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.
At the core of synchronicity is a sense of unity. Why might synchronicity relate to a sense of unity? It is important to note that the universe consists of nonlocal and fractal connection. Nonlocality refers to correlations between spatially separated events (Stapp, 2009). A fractal is a symmetry having a pattern that repeats at different scales (Bak, 1996).
Fractals are thought to be linked to synchronicity experiences (Hogenson, 2005). Significantly, fractal geometry includes the Fibonacci sequence as a unifying theme (Devaney, 1999). The Fibonacci sequence is a recursive series and visualizations of the Fibonacci sequence exhibit self-similarity. For example, the spiral consisting of circular arcs embedded in Fibonacci sized squares:
Another amazing fact is the presence of the Fibonacci sequence in the Mandelbrot set.
Professor Robert Devaney of Boston University has found the Fibonacci numbers in the Mandelbrot set and it's all to do with those buds on the outside of the set! For any two bulbs, the sum of their period is the period of the largest bulb between them. By taking bulbs closer and closer to each other, the Fibonacci sequence is generated.
So synchronicity might relate to nonlocality, fractals, and the Fibonacci sequence generally (Sacco, 2016), and particularly experiences of ultimate meaning, unity, and interconnectedness.
Bak, P. (1996). How nature works: The science of self-organized criticality. New York: Springer.
Hogenson, G. B. (2005). The self, the symbolic and synchronicity: Virtual realities and the emergence of the psyche. Journal of Analytical Psychology, 50(3), 271–84. http://dx.doi.org/10.1111/j.0021-8774.2005.00531.x
Devaney, R. L. (1999). The Mandelbrot set, the Farey tree, and the Fibonacci sequence. The American Mathematical Monthly, 106(4), 289–302.
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://dx.doi.org/10.1111/1468-5922.12204
Stapp, H. (2009). Nonlocality. In Greenberger, D., Hentschel, K., Weinert, F. (Eds.), Compendium of Quantum Mechanics (pp. 405–410). New York: Springer.
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.