Summary: We have introduced the Golden Ratio PHI (Φ) as a universal constant that has a unique property in this universe: it can create HARMONY. Johannes Kepler (1571-1630) called it "a precious jewel". Because PHI can add and multiply itself, its values are compounded in multiple reciprocal harmonic relationships. As a proportional mediator, PHI has the ability to unite the different parts of a whole so that each part preserves its own identity and yet blends into the larger whole.

In brief, no matter how far the multiples & sub-multiples of PHI cascade up and down the scales of magnitude (from the microcosm to the macrocosm), they will always be family: they will always recognize each other, mutually embrace and enter a dance of harmonic resinance, thus creating a cosmic web of LOVE relationships. This is an wonderful skill - by the way, we also have this awesome skill: it is called loving-kindness.

In this SG Lesson #2, we are going to learn how PHI is generated by a special mathematical family of numbers called the "Fibonacci Series".

Leonardo da Pisa aka "Fibonacci" (literally the "son of Bonaccus") was a brilliant Italian mathematician (1170-1240) who helped to introduce to medieval Europe the Hindu-Arabic numerals that we now use. But Fibonacci is mostly famous for his study of the genealogy of rabbits. he found that, in controlled conditions, rabbits were reproducing following a mathematical sequence that is now called the "Fibonacci Series".

Fibonacci and his rabbit teachers.

The Fibonacci Series

Now what is so special about this Fibonacci Series?
In the Fibonacci Series:
1. Each term is the sum of the two preceding terms:
Example: 34 = 21 + 13
2. Each term divided by the previous one gives a progressively
better approximation for PHI,
the Golden Ratio:
Examples: 144 / 89 = 1.617977…
and 233 / 144 = 1.618055
(accurate numerical value for Φ to 5 decimals)

<<< TRY IT FOR YOURSELF!!! >>>

This can be visually seen in the following diagram where the quotient between two Fibonacci numbers fluctuates or alternates above and below the PHI base Line, dynamically seeking harmonic balance and rapidly reaching the limit of PHI = 1.618...

Historical Note: The Fibonacci Sequence was well known in ancient India, where it was applied to the metrical sciences (prosody), long before it was known in Europe. Developments have been attributed to Pingala (200 BC), Virahanka (6th century AD), Gopāla (ca. 1135 AD), and Hemachandra (ca. 150 AD).

Now let us show the direct relationship of PHI and the Fibonacci numbers in a even more direct way. Below is a table of the powers of PHI, in terms of Fibonacci multiples of PHI and the regular Fibonacci Series. There is quite a tie-in and an elegant beauty in the symmetry.

By now, do you get a sense of the deep organic connection between the Fibonacci Numbers  and Phi the Golden Ratio, via the medium of √5?
They generate each other. They are in mutual harmonic inter-connectedness. The Fibonacci Series is the expression in space & time of the archetypal Harmonic Ratio Phi. Conversely,  the Fibonacci Series converges back to the Golden Ratio as it expands.
So, we are seeing here two levels of the same Law of Harmony: an extra-dimensional principle (PHI) inexpressible with exactness in 3D (hence called “irrational” by mathematicians) AND a definite manifestation in space-time (“made manifest”) as sequences of numbers (Fibonacci) whose relationships generate the causal/formative principle PHI.

Above: modeling of a Phi/Fibonacci wave in 3D. (Image credit: Dan Winter).

In conclusion, we can speak about the ubiquity of Fibonacci numbers (and therefore of the Goden ratio PHI) now recognized in nature (on all scales), in mathematics, in culture and very likely soon to be recognized in the structure of consciousness itself.

As an appetizer, below are images of Fibonacci numbers in physics.

Above left: Fibonacci numbers in optics. Above right: Fibonacci numbers in atoms.

And below: the famous Golden Rectangle based on Fibonacci Squares.
From the Golden rectangle, the Golden Spiral can be constructed.

(This is the topic of our next SG Blog Lesson: The Golden rectangle).