Fibonacci Harmonics: A New Mathematical Model of Synchronicity

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DOI: 10.4236/am.2018.96048
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This article aims to provide a brief overview of the relevance of new findings about the Fibonacci Life Chart Method (FLCM) for understanding synchronicity. The FLCM is reviewed first, including an exposition of the golden section model, and elaboration of a new harmonic model. The two models are then compared to illuminate several strengths and weaknesses in connection with the following four major criteria regarding synchronicity: explanatory adequacy; predictability of future synchronicities; simplicity of the model; and generalizability to other branches of knowledge. The review indicates that both models appear capable of simulating nonlinear and fractal dynamics. Hybrid approaches that combine both models are feasible and necessary for projects that aim to experimentally address synchronicity.
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Applied Mathematics, 2018, 9, 702-718
ISSN Online: 2152-7393
ISSN Print: 2152-7385
10.4236/am.2018.96048 Jun. 29, 2018 702 Applied Mathematics
Fibonacci Harmonics: A New Mathematical
Model of Synchronicity
Robert G. Sacco
Fibonacci Lifechart, Toronto, Canada
This article aims to provide a brief overview of the relevance
of new findings
about the Fibonacci Life Chart Method (FLCM) for understanding synchr
nicity. The FLCM is reviewed first, including an exposition of the golden se
tion model, and elaboration of a new harmonic model. The two models are
then compared to illuminate several strengths and weaknesses in connection
with the following four major criteria regarding synchronicity: explanatory
adequacy; predictability of futur
e synchronicities; simplicity of the model; and
generalizability to other branches of knowledge.
The review indicates that
both models appear capable of simulating nonlinear and fractal dynamics
Hybrid approaches that combine both models are feasible and
necessary for
projects that aim to experimentally address synchronicity.
Synchronicity, Dynamical Systems, Brain Dynamics, Fibonacci Life Chart
Method, Golden Section Model, Harmonic Model
1. Introduction
Synchronicity is among the most mysterious experiences, involving a noncausal
connection between mind and matter. Stories involving synchronicity appear in
movies, biographies, theatre, and literature. It has been a perennial challenge to
explain. This article places synchronicity in the context of a mathematical system
in which mind and matter play complementary roles in a reality that has a ma-
thematical structure [1] [2].
In recent decades, two main theoretical approaches have dominated the field
of synchronicity. One of these emphasizes quantum determinants, as shaped by
mind-matter entanglement [3]. The other emphasizes psychological determi-
nants, as shaped by biased cognition [4]. Both have proposed to explain the
How to cite this paper:
R.G. (2018
Fibonacci Harmonics:
A New Mathemati-
cal Model of Synchronicity
Applied M
, 702-718.
May 19, 2018
June 26, 2018
June 29, 2018
Copyright © 201
8 by author and
Research Publishing Inc.
This work is licensed under the Creative
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R. G. Sacco
10.4236/am.2018.96048 703 Applied Mathematics
meaningful coincidence between a thought and related event in the world. The
quantum approach stresses the entanglement induced nonlocal correlations of
quantum physics, conceiving of mind-matter entanglement as the hypothetical
origin of mind-matter correlations and synchronistic phenomena. The biased
cognition approach, generally based on poor probabilistic reasoning, has em-
phasized chance occurrences and our need to make sense of ourselves and the
world. Thus, the disciplines of physics and psychology have been most promi-
nent in guiding how researchers think about synchronicity.
This article turns to a different discipline, mathematics, to explain a theory of
synchronicity. A mathematical approach to synchronicity was defined by Sacco
[5] based on the Fibonacci Life Chart Method (FLCM). The FLCM has four
main assumptions. First, human development is best framed within a nonlinear
dynamical systems paradigm with the Fibonacci sequence driving changes in
developmental timing relative to fixed points in the life cycle. Second, such pat-
terns describe an eight-stage development process. Third, the transition from
one stage to another is characterized by nonlinear dynamics (e.g., self-organization,
emergence, attractors, fractals, complexity, and chaos). Fourth, during transi-
tions, people seek new information and explore more adaptive configurations
until they settle into a new stable state (attractor). In adopting such an approach,
the theory is mathematical in the sense that it looks at principles of pattern for-
mation and change across scientific disciplines and systems as diverse as cells,
neurons, and the economy.
Although applying mathematical principles to synchronicity may seem novel,
the notion that synchronicity might depend on the Fibonacci numbers was an-
ticipated by Jung in a letter on February 9, 1956 [6]. Jung did not specify how
Fibonacci numbers caused synchronicity, but he recognized the conceptual value
of postulating that synchronicity operated based on the Fibonacci numbers be-
cause of their ubiquity in nature. Mathematical models are increasingly being
invoked in psychology. For example, dynamical systems theory helps to analyze
a broad range of cognitive and affective dynamics, interpersonal and group dy-
namics, and personality dynamics [7]. Likewise, fractal patterns are found across
the domains of psychology including the brain, visual search, speech patterns,
memory retrieval, interpersonal relationships, and personality [5]. Thus, the
evidence is accumulating to suggest that mathematics furnishes a useful basis for
making predictions about how people will think, feel, and act.
Previous attempts to apply dynamical systems theory to psychology have neg-
lected one crucial aspect, which will be featured in this article: the points of sys-
tem transition. At any given time, a dynamical system has a state given by a nu-
merical phase space and a rule of evolution specifying trajectories in this space
[8]. The evolution rule of the dynamical system describes what future states fol-
low from the current state, and only when researchers identify this rule, can they
study what is changing and the factors that catalyze this change. To make the
theory predictive, mathematical terms must be defined according to an ordering
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10.4236/am.2018.96048 704 Applied Mathematics
scheme by which emergence and self-organization as principles of unique and
spontaneous order can arise. Here the mathematical framework of Fibonacci
numbers is used to describe precise points at which psychological experiences
will shift.
There are two main parts to this article. The first will shed light on synchro-
nicity by drawing on recent work on the FLCM. The golden section model of
deriving time intervals is described, and an attempt will be made to develop and
elaborate a new mathematical model of synchronicity from a harmonic perspec-
tive. The second section will then compare findings about the two models as a
way of evaluating their capacity to explain and predict synchronicity phenome-
2. Description of the Fibonacci Life Chart Method
Fibonacci Life Chart Method (FLCM) is a theoretical model of human develop-
ment as a nonlinear dynamical system based on the Fibonacci sequence [5] [9].
Central to this conceptual framework is the notion that the Fibonacci sequence
and ratio create fractal patterns in growing self-organizing systems. For example,
this ratio (1.618) is found in phenomena ranging from the human body to snail
shells to the spirals of the galaxies. The spiral cochlea of the ear is a Fibonacci
spiral as is the spiral of the umbilical cord. Moreover, the nucleotide spirals of
our DNA have Fibonacci proportions. The progression of the Fibonacci num-
bers and ratio are well suited to describing organic growth because they have the
property of “gnomonic growth”. Gnomonic growth implies that only the size
changes, while the shape remains constant, as organisms maintain their overall
shape and proportions as they grow. Thus, gnomonic growth relates to the as-
pect of self-similarity [5].
Crucially, FLCM assumes that each Fibonacci number represents a complete
cycle of 24 hours. The value of FLCM depends on this basic assumption, and in
part is based on the empirical observation that the developing human embryo
doubles every 24 hours [10]. By applying this clock cycle to human development,
one can make predictions about how human development will proceed. The
timing of developmental events, for example, will depend on the self-organizing
fractal properties of the Fibonacci sequence. Table 1 shows the FLCM generates
a pattern characterized by eight age groups: Early Infancy (0 - 2 years), Toddler
(2 - 4 years), Early Childhood (4 - 7 years), Middle Childhood (7 - 11 years),
Adolescence (11 - 18 years), Young Adulthood (18 - 29 years), Middle Adult-
hood (29 - 48 years), and Older Adulthood (48 - 78+ years) [9]. These chrono-
logical ages, both higher and lower in frequency, are all golden ratio (1.618) to
the next and previous in the sequence. The golden ratio “Phi” is also called the
“scaling ratio”.
Most notably, because the Fibonacci sequence relates to the well-known pe-
riod doubling in dynamical systems theory [11], a bifurcation with a shift in co-
herence from one FLCM age to another occurs in concert with the onset of
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10.4236/am.2018.96048 705 Applied Mathematics
Table 1. The Fibonacci life chart method.
Fn Date Chronological Age
0 1/01/00 0.00
1 1/02/00 0.00
1 1/03/00 0.01
2 1/05/00 0.01
3 1/08/00 0.02
5 1/13/00 0.03
8 1/21/00 0.05
13 2/03/00 0.09
21 2/24/00 0.15
34 3/29/00 0.24
55 5/23/00 0.39
89 8/20/00 0.64
144 1/11/01 1.03
233 9/01/01 1.67
377 9/13/02 2.70
610 5/15/04 4.37
987 1/27/07 7.08
1597 6/12/11 11.45
2584 7/09/18 18.53
4181 12/19/29 29.99
6765 6/27/48 48.52
10946 6/16/78 78.51
deterministic chaos. The central point to the FLCM analysis of human develop-
ment is that human development depends on the self-organizing properties of
the Fibonacci sequence. When the Fibonacci numbers progressively increase,
therefore, the self-organizing and fractal properties result in biological and psy-
chological phase transition. Phase transitions can result from self-organization
processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, or-
ganism, behavior, and environment) that can influence each other. The
self-organized hierarchy begins at the mathematical and quantum level. This
suggests that the ultimate nature of reality is mathematical [1] [2]. If mathemat-
ics unifies mind and matter, then it is possible to model information transmis-
sion regardless of mental or physical properties.
Although the premise that Fibonacci numbers can explain synchronicity in-
vokes a mathematical model to explain human experience and is therefore es-
sentially a mathematical theory, ironically its most famous advocate came from
Carl Jung, a Swiss psychiatrist. Jung observed that “…synchronicity… is a sec-
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10.4236/am.2018.96048 706 Applied Mathematics
ondary effect of the primary coincidence of mental and physical events (as in the
Fibonacci series). The bridge seems to be formed by the numbers” (p. 288) [6].
By “bridge” he meant the capacity to link mind and matter, although to be sure
he did not specify how Fibonacci numbers could bridge these two worlds. In-
deed, he provided little elaboration for his theory that Fibonacci numbers ex-
plain synchronistic events, a deficiency that this article seeks to address (aided by
the substantial amount of empirical data on Fibonacci numbers in the decades
since Jung’s work). Nonetheless, Jung put forth the observation that synchronic-
ity is essentially something controlled by the Fibonacci numbers.
2.1. Model 1: Golden Section Model
The FLCM goes beyond the simple primary intervals produced by the Fibonacci
sequence. Just as the ratios of adjacent Fibonacci numbers approximate the gol-
den ratio, the golden ratio may be used to compare adjacent primary intervals, to
produce various secondary and tertiary time intervals (for a detailed description
of this model, see [5]). Euclid gave the first written description of the golden ra-
tio in connection with dividing a line segment into two unequal parts, such that
the whole is to the long part as the long is to the short. Dividing a line segment
with these proportions is known as creating a “golden section” or “golden cut”.
Figure 1 shows the interior and exterior golden section procedure based on two
primary intervals (January 27, 2007 and June 12, 2011, see Table 1). After di-
viding the segment, two new secondary intervals appear (Date X and Date Y),
left and right of the original two endpoints.
Insofar as this is a valid procedure, the golden section points (interior and ex-
terior) of the two new endpoints (Date X and Date Y) can be used to calculate
new tertiary and higher intervals. Treating calendar dates as applicable to the
golden section means that this proportion-based system will endow the age dis-
tribution with a self-organized fractal structure [5]. A fractal is a recurring simi-
lar pattern at different scales. Many objects in nature are best described as frac-
tals. The fractal structure is ubiquitous not only in trees, rivers, mountains, isl-
ands, and coastlines but also in human-made artifacts such as cities, streets,
buildings, social media, and the Internet. All these constitute fractals as a set or
pattern with far more small things than large ones. Fractals are said to be
“self-similar”: any subsystem of a fractal system reflects the whole system. Frac-
tals belong to a set of models in which the threshold for activity, rather than be-
ing a function of a purely local variable, depends on nonlocal properties of the
self-organizing structure [12]. Furthermore, all fractal structure involves the
Figure 1. Golden section model of computing secondary intervals shows dividing prima-
ry intervals line segment AB by interior division according to the golden ratio (0.618) to
produce Date X, also dividing primary intervals by exterior division according to the gol-
den ratio (1.618) to produce Date Y.
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10.4236/am.2018.96048 707 Applied Mathematics
golden ratio because of its unique nesting capability, as seen for example in the
Mandelbrot set [13].
Thus, the first prediction based on the golden section model (GSM) is that
synchronistic events associated with nonlocality will be more likely at the golden
section points. People will experience synchronicity near these points in time,
but will not at other points in time (except perhaps in highly unusual circums-
tances). The bottom line is that golden section points have predictive value. Re-
ligious, spiritual, mystical, and synchronistic phenomena and similar expe-
riences will have neural underpinnings triggered by the dynamical interaction
between biology and mathematics [14]. Put another way, the formation and con-
tinuation of a series of synchronicities depends on whether their trajectories
evolve by the rule of Fibonacci numbers than random chance; whether the syn-
chronicity corresponds to nonlinear change than linear change, and so forth.
2.2. Model 2: Harmonic Model
The most fundamental characteristic of natural phenomena is the principle of
cycles. A cycle is a regularly occurring sequence of events. Cycles exist in nature,
economy, and biology. Cycles in nature include the day/night cycle, the four
seasons, and solar activity. The basic business cycle encompasses an economic
downturn, bottom, economic upturn, and top. Cycles are also part of the human
body in the circadian rhythm, menstrual cycle, and brain waves. The harmonic
model (HM) of synchronicity endows those cycles with much power and im-
portance and is an alternative approach to computing secondary time intervals.
One crucial feature of the HM is the time-periodic patterns of the primary in-
tervals. The primary intervals are not divided into secondary (and higher) time
intervals as in the golden section model. Rather, the FLCM primary intervals
become part of a harmonic system, just as harmonics have a periodic series of
cycles repeating in a sinusoidal fashion.
Harmonics relate to standing waves [15]. A standing wave pattern is not ac-
tually a wave, but rather a pattern of a wave. Thus, it does not consist of crests
and troughs, but rather nodes and antinodes. The pattern is the result of the
perfectly timed interference of two waves (sometimes more) of the same fre-
quency with different directions of travel. The waves interfere in a way that
produces points of no displacement at constant intervals. These points have the
appearance of standing still and are referred to as nodes. In a standing wave, the
nodes are a series of locations at equally spaced intervals where the wave ampli-
tude (motion) is zero. At these points, the two waves add with opposite phase
and cancel each other out. This condition is known as resonance. Standing
waves are always associated with resonance. At frequencies other than a har-
monic frequency, the interference of reflected and incident waves leads to a dis-
turbance that is irregular, non-repeating, and non-resonant.
To the extent that human life cycles are a Fibonacci resonance phenomenon,
time-periodic patterns can be described by phase relationships (phase patterns)
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10.4236/am.2018.96048 708 Applied Mathematics
among the primary intervals of the FLCM. Table 2 summarizes the features of
the standing wave patterns for the nine key harmonics of the human lifespan. In
this pattern, the first harmonic or fundamental wavelength is age 78.51 based on
average life expectancy at birth of 78.6 years [16]. The fundamental wavelength
has only one node from the birth date. The number of each successive harmonic
expresses its wavelength as a power of Phi fraction of the fundamental wave-
length: Harmonic 3 = 78.51/29.99 (
, 2.62) of the fundamental wavelength.
Based on a life expectancy of 78.6 years, the harmonic series consists of 240
unique nodal and antinodal points (262 including the birthdate and primary in-
tervals). Note that each consecutive harmonic has a close resonance to the pow-
ers of Phi. The powers of Phi have unusual properties in that they relate not only
exponentially, but are additive as well (e.g., Phi3 = Phi2 + Phi1).
Standing wave harmonics can occur in one, two, and three dimensions. An
example of a harmonic wave in one-dimension is a guitar string. The wavelength
of the standing wave for any given harmonic relates to the length of the string
(and vice versa). These frequencies are related by whole-number divisions of its
length (1/2, 1/3, 1/4, 1/5, 1/6, etc.) producing a series of harmonics whose fre-
quencies are inversely proportional (2x, 3x, 4x, 5x, 6x, etc., where x is the fun-
damental frequency of the string) to those fractional divisions. Unlike standing
waves in one-dimensional media such as waves in a string or sound waves in a
pipe, the resonance frequencies of standing waves in two dimensions are not
simple integer divisions (multiples) of the fundamental frequency. These har-
monics are called “non-integer harmonics” or “interharmonics”. Standing waves
in two dimensions have been studied extensively as Chaldini patterns.
The most significant example of standing waves in three dimensions is the or-
bitals of an electron in an atom. On the atomic scale, it is more appropriate to
describe the electron as a wave than as a particle. Electrons are bound to the
space encompassing a nucleus similar to the way that a guitar string constrains
the waves in the string. The constraint of a string in a guitar forces the string to
vibrate with particular frequencies. In like manner, an electron can only vibrate
Table 2. Numerical patterns associated with standing waves.
Harmonic Phin # of Nodes # of Antinodes Interval (Years)
9th Phi1 = 1.6180 47.01 47.00 1.67
8th Phi2 = 2.6180 29.08 29.00 2.70
7th Phi3 = 4.2361 17.97 18.00 4.37
6th Phi4 = 6.8541 11.09 11.00 7.08
5th Phi5 = 11.0902 6.86 7.00 11.45
4th Phi6 = 17.9443 4.24 4.00 18.53
3rd Phi7 = 29.0345 2.62 3.00 29.99
2nd Phi8 = 46.9787 1.62 2.00 48.52
1st Phi9 = 76.0132 1.00 1.00 78.51
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10.4236/am.2018.96048 709 Applied Mathematics
with particular frequencies. For an electron, these frequencies are determined by
the fine-structure constant denoted by
(the Greek letter alpha). The value of
is approximately 1/137. The fine-structure constant is an expression of Phi:
Phi2/360 = 2.618/360 = 1/137.508 = 0.00727 [17]. Indeed, the golden ratio exists
at the quantum level in the magnetic resonance of atoms [18]. This suggests that
the golden ratio essentially determines how an atom holds together its electrons.
At the deepest level, the human mind may manifest not only local but also
nonlocal characteristics [3]. One source of support for this view was provided by
the quantum holographic model of consciousness proposed by Edgar Mitchell
[19]. Mitchell and Staretz [19] described holographic processing in terms of the
brain’s sensitivity to the phase of the electromagnetic waves of an object. They
specifically drew on a resonance condition interpreted as a standing wave be-
tween the object and the brain. This resonance process is called
phase conjugate
adaptive resonance
(PCAR). With a strong emotional response or focused atten-
tion in a person, some waves emitted from an object (
, subatomic particle,
inanimate or animate thing, cells of human body) initiate into a nonlocal reson-
ance process, whereas the absence of an emotional response or focused attention
does not establish the resonance condition for decoding the object’s nonlocal
information structure. Extrasensory perception has a similar resonance effect,
according to this model.
In short, we may regard Fibonacci harmonic intervals as standing waves in
which the brain and quantum field exchange information across the nodes and
antinodes of the interference. The human brain acts as the medium of informa-
tion exchange with the phase of the receptors in the brain bringing about the re-
sonance that enables the transmission of information from the quantum field to
the brain. The quantum field acts as the source of nonlocal information stored in
the nodes and antinodes of the interference patterns [19]. Both the brain and
quantum field are in phase conjugate adaptive resonance, but whether this re-
sonance involves local or nonlocal information depends on the emotional state
of the person and Fibonacci harmonic intervals.
3. Model Evaluation and Comparison
Having described the two models, we can now evaluate and compare them. In
each section following, the goal is to examine strengths and weaknesses of the
models. The selected criteria for making comparisons between the different
models follow general criteria published in the literature. Model evaluation cri-
teria include: 1)
explanatory adequacy
(whether the theoretical account of the
model helps make sense of observed data; 2)
(whether the model
provides a good predictor of future observations); 3)
(whether the
model’s description of observed data is the simplest possible), and 4)
(whether the model provides a deeper insight or link to another branch of
knowledge). To be preferred, the more criteria satisfied, the better. Although
each criterion can be evaluated on its own, in practice, they are rarely indepen-
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10.4236/am.2018.96048 710 Applied Mathematics
dent of one another. Consideration of all four is necessary to assess the adequacy
of a model.
3.1. Explanation
To explain a phenomenon, three issues need clarification [20]. First, it is impor-
tant to know the causes of a particular phenomenon. Second, one would also
need to determine whether the hypothesized causes correlate with the observed
effect and whether that covariation is deterministic or probabilistic. Third,
whether hypothesized causes precede the observed effects in time requires em-
pirical assessment. Hence scientific models of synchronicity must be able to
meet the demands of these three conditions.
Although few have discussed synchronicity as a scientific concept (For excep-
tions, see [21] [22] [23]), this view is explicit in Jung’s writings. For example,
Jung states that “Synchronicity is not a philosophical view but an empirical con-
cept which postulates an intellectually necessary principle” (p. 96) [24]. At first,
there seems to be a critical gap between the mental and the physical: How can
these categories be compatible? The gap between the two categories is closed by
bridge principles. Jung postulated the Fibonacci numbers as a bridge principle to
explain synchronicity phenomena [6]. It is this bridge principle that creates the
possibility of formulating quantitative predictions. Hence by taking the Fibonac-
ci numbers as a bridge principle into account, both models (GSM and HM) pro-
vide explanatory power with experimentally testable formats in terms of Fibo-
nacci time patterns.
However, there is also an important difference in the models regarding expla-
nation of synchronicity. The GSM secondary (and higher) intervals are conjec-
tured to correspond to synchronicity in terms of bifurcations that accompany
the onset of chaos, where bifurcations are based on the golden ratio. The HM
would, however, propose that the mechanism of synchronicity is via resonant
self-organization, where nodes and antinodes represent periodic attractors based
on Fibonacci harmonics. In the HM model, primary intervals form a harmonic
system (Table 2) based on standing wave resonance patterns with no tertiary
calculations necessary. The HM capitalizes on brain states as harmonic modes of
the brain’s structural connectivity and the synchronicity of cortical activity pat-
terns into these harmonic brain states [25]. The set of all Fibonacci harmonics
thus provides a new functional basis to explain synchronicity, which is based on
the structural connectivity of the human brain. In fact, the brain’s recurring
electromagnetic activities form a highly reproducible harmonic function based
on the golden ratio [26].
To infer a causal relationship between two variables or phenomena, both must
be correlated. In other words, if A is correlated to B, then we should see a rela-
tionship between A and B. Covariation can either be deterministic or probabilis-
tic. Deterministic covariation is when manipulation of only one variable pro-
duces an effect in another variable. Here, the cause is a necessary and sufficient
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10.4236/am.2018.96048 711 Applied Mathematics
condition for the effect. Probabilistic covariation is when an effect is a function
of two or more causes. Here, none of the causes can be inferred from the other,
and none of the causes is sufficient. Paradoxically, in dynamical systems theory,
deterministic and probabilistic descriptions are often complementary [8]. Dy-
namical systems containing bifurcations can have both deterministic (universal
and necessary) and probabilistic (historical happenstance) elements; they can
possess varying degrees of susceptibility to change induced by fluctuations and
they can respond differently to local, regional, and global fluctuations [8].
Finally, temporal ordering demands that to infer a causal relationship between
two variables, the cause must always occur first before the effect. In other words,
if A causes B, then we should see a change in A first, and then a responding
change to B. As regards temporal ordering, it is more difficult to talk about the
models and synchronicity interaction in the sense of causal relations. The notion
of probabilistic causality should be preferred. Probabilistic causality attempts
to explicate causal relationships in terms of probabilistic relationships [27]. This
view is equipped to deal with the inherently stochastic properties associated with
quantum mechanics. The central idea behind probabilistic causality is that caus-
es change the probabilities of their effects. Therefore, no temporal ordering oc-
curs essentially. However, there should be a “match” between Fibonacci time
patterns and synchronicity to some extent. Several other variables (e.g., emo-
tional intensity, focused attention) may raise the probabilities of a synchronistic
event. Empirical studies are needed to verify the proposed relation between Fi-
bonacci time patterns and synchronicity.
3.2. Prediction
Prediction deals with accurate anticipation of future occurrence of an event. By
studying the trend of events regarding a phenomenon, researchers can forecast
or predict what next will happen. The FLCM appears to be a reliable and valid
predictor of biological and psychological change [9]. That is why the dynamical
aspects of FLCM ages are also conjectured to relate to other discontinuous pat-
terns, like synchronicity. A study by Sacco [14] explored the relationship be-
tween FLCM primary ages 11, 18, and 30 and religious and spiritual experience
reports among young adults. Specifically, it was found that age 18 predicted a
higher frequency of religious and spiritual experience. Further analysis of case
reports at age 18 revealed that most involved heightened emotional experiences
associated with crisis events, suggestive that people engage in meaning-making
during times of instability, rather than times of stability [28].
Although both models use Fibonacci measurements, their age predictions are
not identical. The GSM secondary (and higher) points are generated from the
property that any interval between and outside adjacent points of the primary
intervals must be at the golden ratio interval (0.618 and 1.618). In contrast, the
HM describes secondary points as the nodes and antinodes of equally spaced
primary intervals where the wave amplitude (motion) has zero displacement in a
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10.4236/am.2018.96048 712 Applied Mathematics
standing wave pattern. Thus, the GSM and HM differ only in secondary points.
The theoretical exposition noted that when people are at primary and secondary
golden interval points, there are discontinuous forces, such that people may feel
a change in their thoughts, emotions, or behaviors, whereas at other points
people maintain a high sense of stability because the dynamical phase space is
continuous and they are outside the beginning and end points of discontinuities.
As the primary and secondary points exhibit a temporal structure with fractal
and nonlinear features, a person may experience synchronicity as a discontinuity
in the relationship between present conditions and future states [5]. Jung wrote
that “Synchronicity is no more baffling or mysterious than the discontinuities of
physics” (p. 102) [24]. In physics, a process becomes discontinuous in space and
time at the very smallest of length or time scales. Discontinuities are, therefore,
expected to involve events in space-time at the Planck length. The Planck length
is the length scale at which the structure of space-time becomes dominated by
quantum effects. The golden ratio, which to two decimal places is 1.62, corres-
ponds to the digits of the Planck length, 1.62 × 1033. Thus, on theoretical
grounds, golden ratio time intervals could predict discontinuities in space and
time, mind-matter correlations, and synchronistic phenomena. One problem to
understand is the passage from a continuous to a discontinuous state. Is there an
abrupt change or gradual transition?
Regarding the HM, a useful basis for making predictions about how people
will think, feel, and act is the energy transfer and amplitude of the standing wave
pattern. The total energy of a standing wave oscillates between a maximal value
at zero displacement (node) and zero value at maximal displacement (antinode).
This conclusion presents us with an immediate prediction: Energy transfer that
changes the entire phase space may be amplified at nodes and antinodes, and the
most important phase space changes may occur at these points. As described
earlier, many factors may become intertwined in a resonance condition between
the world and the brain, as the standing wave accumulates experiences, emo-
tions, and commitments (some of which may be more immune to change). Of
particular interest is that standing wave patterns can have varying amplitude:
Amplitude measures how much energy is transported by the wave. Specifically,
the amplitude of waves in the harmonic model increase according to powers of
Phi. Put another way, as the amplitude of the waves get larger as the person ages
there may be more energy associated with them.
3.3. Simplicity
To consider the GSM first, for simplicity, this model extends the original prima-
ry intervals (21 dates) to two iterations of measured points (112 dates). The
model computes measured points from the distance of the primary intervals
(internally and externally) by multiplying adjacent primary intervals by the Fi-
bonacci ratios 0.618, 1.618, 0.786 (square root of 0.618), and 1.272 (square root
of 1.618). The method halts after two iterations for simplicity as a global con-
R. G. Sacco
10.4236/am.2018.96048 713 Applied Mathematics
straint [5]. The usefulness of the square root ratios (
, 0.786 and 1.272) could
be called into question since the main idea was to come up with a general golden
ratio based parameterization of the primary intervals. Although, the square root
of the golden ratio (1.27) has astronomical significance in the ratio of the di-
ameters of earth and moon and biological significance in the median insect
growth ratio [29].
The HM presents a novel general closed-form, continuous forward model of
the primary intervals. This leads to phase-constant temporal patterns. Contrary
to the GSM, no external constraints within the model are required, except the
standard assumption of halting temporal patterns at the average human life ex-
pectancy. For simplicity, the fewer assumptions imposed by the model, the bet-
ter, and so the HM may be considered simpler than the GSM. Interestingly, the
Fibonacci sequence itself is a sinusoidal waveform with the ratios of adjacent
terms progressing in an alternating bigger (+) and smaller (−) pattern, and the
24 repeating period of the Fibonacci numbers is approximately sinusoidal [9].
Harmonics, which causes resonance in the sinusoidal waveform, is usually con-
sidered as an integer multiple of the fundamental frequency. This form of har-
monics is called “integer harmonics” or simply “harmonics” since it is common.
However, the present research deals with a slightly more complicated form of
harmonics called “non-integer harmonics.” Non-integer harmonics are harmonics
that are a non-integer multiple of the fundamental frequency. Specifically, the
harmonics frequencies observed in this research are powers of Phi fractions of
the fundamental frequency.
3.4. Generality
The models carry several advantages for generality, including drawing from
scientific disciplines ranging from mathematics and physics to biology and psy-
chology. There are multiple sources of the widespread appearance of the Fibo-
nacci numbers and ratio in the natural universe. For example, an overall opti-
mum healthy heart function occurs when there is convergence of the Fibonacci
numbers and Phi relationship between the waves on the electrocardiogram [30].
It is dramatically seen in the shape of spiral galaxies. Fibonacci resonance can
explain why the orbital distances of planets and hence by Kepler’s laws their or-
bital periods are found at golden ratio distances from the Sun [31], and why
electrons abide by quantum levels [18]. Findings based on the study of data from
cosmic background radiation reveal that the universe is shaped like a dodecahe-
dron, a geometric shape based on pentagons, which are based on Phi [32]. Fibo-
nacci resonance is also found in Mandelbrot’s fractal [13], suggesting it to be the
basic dynamic fractal of the universe.
Both models have several clinical implications. One possible avenue for psy-
chotherapy practice could be to assist therapists in their work with adults coping
with life transitions. Insofar as Fibonacci numbers predict attractors, fractals,
and chaos [11] [13], it can reduce the risks of counsellors missing crucial op-
R. G. Sacco
10.4236/am.2018.96048 714 Applied Mathematics
portunities or focusing on irrelevant issues, such as by charting clients’ adaptive
landscapes, which allows the counsellor to attend to system self-organization
and emergent dynamics. Also, it allows both subjective and objective informa-
tion to be portrayed; permits an improved understanding about the self by re-
lating past experiences; and allows clients to experience improved understanding
of synchronicity experiences that occur during phase transitions. This dynamical
systems approach allows compatibility with a wide range of counseling ap-
proaches, styles, or theories. For example, this kind of cognitive reflection may
be combined with mindfulness-based practices [33].
The models proposed in this paper apply best to synchronicity broadly con-
strued. They are less applicable to more ordinary spiritual experiences (because
of the lack of nonlinear dynamics, and because of their wide variability). As Wil-
liam James noted [34], spiritual experiences are diverse. Many spiritual expe-
riences (e.g., awe and joy, watching a sunset) do not infringe “everyday” causal
principles, and may not emphasize the sense of dramatic change. In contrast, the
basis of synchronicity is altered perceived causality and meaningfulness that fa-
cilitates a feeling of sudden change. Often there is a sense in which the synchro-
nicity occurs as part of a sequence of events with a central theme. Viewed in this
light, synchronicity can be seen as a configuration in neural networks in which a
sequence of events pushes the system into a new emergent state leading to a
phase transition. Therefore, the models are a preliminary comprehensive model
of synchronicity based on dynamic systems principles.
4. Discussion and Conclusion
This article had two purposes. First, it sought to elaborate a harmonic model of
synchronicity based on the Fibonacci Life Chart Method (FLCM). Second, it
sought to compare the golden section model (GSM) and harmonic model (HM).
The development of the HM was treated as a separate task from the model com-
parison, and some explanations and predictions were developed that can be
tested in future data.
The analysis of synchronicity as Fibonacci time patterns appears capable of
supporting a broad range of testable predictions. It is also a useful link between
mathematics and synchronicity. That is, the author followed Jung [6] in sug-
gesting that Fibonacci numbers can furnish a plausible explanation for synchro-
nicity, but the theory itself is essentially dynamical in the sense that it specifies
how change will be meaningfully interlinked and organized. Like modern phys-
ics research itself, the FLCM draws on dynamical systems theory and its under-
standing of change dynamics, self-organization, emergence, and fractal dynam-
ics. Thus, the present theory is dynamical, because it proposes that synchronicity
will operate based on Fibonacci numbers controlling the interplay of genes and
brain development, which change coherence of brain waves, which cycle around
again to influence brain development.
The second part of the article compared the two models, and they were gener-
R. G. Sacco
10.4236/am.2018.96048 715 Applied Mathematics
ally consistent with many aspects of dynamical systems theory. Both models
have several essential features such as Fibonacci time patterns and nonlinear
dynamics consistent with the view that synchronicity is often a sudden change in
the lives of many people. Thus, in dynamical systems theory complex systems
are prone to exhibit extreme sensitivity to perturbations, so that nonlinear cau-
sality among interacting components allows small differences to produce large
effects over time. This classic feature of dynamic systems is known as the “but-
terfly effect”: the butterfly’s wings can influence air currents that, over many ite-
rations, result in a thunderstorm. But if self-organizing cognition-emotion inte-
ractions are so sensitive, how do they attain the coherence and resiliency that
characterize personality? And how do they attain consistency within, and some-
times across, individuals, as highlighted by conventional norms in society and
culture? These questions lead to a reconsideration of the sources of orderliness
in development and personality (see Table 1). These seemingly contradictory
observations can be reconciled through an understanding of Fibonacci time pat-
terns, which show that attractor dynamics are insensitive to perturbation, whe-
reas the dynamic trajectory of the system is extremely sensitive to conditions
near fractal boundaries.
On any given state in the phase space diagram, human development gravitates
toward attractor states that are codetermined by genetic, cultural, and experien-
tial histories. Biological and cultural constraints influence the way cognitive
elements cohere together and the way cognitions and emotions reciprocally ac-
tivate each other, but these constraints are continuously modified by the emer-
gent structure of biological unfolding of events involved in a person changing
gradually from a simple to a more complex level. Both universal and idiosyn-
cratic constraints thus guide human development, allowing for normative
themes and individual variations in cognition-emotion interactions.
In general, the data suggest that FLCM ages have strong predictive value as
points of discontinuity, whereas non-FLCM ages command a significant source
of continuity. Most of the findings pertain to biological and psychological
changes, and it seems reasonable to conclude that Fibonacci time patterns pre-
dict the evolution of trajectories in phase space [11]. There is less evidence to in-
dicate that Fibonacci time patterns continue to be relevant to synchronicity ex-
periences, though some findings (such as involving religious and spiritual expe-
riences) do suggest that the dynamical effects of age 18 are a predictor of spiri-
tual experience [14]. FLCM ages specify exclusive discontinuity and change, as
the person is removed from points of stability in dynamical phase space, and the
psychological processes of that phase space are more relevant to nonlinear and
fractal dynamics.
Synchronicity is a phenomenon that disrupts conventional notions of tempor-
al ordering, and so it would be surprising if mathematics and dynamical prin-
ciples were absent. I do not seek to replace all other theories of spiritual expe-
rience with this mathematical one, and I have noted that the mathematical anal-
R. G. Sacco
10.4236/am.2018.96048 716 Applied Mathematics
ysis does not apply equally well to all spiritual experiences. I propose only that
the mathematical analysis deserves to be included as one important dimension
of synchronicity experience and one useful approach for understanding the non-
linear dynamics of synchronicity, especially the large-scale, personality trans-
forming experiences of synchronicity. Considering how well the FLCM model
fits the developmental data, I think it appropriate that researchers interested in
the mathematical aspects of synchronicity can begin conducting more prospec-
tive and explicit tests of it.
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