Seeking meaningful connections in life is an innate emotional drive, a uniquely human characteristic. Human lives are rich with meaning, and a useful way to capture those meanings is with a personal timeline. This post introduces an innovative personal timeline method—the Fibonacci Lifechart—in research on synchronicity experiences. This method integrates mathematical modelling to facilitate the understanding of meaningful coincidences within a fractal resonance framework. It allows an in-depth understanding of synchronicity with adherence to methodological and scientific rigor.
I will discuss the results of an empirical research project on “The Predictability of Synchronicity” which was accomplished in the years between 2018 and 2019. The project analyzed the synchronicity dates of Jungian analysts in proximity to Fibonacci time patterns. The theoretical prediction was that synchronicity dates would be correlated to Fibonacci time patterns based on the idea that synchronicity is a fractal resonance phenomenon between the brain and the environment. The findings have been published in the International Journal of Psychological Studies and may help explain meaningful coincidences as a fractal resonance phenomenon (Sacco, 2019).
Fractals are symmetrical patterns generated from immense complexity within the resonant frequencies of the universe. Such frequencies result in the exchange of energy and the coupling of informational systems at various levels and scales. At a fundamental level communication happens via “resonance”, and this, in turn, manifests at a physical level as self‑replicating and self‑resonating fractal patterns. The transference of resonant frequency from beyond space-time to space-time structure is proposed in the mathematics of the Fibonacci series, golden ratio, and quantum resonance. The issue is the relatively simple conceptual one of the transference of information between a number of scales.
This could be the reason why at a physical level every structure (associated with both animate and inanimate entities) carries the Fibonacci series and golden ratio as a fractal imprint, making it the cosmic signature (Livio, 2008). Thus, it is hypothesized meaningful coincidences could emerge as a fractal-resonant phenomenon, resulting from the interplay between material and non‑material aspects. In this regard, frequencies or vibrations are fundamental to life, and consciousness is made up of frequencies that exist in fractal harmonic form based on the golden ratio (Pletzer, Kerschbaum, & Klimesch, 2010). This calls for scientific studies that aim at understanding the effect of various frequencies and vibrations on the experience of meaningful coincidence.
The Fibonacci Lifechart
The Fibonacci sequence is a recursive sequence yielding wave-like phenomena and self-organized patterns found throughout nature. From nautilus seashells, pine cones, plants and sunflowers, to DNA molecules, human brain waves, the stock market, and the shape of galaxies, the Fibonacci sequence and golden ratio make its presence known. The Fibonacci sequence and golden ratio are mysterious because they appear in so many forms of nature. But could they also be useful to guide the scientific study of synchronicity? This crucial question was explored by Carl Jung in a letter he wrote in February 1956.
From the point of view of Carl Jung, the Fibonacci sequence served as a bridge between mind and matter (Jung, 1976). For Jung, it was therefore essential for the interpretation of synchronicity. This view was no doubt influenced by his friend and collaborator Wolfgang Pauli, a Nobel laureate and one of the founders of quantum physics. However, the Fibonacci sequence was completely ignored in synchronicity research. Only very recently efforts were begun to explore the practical possibilities of the Fibonacci sequence in human development.
Searching for psychotherapy’s role in understanding synchronicity led me to a new mathematical approach to human development: The Fibonacci Lifechart. I learned from John Waskom and Norman Rose who postulated that stages of human development followed the Fibonacci sequence to see individuals in relation to their larger environmental field. I designed the Fibonacci Lifechart based on mathematics, world religions, philosophies, and psychology looking for the underlying mathematical structure of life. Examples of classic works I studied included: Taoism, Buddhism, Hinduism, Islam, Judeo-Christian traditions; philosophies, including those by Pythagoras and Plato; and the work of psychologists including Jung and Erikson.
In 2013, I showed that the mathematical modeling of the Fibonacci sequence can predict important human life outcomes (Sacco, 2013). These include brain development, puberty, optimal fertility, personality stability, happiness levels, menopause, spiritual experience, and average life expectancy. Following this line of research, I discussed and compared two methodologies for the mathematical modeling of synchronicity. The basic idea underlying this research is that synchronicity reflects the circular causality or positive feedback loop between microscopic and macroscopic dynamics.
Both serial order and circularity overlap can be represented by the nonlinear exponential growth properties of the Fibonacci sequence. Indeed, circularity is mysteriously encoded into the Fibonacci sequence itself. For example, the 5th number in the Fibonacci sequence is the number five. The 10th number is 55 (5 + 5). And the 12th number is 144 (12 x 12). According to the principles of The Fibonacci Life Chart Method (FLCM), the human brain and environment exhibit holistic synchronicity effects through their resonance on multiple levels of organization. Holistic resonance depends on the shared fractal geometry of the golden ratio. In short, a “shared” fractal geometry of human brain waves and the environment could create meaningful coincidences.
World’s First Scientific Proof of Synchronicity?
As a way to move the scientific study of coincidence and synchronicity forward, in January 2018 I discussed with Bernard Beitman the possibility of creating a new journal of coincidence. When Bernard and I talked about creating a new journal that focuses on the science of synchronicity, I suggested that we perhaps did not need a new journal as there were not enough scientific papers to go around the existing journals already. Instead of creating a new journal, I decided that I wanted to focus my efforts on testing a question I had about the relationship between Fibonacci time patterns and synchronicity.
In February 2018, I sent a Synchronicity Survey to members of the International Association for Analytical Psychology (IAAP). The Synchronicity Survey was meant to investigate the relation between Jungian analyst experiences of synchronicity and Fibonacci time patterns. The result was, for the first time, I finally got the chance to answer my question about the relationship between Fibonacci time patterns and synchronicity: Fibonacci time patterns seem to validate the theoretical prediction of synchronicity to ±34 days.
Simple survey methods that we take for granted have illuminated our understanding of synchronicity. However, caution should be exercised in the interpretation of the results. First, the sample size was small (41 instances of synchronicity from 18 subjects) and statistical significance was reported at the 10% statistical significance level. This is a bit less orthodox because usually, a significant finding is less than 5% statistical significance. Generally, a 10% confidence level is not considered statistically significant. But in some situations, it can be argued it may be significant (e.g., because of the small sample size). Follow-up studies remain to be done and should expand the sample size.
Second, the population sample may also have affected the research results. The Jungian analysts surveyed are more familiar with the subject of meaningful coincidences than the general population. Whether the study findings can also apply to the general population is questionable. It is conceivable that personality traits may also relate to the population dimension. For example, characteristics that suggest proneness to synchronicity include self-consciousness, high negative affect, openness to experience, intuition, and the search for meaning. Therefore, identifying people who may be more prone to experience synchronicity may improve the predictive value of the Fibonacci Lifechart. Even so, with the help of the Fibonacci sequence and simple survey research, a revolution is underway in the scientific study of synchronicity.
At present, quantum computing, quantum communication, and quantum biology offer a new understanding of the universe in terms of information processing, suggesting possibilities in which science and spirituality converge. It’s been a long journey creating it, but I’ve accelerated my own growth with the Fibonacci Lifechart and am happy to have built it up from ground zero to help others do the same. The Lifechart is also available as a completely free download at www.fibonaccilifechart.com.
Jung, C. G. (1976). Letters of C.G. Jung (Vol. 2). London: Routledge and Kegan Paul.
Livio, M. (2008). The golden ratio: The story of Phi, the world’s most astonishing number. New York, NY: Broadway Books.
Pletzer, B., Kerschbaum, H. & Klimesch, W. (2010). When frequencies never synchronize: The golden mean and the resting EEG. Brain Research, 1335, 91-102. https://doi.org/10.1016/j.brainres.2010.03.074
Sacco, R.G. (2013). Re-envisaging the eight developmental stages of Erik Erikson: The Fibonacci Life-Chart Method (FLCM). Journal of Educational and Developmental Psychology, 3(1), 140–146. https://doi.org/10.5539/jedp.v3n1p140
Sacco, R. G. (2019). The predictability of synchronicity experience: Results from a survey of Jungian analysts. International Journal of Psychological Studies, 11(3), 46-62. https://doi.org/10.5539/ijps.v11n3p46
The question “What is truth?” may sound profound.
In reality, I think we all know the answer to this age-old question. That is because we presuppose a certain definition of truth in our speech and actions every day of our lives.
Perhaps the problem is not that we do not know what truth is but rather that we do not know that we know. In other words, we struggle to articulate a definition of truth. But this is easily remedied if we take a few moments to reflect on the nature of truth.
Three Views on Truth
There have been three dominant theories of truth.
First, there is the social construction view that truth is relative to what people want. This makes truth relative to personal and social goals. But, if we look further, we see that this proposition is not logical. In fact, it is self-refuting. If all truth is relative, then the statement "All truth is relative" would be absolutely true. If it is absolutely true, then not all things are relative and the statement that "All truth is relative" is false.
Second, there is the coherence theory of truth: truth is logical consistency (coherence) among a set of beliefs an individual holds. However, coherence is better described as a criterion for truth, not truth itself. Identifying truth with coherence fails because there are opposing views that are each internally coherent even though they do not agree with each other. For example, two different religions may each have complex worldviews full of internal coherence, but they are not compatible with each other and with scientific findings.
Finally, there is the correspondence theory of truth: truth is when an idea, belief, or statement matches (or corresponds with) the way the world actually is (reality).
This may rightly be labeled the “common sense” view of truth. The correspondence theory of truth states that an idea, belief, or statement is true if it matches, or corresponds with, reality. In this sense, reality is the truth-maker, and the idea, belief, or statement is the truth-bearer. When the truth-bearer (an idea) matches the truth-maker (reality), they are said to stand in an “appropriate correspondence relationship,” and truth obtains.
Synchronicity: A Case of Correspondence
I have advanced a scientific argument in favor of the correspondence theory of truth regarding synchronicity experience. In this view, a synchronicity is not true simply because it works for us (the relativist view) nor because it is consistent with a web of scientific theory (the coherence view). The scientific theory of synchronicity is true because it is an objective fact that corresponds with reality!
For example, an individual who has a synchronicity, actually experiences truth, a correspondence relation between his thought and reality. Again, this is the “common sense” definition of truth since it is the view we all presuppose in our daily actions and speech (i.e., everyone assumes the correspondence theory of truth when reading a medicine label).
In other words, either a synchronicity corresponds to reality or it does not. If it does, the person presupposes the correspondence view. On the other hand, if a synchronicity does not correspond to reality, then we have no reason to accept it.
The Fibonacci Lifechart is based on the correspondence theory of truth. It is a tool that can help you to decide if one’s synchronicity should be taken seriously or not, for only a synchronicity which matches the way things really are is worthy of our attention and belief.
Contrary to the postmodern adage, truth is not “relative.” Rather, truth is what we have taken it to be all along, what we assume it to be every day.
The most revolutionary discovery of quantum mechanics is the paradox of quantum nonlocality. This draws attention to the phenomenon known as quantum entanglement.
As a result, quantum mechanics violates the principle of locality. This principle states that changes in one physical system should have no immediate effect on another separated system.
Our “local” view of the world assumes phenomena are separated by time and space. Thus, no influence can travel faster than the speed of light.
Quantum nonlocality proves these assumptions are mistaken.
A principle of holistic interconnectedness exists operating at the quantum level. And this contradicts the local assumptions of classical physics.
At the quantum level, instantaneous actions can occur.
Science does not appear ready to equate nonlocality with the spiritual realm. However, the nonlocal reality that we live in tends to mimic the properties often associated with the spiritual.
Carl Jung developed an acausal understanding of synchronicity. His concept of synchronicity was inspired by quantum theory and through his association with the great physicist Wolfgang Pauli,
Science is an approach to knowledge based on causality. Causal explanations emphasize one thing affecting another through an exchange of energy, forces, or information.
In contrast, synchronicity involves acausal explanations with no identifiable governing forces or physical energy exchange.
With synchronicity, there appear no causal connections between inner and outer events. Instead, they are symbolic expressions of transformative meaning. Thus, they challenge fundamental axioms of science.
In June 2018 (Sacco, 2018) I described a method useful in predicting synchronicity phenomena for use in counseling sessions or for individual purposes.
The method includes such steps as calculating Fibonacci time intervals obtained from adding the Fibonacci numbers to a birthdate, and chronological charts and statistics based on Fibonacci time intervals.
This method promises to provide everyone with the ability to enjoy more meaningful understanding of their past and present synchronicities.
In this blog post, I will delve deeper into this method, termed the "Harmonic Model."
Synchronicity is a concept advanced by Carl Jung (1952) that involves the meaningful coincidence between outer and inner events. Jung developed his ideas of synchronicity in close collaboration with the quantum physicist Wolfgang Pauli.
Synchronicities are acausal phenomena in the sense that they cannot be reduced to a cause-and-effect explanation. The term “synchronicity” served as an umbrella for Jung, under which he grouped many paranormal events. People also use words such as superstitious, magical, and supernatural to refer to the disruption of “every day” causal principles.
Synchronization is a universal phenomenon in nature and society (Pikovsky, Rosenblum, & Kurths, 2001). The term “synchronization” is used in nonlinear dynamics to mean adjustment of the rhythms of oscillatory processes because of their interaction. Two or more objects are said to be synchronized, or in “synchrony,” when there exists a fixed phase relation between them.
Synchronization represents a general mechanism of self-organization in complex systems, which involves the emergence of spontaneous order often with a regular geometric pattern and quasi-periodic structure. Several recent findings point to Fibonacci numbers and the golden ratio as crucial to synchronization.
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, etc.) is a recursive sequence in which every term is the sum of the previous. The mathematics of the Fibonacci sequence and golden ratio (about 1.618034) interrelate in that the ratios of the successive numbers in the Fibonacci sequence converge on the golden ratio.
Both Fibonacci numbers and the golden ratio are found widely in nature. In particular, harmonic proportions related to the golden ratio explain the synchronization of spiral galaxies and orbital periods (Sacco, 2019), magnetic resonances of atoms (Coldea et al., 2010), and also brain waves (Pletzer, Kerschbaum, & Klimesch, 2010). In short, Fibonacci numbers and the golden ratio are a powerful source of synchronization.
To increase scientific understanding of synchronicity requires developing theories that explain the sources of these experiences. Since Jung introduced his theory of synchronicity (Jung, 1952), theorists have struggled to formulate a model for this phenomenon.
I have suggested that the Fibonacci numbers can provide a basis for predicting synchronicity (Sacco, 2016, 2018). I will now describe the method that has been constructed to fulfil this purpose.
Summary of Harmonic Model
The technical problem addressed by the harmonic model is to provide a process by which to predict synchronicity phenomena. The implication is that understanding the origin and nature of synchronicity phenomena is essential since they permit researchers and clinicians to utilize change processes more effectively.
The solution to this problem is based on the idea that synchronicity phenomena can be predicted by the Fibonacci numbers. Fibonacci numbers are present at the quantum level, the DNA molecule, biological cell division, and self-organizing systems, and offer a reliable mechanism for predicting physical and psychological change. In particular, age intervals identified by the Fibonacci numbers can indicate an increase in the experience of synchronicity.
The Harmonic Model is a computer-based method for predicting synchronicity in the human lifespan. The method comprises the following steps:
a. selecting a birthdate;
b. calculating primary intervals obtained by adding the first 21 Fibonacci numbers to the birthdate with Fibonacci numbers representing a time scale of 24-hours;
c. calculating secondary intervals wherein dates generated from step b can be used to calculate standing wave harmonics, which occurs when the primary intervals are repeated at constant intervals up to age 78.51 or another terminal period.
d. displaying the time intervals on a chart or graph.
The Harmonic Model can also be used in counseling by:
a. displaying the time intervals on a chart or graph;
b. analyzing the displayed results; and
c. counseling or advising the person based on the results.
Advantageously, the Harmonic Model provides a practical system of counseling that includes diagnostic elements, plus distinct principles to guide counselor interventions.
These and other benefits will become evident from a consideration of the accompanying figures.
Figure 1: Primary interval calculations. Fig. 1 is a view of the primary interval calculations for a birthdate of January 1, 2000 (102). The first 21 Fibonacci numbers (100) in the Fibonacci sequence are: F1=1, F2=1, F3=2, F4=3, F=5=5, F6=8, F7=13, F8=21, F9=34, F10=55, F11=89, F12=144, F13=233, F14=377, F15=610, F16=987, F17=1597, F18=2584, F19=4181, F20=6765, F21=10946. The step depicted by 101 are the first 21 Fibonacci numbers in the Fibonacci sequence expressed in terms of a time scale of 24-hours. At the step depicted by 102, the Fibonacci numbers are added to an individual birthdate. The step depicted by 103 shows the accrual age in years.
Figure 2: Secondary interval calculations. Fig. 2 is a view of the secondary interval calculations of the harmonic model based on primary interval calculations (103). The last nine primary interval calculations (104-112) are used for secondary date calculations (200-208). Secondary intervals are derived, whereby primary intervals are added together starting from the birthdate . For example, the formula for deriving the secondary interval 2003-05-03  = 2001-09-1  + 1.67 years (104). These calculations represent the nodal points of standing wave harmonics. The antinodes  are calculated from the averages of adjacent nodes.
The Fibonacci Lifechart displays the secondary interval calculations in both cycle plot (displayed in Figure 2) and chronological listing. Both the cycle plot and chronological listing of dates can be compared to the seven life domains (residence, cohabitation, intimate relationships, family, occupation, health, and spiritual experience) to assess for change processes.
Click here to get a copy of the Fibonacci Lifechart.
Coldea, R., Tennant, D. A., Wheeler, E. M., Wawrzynska, E., Prabhakaran, D., Telling, M.,... Kiefer, K. (2010). Quantum criticality in an Ising chain: Experimental evidence for emergent E8 symmetry. Science, 327(5962), 177-180. http://sci-hub.tw/10.1126/science.1180085
Jung, C. G. (1952). Synchronicity: An acausal connecting principle. CW 8.
Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization: A universal concept in nonlinear sciences. Cambridge, UK: Cambridge University Press.
Pletzer, B., Kerschbaum, H., & Klimesch, W. (2010). When frequencies never synchronize: The golden mean and the resting EEG. Brain Research, 1335, 91-102. http://sci-hub.tw/10.1016/j.brainres.2010.03.074
Sacco, R. G. (2016). The Fibonacci Life-Chart Method (FLCM) as a foundation for Carl Jung’s theory of synchronicity. Journal of Analytical Psychology, 61(2), 203-222. http://sci-hub.tw/10.1111/1468-5922.12204
Sacco, R. G. (2018). Fibonacci harmonics: A new mathematical model of synchronicity. Applied Mathematics, 9, 702-18. http://sci-hub.tw/10.4236/am.2018.96048
Sacco, R.G. (2019). Modeling celestial mechanics using the Fibonacci numbers. International Journal of Astronomy, 8, 8-12. http://sci-hub.tw/10.5923/j.astronomy.20190801.02
There has been much discussion of the stages of mystical experience.
Indeed, William James, a pioneer in the study of mystical experience, first characterized the process as one of “organic ripening.”
There are some minor differences in the stages proposed by various scholars. But all the phase-models share similarities to the experience of a psychological crisis.
For example, all the models proposed by scholars describe
In my book Mystical Experience: A Psychological Perspective, I described a five-phase crisis model. This model was based on the five major phases of a crisis experience: 1) pre-crisis, 2) impact, 3) crisis, 4) resolution, and 5) post-crisis.
Here I will provide an overview of these five phases.
Of course, there are limits of any phase-model to capture the essence of mystical experience. Some phases can succeed each other more quickly, some phases can appear to be bypassed, or the process can become locked into a particular phase.
However, the usefulness of a crisis model is that it produces testable predictions for empirical research.
Phase 1: Pre-Crisis Phase
During the pre-crisis stage, the person maintains an equilibrium.
The stress they encounter is balanced by their coping mechanisms.
To maintain equilibrium people are always solving problems. As long as people believe their problems can be solved using their available coping skills, equilibrium is maintained.
Phase 2: Impact Phase
A psychological crisis begins with a sudden disruption to important life goals. These disruptions cannot be overcome with usual problem-solving methods.
Examples of stressful events precipitating a crisis include:
• troubled personal relationships
• loss of jobs
• financial problems
• life transitions
• terminal illnesses
The first 1-3 days of a crisis are spent in shock, feeling confused, weak, and helpless, while everything seems to be in chaos.
Not all stressed persons move into a crisis state.
The external events themselves do not define the crisis, but instead, the person’s responses and reactions to the event do.
Phase 3: Crisis Phase
The initial impact period brings a 1-3 month crisis period. This is characterized by anxiety, hypersensitivity, irritable mood, tiredness, depression, and sleeping disorders.
In the crisis phase the person confronts the problem. The person’s usual strategies have failed to solve the problem brought about by the precipitating event.
From this point, the person is in a downward spiral becoming increasingly disorganized, anxious, tense, and hopeless.
Sooner or later the person confronts deep-rooted existential concerns about the meaning of life.
This is the sense one must, and yet one cannot find any real “truth.” This leads to non-rational guidance.
Phase 4: Resolution Phase
The resolution phase includes all the new attempts directed toward easing tension. The decision-making processes, however, are based on intuitive gut reactions, instead of analytical thinking.
Resolutions have the potential to be adaptive or maladaptive.
William James acknowledged mystical experiences were not always positive. He described some that were downright, harmful, evil, and frightening.
A comprehensive model of mystical experience needs to consider the potential for both positive and negative outcomes.
Phase 5: Post-Crisis Phase
The post-crisis phase is the 2-12 month period after which the above phases have passed.
This last phase is a period of restoration, rebuilding, and readjustment. It is a period of significant, sometimes radical, transformation in the person.
Some equilibrium is restored during the post-crisis phase. Yet this equilibrium can be lower, the same, or higher than the one previous to the crisis.
Figure 4 illustrates four potential consequences after encountering an adverse event.
Figure 4. Four possible outcomes of adversity
Mystical experiences often cause positive, lasting effects. The positive change has been classified into four areas toward self, others, life, and the mystical experience itself.
First, the mystical experience induces a powerful transformation in one’s personality. The personality changes are not so much at the level of core personality traits. Instead, goals, feelings, attitudes, behaviors, and life meaning are radically transformed.
The second change following mystical states is the person acts more open, authentic, and tolerant of others.
The third change is about attitudes toward life. Mystical experience often facilities the emergence of hope. Purpose and meaning become more prominent in everyday life.
The last change involves attitudes toward the mystical experience. There is a new, more profound understanding of the role mystical experiences play in the unfolding of life.
So, the next time you have a stressful life event occur, and seem to despair about life’s meaning, it could be the first step to transformation. Which could lead you to a greater experience of meaning than you could ever imagine.
One may say truly, I think, that personal religious experience has its root and center in mystical states of consciousness.
—William James (1902, p. 379)
Have you ever had a mystical experience?
What separates a mystical experience from an ordinary one?
The Core of Mystical Experience
Researchers agree the one essential feature of mystical experience is an experience of unity.
Indeed, debates on mystical experience are best understood in terms of how this unity is to be interpreted.
Much of the Western study on mystical experience begins with reference to William James’s book Varieties of Religious Experience.
James tells us that mystical experience is a relatively rare event in which individual consciousness merges in radical unity with the universe.
James identified four characteristics of all mystical experiences:
What is essential from James’s investigation is the notion that mystical experience is different from the everyday world of experience.
James set the standard for the modern study of mystical experience.
Introvertive and Extrovertive Unity
One of the questions researchers have asked since James work is: What is the mystical experience of unity all about exactly?
Stace (1960) contended that the mystical experience of unity is expressed in one of two ways: “introvertive” and “extrovertive.”
In the introvertive type of unity the subject “looks inward into the mind” to achieve “pure consciousness.” This type of unity is experienced during meditation with one’s eyes closed and is perceived as an experience of nothingness.
In the extrovertive type of unity the subject “looks outward through the senses” (Stace, 1960, p. 110) and grasps the unity of all things in or through the multiplicity of the natural world. In this type of experience all things are felt to be alive (e.g., stones, trees, the sky).
Synchronicity, a term coined by the late psychologist Carl Jung, is said to occur when an objective event meaningfully relates to one’s inner state.
In a person’s synchronistic encounter, there is no causal connection between the objective event and one’s inner state of being. In other words, this phenomenon can capture the extrovertive sense of mystical experience.
The Fibonacci numbers and ratio have been studied by mathematicians of all ages. It is also familiar to those in art, biology, architecture, music, botany, and finance.
Is this the most astounding number sequence in the world? Or are we distorting reality to see mathematics where none exists?
Let’s take a closer look at the mathematical phenomenon that has attracted the attention of thinkers from all disciplines and periods.
1. Does Architectural design reflect the golden ratio?
It has been claimed the golden ratio appears in several ancient and modern architectures: The great pyramid of Giza, Parthenon, Notre Dame Cathedral, Tajmahal, Eiffel Tower, Toronto’s CN tower, the United Nations Secretariat building, and more.
The golden ratio may or may not have been included intentionally among all of these buildings.
The architects of the Parthenon could have been aware of the golden ratio. Though the Parthenon has an intricate design and it is not clear that the golden ratio was intentionally represented.
However, it seems that the golden ratio was intentionally included in the design of Toronto’s CN tower. The ratio of the total height (553.33 meters) to the height of the observation deck (at 342 meters) is 1.618.
The CN Tower is a communications tower built in 1976. It was the world’s tallest free-standing structure at the time.
2. Is the spiral of the Nautilus shell based on the golden ratio?
Yes. But the truth is a bit different than what you often hear.
The traditional “golden spiral” is constructed from adjacent golden rectangles. This creates a spiral that increases in dimension by the golden ratio with every 90-degree turn of the spiral.
The spiral of the Nautilus shell does not match this golden spiral.
The spiral constructed from a Golden Rectangle is NOT a Nautilus Spiral.
But here is the crucial issue: There is more than one way to create a spiral with golden ratio proportions.
You can create a spiral that expands by the golden ratio with every 180-degree turn of the spiral. This spiral is a closer match for the spirals of many Nautilus shells.
A spiral expanding by the golden ratio at every 180-degree turn is a closer match to some Nautilus shells for the first few rotations
You see the difference, I hope.
Here is another scientific fact: The mathematical proportion of growth of the nautilus shell is the number 108 (see The Number 108). This is the number of the pentagram that is based on the logic of the golden ratio.
So we see that the Nautilus spiral can exhibit proportions close to Phi.
3. Did renaissance artists use the golden ratio in their paintings?
Yes. Golden ratios are quite easy to see in Leonardo Da Vinci’s “Last Supper”.
Following his lead, in 1955, Salvador Dalí painted The Sacrament of the Last Supper, with golden ratio proportions.
Michelangelo’s “Creation of Adam” also has God’s finger touching Adam’s finger at the precise golden ratio point of the width of the area in which they are framed.
Those are just a few examples of how Renaissance artists used the golden ratio in their paintings.
4. Are the spirals seen in nature based on the golden ratio?
You may have heard that spirals found in nature are based on the golden ratio. While that might not always be 100% accurate, it’s definitely true in some cases.
Sunflower seeds are well-known for being clustered in a pattern of interconnecting spirals based on Fibonacci numbers. Because patterning based on Fibonacci numbers allow for the highest number of seeds on a seed head.
This optimization behavior is not just found in sunflower seeds. Leaves, branches, and petals can grow in spirals based on the golden ratio. This allows the most sunlight to reach older leaves as new leaves grow.
Logarithmic spirals occur commonly in nature, for example in the arms of spiral galaxies, ram horns, hurricanes, whirlpools, and many more.
The golden spiral is one particular case of logarithmic spirals and can appear in the examples mentioned above.
Although, the logarithmic spirals that appear in nature do not have to be golden spirals necessarily.
5. Is there a new algorithm based on the golden ratio that can predict spiritual experience?
Yes, it appears that mathematics may have a role in spiritual experience. Sacco found in a 2017 study that the dynamical effects of age 18 predicted increased spiritual experience. This was as predicted by the FLCM. This supports the link between the relationship of the golden ratio and spiritual experience.
However, the dynamical effects of age 11 and 30 did not support the hypothesis of increased spiritual experience as predicted. This result required an alternative explanation.
It should be evident that children at age 11 may not be able to communicate the complexities of some spiritual experiences. And it’s entirely reasonable that spiritual experiences could be linked with the instability that is less likely at age 30 when personality becomes stabilized.
In the real world, there are many compelling examples of the golden ratio in natural phenomena. But not all phenomena in nature are based on the golden ratio.
You can find plenty of examples where the golden ratio is not found in nature as might be expected. But you can’t deny the power of data and evidence to prove the inherent power of the golden ratio in reality.
The critical question you need to ask yourself is: Why does the golden ratio show up so commonly in nature?
1. The Golden Ratio
The Fibonacci sequence and the golden ratio are interconnected. If you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio (Φ) which is approximately 1.618034...
By taking 2 sides of a rectangle in which the ratio of length and width is 1.618 we get a golden rectangle. This shape results in a fractal process that can be repeated into infinity — and which takes on the form of a spiral.
3. Cyclic Patterns
There are cyclic patterns in the final digits of the Fibonacci numbers. If you look at the final digit of each Fibonacci number there are 60 numbers that repeat. We say the series of final digits repeats with a cycle length of 60.
Suppose we look at the final two digits in the Fibonacci numbers. Do they have a pattern? Yes, there is a pattern here too. The last two digits repeat the same sequence again and again. The cycle length is 300 this time.
So what about the last three digits? And the last four digits? And so on? For the last three digits, the cycle length is 1,500. For the last four digits, the cycle length is 15,000, and for the last five digits the cycle length is 150,000 and so on...
We see examples of the Fibonacci sequence and the golden ratio spiral all around us in nature on a daily basis. For example, the number of petals in a flower, the way tree branches split, snail shells, or the human body, are all examples of the Fibonacci sequence and golden ratio.
5. Energy Flow
The Fibonacci sequence and golden ratio tend to maximize energy flow. This can be seen in the path falcons fly when approaching their prey and the way plants grow. It seems the way nature works is by using the golden ratio as optimal energy flow in time.
6. The Human Body
The golden ratio appears in several human body parts:
7. Body Temperature
Body temperature is a marker for measuring the timing of the circadian rhythm. Body temperature decreases to a minimum during sleep at around 04:00 hours and increases until the maximum of the rhythm is reached at about 18:00.
The hours 04:00 and 19:00, on a 24-hour clock, would be at an angle of 137.5° (the golden angle). So: the golden ratio is controlling our circadian rhythm, which means the 24-hour day/night cycle is intimately linked to the golden ratio.
Also, if we take the golden ratio of the body temperature (i.e., 37C * 0.618 = 23C), we get the average room temperature.
8. Human Development
The Fibonacci sequence represents a new paradigm in human development modelling. In 2013, I showed that the Fibonacci sequence results in an eight-stage model of human development.
The eight stages are:
• 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 Adulthood (29–48 years)
• Older Adulthood (48–78+ years)
9. Art and Music
The Fibonacci series and golden ratio are found in art and music. Leonardo Da Vinci’s “The Last Supper” was based on the Golden Ratio. It is also claimed that classical composers like Mozart and Bartok used the Fibonacci series in some of their pieces.
10. Spiritual Experience
The Fibonacci sequence represents a new paradigm in predicting spiritual experience. In a recent study on the topic, I found that the FLCM algorithm predicted spiritual experiences at age 18 (Sacco, 2017).
As more data becomes available, more understanding of the complexity of spiritual experience can be tackled. As a result, the FLCM is a new paradigm in studying spiritual experience.
If you want to learn more about the ever-present golden ratio check out our next blog post where we will look at the surprising truths and myths about the golden ratio.
In his book Out of Control, Kelly (1994) describes the "Nine Laws of God" as universal laws that govern all complex systems in nature. Kelly recognizes that the nine laws he offers are not the only laws necessary in complex adaptive systems; but he suggests that these
principles are the broadest and most representative of all the observations noted in the science of complexity.
Kelly defines the Nine Laws of God as:
Kelly, K. (1994). Out of control: The rise of neo-biological civilization. Addison-Wesley.
The Golden Ratio: A Principle of Energy Flow
In nature, the golden ratio governs objects as large as galaxies and as small as the DNA. The golden ratio has for centuries represented perfect harmony, or the most attractive proportion in almost all things. Adrian Bejan, a Duke University engineer, has found it to be a compelling ratio for a single law of nature's design. In numerous papers and books, Bejan has demonstrated that the constructal law (www.constructal.org) shows how all shape and structure in nature arise to facilitate flow. Furthermore, Bejan mentions how his constructal law provides a greater scientific context for nature’s efficiencies such as the golden ratio (Bejan, 2009). The designs in nature accomplish specific goals with the minimum of resources and energy, and the golden ratio is the form of the natural movement of energy.
The golden ratio creates a form which can increase in size indefinitely without altering its shape. This way, across size scale, from very small to very large, the same form arises again and again. The golden ratio enables flow optimization because it follows a path of least resistance, so that a maximum result can be achieved with the least amount of effort. The golden ratio explains why falcons fly in a golden spiral path when approaching their prey (Tucker, 2000). For falcons, the golden spiral is the energy efficient flight path of least resistance. Fibonacci spirals are the configuration of least energy and experimental results also provide a vivid demonstration of this energy principle in phyllotaxis (Li, Ji, & Cao, 2007)
The Purpose of Life: Flow
The Constructal Law shows there is indeed an actual purpose to all life as well as meaning of individual life something that reconciles modern science with ancient scriptures and spiritual writings. It suggests we consider how everything in the universe is ultimately comprised of energy. One of the most important principles of energy is that it doesn't like differences and works out ways to reduce and balance them. This is why energy flows from where it is concentrated (like the sun) out into the colder universe.
The Constructal Law shows the ultimate purpose of all life is to help energy flow and balance. The same laws of energy indicate that a meaning of your own life is to find how your energy flows best. While the 25 chemicals that comprise your body are the same as those in everyone else, the way energy is mixed with them is different in each of us. We all have bodies with similar brains with a similar number of nerves in each, but the way those nerves are connected is different in each of us. The experiences, learnings and resulting nerve connections are unique and are what makes you who you are - makes your character and personality.
When your energies are flowing together, focused in one direction, you may experience what in the psychological literature are called flow or peak experiences (Csikszentmihalyi, 1990; Maslow, 1964). This sensation is like being carried along by the flow of an effortless current of some type. The elements associated with the flow state can be classified into the three areas: Causes of Flow, Characteristics of Flow, and Consequences of Flow.
1. Causes of Flow
2. Characteristics of Flow
3. Consequences of Flow
What does your energy enable you to do best? This can be as simple as discovering what you are truly passionate about or your individual talents. How you use energy best varies for everyone. When you sense your energy flowing well, this can provide a good indication of who you really are and what you do best—and your individual meaning in life. We each have unique energy patterns, as individual as your fingerprints. Science, and your own personal experience, shows that when your energy flows well you perform best, are happiest, most passionate, most content and even at your healthiest. This is your energy reinforcing your individual purpose in life.
Bejan, A. (2009). The golden ratio predicted: Vision, cognition and locomotion as a single design in nature. International Journal of Design & Nature and Ecodynamics, 4(2), 97-104.
Csikszentmihalyi, M. (1990). Flow: The Psychology of Optimal Experience. Harper and Row, New York, NY.
Li, C., Ji, A., & Cao, Z. (2007). Stressed Fibonacci spiral patterns of definite chirality. Applied Physics Letters, 90(16), 164102.
Maslow, A. H. (1964). Religions, values, and peak-experiences. Columbus, OH: Ohio State University Press. (Original work published 1940)
Tucker, V. A. (2000). The deep fovea, sideways vision and spiral flight paths in raptors. Journal of Experimental Biology, 203(24), 3745-3754.