Derivation of the Timewave from
the King Wen Sequence of Hexagrams
by Terence McKenna

Figure 1
The idea that time is experienced as a series of identifiable elements in flux is highly developed in the I Ching. Indeed the temporal modeling of the I Ching offers an extremely well-developed alternative to the "flat-duration" point of view. The I Ching views time as a finite number of distinct and irreducible elements, in the same way that the chemical elements compose the world of matter. For the Taoist sages of pre-Han China time was composed of sixty-four irreducible elements. It is upon relations among these sixty-four elements that I have sought to erect a new model of time that incorporates the idea of the conservation of novelty and still recognizes time as a process of becoming.

The earliest arrangement of the hexagrams of the I Ching is the King Wen Sequence. It was this sequence that I chose to study as a possible basis for a new model of the relationship of time to the ingression and conservation of novelty. In studying the kinds of order in the King Wen Sequence of the I Ching I made a number of remarkable discoveries. It is well known that hexagrams in the King Wen sequence occur in pairs. The second member of each pair is obtained by inverting the first. In any sequence of the sixty-four hexagrams there are eight hexagrams which remain unchanged when inverted. In the King Wen Sequence these eight hexagrams are paired with hexagrams in which each line of the first hexagram has become its opposite, (yang changed to yin and vice-versa).

The question remains as to what rule or principle governs the arrangement of the thirty-two pairs of hexagrams comprising the King Wen Sequence. My intuition was to look at the first order of difference, that is, how many lines change as one moves through the King Wen Sequence from one hexagram to the next. The first order of difference will always be an integer between one and six. When the first order of difference within pairs is examined it is always found to be an even number. Thus all instances of first order of difference that are odd occur at transitions from one pair of hexagrams to the next pair. When the complete set of first order of difference integers generated by the King Wen Sequence is examined they are found to fall into a perfect ratio of 3 to 1, three even integers to each odd integer. The ratio of 3/1 is not a formal property of the complete sequence but was a carefully constructed artifact achieved by arranging hexagram transitions between pairs to generate fourteen instances of three and two instances of one. Fives were deliberately excluded. The fourteen threes and two ones constitute sixteen instances of an odd integer occurring out of a possible sixty-four. This is a 3/1 ratio exactly.

Figure 1 shows that when the first order of difference of the King Wen Sequence is graphed it appears random or unpredictable. However when an image of the graph is rotated 180 degrees within the plane and superimposed upon itself it is found to achieve closure at four adjacent points as shown below:

Figure 2

Figure 3
While closure might logically be expected anywhere in the sequence, it in fact occurs at the conventional beginning and end of the sequence. While an arrangement with closure might have placed any two hexagrams opposite each other, what we in fact find is that the hexagrams opposite each other are such that the numbers of their positions in the King Wen Sequence when summed is always equal to sixty-four. These facts are not coincidences, they are the artifacts of conscious intent.

Over 27,000 hexagram sequences were randomly generated by computer (all sequences having the property possessed by the King Wen sequence that every second hexagram is either the inverse or the complement of its predecessor). Of these 27,000 plus sequences only four were found to have the three properties of a 3/1 ratio of even to odd transitions, no transitions of value five and the type of closure described above. Such sequences were found to be very rare, occurring in a ratio of 1 in 3770. Here is the complete graph of the King Wen first order of differnce with its mirror image fitted against it to achieve closure, as shown in Figure 3.
For these reasons I was led to view the King Wen Sequence as a profoundly artificial arrangement of the sixty-four hexagrams. Look carefully at Figure 3 at left. Review in your mind the steps from the King Wen sequence that led to it. Notice that it is a complete set of the sixty-four possible hexagrams, running both sequentially forward and backward. Since it is composed of sixty-four hexagrams of six lines each it is composed of 6 x 64 or 384 lines or yao. One might make an analogy and say Figure 3 is to the King Wen sequence as a cube is to a square; it is composed of the same elements as the King Wen Sequence but it has more dimensions.

It is my assumption that the oracle building pre-Han Chinese viewed the forward-and backward-running double sequence of Figure 3 as a single yao or line and that it is therefore open to the same treatment as lines are subject to in the I Ching, namely multiplication by six and sixty-four.

Since a hexagram has six lines I visualized six double sequences in a linear order. But a hexagram is more than lines; a hexagram also contains two trigrams. Thus over the six double sequences I overlaid two double sequences, each three times larger than the six double sequences. A hexagram also has an identity as a whole; thus over the six and the two double sequences a single, larger double sequence is projected.

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