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.
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 [
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 [
This article turns to a different discipline, mathematics, to explain a theory of synchronicity. A mathematical approach to synchronicity was defined by Sacco [
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 [
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 [
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.
Fibonacci Life Chart Method (FLCM) is a theoretical model of human development as a nonlinear dynamical system based on the Fibonacci sequence [
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 [
Most notably, because the Fibonacci sequence relates to the well-known period doubling in dynamical systems theory [
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 development 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 psychological phase transition. Phase transitions can result from self-organization processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, organism, 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 [
Although the premise that Fibonacci numbers can explain synchronicity invokes a mathematical model to explain human experience and is therefore essentially a mathematical theory, ironically its most famous advocate came from Carl Jung, a Swiss psychiatrist. Jung observed that “…synchronicity… is a secondary 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) [
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 [
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 [
golden ratio because of its unique nesting capability, as seen for example in the Mandelbrot set [
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 [
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 importance and is an alternative approach to computing secondary time intervals. One crucial feature of the HM is the time-periodic patterns of the primary intervals. 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 [
To the extent that human life cycles are a Fibonacci resonance phenomenon, time-periodic patterns can be described by phase relationships (phase patterns) among the primary intervals of the FLCM.
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 frequencies are inversely proportional (2x, 3x, 4x, 5x, 6x, etc., where x is the fundamental 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 harmonics 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 orbitals 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
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 |
with particular frequencies. For an electron, these frequencies are determined by the fine-structure constant denoted by α (the Greek letter alpha). The value of α is approximately 1/137. The fine-structure constant is an expression of Phi: Phi2/360 = 2.618/360 = 1/137.508 = 0.00727 [
At the deepest level, the human mind may manifest not only local but also nonlocal characteristics [
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 [
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 independent of one another. Consideration of all four is necessary to assess the adequacy of a model.
To explain a phenomenon, three issues need clarification [
Although few have discussed synchronicity as a scientific concept (For exceptions, see [
However, there is also an important difference in the models regarding explanation of synchronicity. The GSM secondary (and higher) intervals are conjectured 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 (
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 relationship between A and B. Covariation can either be deterministic or probabilistic. Deterministic covariation is when manipulation of only one variable produces an effect in another variable. Here, the cause is a necessary and sufficient 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 [
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 [
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 [
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 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 [
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.
To consider the GSM first, for simplicity, this model extends the original primary 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 Fibonacci 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 constraint [
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 expectancy. For simplicity, the fewer assumptions imposed by the model, the better, 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 [
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 [
Both models have several clinical implications. One possible avenue for psychotherapy practice could be to assist therapists in their work with adults coping with life transitions. Insofar as Fibonacci numbers predict attractors, fractals, and chaos [
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 [
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 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 [
The second part of the article compared the two models, and they were generally 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 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
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 [
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 analysis 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 nonlinear dynamics of synchronicity, especially the large-scale, personality transforming 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 prospective and explicit tests of it.
Sacco, R.G. (2018) Fibonacci Harmonics: A New Mathematical Model of Synchronicity. Applied Mathematics, 9, 702-718. https://doi.org/10.4236/am.2018.96048