S-Law, Golden Ratio & Angle and Fibonacci - Hồn Trống Việt
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S-Law, Golden Ratio & Angle and Fibonacci

Golden ratio, angle & Fibonacci numbers are related to each other. They are present on the Drum’s diagrams & laws, especially the S-Law or the law of S curved motion, a fundamental law of the Drum’s law of changes.

The S-Law or the Law of S curved motion

On the Drum, people, birds, chickens, deer & boats move in a counterclockwise direction, no matter which direction the drum is rotated. There are many explanations for that, some are partly true such as:

Figure 1: The Law of S shaped motion
  • Since the sun moves from east to west, going in the CCW direction means following the sun. 
  • The ancient Vietnamese respect the left side, their shirt is laced to the left, the left side is the side of the heart.

The former explanation is correct when we face the north, but it would be wrong if we turn to the south.

For the later, it is important to point out that the Drum talks about the permanent & universal laws, common to all, and the CCW direction indicates the return to Tao or God. 

Although significant, the returning is only a part of a fundamental law of the Drum, the Law of the S shape or spiral motion.

Figure 2: S shapes on “Ngoc Lu” bronze drum

In addition to the CCW return to the center, the S law also includes the clockwise outward movement, and the most important element, the Center. It is in the Center that Am & Duong (Yin & Yang), representing all opposite but complementary balance, meet & exchange to resolve the conflicts & give birth to everything.

Probably, due to its important, the S-law is present at many places on the Drum & its diagrams such as BQLD, BQHT & TCV. On the Thai Cuc Viet, the two intertwining of Am Duong (Yin Yang) is its most inclusive expression.

Fibonacci & Golden ratio on Ha thu

The golden ratio, denoted by Φ (Phi) has a value of about 1.618. It is the ratio of the Fibonacci’s two consecutive numbers starting from 21 to infinity, for example 34/21 or 55/34.

Figure 3: The Golden Ratio and the Fibonacci numbers

According to Wikipedia

It is not known when the golden ratio existed. Previously, it was still thought that a Roman, Vitruvius, who lived nearly 2100 years ago, discovered the golden ratio. Recently, archaeologists found writings about the golden ratio in the  pyramids  in  Egypt . That proves that the golden ratio appeared very early (about thousands of years ago).

Euclidean, the mathematician of all time, once mentioned the golden ratio in his immortal work entitled  “Fundamentals” …

We have seen that the motion on the Ha thu is spiral, which is a combination of 2 movement, vertical and horizontal. This spiral motion is shown on Ha thu as a ratio of 2 Fibonacci numbers 31 & 24. It is the golden ratio Φ, 1.619.

Fig. 4: Ha thu, fibonacci & Golden ratio

Golden angle & S-Law

Figure 5: Golden angle

The golden angle or golden spiral is a movement based on the golden ratio and is a special form of the S curved motion.

Golden angle spiral movement abounds in nature, architecture, art, the human body, and in the DNA. It is not difficult to recognize their similarity with the S curved motion shown on the Drum’s diagrams.

Thai Cuc Viet

Figure 6: Golden Angle Motions on TCV & in Nature

Analysis of the relationship or the motion from BQLD to BQHT demonstrates clearly that it is in fact, the golden angle movement. When the trigram Càn on BQLD is taken as a reference for the starting point of the motion, the relationship can be expressed as the following formula:

Fig 7: Golden Angle Movement of Trigram Càn

Trigram Càn (on BQHT) / Trigram Càn (on BQLD) = 1 + 1/(Phi^2).

Replacing 1 with 360 degrees (Circle), we’ll get:

Càn (BQHT) / Càn (BQLD) = 360 degrees + 137.5 degrees.

Càn (BQHT) / Càn (BQLD) – 360 degrees = 137.5 degrees or Golden Angle.

Since other trigrams move at the angles equal to 1/3, 2/3 … of the Golden Angle, their motion angles are derivatives of the Golden Angle.

So, the Drum, via the numbers, images, ratios & angles has displayed & described the golden angle S-shaped motion.

Drum’s Law of Changes & S-Law

Like the S movement, when there is a vortex, there will be another in the opposite direction, representing the unity of yin and yang, also known as “Bipolar Unity” or “One yin and one yang is the Tao”. In the center of the vortex, the currents are moving faster than on the outside, and the energy generated is stronger. Paradoxically, the center is the most peaceful and balanced place, therefore it is called the point of peace, the eye of the storm or the Tao.  

Thanks to the golden angle S shaped movement & the Center or Tao, 2 opposite Yin & Yang (Am & Duong) can meet, complement & exchange to give birth to all things in the space-time, and develop with the same geometry at all levels i.e. they just become bigger but still keep their original initial shape. We can see this when we continue to add more concentric circles & connect the numbers according to Ha Thu’s instructions. Doing so will give us the same but bigger TCV.

Figure 8: Mutable but Immutable. Changes without change.

In addition, the golden angle S curved motion is a form of continuous movement but does not change its character. As such, it reveals another aspect of the Drum’s law – changes without change i.e. the changes always follow the laws that are unchanged.

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