## King Wen: Change And SymmetryUpdating and expanding, with corrections.
The King Wen sequence contains a perfect balance of changing lines, with some balances deriving from the pair-set structure and others apparently related to King Wen's specific grouping of hexagrams in rows. Those who use the i-Ching for divination know that there are numerical values for "changing lines" -- when a line in a reading is becoming its opposite. The values are:
But you can also write those values into the King Wen sequence. For instance, consider the following two consecutive hexagrams. You can assign the changing line values to describe how hexagram 19 is transforming into hexagram 20. If you assign the changing line values to the entire sequence, it looks like the chart below. Gray boxes are yang lines, white are yin, and hexagram 64 is treated as if it is transforming into hexagram 1. Author's note: Please feel free to proofread the charts and send in corrections. I belive this is finally right, but I have no editor. I am doing the best I can, but this chart is the stuff of migraines. Changing even one number has massive snowballing effects on the patterns (or lack thereof). Your patience and understanding are appreciated.
If you add all those values together, the total is 2,880. That number is not random. When you look at the changing line values, there's a numerical correspondence. Changing yin (6) plus changing yang (9) equals 15. Static yin (7) plus static yang (8) also equals 15. Split the difference and you get 7.5. Multiply 7.5 by the 384 lines of the i-Ching and you get 2,880. This gets confusing and made the first run of charts into a very frustrating mess. So I thought about how to do it better, and realized the 6,7,8,9 values could be reformatted for a more intuitive chart. I converted the numbers into a deviation from zero, which is probably more easily grasped in the table below.
Mathematically, this is the same difference -- but from the perspective of intuitively looking for symmetry, it's much easier to work from zero as the midpoint. Instead of talking about a sum of 2,880 the values in the chart now total zero, and it's now apparent what that means. All the changing and static yin and yang lines are balanced and therefore cancel each other out. The chart now looks like this: There are 107 changing yang lines, 107 changing yin lines, 85 static yang lines and 85 static yin lines. This balance appears to be the result of the pair structure (pending more tests). Swapping pairs changes the proportion of changing lines to static lines, but the balance remains. Breaking the pair structure appears to unbalance the lines (pending more tests). You will see in the chart above that I have added the totals of each row, and each column. The column totals are all even, and the row totals are all odd. The row totals are always even because of the pair structure, but the column totals are odd for reasons specific to King Wen's sequence. However, King Wen's sequence isn't the only way to get that result. Swapping pairs out of sequence sometimes disrupts the columns, but not always. The row totals have another feature. The top and bottom halves are mirrored, i.e., if you add the totals for the mirror rows, they always add up to zero (Rows 1 and 8, 2 and 7, 3 and 6, 4 and 5). Again, this isn't unique to King Wen, but it's also not inherent in the pair structure. If you swap a pair from the top half with a non-mirror pair from the bottom half, the pattern breaks. Most pair-based charts will also be balanced top and bottom, and left and right, and King Wen is too. If you add the values on the top half of the chart, it comes out to eight and the bottom comes out to negative eight. The left half totals four and the bottom half totals negative four. If you break it down into quadrants, the totals look like this:
Below, you get basically the same pattern if you designate the regular sequence lines with yin as 1 and yang as -1. However, the odd/even pattern carries over if you designate yin as zero and yang as one (even and odd), with no changing lines. The rows in either case do seem to form some kind of pattern, but it's not as easy to describe as the changing line pattern. Finally, here's a look at the changing lines with a color scheme to highlight the distribution. The pattern isn't apparent, but the balance is there. Changing yang is red, changing yin is yellow, static yin is white, and static yang is black. All text, images and Web design (C) 2006, J.M. Berger, all rights reserved. for permissions. |