
are parallelograms;
3) the point А is the vertex of the hyperbola yx = 1,
that is, xA = 1, yA = 1, therefore 2OA .
Let us evaluate the abscissa of point M2 denoted
. Taking into consideration a symmetry of the
points M1 and M2, we can write: 1
. It follows
from the symmetry condition of these points what the
line segment M1M2 is tilted to the coordinate axises un-
der the angle of 45˚. The line segment M1M2 is parallel to
the line segment ОN1; this means that the line segment
ОN1 is tilted to the coordinate axises under the angle of
45˚. Therefore, the point N1 is a top of the lower branch
of the hyperbola; here 11
x, 11
y, 12ON OA.
It is clear that 112
2ONM M. And now it is ob-
vious, what the remainder between the abscissas of the
points M1 and M2 is equal to 1.
These considerations resulted us in the following equ-
ation for the calculation of the abscissa of the point M2,
that is, 2
1or 10,xxx x
  (2.3)
This means that the abscissa 2
x is a positive
root of the famous “golden” algebraic equation:
 . (2.4)
Thus, a study of the metric properties of the phyllo-
taxis lattice in Figures 5 and 6 unexpectedly resulted in
Figure 5. A general scheme of the phyllotaxis lattice trans-
formation in the system of the equatorial hyperboles.
Figure 6. The analysis of the metric properties of the phyl-
lotaxis lattice.
the golden mean. And this fact is the second key outcome
of Bodnars geometry. This result was used by Bodnar
for the detailed study of phyllotaxis phenomenon. By
developing this idea, Bodnar concluded that for the ma-
thematical simulation of phyllotaxis phenomenon we
need to use a special class of the hyperbolic functions,
named “golden hyperbolic functions [1]:
Copyright © 2011 SciRes. AM
The “golden” hyperbolic sine
The “golden” hyperbolic cosine
In further, Bodnar found a fundamental connection of
the “golden” hyperbolic functions with Fibonacci num-
 
21 21
Fk Gchk ; (2.7)
kGshk. (2.8)
By using the correlations (2.7), (2.8), Bodnar gave
very simple explanation of the “puzzle of phyllotaxis”:
why Fibonacci numbers occur with such persistent con-
stancy on the surface of phyllotaxis objects. The main
reason consists in the fact that the geometry of the “Alive
Nature”, in particular, geometry of phyllotaxis is a non-
Euclidean geometry; but this geometry differs substan-
tially from Lobachevsky’s geometry and Minkovsky’s
four-dimensional world based on the classical hyperbolic
functions. This difference consists of the fact that the
main correlations of this geometry are describ ed with the
help of the “golden” hyperbolic functions (2.5) and (2.6)
connected with the Fibonacci numbers by the simple
correlations (2.7) and (2.8).
It is important to emphasize that Bodnar’s model of
the dynamic symmetry of phyllotaxis object illustrated
by Figure 4 is confirmed brilliantly by real phyllotaxis
pictures of botanic objects (see, for example, Figures 1
and 2).
2.5. Connection of Bodnar’s “Golden”
Hyperbolic Functions with the Hyperbolic
Fibonacci and Lucas Functions
By comparing the expressions for the symmetric hyper-
bolic Fibonacci and Lucas sine’s and cosines [3] given
by the formulas
Symmetric hyperbolic Fibonacci sine and cosine
 
sFs xcFsx
Symmetric hyperbolic Lucas sine and cosine
sLs x
 ;
cLs x
  (2.10)
with the expressions for Bodnar’s “golden” hyperbolic
functions given by the Formulas (2.5), (2.6), we can find
the following simple correlations between the indicated
groups of the formulas:
Gsh xsFsx (2.11)
Gch xcFsx (2.12)
2Gsh xsFsx (2.13)
2Gsh xcFsx (2.14)
The analysis of these correlations allows to conclude
that the “golden” hyperbolic sine and cosine introduced
by Oleg Bodnar [1] and the symmetric hyperbolic Fibo-
nacci and Lucas sine’s and cosines, introduced by Stak-
hov and Rozin in [3], coincide within constant factors. A
question of the use of the “golden” hyperbolic functions
or the hyperbolic Fibonacci and Lucas functions for the
simulation of phyllo taxis objects has not a particular sig-
nificance because the final result will be the same: al-
ways it will result in the unexpected appearance of the
Fibonacci or Lucas numbers on the surfaces of phyllo-
taxis objects.
Concluding Part II of this article, we emphasize a sig-
nificance of Bodnar’s geometry for modern theoretical
natural sciences:
1) Bodnar’s geometry discovered for us a new “hy-
perbolic world”—the world of phyllotaxis and its geo-
metric secrets. The main feature of this world is the fact
that the basic mathematical properties of this world are
described with the hyperbolic Fibonacci and Lucas func-
tions, which are a reason of the appearance of Fibonacci
and Lucas numbers on the surface of phyllotaxis objects.
2) It is important to emphasize that the hyperbolic Fi-
bonacci and Lucas functions, introduced in [3,4], are
“natural” functions of Nature. They show themselves in
different botanical structures such, as pine cones, pine-
apples, cacti, heads of sunflower and so on.
3) As is shown in Part I, the hyperbolic Fibonacci and
Lucas functions, based on the golden mean, are a partial
case of more general class of hyperbolic functions–the
hyperbolic Fibonacci and Lucas
-functions (
> 0 is a
given real number), based on the metallic means. As
Bodnar proves in [1], the hyperbolic Fibonacci and Lu-
cas functions underlie a new “hyperbolic world”—the
world of phyllotaxis phenomenon. In this connection , we
can bring an attention of theoretical natural sciences to
the question to search new hyperbolic worlds of Nature,
based on the hyperbolic Fibonacci and Lucas
This idea can lead to new scientific discoveries.
3. References
[1] O. Y. Bodnar, “The Golden Section and Non-Euclidean
Copyright © 2011 SciRes. AM
Geometry in Nature and Art,” In Russian, Svit, Lvov,
[2] V. G. Shervatov, “Hyperbolic Functions,” In Russian
Fizmatgiz, Moscow, 1958.
[3] A. P. Stakhov and B. N. Rozin, “On a New Class of
Hyperbolic Function,” Chaos, Solitons & Fractals, Vol.
23, No. 2, 2004, pp. 379-389.
[4] A. P. Stakhov and I. S. Tkachenko, “Hyperbolic Fibonac-
ci Trigonometry,” Reports of the National Academy of
Sciences of Ukraine, In Russian, Vol. 208, No. 7, 1993,
pp. 9-14.