The I Ching reveals more symmetries

The classic Chinese text Zhouyi, known in the west as the Book of Changes or I Ching, has existed for three millennia. It is organized around 64 six-line hexagrams, each constructed from pairs of the eight three-line trigrams.  Around this core, layers of verbal content accreted over the centuries: first proverb-like divinatory messages, then philosophical commentary.

Apart from the traditional ordering of the 64 hexagrams attributed to King Wen, another sequence may be produced by reading each hexagram from the top down as a six-place binary number, reading unbroken lines as 1 and broken lines as 0.  The 64 hexagrams are found to code uniquely for the numbers 0 to 63. So kūn is the first hexagram corresponding to the number 0; bō is the second, corresponding to 1; bǐ, 2 through to qián, 63.  This is not a new idea; the understanding of the hexagrams as binary numbers goes back to Shao Yong during the Song dynasty period, and the King Wen sequence was not universally used in early Chinese manuscripts of the Zhouyi (Redmond, 2017).

Since the work of Spencer-Brown (1969) we can now reduce each hexagram to one of two states called marked or unmarked by considering them as arrangements in his calculus of indications.  An unbroken or ‘positive’ line is read as the mark and a broken or ‘negative’ line is read as absence of a mark. The whole hexagram is simplified by the rule that when one mark is placed inside another they cancel out to the unmarked state.  Consequently, the 32 hexagrams with an even number of unbroken lines take the unmarked state; the others with an odd number are marked. However, we would obtain the same result if we chose to identify unbroken lines as marked and broken as unmarked because a hexagram with an odd number of broken lines must also have an odd number of unbroken lines.

Taking the hexagrams in the binary sequence outlined above and counting unmarked as 0 and marked as 1, we get the following string.

0110100110010110100101100110100110010110011010010110100110010110

This string is a palindrome consisting of the two shorter palindromic units, 0110 and 1001, each occurring eight times.  It can be seen that these two units are complementary. Further, if the first half of the string is written alongside the second half, these are also seen to be complementary, i.e. 

01101001100101101001011001101001
10010110011010010110100110010110

Looking more closely at how the two shorter four-digit units are arranged, replace 0110 with ‘A’ and 1001 with ‘B’. This makes it easier to recognise another palindrome ABBABAABBAABABBA, which consists of two complementary palindromic units ABBA and BAAB, each occurring twice.  Naturally, the second half is the complement of the first.

Incidentally, when the eight trigrams are treated in the same way by arranging them according to the three-place binary numbers they represent and simplifying each to marked or unmarked, they yield a string containing one instance of each of the two complementary palindromic units, i.e. 01101001, but not an overall palindrome.

Of course, this structure was not arbitrarily designed by the old sages. Not does it necessarily have any connection to the verbal content of the Zhouyi or its original use as a divination tool. Rather it is an inherent property of the binary numbers the hexagrams represent, and is evidence in support of Shao Yong’s binary sequence. It is also another example of the power of Spencer-Brown’s calculus of indications to penetrate beyond words and numbers to the simpler deep structure of things.

-oOo-

References

Redmond, G. (2017) The I Ching (Book of Changes). (Bloomsbury: London).

Spencer-Brown, G. (1969) The Laws of Form. (Allen & Unwin: London).

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