MANAGEMENT FROM A NATURAL PERSPECTIVE: DISCOVERING THE
MEANING OF FIBONACCI NUMBERS FOR MANAGEMENT
[WORK IN PROGRESS]
Vlado Dimovski, Ph.D.
University of Ljubljana
Faculty of Economics
Kardeljeva ploščad 17, SI-1000 Ljubljana, Slovenia; Phone: (+386) 1 5892 558
[email protected]
Miha Uhan, M.A.
University of Ljubljana
Faculty of Economics
Kardeljeva ploščad 17, SI-1000 Ljubljana, Slovenia; Phone: (+386) 1 5892 689
[email protected]
MANAGEMENT FROM A NATURAL PERSPECTIVE: DISCOVERING THE
MEANING OF FIBONACCI NUMBERS FOR MANAGEMENT
[WORK IN PROGRESS]
Abstract: The golden spiral, based on Fibonacci numbers, occurs in plants, seashells,
galaxies, and the DNA helix. Golden ratio represents a simplicity of supreme significance, for
it spans the largest and smallest of natural phenomena, with human design in between.
Human development was therefore intended to match the natural order of the material
universe. If managers would be able to organize the businesses according to the golden ratio,
the organizations they manage would be able to live "in sync" with the universe, which would
contribute to the prosperity of those organizations. This article discusses the possible
applications of Fibonacci numbers or golden ratio to management.
Keywords: Fibonacci sequence, golden ratio, phyllotaxis, natural design, development,
management
JEL classification: C00, M00
The ratio is always the same. 1 to 1.618 over and over and over again. Patterns are hidden in
plain sight, you just have to know where to look. Things most people see as chaos actually
follow settled laws of behavior; galaxies, plants, seashells. The patterns never lie, but only
some of us can see how the pieces fit together.
Jake Bohm, Touch
1 INTRODUCTION
A good hundred years have passed since Frederick Winslow Taylor published his influential
monograph The Principles of Scientific Management (1911). It was one of the earliest
attempts to apply science to the engineering of processes and to management and a theory of
management that analyzed and synthesized workflows, and whose main objective was
improving economic efficiency, especially labor productivity,  taylorism  was later
developed based on this and his other works.
In our paper we want to shed light on the possible meanings or applications of a mathematical
sequence to the science of management. This has to our best knowledge up to date never been
done before. We too argue that through this “natural management” some economic benefits
can be pursued, but contrary to Taylor, we speak not only of economic efficiency (doing
things right), but also of economic effectiveness (doing the right things). We argue that
through application of Fibonacci numbers to the management practice, the desired business
results can be easier achieved due to the positive effect on the employees and the optimal or
“natural” functioning of the organization as a whole.
Ian Stewart in his book on the unreal reality of mathematics (1995, p. 127) makes a remark
that chaos teaches us that systems obeying simple rules can behave in surprisingly
complicated ways, and that it offers important lessons for everybody  managers who imagine
that tightly controlled companies will automatically run smoothly, politicians who think that
legislating against a problem will automatically eliminate it, and scientists who imagine that
once they have modeled a system their work is complete. However, he also makes another
V. DIMOVSKI, M. UHAN | FIBONACCI NUMBERS AND MANAGEMENT [WORK IN PROGRESS]
remark that the world cannot be totally chaotic, otherwise we would not be able to survive in
it. In fact, one of the reasons that chaos was not discovered sooner is in his opinion that in
many ways our world is simple. Stewart (1995, p. 136) exemplifies this simplicity of our
world by the numbers that arise in plants (flowers, pineapples, artichokes etc.) and that
display mathematical regularities. They form the beginning of the so-called Fibonacci series:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,….
Leonardo Fibonacci, in about 1200, invented this series in a problem about the growth of a
population of rabbits. It was a very interesting piece of mathematics as it was the first model
of its kind and as mathematicians find Fibonacci numbers fascinating and beautiful in their
own right. At first sight one may wonder what makes this sequence of numbers so revered. A
quick inspection shows that this sequence of numbers can go on infinitely, as it begins with
two 1s and continues to get succeeding terms by adding, each time, the last two numbers to
get the next number (i.e. 1 + 1 = 2; 1 + 2 = 3; 2 + 3 = 5; and so on). By itself, this is not very
impressive. Yet, there are, as Posamentier and Lehman (2007, p. 13) state, no numbers in all
of mathematics as ubiquitous as the Fibonacci numbers.
2 THE OCCURENCE OF FIBONACCI NUMBERS
The use of Fibonacci numbers and the Golden Mean through-out nature and by man is well
established (Bachmann & Bachmann, 1979, p. 73). Fibonacci numbers appear in geometry,
algebra, number theory, and many other branches of mathematics. However, even more
spectacularly, they appear in nature; for example, the number of spirals of bracts on a
pinecone is always a Fibonacci number, and, similarly, the number of spirals of bracts on a
pineapple is also a Fibonacci number. The appearances in nature seem boundless. The
Fibonacci numbers can be found in connection with the arrangement of branches on various
species of trees, as well as in the number of ancestors at every generation of the male bee on
its family tree. According to Krebs (2008, p. 185), petals on flowers, seeds on sunflowers
(Figure 1), and the ratio of your height to the distance from your belly button to the ground
provide for more of the examples.
Figure 1: The double spiral pattern of phyllotaxis in a sunflower head
Source: Ball (2009, p. 231).
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Fibonacci numbers can also be found in areas of science, which, besides mathematics, include
symmetry as a constant in physical laws (e.g. cosmology), fiber optic networks (Dotson et al.,
1993), and statistical control of manufacturing systems (Tagaras & Lee, 1988). Günduz
(2000, p. 320) even notes that the fractal dimension of the growth of Ottoman Empire is equal
to golden number and that the rate of growth of the total area scales with golden number while
the area scales with the square of golden number. Fibonacci sequnce even found its way into
the world of fassion (Cyranoski, 2002). There is practically no end to where these numbers
appear (Posamentier & Lehman, 2007, p. 13). Kramer (1973, p. 112-113) writes:
A lore has grown up around Fibonacci numbers, not only because of their many
surprising and elegant mathematical properties, but also because they continually appear
in the physical world in seemingly unrelated contexts. There is experimental
psychological data, for example, that golden mean proportions in cards, mirrors,
pictures, etc., (which would be provided by Fibonacci lengths) appeal to our sense of
symmetry (Bicknell & Hoggatt, 1969); the Fibonacci series was apparently used to give
golden mean proportions in Greek vases and in the poetry of Vergil (Horadam, 1963); it
appears in Minoan architecture (Preziosi, 1968) as well as that of the ancient Greeks and
in Gothic cathedrals; it was apparently used to determine patterns in mosaic designs in
several ancient countries (Moor, 1970); it is found to determine patterns of shell growth
(Horadam, 1963) and in phyllotaxis (leaf and petal arrangement) - various types of
flowers tend to have on the average Fibonacci numbers of petals, and in trees with spiral
arrangements of leaves or branches the Fibonacci series determines the number of
rotations before we find a leaf or branch directly above a given one, and there is a
tendency for Fibonacci numbers of leaves or branches to be in such spirals, etc. (de Sales
McNabb, 1963); the Fibonacci series tends to determine the number of spirals of eyes
(fruitlets) in pineapples (Onderdonk, 1970); the series determines the number of
ancestors a bee will have (since male bees hatch from unfertilized eggs while female bees
come from fertilized eggs) (Basin, 1963); the Fibonacci numbers appear in certain
electrical networks (Basin, 1963); Fibonacci numbers are approximated in the structures
of atomic and subatomic particles (Huntley, 1969); the ratios of the distances of the
moons of Jupiter, Saturn, and Uranus from their parent planets follow Fibonacci (i.e.,
golden mean) ratios, as do (in a poorer approximation) the distances of the planets,
including the asteroid belt, from the sun (Read, 1970); Fibonacci numbers have played a
part in the branch of cancer research that attempts to construct a mathematical model for
the movement of malignant cells (Blumenson, 1972); the Fibonacci numbers have been
useful in water pollution control in determining where best to place sewage treatment
plants for cities on the same river (Deininger, 1972); there is some evidence that
Fibonacci numbers relate to the number of years in the cycles between peaks and peaks,
peaks and lows, lows and peaks, and lows and lows of the stock market (Faulconbridge,
1964); it has even been suggested that Fibonacci numbers are involved in the cycles of
the extremes of such diverse events as grasshopper abundance, automobile factory sales,
the ratio of male to female conceptions, advertising effectiveness, sunspots, tree ring size,
rainfall in India, Nile floods, financial panics, and furniture production (Faulconbridge,
1965).
According to Birken and Coon (2008, p. 59), Fibonacci numbers may appear also in the
sphere of music. An octave consists of 8 notes and is represented on the piano by 8 keys. If
we include sharps and flats, we add 5 black keys to the 8 white keys for a total of 13 keys,
often referred to as the chromatic scale. The black keys themselves are positioned in groups of
2 and 3. All the numbers mentioned — 2, 3, 5, 8, and 13 — are Fibonacci Numbers. Another
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example of the occurence of Fibonacci numbers in music are the proportions of the famous
Stradivarius violin design (Figure 2). Some authors have found Fibonacci-like patterns or
Fibonacci ratios in the works of composers as diverse as Mozart, Beethoven, Bach, Schubert,
Debussy, and Bartok (Bachmann & Bachmann, 1979; Birken & Coon, 2008; Kramer, 1973;
Locke, 1987); Chopin and Wagner (Posamentier & Lehman, 2007); and Stockhausen, Nono,
Bunger, Walker, Schillinger, Eloy, Krenek, and Norden (Kramer, 1973). Golden ratio
(dividend of the two consecutive Fibonacci numbers) has been successfully employed by
composers mainly in two areas of the compositional process. The first relates to the location
of the climax and the second relates to form (Posamentier & Lehman, 2007, p. 272).
Figure 2: The distinctive Fibonacci proportions of the Stradivarius violin design
Source: Posamentier & Lehmann (2007, p. 291).
In poetry, too, occurrences of Fibonacci Numbers are linked to familiar patterns (Birken &
Coon, 2008, p. 59). The common limerick is composed of 5 lines with 13 stressed beats.
These beats are found in patterns of 2 and 3. Turning from the limerick to more serious
poetry, scholars have studied the appearance of the Fibonacci sequence, or ratios of
successive Fibonacci Numbers, in epic works including Virgil’s Aeneid, written in the first
century B.C.E., and in Dante’s The Divine Comedy, written in the thirteenth century C.E.
(Birken & Coon, 2008, p. 60). Also, Hill (2010, p. 166) proposes that key moments in the
rhetorical and emotional development of the Old English heroic epic poem, Beowulf,
correspond to a Fibonacci sequence, although there is nothing to indicate widespread
knowledge of this sequence in Europe prior to the thirteenth century.
The rectangle whose sides fit the golden ratio or “phi” (φ) is said to be the “most pleasing” in
form of all rectangles (Birken & Coon, 2008, p. 63). Claims have been made that it is found in
many architectural examples, from the ancient Parthenon and Great Pyramid of Giza, to the
more modern work of Le Corbusier. Close approximations of Golden Rectangles also may
exist in painting and sculpture, including works by Leonardo da Vinci, Michelangelo,
Georges Seurat, and Piet Mondrian (Birken & Coon, 2008, p. 63), and Dürer, Raphael, Giotto,
Boticelli, and Dalí (Posamentier & Lehman, 2007, pp. 258-268). Posamentier and Lehman
(2007, p. 231) too agree that the golden ratio seems to be ever present in art and architecture.
According to them, perhaps the most famous structure that exhibits the golden rectangle is the
Parthenon on the Acropolis in Athens. Spectators have also noted the golden section on
Egyptian and Mexican pyramids, Japanese pagodas, and even Stonehenge (cca. 2800 BCE) in
England. There is, moreover, speculation that the Arch of Triumph in ancient Rome was built
on the basis of the golden ratio (Posamentier & Lehman, 2007, p. 237). Rose (1991) makes an
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interesting remark about the work of his friend Waskom: “He also noted that the Greeks used
phi (or the golden mean) all through their arts. He reasoned that if an object were composed
of the same proportions as the viewer, it would be perceived as balanced and natural, which
is what observers say about Greek creations. To explain this preference, Waskom drew upon
Huntley, who suggested that as the eyes view the sides of a golden rectangle, they note a scantime ratio of phi, which is then interpreted as pleasing. Studies have established the aesthetic
preference for phi-proportioned objects.” Indeed, Benjafield & Green (1978, p. 25) note that:
… over the last century, a large number of experimental investigations have been
devoted to determining the psychological properties of the golden section (Zusne, 1970).
Fecnhner (1876) provided the first experimental evidence that the golden section is the
most pleasing proportion, and current research has tended to confirm Fechner’s findings
(e.g. Benjafield, 1976; Segalowitz & Benjafield, 1976). Recently, Benjafield & AdamsWebber (1976) have suggested that the golden section may play an important role in
interpersonal as well as in aesthetic judgement.
3 THE MEANING OF FIBONACCI NUMBERS FOR MANAGEMENT
There have been some applications of Fibonacci to business sphere in the past, but most of
them dealt with predicting the markets in trading (e.g. Boroden, 2008; Brown, 2008; Evans &
Sheean, 2005; Fischer, 1993; Greenblat, 2007; Jardine, 2003; Pesavento & Shapiro, 1997),
with the most famous one certainly being the Elliot’s influential work named The Wave
Principle (1938). Elliot’s waves have further been applied to the field of sociology in 1999,
when Prechter published his book The Wave Principle of Human Social Behavior and the
New Science of Socionomics, in which he defines the whole new area of science –
socionomics. This work was later supplemented by the Pioneering Studies in Socionomics
(Prechter, 2003).
Maybe the most useful application of Fibonacci numbers for the field of social sciences to
date is the unique paradigm of human development developed by John D. Waskom (Rose,
1991), who has sensed the possibility that human development was intended to match the
natural order of the material universe. Waskom loved to relate it to the spiral and its
occurrence in plants, seashells, galaxies, and the DNA helix. To Waskom, phi was a
simplicity of supreme significance, for it spanned the largest and smallest of natural
phenomena, with human design in between. When he called attention to the fact that young
children unconsciously used phi proportions in their artwork, Waskom was affirming that
unspoiled humans possessed a natural genius for living "in sync" with the universe. He
sensed that if he could establish a relationship between phi and human psychological growth,
he could begin to describe a natural pattern of developmental genius throughout the lifespan
(Rose, 1991).
What is interesting about the Waskom's paradigm of human development is his interpretation
of the sequence or progression of Fibonacci numbers. He saw each Fibonacci number as the
beginning point for a new subset or cycle of numbers. Each whole number preceding a
Fibonacci number was then the completion point of the previous subset or cycle (Rose, 1991).
Visually, this can be represented as in Figure 3.
Figure 3: Waskom's interpretation of the sequence or progression of Fibonacci numbers
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End of old cycle:
Start of new cycle:
0
1
1
2
3
4
7
12
20
33
54
88
...
5
8
13
21
34
55
89
...
Source: Rose (1991, p. 6)
To illustrate, Waskom pointed out human applications that have become common in our
culture. For example, a dozen (12) is considered a complete set and a deck of cards is
complete with fifty-four (including the jokers). In music, the octave scale actually has only
seven notes, for the eighth is the beginning of a new octave. Likewise, the chromatic scale's
thirteenth note begins the next octave. Even the 88-note piano keyboard can be justified
(Rose, 1991).
According to Rose (1991), using the Fibonacci sequence, Waskom found a natural
progression of human development revealing itself. He simply looked at the number chart
above and imagined those numbers as ages in a person's life. The numbers in the bottom row
would represent age markers for the beginning of cycles or stages of development. The
numbers on the top row would represent the end points of each cycle or stage. Practiced eyes
in the developmental field will see familiar numbers, numbers which have been traditionally
used to mark stages of the life cycle, both physiologically and socially. Thus it could be
asserted that the numbers that mark the milestones of human life are the same numbers that
express the perfect proportion of nature. Reduced to an assertion of simplicity and unity,
under proper conditions design and development are one, since both can be described with the
same natural progression. This, then, was to become the premise underlying Waskom's
paradigm for thinking about development and learning (Rose, 1991).
Applying mathematics and Fibonacci to management science, although it might seem
somewhat unusual at first sight, therefore seems as an obvious natural next step. The possible
applications of Fibonacci numbers or golden ratio to the management, or broadly put,
business sphere are presented below. It is naturally up to individual organizations to figure out
how exactly to apply the Fibonacci numbers to their operations. The below possible
applications might therefore serve them only as an idea for their own thinking with regard to
the organizational design and other important managerial issues. Our presentation of the
possible applications of Fibonacci numbers to management is based on the four major
management functions, as defined by Daft and Marcic (2006, p. 8) and presented in Figure 4.
Figure 4: The Four Management Functions
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Source: Daft & Marcic (2006, p. 8)
3.1 Fibonacci and planning
We have identified two examples of application of Fibonacci sequence to the management
function of planning – inventory policy optimization (Wagner, O’Hagan & Lundh, 1965) and
statistical control of manufacturing systems (Tagaras & Lee, 1988). Other possible
applications include (but are not limited to):
 Business estimating and planning – how to estimate the future profits and how to plan
future business accordingly,
 Investments (when to invest, how much to invest, etc.),
 Risk management,
 M&As, MBOs, entry/exit strategies (when to perform a merger, acquisition or an
MBO; when to enter or exit a company if you are a venture capital company),
 Market entries, expansions (when to enter new markets, and how many at the same
time),
 Debt/Equity ratio (it might be that the optimal solution might be close to the Golden
Ratio, which is app. 1.618, which would mean that the optimal capital structure for a
company would be around 62% debt and 38% capital),
 Ownership structure (private/public, etc.).
3.2 Fibonacci and organizing
The second management function, organizing, is a process that leads to the creation of
organization structure, which Daft & Marcic (2006, p. 256) define as (1) the set of formal
tasks assigned to individuals and departments; (2) formal reporting relationships, including
lines of authority, decision responsibility, number of hierarchical levels, and span of
managers’ control; and (3) the design of systems to ensure effective coordination of
employees across departments. This second function also deals with the questions such as
what is the optimal size of a particular institution. The American Journal of Sociology has
published an interesting research by Chapin (1957), who studied the optimum size of the
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institutions. In his article, Chapin took the Fibonacci proportion as a measure of integration,
harmonious balance of parts, and equilibrium of structure, and suggested a mathematical
model with the logarithmic spiral as the principle of growth. He found out that the optimum
size of the two largest subgroups (membership and Sunday-school enrolment) of eighty
churches seems to be that particular relative size which approaches most nearly the Fibonacci
number series and proportions (0.618). What is also interesting is that although Chapin has
performed his study with the eighty Minneapolis churches, the Fibonacci numbers and
proportions have, according to him (p. 459) no special applicability to religious institutions
but are equally applicable to the study of optimum size of large secular groups.
The other possible application of Fibonacci sequence to the management function of
organizing includes (but is not limited to) the fluctuation policy (predicting and organizing the
human resource needs).
3.3 Fibonacci and leading
The possible applications of Fibonacci sequence to the management function of leading
include (but are not limited to):





Motivation of the employees,
Trainings for the employees (when to employ their trainings – in what time intervals),
Leadership development (when to start developing the successor),
Corporate universities educational design,
Promotions (in years 1, 2, 3, 5, 8, 13, 21, …).
3.4 Fibonacci and controlling
The possible application of Fibonacci sequence to the management function of controlling
includes (but is not limited to):
 Controlling – when (in what time intervals) to monitor the activites of your employees
(after 1, 2, 3, 5, 8, 13 etc. weeks), etc.
4 DISCUSSION AND IMPLICATIONS
In his paper, where he utilized the concept of a fractal dimension that characterises the
nonlinear growth of very intricate dynamics of biological, chemical, and physical systems
to model model the growth of the Ottoman Empire as a social system, Gündüz (2000, pp. 303304) writes:
The description of natural events with fractal dimensions is one of the most important
concepts of science and has a vast area of applications. After the pioneering work of
Mandelbrot (1983) it has been believed that a fractal dimension can be assigned to very
complicated events, even to an economy, psychology, and a society. Although the para­
meters of social events are too many, they can be very well evaluated and successful
forecasting can be done in certain cases. The social strategy evaluates the available
information usually on a statistical basis and ends up with conclusions about what to do
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in future. In forecasting, the important social parameters, which prevail in future, need to
be known. The social parameters more or less have some correspondences with physical
parameters. For instance the distribution of national income among the people and the
distribution of energy in a stochastic system among particles are analogically similar.
The physical world in a broad sense covers the physical principles of the universe and
nothing is exempt from it. If the parameters of the social life could be well analysed we
could use mathematical tools to achieve precise conclusions about the social life.
However the application of mathematics to complex human activities is still at a
premature level. If the dynamics underlying social events can be well defined, then it may
be possible to set up the mathematical model.
5 LIMITATIONS AND FUTURE RESEARCH OPPORTUNITIES
Let us rephrase the Stewart’s (1995, p. 127) thought about the managers’ perception that
tightly controlled companies will automatically run smoothly. Not only do managers 
similarly to the Daoist concept of “wu wei” or non-doing  need to be aware that the more
control is not necessarily the better, but we might (among other things) even be able to predict
in the future how much control is optimal for a smooth running of the organization and
exactly when should the managers exercise the control. This is due to the fact that nature is, in
its own subtle way, simple, and we can learn a lot, simply by observing and studying it.
Benjafield & Green (1978, p. 34) are of opinion that “most research on the golden section has
been concerned with whether or not it is, in fact, a preferred proportion (Zusne, 1970)” and
that only “few studies have explored cross-cultural and developmental determinants of golden
section preferences.” They state that “given the growing evidence for the psychological
reality of the golden section, such [cross-cultural and developmental] studies are increasingly
appropriate, and should shed light on how preferences for golden section relations arise and
are maintained in everyday life.” However, as Freeman (1981, p. 22) warns, the study of the
Fibonacci sequence must not become the collection of individual interpretations of the
“Rorschab blob”.
We want to finish our article with the following quote from Bachmann and Bachmann (1979,
p. 73): “Except for an interpretation based on aesthetics, the philosophical meaning of the
universe constructed on the Fibonaccian sequence at present eludes researchers, although a
dim significance can be ascertained. It is hoped that a pragmatic explanation will become
available in the near future through the efforts of philosophers and research scientists.”
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V. DIMOVSKI, M. UHAN | FIBONACCI NUMBERS AND MANAGEMENT [WORK IN PROGRESS]
About the authors
Vlado Dimovski is a full professor of management and organization at the Faculty of
Economics University of Ljubljana. His research interests are: strategic management,
organizational learning, competitiveness, developing knowledge-based organization, and
labor market issues. Dimovski has received his B.A. degree in economics and philosophy, and
M.A. in Economics from University of Ljubljana, and Ph.D. in Management and Finance
from Cleveland State University. He was State Secretary for Industry in the Government of
Slovenia (1995-1997), President of the Center for International Competitiveness (1997-2000),
and Minister for Labor, Family, and Social Affairs (2000-2004). Dimovski has also wide
experience in consulting for numerous companies, institutions, and governments, particularly
on the issues of strategic management, labor market, mergers and acquisitions, and the EUrelated issues. As an academician Dimovski has taught and researched at the various
universities and institutions, and has published in recognized journals.
Miha Uhan graduated in Management at the Faculty of Economics, University of Ljubljana
in 2008. He attended the JOSZEF Study Program at the Vienna University of Economics and
Busines in 2009 and received his master's degree at the Faculty of Economics, University of
Ljubljana in 2010. He is currently employed as a young researcher at the Faculty of
Economics, University of Ljubljana, where he is also a doctoral candidate.
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