The Rectangular Predecessors of the King Wen Sequence The King Wen Table and the King Wen Array

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The Rectangular Predecessors of the King Wen Sequence

The King Wen Table and the King Wen Array

For over the last two thousand years, scholars have been unable to come into accord on the meaning of the arrangement of the hexagrams in the traditionally received (King Wen, KW) sequence. In fact, besides the connection of the odd and the subsequent even numbered hexagrams, it is hardly possible to find any relation between the content of a given sign and its place in the row. Beginning with the ninth wing of the Yijing (the Xu Gua), many different explanations have been suggested on the possible meaning of this arrangement. In the opinion of many scientists, the sequence hides some kind of ancient astronomical, physical, mathematical, or other knowledge. Many others say the sequence was generated randomly, and it is useless even to think about any order in it. In short, there is no generally accepted theory that would explain the place of each hexagram in the KW sequence.

In this short study a two-dimensional arrangement will be shown, the King Wen Table. It has been generated from the KW sequence by the simplest way. In this table, certain groups of related hexagrams can be discovered, having regular positions there. In the sequence, however, these groups and their structure are not recognizable. Consequently, the well structured KW Table can be rightly considered the direct predecessor of the irregular sequence. If somebody wants to find any meaning or significance in the arrangement of the hexagrams, it is also well worth to study this table.

Still, based on the structure of the KW Table, a good guess can be made about an earlier arrangement, the King Wen Array.

Throughout this argumentation, there are facts based on observations, and there are assumptions based on these facts: The final hypothesis has at least one advantage over the other explanations: it is in accordance with the rule of Ockham’s razor, using the fewest and simplest assumptions.

In a former work, the author set forth a theory on the spherical arrangement of the hexagrams, the Yi-globe.[1] There, among others, he dealt with the development of the KW sequence from the structural elements of this sphere. The Yi-globe, however, was only a hypothetical form and originating the KW sequence from might also seem conjectural. In this article, a more general solution will be demonstrated, based on the positions of the hexagrams in the KW sequence alone, without any reference to the Yi-globe at all.

Regularities in the King Wen Sequence

In the traditional sequence, there are two well-known regularities:

a)      In the sequence, each even numbered hexagram, except the eight symmetrical ones, is followed by its reverse sign (twenty-eight pairs altogether).

b)      The eight symmetrical hexagrams are in pairs with their complements (four pairs).

With respect to the order of the pairs, opinion varies considerably. The main views can be classified as follows. There are only two or three works given as examples in each group. They are subjectively picked out from the vast literature, from scholarly books to metaphysical websites, simply to demonstrate the diversity of opinions. A more systematic and scholarly work on this this subject was published by Steve Moore.[2]

a)      The traditional sequence comes from King Wen, and it is the only authentic arrangement. There are close relations between the contents of the adjacent hexagrams and pairs, and/or there are coherent groups in the sequence. For example:

-   The ninth wing of the Yijing contains explanatory notes on the connection of each hexagram with the previous one in the sequence.[3]

-   Liu I-ming took the hexagrams as successive steps along the way towards perfect enlightenment.[4]

-   Frank Kegan identifies six sets of ten successive hexagrams, each with a particular function. In the sets, a special meaning is given to each place.[5]

b)      There has to be some kind of regularity in the sequence but it is unknown and waits for discovery. The representatives of this view usually propound a rule, disclosed by them. For example:

-   The monograph of Richard S. Cook “resolves the classical enigma. It provides a comprehensive analysis of the hexagram sequence, showing that its classification of binary sequences demonstrates knowledge of the convergence of certain linear recurrence sequences.”[6] (Citation from the editor’s note on the book.)

-   Based on the principles of his “Novelty Theory”, Terence McKenna constructed a mathematical function, using numerical values derived from the King Wen sequence. Using this “Timewave Zero” formula, he predicted the “end of time” at December 21, 2012.[7]

c)      There are different, new arrangements; some developed long ago, others more recently, with additional meanings. They in many ways seek to compensate for the lack of universality in the traditional sequence. Two of these inventions are as follows:

-   In the “natural” system of Shao Yong (1011–1077), the sequence starts with the Creative. In the following signs the lines gradually change, beginning from the upper line and down to the lowest one, according to a definite rule. In fact, this arrangement is a kind of transcription of the binary numbers (derived much later) from 111111 to 000000.[8]

-   Andreas Schoter arranged the hexagrams in a lattice “by energy level, from the least energetic at the bottom, to the most energetic at the top. … In mathematical terms, this is a six-dimensional hypercube.”[9]

d)     There is no order in the sequence, nor should we try to create one. This is the view of most users and readers. They say the essence of the book is not in the arrangement of the hexagrams but in their meaning separately and together. The meaningful pairs are enumerated in the sequence at random, without any rule or order among them.

-   Bradford Hatcher thinks the pairs have been scrambled into a random sequence, and the efforts of finding some order in this sequence are simply a useless expenditure of time.[10]

The argument made in this paper partly belongs to group b above because it discloses some kind of regularity in the traditional sequence. However, this order comes from another, geometrical arrangement that has not been known up to the present.

Three regular groups in the King Wen sequence

In the Yijing literature, there are well-known short or longer sequences and groups where the hexagrams are categorized according to the meaning or the composition of the hexagrams. The following gives examples of such groups:

-   The eight doubled trigrams, composed from two identical trigrams;

-   Nine hexagrams that show the development of character;[11]

-   The twelve sovereign (waxing and waning) hexagrams;

-   Jing Fang’s “eight palaces” (or eight houses), with eight hexagrams in each.

Observing the twelve sovereign hexagrams and two other groups, so far unknown relations can be found among their ordinal numbers in the KW sequence.

1.      The ordinals of the sovereign hexagrams are: 1–2, 11–12, 19–20, 23–24, 33–34, 43–44.

2.      The ordinals of the hexagrams in the sequence, created by exchanging the trigrams in the sovereign hexagrams, are: 1–2, 11–12, 9–10, 15–16, 25–26, 45–46.

3.      A group where the hexagrams are composed from two opposite (complementary) trigrams, in other words, from the opposite pairs of the Earlier Heaven: 11–12, 31–32, 41–42, 63–64.

As it is easy to see, the differences between the ordinals in these groups are in connection with the number ten (see the bold-faced numbers). These connections are frequent enough to see them as an indication of some kind of regularity.

It will be useful to demonstrate this regularity in a table (Figure 1).

Figure 1. Three groups in tabular form

There are three rows in the table, one for each group:

-   The pairs of the Earlier Heaven are in the first row.

-   The sovereign hexagrams are in the second row. The Creative and the Receptive (#1 and #2) are not included here because their places have been assigned to the leading positions, probably in the long past. As for the pair #11–12, it has been already placed in the first row.

-   The third row contains of the opposite hexagrams of those in the second row (with the exchanged trigrams).

In this arrangement, it is easy to find regularity among the elements. Each of the four columns consists of the first three pairs of four consecutive decades respectively. There are only three pairs not following this order; they have two asterisks (**) on them.

-   Column 1: #11–12, #19–20 (instead of #13–14), and #15–16.

-   Column 2: #63–64 (instead of #21–22), #23–24, and #25–26.

-   Column 3: #31–32, #33–34, and #9–10 (instead of #35–36).

-   Column 4: #41–42, #43–44, and #45–46.

Putting to one side the three anomalies, Figure 1 seems to be the part of a greater table in which all the hexagrams would be found (Figure 2).

Figure 2. The King Wen sequence in table form

The numbers in the cells proceed one another in order, ten (five pairs) in each columns, and indicate the position of the given cell in the table. In the frame of double order, the arrangement of the ordinal numbers visibly agrees that of the hexagrams in Figure 1. That is, the cell No 11–12 holds the hexagrams #11–12, the cell No 31–32 holds #31–32, and so on. The cells No 21–22, 13–14, and 35–36 are exceptions, corresponding to the three misfits in Figure 1.

The cells of the three hexagram sequences dealt with above are marked with different colors. It can be well seen that each of these sequences has one definite region in the table: they are the three rows in the frame. That is, these groups of hexagrams have their own regions in the table, though one pair in each group stands separately and there is an ‘alien’ hexagram on their places. Still, the regularity of this arrangement is apparent and can be formed as follows.

There are three groups of hexagrams where the elements belong together in a definite way (by form and content). Arranging the hexagrams into the table of Figure 2 and in the cells corresponding to their ordinals, the elements of each group remain together in adjacent cells, forming a separate region for the group.

 The question is whether could be this regularity applied to other groups of the hexagrams. First, with a view to answering this question, it is necessary to examine more exactly the common characteristics of the hexagrams in the three groups mentioned above.

Group of the eight directions (DIR8)

Previously, the hexagrams of the first group were defined as comprising the opposite pairs of the Earlier Heaven. They can be seen in Figure 3, in detail.

Figure 3. The four pairs of the Earlier Heaven (Group DIR8)

In this diagram, the hexagrams are in the order as they are placed in the cells of Figure 2 (the KW sequence in table form), from No 11–12 to 41–42. Under the images of the hexagrams, there are the names of the upper and the lower trigrams. The numbers below are the ordinals in the KW sequence. The double lines indicate the boundary of the region. Three pairs in the region contain the opposite (complementary) trigrams of the Earlier Heaven; they are individually framed and shadowed. In the second cell, however, the hexagrams (#21 and #22) are different. The missing pair of the group is #63–64; it is shown outside the region.

In the eight hexagrams of this group, i.e. the hexagrams in the grey cells, the upper and the lower trigrams are the complements of each other. In the circular diagram of the Earlier Heaven, these trigrams are in opposition, and their connecting lines show the eight directions of space. Hence, this is the group of the eight directions, DIR8 in short.

Group of the sovereign hexagrams (SOV8)

The second row belongs to the group of the eight sovereign hexagrams. (As it was noted above, two pairs of this sequence, #1–2 and #11–12 have already been placed in other cells.) In this group, one trigram in the hexagrams is always the Qian or Kun, and the other is one of the children, excepting the Li and Kan. The group is named SOV8 after the eight hexagrams of the sovereigns.

In Figure 4, the hexagrams of this row are shown.

Figure 4. The eight sovereign hexagrams (Group SOV8)

The arrangement of this diagram is similar to that of Figure 3. Three pairs of the eight sovereigns, #23–24, #33–34, and #43–44 are settled in the frame. The pair #13–14, however, does not belong to the sovereign hexagrams but occupies a cell in the row. The fourth pair of the sovereigns is #19–20, and it is outside this region, in the cell 19–20. It is shown beside the region.

Group of the sovereign hexagrams with exchanged trigrams (SOVX)

The third sequence contains the opposites of the eight sovereign hexagrams in the previous group (SOV8); the upper and the lower trigrams are exchanged in them. They are shown in Figure 5. The pair #35–36 is the alien and the missing pair is the #9–10; the latter stands outside the region.

The constitutive trigrams are the same as those in group SOV8: the Qian or Kun, and one of the children, excepting the Li and Kan. This is the group of the eight sovereigns with exchanged trigrams, SOVX in short.

Figure 5. The eight sovereigns with exchanged trigrams (Group SOVX)

Based on the above observations, it seems to be worthy to look for other regular groups in the table. As it was formulated above, regularity means some groups of hexagrams being coherent according to their upper and lower trigrams and the distinct regions for these in the table.

The next four regular groups in the King Wen sequence are as follows.

Group of the four cardinal trigrams (CPT4)

It was found that in the groups SOV8 and SOVX, in the majority of the hexagrams, one trigram was the Qian or Kun, and the other was one of the ‘children’, but the Li and Kan are excepted. A new regular group can be created from the complementary hexagrams, where beside the Qian and Kun, the Li and Kan should be the second trigrams.

In the first column of Figure 2, the first pair (#1–2), has  an exceptional and distinguished place, but below them the hexagrams #5–6 and #7–8 have the above required qualities. The one is composed of the trigrams Qian and Kan, and the other from the Kun and Kan, but the Qian–Li and the Kun–Li pairs are missing.

There are two pairs, however, with the combinations sought, in the cells No 13–14 (Qian and Li) and in No 35–36 (Kun and Li), where apparently they do not belong to the given group (see Fig. 4 and Fig. 5). These hexagrams (#13–14 and #35–36) will make the first column complete, replacing #3–4 and #9–10 in their present cells. As a matter of fact, the pair #9–10 obviously belongs to group SOVX, to the cell No 35–36. Also #3–4 will find its correct place in another group and region.

All these pairs can be seen in Figure 6.

Figure 6. The hexagrams of the four cardinal points (Group CPT4)

After the necessary replacements, one trigram in these hexagrams always will be Qian or Kun, and the other the Li or Kan. These characteristics make these four cells the region of the four cardinal points of space. Thus, the group is termed CPT4.          

Group of the children of same sex (SAMX)

Following the three rows in the middle of the table, the hexagrams in the fourth and fifth row are shown (Figure 7).

Figure 7. The pairs of the children of same sex (Group SAMX)

Examining the eight pairs here, four of them can be found that are composed of the same rule: #27–28, #37–38, #39–40, and #49–50. They are shadowed in the frame. Their composing trigrams are the pairs of two different ‘sons’ (from the Zhen, Kan, and Gen) or two different ‘daughters’ (from the Xun, Li, and Dui); there are no ‘mixed’ pairs. In addition, there are two pairs that correspond to this rule and belong to this group but they are at other places in the table, in the cells No 3–4 and 61–62 (the hexagrams #3–4 and #61–62). The irregular pairs are in the cells No 47–48 and 29–30, and make the recognition of the regularity difficult.

The hexagrams #17–18 and #19–20 remain outside this group and the region.

This group will be marked SAMX, according to the combination of the hexagrams from two children of same sex.

Group of the children of opposite sex (OPPX)

The complementary part of group SAMX contains the hexagrams with the trigrams of opposite sex. These are at different places of the table as follows:   

-   Three pairs, #53–54, #55–56, and #59–60 are together in the sixth column, with the double trigram #57–58 sandwiched between them.

-   The pair #21–22 would join with these, in the cell 63–64. There, the pair #63–64 is alien because it belongs to DIR8 and its place is the cell No 21–22.

-   The pair #17–18 resides separately, in cell 17–18.

-   Also the pair #47–48 remains without a dwelling place. At present, the only empty cell in an unfilled region is at 19–20, and it may go there.

-   Still, three pairs in the first row (#11–12, #31–32, and #41–42) are composed from the trigrams of opposite sex but they belong to a separate group of higher rank, to the group of the Earlier Heaven (DIR8).

The six pairs enumerated above (the three pairs in group DIR8 not included) are shown in Figure 8. Four of them are in the region of the last columns and two are in the adjacent cells at the lower end of the second column.

As was the case with the other six groups, these hexagrams have common qualities in terms of their composition. The twelve hexagrams contain the son–daughter combinations of the trigrams. They will be named group OPPX, after the children of opposite sex.

Figure 8. The pairs of the children of opposite sex (Group OPPX)

Group of the eight double trigrams (DBL8)

In terms of their composition, the double trigrams obviously belong together, even though they are scattered in the table of the KW sequence. Only two pairs, #51–52 and #57–58 are together in the sixth column, although not in adjacent cells.

The places are shown in Figure 9.

Figure 9. The eight double trigrams (Group DBL8)

The reason for the separate place of the Creative and the Receptive (#1 and #2) has already been offered. With respect to the fourth double pair, the Abysmal and the Clinging (#29 and #30) do not have regular place in cell No 29–30, in the region of group SAMX. Instead, their expected cell would be No 61–62, beside #51–52, at the top of the last column. There, the pair #61–62 is alien and belongs to the group SAMX. In this way, three pairs of the double trigrams would be close to each other, but only two of them will be located in adjacent cells, in No 51–52 and 61–62. Their common region is in the last two columns.

The group of the eight double trigrams will be named as DBL8.

The composition of the seven groups is shown in the next table (Figure 10). The boldfaced numbers indicate the hexagrams that are within their corresponding region, the other hexagrams are the aliens.

Figure 10. The composition of the seven groups

Now, the table of the KW sequence can be completed with all the groups and regions (Figure 11).

Figure 11. The table of the KW sequence with the groups and regions

Here, the color of each cell designates the group to which the two hexagrams in the given cell belong. Apparently, the majority of the hexagrams belong to definite regions of adjacent cells in the table except for one or two alien (misplaced) pairs. In detail:

-        The table contains six regions, regularly arranged in the rows and columns. (The sixth region consists of two separate parts: the last two columns and two adjacent cells at the end of the second column.)

-        In each of these regions, a particular rule can be applied for the composition of the majority of the hexagrams. In such a way, seven groups of hexagrams have been created (two groups in the sixth region).

It is important to note that in the individual groups, all the possible opposite hexagram pairs are present, i.e. the reverse and the inverted hexagrams, and the hexagrams with exchanged trigrams. (The exchanged trigrams of group SOV8 are separately in group SOVX.)

Based on these regularities, it can be supposed with reason that this tabular form (or a very similar one) had been deliberately developed it was in use before the linear KW sequence.

At some unknown time in the past, however, the table was extended in one row, following the rules of Chinese reading (reading the columns from the top down). In such a way, the adjacent hexagrams in the horizontal rows departed from each other, and their connections became unrecognizable.

This kind of transformation of the table may have happened when the demand occurred to make records of the hexagrams together with the associated judgments and line texts. In the course of recording, the hexagrams were written at the top of a bamboo slip (or other material) and the corresponding texts below. Each hexagram had one or more bamboo slips and they all were tied together, one beside the other in a row. The succession of the hexagrams mechanically followed their order in the table, i.e., it went according to the rules of Chinese reading, from the top down and consecutively in the columns. That is to say, the linear sequence of the hexagrams was only a formal necessity, determined by the form of writing. By that time (before the second century BC), the original sense of the arrangement had probably been forgotten and might not be taken into account. Afterwards, the text and the linear arrangement served as the basis of the canonized Yijing classics, and the sequence has remained unchanged until today.

A similar case would occur if somebody read an English poem according to Chinese practice, beginning with the first word of each line, then the second word, and so on. In the end, the sense of the verse would completely disappear. This might happen with the original tabular form of the hexagrams.

Changes in the table of the KW sequence

The hypothesis about the transformation of the table into the KW sequence would be more easily acceptable if an explanation was found for the positions of the hexagrams that lie outside the region in which one would expect to find them.

It is rather easy to find reasons for two changes that refer to the pair #29–30 (the Abysmal and the Clinging) and #63–64 (After Completion and Before Completion). In the Yijing, the hexagrams are divided into two parts, from 1 to 30 (Upper Canon) and from 31 to 64 (Lower Canon). It can be supposed with reason that the two pairs above were intentionally removed from their original positions in consequence of this separation, on the occasion of the transformation of the table into the sequence.

In the table, the Abysmal and Clinging pair originally had to be in cell 61–62, and not in 29–30, as they were shown in Figure 11. In the tabular arrangement, these hexagrams might be among the other double trigrams, in the last columns. Similarly to the Creative and the Receptive (#1–2) at the head of the first column, the Abysmal and Clinging had to be at the top of the last one. These four hexagrams, in the diagram of the Earlier Heaven, in their simple trigram form, represented the four cardinal points of the universe. In the table form, they might have the same role; they marked out the limits of the created world. In the linear arrangement and at its separation in two parts, the Abysmal and the Clinging would have lost their distinguished position, ending up in the insignificant penultimate place in the second part of the sequence. Not willing to allow this to happen, they were removed to a similar, important position, at the end of the Upper Canon, changing place with the Inner Truth–Preponderance of the Small pair. Thus, in the sequence, they received the ordinal numbers 29–30 according to their new positions.

A similar change-over may have happened at the places 21–22 and 63–64. In the table, the functionally correct place of the After Completion and Before Completion pair was in the cell 21–22 (see Figure 3). After the transformation and the partition, it was also necessary to close the Lower Canon with one of the cardinal hexagrams, as happened in the first part. For this reason, the two signs of Completion were removed to the end of the sequence, changing places with the less importantBiting Through–Grace pair in cell 63–64. Thus, the two pairs, the Creative–Receptive and the After Completion–Before Completion, as the symbols of the Heaven and the Earth, and the Beginning and the End, provided a symbolic frame to the whole material, and got the last two ordinals, No 63 and 64.

In such a way, a reasonable explanation may be given for the location of these four misplaced or “deviant” pairs. Before these changes, the table might have had the form of Figure 12. Here, in the cell 21–22, the hexagram pair #63–64 occupies its correct place in the region of group DIR8, and the pair #21–22 is in the last column, among the members of its own group OPPX. Similarly, the pair #29–30 is in its original cell 61–62 at the top of the last column, and #61–62 occupies the cell 29–30 in the region of group SAMX.

Figure 12. The table of the KW sequence just before the transformation

Replacing the ordinal numbers with the images of the hexagrams, the following table will be developed (Figure 13). According to the hypothesis, this was the arrangement just before the transformation. Also, it may be perfectly right to call it the King Wen Table (after the King Wen Sequence).

Figure 13. The King Wen Table

Notes to Figure 13:

1. The numbers on the upper row denote the limits of the present KW ordinals of the hexagrams in the corresponding column. (The exceptions are the #63–64 and #21–22 pairs, and the #29–30 and #61–62 pairs, as was shown in Figure 12 above.)

2. In the diagram, the misplaced pairs are marked with small circlets. According to the ordinal numbers, each of these pairs has a given position in the array but by composition it belongs to another cell in another region. The color of the circlets shows the proper region of the pair.

3. The vertical line X---X shows the plausible place where the table might have been cut in half before the extension in the row.

Here, there are only six pairs that stand in wrong cells; each of them takes the place of another pair of the six. It is rather easy to see the misplacements: the pair #9–10 has changed place with #35–36. The other four pairs have changed places successively, from cell 3–4 to 13–14, from 13–14 to 19–20, from 19–20 to 47–48, and from 47–48 to 3–4. At present, no reasons can be found for these changes. These anomalies, however, are so few in number and in proportion that the whole table can be regarded as a regular design with a few incidental errors and not a random array with so many uniformities.

The series of the latter misplacements in the table might happen in another way. The hexagram pair #35–36 originally might be in the cell 3–4 and not in cell 9–10 as it was supposed in the above example. This meant six successive misplacements; the hexagrams in the cell 3–4 were moved to the place 35–36, from 35–36 to 9–10, from 9–10 to 13–14, from 13–14 to 19–20, from 19–20 to 47–48, and from 47–48 to 3–4. That is, the whole process of the changes was carried out not at random, but in a cyclic way.

 Still, there is a disturbing element in the KW matrix, where two pairs stand separately, in the cells 17–18 and 19–20. They belong to group OPPX, according to the composition of the hexagrams. It might have been the case that in an earlier variant of the KW matrix, they were together with the other members of this group in the last column, and their present cells were empty. This question will be discussed later.

To sum up, the formation of Figure 13 could be and indeed had to be the arrangement of the hexagrams just before the development of the KW sequence. After the extension of the columns in a row, the sequence was cut in half, and, at last, the pairs #29–30 and #63–64 occupied their distinguished places at the ends of the Upper and the Lower Canon respectively (see the arrows on Figure 14). This sequence has remained unchanged until now, and has become the traditionally received, so-called King Wen sequence (Figure 14). Later still, the ordinal numbers were added.

Figure 14. The King Wen sequence, just after the transformation

It is often said that the arrangement of the hexagrams in this sequence has been made randomly. In the demonstration above, however, we have seen that each movement of the hexagrams was a conscious choice, following their order in the table and placing the hexagrams successively in the linear arrangement. From this point of view, the KW sequence may be considered the regular extension of the King Wen Table.

Conclusion

At the beginning of this article, it was shown that the arrangement of the hexagrams in the KW sequence contained some traces of an order. This order became visible when the hexagrams were arranged in a rectangular array of five rows and seven columns. The transformation of the sequence into this form was very simple: ten consecutive hexagrams (five pairs) went to each column. The last three cells remained empty (Figure 2). Observing the compositions of the hexagrams and their positions in the array, two rules might be established:

-   The hexagrams were distributed in classes (groups), collating their ordinal numbers and the constituent trigrams.

-   A definite region (a rectangular area of adjacent cells) in the array was associated with each group (Figure 13).

-   There were only six pairs out of these rules.

The classification of the hexagrams, the arrangement of the regions, and the strong relationship between the groups and the regions likely have resulted from the conscious design of this table by one or more intelligent person at some point in the distant past. In contrast, the hexagrams in the traditional King Wen sequence apparently do not have any order other than the traces of the rectangular array.

The irregular KW sequence, in great probability, has been preceded by the regular, two-dimensional King Wen Table.

Accepting the above hypothesis, the further guessing will be easier in connection with the significance of the KW sequence. The sequence only seems to be the simplified, mechanically extended variant of the KW Table. If somebody should want to find some hidden meaning in the arrangement of the hexagrams or in their mutual relations, he/she will have to search in this tabular form for it.

Supplement

On the basis of the conclusion above, I have made some speculations on the possible arrangements of the hexagrams before the existence of the King Wen Table. I supposed that this table had had an earlier variant where all the hexagrams were regularly arranged and there were not any ‘deviant’ pairs.

Earlier arrangements

At present, there are rules for the classification of the hexagrams and there are rules for the placements of the groups in the array. In the past, there had to be a conscious mind that planned and elaborated these rules and they certainly were applicable for all the hexagrams, without exceptions. The misplacements might be made intentionally or at random but they had to happen in an originally regular arrangement.

Thus, knowing the rules, the original, regular array can be reproduced by removing the deviant pairs to their determined correct positions (Figure 15).

Figure 15. An early variant of the King Wen Table

Here, in accordance with the (supposed) intentions of the creator, still two modifications have been made. The three pairs of the double trigrams in the last columns were united in one region, and the six separated pairs of group OPPX received a common area. The necessary changes were as follows:

-   The hexagram pair #57–58 went up to the cell 53–54, adjoining #51–52. At the same time, #53–54 and #55–56 got one step lower, to the cells 55–56 and 57–58 respectively.

-   The pair #21–22 from the cell 63–64 moved to the next cell 65–66, and the two pairs from cells 17–18 and 19–20 went to the empty cells at the end of the last column. Thus, all the members of group OPPX came together in a rectangular region at the end of the table.

-   Three cells remained empty: 17–18, 19–20, and 63–64.

The result of these modifications was an arrangement where every group, including the DBL8 and OPPX groups, had its own, rectangular region in the array.

At this moment, one has to think about the occurrence of the arrays in real life. The above tables (Figures 14 and 16) represent the sixty-four hexagrams in the plane as they might have been arranged in the ancient past. Here, they are shown in the form of modern drawings but in the past they could not been easily realized with the contemporary tools. In fact, these arrangements probably were composed from discrete objects (such as pieces of oracle bones, tortoise shells, bamboo slips, etc.) and laid out on a table or on the floor. There, it was not necessary to have sixty-four pieces of them because the reversed hexagrams could be simply shown by inverting the corresponding hexagram upside down. Only the four symmetrical pairs (#1–2, #27–28, #29–30, and #61–62) had to be made from two pieces each.

On this subject Larry J. Schulz referred to the essay of Lai Zhide (1525–1604). He wrote:

Lai offered an explanation for the division of the Zhouyi into two sections of unequal length. In his opinion, Wen Wang, the progenitor of the Zhou dynasty, treated the inverted gua pairs as single six-line units when he established the Zhouyi order. That is, if the second pair of gua Zhun and Meng, and all others in invert pairs are counted as one unit and the eight linear opposites … are counted individually, the result will be 18 units in the “Former Section” and 18 in the “Latter”.[12]

Using this form of representation, the most expressive and the most probable variant of the rectangular arrays can be visualized (Figure 16). Here, each cell only contains one single hexagram that represents both members of the corresponding reversed pair. The symmetrical pairs occupy two adjacent cells. Now it becomes seemingly apparent why have the three cells remained empty just beside the symmetrical pairs in the former table (Figure 15): those are the places of the second members of these pairs.

Figure 16. The King Wen Array

Looking at this array, I have to think about a wise man sitting on the floor, taking the wood tables one by one out of a pile, and arranging them in rows and columns before him. He carefully examines the images, contemplates the meanings, and decides their places in the layout. If such a man had ever lived, he might have been Ji Chang himself, the later King Wen, or somebody else whose work would have been attributed to the King. Anyhow, I should call this arrangement the King Wen Array because it is worthy of this name.

This array might be very convenient for the daily use. It was easy to keep in mind and to reconstruct it from memory. Owing to a lack of any recorded form, however, in the course of centuries (from the era of King Wen to the first manuscripts), the meaning of the arrangement might have been forgotten and the pieces of the hexagrams were removed from their original places through ignorance or by accident. The last corrupted variant might be the array of Figure 13, with relatively few changes in it, in comparison with the elapsed time. Then, from that variant the known sequence has been evolved.

When contemplating the King Wen Array, one also should find a rich symbolism. In my imagination, for example, the rectangular array may represent a house or a palace, supported by two pillars on the sides. The hexagram groups may be the main building blocks and the cells are the bricks. Each element has its own meaning, but its position in the array, the group to which it belongs, and the surrounding hexagrams add much more significance to it. Others may find different interpretations; all may lead to a better understanding of this old mystery.

Still, I thank Steve Marshall for reading my early manuscript and offering useful and valuable remarks that changed my attitude to the subject.

Endnotes

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[1] Drasny Jozsef, A Ji King elfeledett vilagkepe – A Ji-gomb (The forgotten worldview of the I Ching – The Yi-globe). (Budapest: Szenzar Kiado, 2005).

[2] Steve Moore, Structural Elements in the King Wen Sequence of Hexagrams. Oracle Paper No. 1. February 2005. Online: http://www.biroco.com/yijing/Moore_Structural_Elements.pdf. Retrieved on January 11, 2013.

[3] The Yijing. The Sequence of the Hexagrams (Xu Gua).

[4] Liu I-ming, The Taoist I Ching, trans. Thomas Cleary (Boston: Shambhala, 1986).

[5] Frank R. Kegan, “King Wen Sequence Explained for First Time since 1100 B.C.E.”, Stars-n -Dice. Online: http://www.stars-n-dice.com/fluxtome.html. Retrieved on January 11, 2013.

[6] Richard S. Cook, Classical Chinese Combinatorics: Derivation of the Book of Changes Hexagram Sequence (Berkeley: STEDT Monograph 5, 2006).

[7] Terence McKenna, “Derivation of the Timewave from the King Wen Sequence of Hexagrams”, Levity. Online:http://www.levity.com/eschaton/waveexplain.html. Retrieved on January 11, 2013.

[8] Hellmut Wilhelm, “Change: Eight Lectures on the I Ching.” in Understanding the I Ching. The Wilhelm Lectures on the Book of Changes, eds. Hellmut Wilhelm and Richard Wilhelm (Princeton: Princeton University Press, 1995), pp. 114–119.

[9] Andreas Schoter, “The Yijing as a Symbolic Language for Abstraction”, Yijing.  Online:  http://www.yijing.co.uk/downloads/LoA.pdf. Retrieved on January 11, 2013.

[10] Clarity – I Ching Community, Clarity. http://www.onlineclarity.co.uk/friends/showthread.php?t=12850&page=2. Retrieved on January 11, 2013.

[11] The Yijing,  Xi Ci Zhuan, Part II, Chapter VII.

[12] Schulz, L. J. (1990), “Structural motifs in the arrangement of the 64 gua in the Zhouyi. Journal of Chinese Philosophy”,  Journal of Chinese Philosophy, vol. 17: pp. 347–348. Online: http://www.biroco.com/yijing/Schulz_Structural_Motifs.pdf. Retrieved on January 11, 2013.

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