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I Ching and binary algorithm

Source: Shanghai Daily | March 27, 2016, Sunday | Print Edition

GUA and their sequences in I Ching by binary method

Is the mathematical logic underpinning I Ching gua a precursor to the binary system? The answer is still the subject of ongoing discussions among experts. We incline to say that the binary arithmetic invented by German great thinker Leibniz (1646-1716) is of the same origin with Chinese master scholar Shao Yong’s (1010-1077) method, as witnessed by Leibniz himself when he calculated the sequential numbers for gua.

Following Shao Yong’s approach, Leibniz applied the binary arithmetic to come up with the ID numbers for all the I Ching gua (see Part 3).

We may therefore call the ID sequence in Picture 1 as “Shao Yong Sequence”, short for “Shao Yong-Leibnitz Sequence.” Wherein yin yao is defined as 0 and yang yao as 1. The ID is calculated from top yao to bottom yao with the weights doubled: 1, 2, 4, 8, 16, and 32 (top-down). Picture 1 shows the original sequential number in Zhouyi. For illustrations, the ID of 未济(64) is, from top to down, 1*1+0*2+1*4+0*8+1*16+0*32 = 21; (28)大过gets its ID equal to 30 (0*1+1*2+1*4+1*8+1*16+0*32 = 30). To test your understanding of Shao Yao Sequence, you may try to calculate the ID numbers for 63既济and 27颐. They are respectively 42 and 33.

Alternatively, you may use a simple method of Shao Yong Sequence:

For a gua positioned at (a, b) in the table, its Shao Yong ID = a*8+b (where a is the row number and b the column number). For example, 3屯 is located in row Thunder-4 and column Water-2, its Shao Yong ID is then 4*8 +2= 34; and 4蒙, 3屯’s pair in Zhouyi, is Mountain-1 below and Water-2 below, so its Shao Yong ID is 2*8 +1= 17.

If using the regular binary arithmetic, calculate from the bottom yao to the top yao, the resulted ID in the Digital Sequence are different. Then, the digital ID of 未济, from bottom up, is 0*1+1*2+0*4 +1*8+0*16+1*32 = 42; the digital ID for 大过, 既济, 颐 are 30, 21 and 33, respectively.

The ID from both sequences are shown in Picture 3.

As we previously explained, when calculated the Bagua sequence, Leibniz obtained Shao Yong Sequence (top-down), which is Sky-7, Valley-6, Fire-5, Thunder-4, Wind-3, Water-2, Mountain-1, and Earth-0.

It should have been Sky-7, Wind-6, Fire-5, Mountain-4, Valley-3, Water-2, Thunder-1 and Earth-0, if Leibniz used the regular binary method (bottom-up).

Using the simple alternative, the digital ID for each gua is then (a+b*8) where a stands also for the row number and b the column number, but in the Digital Sequence of Bagua. For example, 3屯 is located in column Water-2(above) and row Thunder-1(below), its Digital ID is then 1+2*8 = 17; 4蒙 is Mountain-4 above and Water-2 below, so its Digital ID is 2+4*8 = 34. In other words, the ID numbers swap in the two sequences.

In Picture 1, the colored 32 gua in Picture 1 are called “primary Gua” due to their particularly meaningful structure of yao. The remaining 32 gua (no color) are called “secondary Gua.”

Picture 3 displays two sequences: the Shao Yong ID (in red) and the Digital ID (green) in the same order—from 0 down to 63. Either sequence is solely determined by the yao structure of hexagrams. The gua listed side by side are originally in pair in Zhouyi, with the exception of only eight gua as highlighted in yellow. They form four pairs of special kind. Those eight gua in blue are also special. We will elaborate the underlying reason in the next issue.

I Ching, binary system and “0”

It’s improbable to come to the conclusion that our ancestors, including Shao Yong, developed a prototypical binary algorithm. A primary reason of improbability is that the symbol of “0” was unknown to Chinese at the time. As a binary number, binary coding or positional notation, “0” has different faculties in calculation and counting. The long journey for the Ancient Indian-Arabian “0,” an eastern concept of “nothingness”, to be introduced and adopted in the West was extremely tough.

Ancient Greeks exceled in logic analysis in their explorations of mathematics and astronomy. However, the concepts such as “nothingness”, “nil”, or “void” were unthinkable in their numeric system. In short of “0”, the decimal system was sadly incomplete in dealing with nature and deficient in computation. Which in turn slowed down substantially the development of European civilization. Indeed, as late as in the beginning 13th century, Arabian mathematicians had advanced way ahead of their European peers.

It was Italian Leonardo Fibonacci (1170-1250) who ushered to Europe the symbol and functions of “0”. Over the years as a business apprentice in Northern Africa, he learned from Moors and honed the use of 0. Having returned to his hometown Pisa, Fibonacci published his monumental work “Liber Abaci” in 1202, which was widely regarded as a historic milestone leading to the full blossom of Renaissance. Fibonacci applied “0” and solved numerous puzzles at that time, including the renowned Fibonacci Series deciphering the Golden Ratio, the mythic value of 1.618.

Back to Chinese tradition, “0” and the nothingness is always understood as an ultimate origin of universe and a fundamental component of constant changes. This idea has in fact rooted in Chinese culture all along, reflecting as yin and yang and the paradigm of “opposites yet complementary.” This mental model is deeply established among Chinese, and surely essential in accessing Chinese civilization, its standpoint in observing the universe and its values in dealing with neighbors and transactions.

Now and Then


 

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