Fibonacci Harmonics: A New Mathematical Model of Synchronicity

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.


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 mathematical 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 determinants, as shaped by biased cognition [4]. Both have proposed to explain the 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 emphasized chance occurrences and our need to make sense of ourselves and the world. Thus, the disciplines of physics and psychology have been most prominent 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 patterns 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 transitions, 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 formation 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 anticipated 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 because 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 dynamics, 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 neglected one crucial aspect, which will be featured in this article: the points of system transition. At any given time, a dynamical system has a state given by a numerical phase space and a rule of evolution specifying trajectories in this space [8]. There are two main parts to this article. The first will shed light on synchronicity 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 perspective. The second section will then compare findings about the two models as a way of evaluating their capacity to explain and predict synchronicity phenomena.  [9]. These chronological 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".

Description of the Fibonacci Life Chart Method
Most notably, because the Fibonacci sequence relates to the well-known period doubling in dynamical systems theory [11], a bifurcation with a shift in coherence from one FLCM age to another occurs in concert with the onset of 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. Indeed, he provided little elaboration for his theory that Fibonacci numbers explain 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 synchronicity is essentially something controlled by the Fibonacci numbers.

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 golden 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 ratio 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".  Table 1). After dividing 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 exterior) 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 distribution with a self-organized fractal structure [5]. A fractal is a recurring similar pattern at different scales. Many objects in nature are best described as fractals. The fractal structure is ubiquitous not only in trees, rivers, mountains, islands, 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. Fractals belong to a set of models in which the threshold for activity, rather than being a function of a purely local variable, depends on nonlocal properties of the self-organizing structure [12]. Furthermore, all fractal structure involves the 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 circumstances). The bottom line is that golden section points have predictive value. Religious, spiritual, mystical, and synchronistic phenomena and similar experiences will have neural underpinnings triggered by the dynamical interaction between biology and mathematics [14]. Put another way, the formation and continuation of a series of synchronicities depends on whether their trajectories evolve by the rule of Fibonacci numbers than random chance; whether the synchronicity corresponds to nonlinear change than linear change, and so forth.

Model 2: Harmonic Model
The Harmonics relate to standing waves [15]. A standing wave pattern is not actually 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 frequency 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 amplitude (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 harmonic frequency, the interference of reflected and incident waves leads to a disturbance 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)    [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 between the object and the brain. This resonance process is called phase conjugate adaptive resonance (PCAR). With a strong emotional response or focused attention in a person, some waves emitted from an object (i.e., subatomic particle, inanimate or animate thing, cells of human body) initiate into a nonlocal resonance 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 information exchange with the phase of the receptors in the brain bringing about the resonance 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 resonance involves local or nonlocal information depends on the emotional state of the person and Fibonacci harmonic intervals.

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 criteria include: 1) explanatory adequacy (whether the theoretical account of the model helps make sense of observed data; 2) predictability (whether the model provides a good predictor of future observations); 3) simplicity (whether the model's description of observed data is the simplest possible), and 4) generalizability (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-

Explanation
To explain a phenomenon, three issues need clarification [20]. First, it is important 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 empirical 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 exceptions, 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 concept 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  (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 patterns 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].

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 patterns, like synchronicity. A study by Sacco [14] explored the relationship between 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 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 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, emotions, 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.

Simplicity
To consider the GSM first, for simplicity, this model extends the original prima- Harmonics, which causes resonance in the sinusoidal waveform, is usually considered as an integer multiple of the fundamental frequency. This form of harmonics 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.

Generality
The models carry several advantages for generality, including drawing from scientific disciplines ranging from mathematics and physics to biology and psychology. There are multiple sources of the widespread appearance of the Fibonacci numbers and ratio in the natural universe. For example, an overall optimum 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 orbital 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 dodecahedron, a geometric shape based on pentagons, which are based on Phi [32]. Fibonacci resonance is also found in Mandelbrot's fractal [13], suggesting it to be the basic dynamic fractal of the universe. landscapes, which allows the counsellor to attend to system self-organization and emergent dynamics. Also, it allows both subjective and objective information to be portrayed; permits an improved understanding about the self by relating 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 approaches, 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 construed. They are less applicable to more ordinary spiritual experiences (because of the lack of nonlinear dynamics, and because of their wide variability). As William James noted [34], spiritual experiences are diverse. Many spiritual experiences (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 facilitates a feeling of sudden change. Often there is a sense in which the synchronicity 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.

Discussion and Conclusion
This article had two purposes. The development of the HM was treated as a separate task from the model comparison, 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 suggesting that Fibonacci numbers can furnish a plausible explanation for synchronicity, but the theory itself is essentially dynamical in the sense that it specifies how change will be meaningfully interlinked and organized. Like modern physics research itself, the FLCM draws on dynamical systems theory and its understanding of change dynamics, self-organization, emergence, and fractal dynamics. 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- are prone to exhibit extreme sensitivity to perturbations, so that nonlinear causality among interacting components allows small differences to produce large effects over time. This classic feature of dynamic systems is known as the "butterfly effect": the butterfly's wings can influence air currents that, over many iterations, result in a thunderstorm. But if self-organizing cognition-emotion interactions are so sensitive, how do they attain the coherence and resiliency that characterize personality? And how do they attain consistency within, and sometimes 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 patterns, which show that attractor dynamics are insensitive to perturbation, whereas 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 experiential histories. Biological and cultural constraints influence the way cognitive elements cohere together and the way cognitions and emotions reciprocally activate each other, but these constraints are continuously modified by the emergent structure of biological unfolding of events involved in a person changing gradually from a simple to a more complex level. Both universal and idiosyncratic 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 predict the evolution of trajectories in phase space [11]. There is less evidence to indicate that Fibonacci time patterns continue to be relevant to synchronicity experiences, though some findings (such as involving religious and spiritual experiences) do suggest that the dynamical effects of age 18 are a predictor of spiritual 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 temporal ordering, and so it would be surprising if mathematics and dynamical principles were absent. I do not seek to replace all other theories of spiritual experience with this mathematical one, and I have noted that the mathematical anal-