Structure In King Wen's SequenceBy J.M. Berger
This is a draft version of this article. Comments and corrections welcome at . You may want to check out the diagrams on pair symmetry, which has less textual analysis but much clearer significance.
In The Hidden 108 In King Wen's Sequence, I wrote about a grid I created for the i-Ching based on King Wen's circular arrangement, also known as the lo-shu or sometimes ba gua. I used the order of the trigrams in the Lo Shu to create a chart of all 64 hexagrams laid out in an 8x8 grid.
As part of my continuing exploration of the numbers, I created a small map that traces a line between each hexagram in its numerical order off of the 8x8 grid. An animation of the lines being drawn can be seen at right. Each step advances one octet -- a series of eight consecutive hexagrams.
I discovered that the map had some interesting characteristics. No line ever doubled back on itself, and at every consecutive hexagram, the line changed direction (i.e., 21, 22 and 23 were not all along one straight line).
These traits appear fairly unusual. I tried nearly a dozen variations using different circular arrangements to order the hexagrams eight rows by eight columns and several rearrangements without a trigram schene. In all of the other grids, lines doubled back on themselves, and in most, lines also formed continuous lines across three (or even more) numbers (i.e., 21, 22 and 23 could all be in a single row).
There are 1.27e+89 permutations of 64 numbers in sequence. Since the Wen pairs are fixed according to specific rules, there are 2.63e+35 possible sequences of pairs. I think that works out to 2.6 trillion trillion trillion (give or take an 'illion). I'm trying to work out the permutations to give a sense of exactly how rare the absence of doubling-back and continuous lines would be, but this isn't my forte. Drop me a line if you're the sort of person who can quickly and easily figure that out. Otherwise, I will update once I plod through it.
UPDATED: I'd like to throw out one possibility here. In my examination of Fu Xi pair symmetry I found hints that sequential time (and thus sequential ordering) is strictly a yin phenomenon and that it cannot be applied to yang hexagrams. In light of this, I now suspect that the ordering of the overall King Wen sequence must therefore be dependent on yin alone. That means the structure analyzed in this piece, while interesting and possibly useful, may not reflect the key element of King Wen's ordering.
There are a handful of overlapping lines in this design -- most prominently one section of the line from 57 to 58, 30 to 29 and 1 to 2. I'm torn about the significance. There is no doubling back, however, and all lines turn at each hexagram. The full diagram below shows the upper canon in red, the lower canon in blue, and overlapping lines in yellow and orange.
The structure is asymmetrical. The most intense concentration of intersecting lines coincides with Hexagram One, Heaven, but the structure is sufficiently chaotic to defy a quantitative analysis (by me, with the amount of time I currently possess, at least) of how and why the lines intersect.
A rendering of the structure in black and white may help give a better sense of its overall shape and structure.
The edges reflect the 8x8 grid. There are two flat edges which reflect consecutive numbers in the same column or row. The center, not unexpectedly, is more dense with connections than the edges. You can easily tile the image, but that's also consistent with its grid origin, so it's not particularly significant.
Tiling the image irregularly (rotated 90 degrees and with edges out of alignment) does produce some interesting shapes, and it could be that the design was somehow intended or otherwise rendered appropriate for such a use, but there isn't any apparent correlation to any real world phenomenon that I can identify with the naked eye. Obviously, that means there are plenty of possible correlations I've missed. Feel free to drop me a line () with any thoughts you might have. I did find that the image reminded me, perhaps misleadingly, of protein-folding. In addition to its obvious relevance to genetics (see this site for an introduction to the i-Ching and genetic code), protein-folding may play a role in organismic intelligence, according to some authors.
Finally, I should note I followed Western convention in arranging the trigrams in the grid (left to right, top to bottom). So the shape, if it was intended by design, should probably be mirrored both horizontally and vertically, as in the image at right. In this color illustration, the different line colors indicate each of the eight hexagram octets. White lines indicate connections between octets. As the color differentiation shows, there does not appear to be any repeating pattern in how the lines are arranged by octet.
The octets are broken out into individual arcs below:
I also used the structure as the basis for a series of i-Ching mandalas. I find this structures fascinating. I created the one depicted below (click image for larger picture) by rotating the King Wen structure eight times, to correspond to the eight directions laid out on the ba gua. I don't know the total significance of this mandala, but it is a direct visual depiction of the i-Ching, and that counts for something. I will be posting some artistic interpretations and variants on this mandala (and other related structures from my King Wen widget), at a later date. I have some intriguing thoughts about what they might depict in relation to real world phenomena.
All text, images and Web design (C) 2006, J.M. Berger, all rights reserved. for permissions.
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I'm J.M. Berger. I'm a freelance writer covering a wide variety of topics, including religion, spirituality and religion-science convergence. This site is devoted to a personal obsession of mine -- figuring out the logic that informs the King Wen sequence of the i-Ching. I hope you enjoy it. I have other Web sites relevant to this material, which are linked above.