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
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.
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.
In a previous post it was pointed out that the digital roots of the Fibonacci sequence produce an infinite series of 24 repeating numbers. This post discusses how the 24 repeating numbers relate to the dimension of time.
The Fibonacci 24 Repeating Pattern
First, the 24-repeating pattern follows an approximate sinusoidal pattern (Figure 1). The 24 repeating digital roots of the F sequence are: 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9. Numbers 7, 8, 2, and 1 are the only numbers not fitting the sinusoidal pattern.
Figure 1. Digital roots of the 24-repeating pattern
The 24-hour day/night cycle also has a 24-repeating sinusoidal pattern, since one day is equal to 24 (2 x 12) hours.
The Calendar and the Golden Ratio
Second, the modern calendar, which originated in Egypt, is based on four numbers: 12, 30, 60, and 360. There is a logical reason for the use of these numbers. The Egyptians made their calendar and time systems correspond with the dodecahedron and golden ratio. To the Egyptians the golden ratio and number 5 were very sacred. The Egyptians were aware of the 5 platonic solids and considered the dodecahedron with 12 faces, 30 edges, and 60 planar angles on its surface to be very important. Further, a dodecahedron’s faces are pentagons (5 sides).
Figure 2. Dodecahedron
Figure 3. Egyptian Pharaoh Mask (Tutankhamun)
The ancient Egyptians were aware that Solar system cycles reveal the same numerical parameters of the dodecahedron (i.e., the cycle lengths of Jupiter, Saturn, and Solar system are 12-years, 30-years, and 60-years). Thus, a profound mathematical relationship exists between the Solar system and dodecahedron. This explains why the Egyptians and Plato chose the dodecahedron as the geometric symbol of the Universe.
The Egyptians were the first to adopt a solar calendar in 4,000 B.C. The solar year was 365 days and divided into 12 months consisting of 30 days. Although the twelve 30 day months equaled 360 days, five holidays were added at the end of the year, thus totaling 365 (365 = 12 x 30 + 5).
The Egyptians also divided time into the Hour, Minute, and Second. The unit of hour was chosen so 1-day=24 (2 x 12) hours. Further, 1-hour = 60 minutes, and 1-minute=60 seconds.
In fact, the origin of the word hour comes from the Egyptian god Horus. Horus in Egyptian mythology is symbolic of the sun. The word Horizon also comes from Horus.
In summary, an intimate connection exists between time and the golden ratio. Firstly, in terms of the 24-repeating sinusoidal pattern, and secondly in terms of the roots of our modern calendar in the 12-sided dodecahedron.
The number "108" has long been considered a sacred number. In Hindu/Buddhist beliefs, there are 108 virtues to cultivate and 108 defilements to avoid, which speaks to the value of balance. Hindu/Buddhist malas (rosary) also contain 108 beads. In Islam, the number 108 sometimes is used in place of the name of God. There are also 108 movements in tai chi. The frequency with which the number 108 appears across religious disciplines is too great to be dismissed as coincidence.
108 and Growth
It may be pointed out that the number 108 connects the Sun, Moon, and Earth: The average distance of the Sun and the Moon to Earth is 108 times their respective diameters. While this is an astonishing example of the number 108 in the natural world, an even more amazing example is in the constant growth of a nautilus shell.
A nautilus shell is found to illustrate the 1.08 progression, small to large, of the rooms or segments of its shell. In other words, as the sea creature grows, each chamber of its shell's growth is 1.08 times larger than the last chamber, creating a logarithmic spiral within the exterior of the shell.
Marcus du Sautoy, Professor of Mathematics, at the University of Oxford, reveals how the nautilus uses a 1.08 constant growth rate to build an elegant spiral shell. Part of the BBC Two documentary The Code, he explains how spirals and the power of math can be found throughout the natural world.
108 and The Number 5
The number 108 is related to the number 5. The clue is related to the angle of the Pentagon, being 108˚.
108 and The Fibonacci Sequence
To fully understand the significance of the number 108, it is necessary to understand the numerical science of decimal parity.
In many ancient cultures (e.g., Egypt and India) decimal parity was used as a way to understand the truth of numbers.
Decimal parity is a method for breaking numbers down into single digits. For example, the decimal parity equivalent of the number 361 is 3 + 6 + 1 = 10 and 1 + 0 = 1. So the decimal parity equivalent of 361 is 1.
The first 24 numbers of the Fibonacci Sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657.
When decimal parity is applied to the Fibonacci sequence it is found there is a repeating series of 24 digits being the following: (0), 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1.
If we add these 24 digits up, we find the number 108.
0 + 1 + 1 + 2 + 3 + 5 + 8 + 4 + 3 + 7 + 1 + 8 + 9 + 8 + 8 + 7 + 6 + 4 + 1 + 5 + 6 + 2 + 8 + 1 = 108
Amazingly, the 1.08 constant growth rate the nautilus uses to build its spiral shell is the same pattern that repeats every 24 numbers in the Fibonacci sequence.