The 108 in King Wen's Sequence

By J.M. Berger

King Wen is a figure shrouded in myth. We know almost nothing about him except that someone using his name created a famous sequence, an ordering, for the i-Ching. Over time, King Wen's sequence has proven impenetrable to any mathematical explanation of why the hexagrams are in presented in this particular order.

King Wen's sequence orders the i-Ching hexagrams into 32 pairs. Each hexagram in a pair is structurally related to the other hexagram in the same pair, but the sequential ordering of the pairs appears to be random.

King Wen's circular arrangementThe i-Ching is inherently numerical and any sequence of hexagrams can be made to perform a variety of mathematical or narrative tricks, but the underlying mathematical-structural intention of King Wen's ordering is nothing short of baffling -- to the point that most scholars fall back to the "random" theory. While some commentators have advanced theories about the ordering, most of these are unconvincing and somewhat contrived.

I've been looking into the ordering of the King Wen sequence for a while now. Although I haven't solved the underlying logic, I recently uncovered a set of correspondences tied to King Wen which has some interesting implications as far as arranging and ordering the hexagrams. I am increasingly certain that the King Wen sequence is not random, although the exact pattern still eludes me.

I began with the King Wen circular arrangement of the ba gua (eight trigrams). The King Wen ba gua contants a "magic square" number scheme -- if you add the numeric values of each position in a row, column or diagonal, the total is always 15. I was curious about how far the properties of this square could be extended, so I created a diagram running the numbers out in a 9x9 grid. 8x8 for the normal grid used to find an i-Ching hexagram, plus a column in the center to correspond to the five in the center of the ba gua (which symbolizes the Tai Chi, or yin-yang symbol of totality).

I added the values of the vertical and horizontal columns, then added the totals of the columns and rows. The result was not a magic square, but it was surprising.

The 108 Cross in King Wen's ba gua

The rows and columns corresponded horizontally and vertically, but the 108 especially jumped out at me, because of its importance in other Eastern systems not immediately linked to the numerology of the i-Ching. (The number recently became very prominent to the wider public due to its recent use on ABC's LOST.)

Within the four quadrants created by the "108 cross" the summed values alternate perfectly between even and odd. While not conclusive, this suggests an element of design, rather than chance. This suggestion becomes far more concrete when you analyze how the "108 cross" interacts with the full 64-number King Wen sequence.

When you plug the King Wen numbering into the appropriate slots for the hexagrams (collapsing the "five" line into an axis, since there are no corresponding hexagrams), you get the following arrangement:

King Wen ba gua grid symmetry

The blue-shaded hexes are symmetrical -- yin-yang opposites are reflected diagonally around the axis of the "108 cross." The reflected hexagrams aren't just any hexagrams either -- the set includes the first two hexagrams in the King Wen series, Nos. 1 and 2, Pure Yang and Pure Yin; Nos. 29 and 30, Fire (strong yang) and Water (strong yin); Nos. 11 and 12, Advance and Obstruction; and the final two hexagrams in the King Wen series -- Nos. 63 and 64, Unsettled and Settled.

The hexagrams represented in the set of 16 symmetrical hexes are the purest yin and yang hexagrams. All of them represent major forces. But the symmetry doesn't end there.

Although you can easily build a narrative for the 108 cross (or any other sequence of hexagrams), there is no apparent rhyme or reason to the numbers within the cross, in terms of hexagram structure or numerical order.

King Wen numerical values in 108 grid

When the numbered hexagrams are divided into simple 4x4 quadrants, again, there is no pattern in hexagram structure or numerical order -- at first glance. But a deeper analysis shows that there is an underlying order to the arrangement. I found order in the "nuclear" hexagrams that correspond to the hexagrams in the square. A nuclear hexagram is found by taking the second, third and fourth lines (counting from the bottom) as the lower trigram, and the third, fourth and fifth lines as the upper trigram -- forming a new hexagram. This is a common practice for advanced i-Ching practitioners.

If you derive the nuclear trigram for every hexagram in the i-Ching, you end up with a set of 16 hexagrams that repeats four times.

King Wen-108 grid with nuclear hexagrams

This time, the chart does reveal an ordering, and a symmetry -- a complete set of nuclear trigrams is found in each distinct quadrant. The same number never appears twice in the same quadrant. The yin-yang mirroring in the clusters of four hexes surrounding the 108 cross is also maintained after the transformation, again providing a central cluster of symmetry -- but still off center from the overall diagram.

Additionally, the two vertical columns down the center axis are perfectly symmetrical, and each row and column has a perfect reflection across the horizontal and vertical axes, although they are not symmetrically arranged.

Clearly, this division of the hexagrams is not random, but it is also lacks a simple, obvious pattern. In fact, the table is similar to the King Wen numerical sequence, because there is clearly a grouping (quadrants in the chart, pairs in the sequence) but the pattern that dictates the grouping remains impenetrable. It may be fractal in nature, in which case a final explanation could prove impossible to discern.

I'm not giving up yet. I will update with whatever additional correspondences I turn up after some additional study of these charts. Based on the complexity of the relationships, I am increasingly certain there is an underlying mathematical-structural explanation for the ordering of the King Wen sequence. I suspect the answer lies in the ba gua diagram...

Oh, yes, the square at the center of the 108 cross? "Joy," No. 58, also known as "Lake." An appropriate hexagram for 108, a number so closely associated with transcendence and nirvana.

I-Ching Links

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By J.M. Berger

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(As Contributor)

(Fall 2007)

About this site

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.