The 64 Hexgrams are Z2 x Z4 x Z8

64 Hexgrams Grouped into Eight Houses

As seen in the previous entry, the eight Bagua trigrams have interpretations in each of the possible groups of order eight. Nevertheless, one must consider the vital fact that the I-Ching never directly refers to trigrams, but instead to hexgrams.

Traditionally, hexgrams are interpreted as two trigrams stacked on top of one another -- and the names of these two trigrams are used as memory aids to the individual names of each hexgram (e.g. 風山漸 means that the two trigrams 風 (巽) and 山 (艮) together form the hexgram named 漸, with 風 on top of 山).

In the Han dynasty (around 50 BC), a scholar of the name Jing Fang (京房) revolutionized the study of the I-Ching by merging the hexgrams with the concepts of the Five Elements as well as the Gan (干, or "trunk") and Zhi (枝, or "branch") cycles. His method formed the basis of the current standard fortune telling technique that yield much more detailed information than previously using only the 64 hexgrams.

The first step in Jing Fang's method is to divide the 64 hexgrams into eight houses. His method of dividing the hexgrams is novel and sheds light on the structure of the hexgrams themselves.

The houses are formed by first taking the eight hexgrams with the same upper and lower trigrams -- these eight hexgrams can potentially form a sub-group that is isomorphic to the Bagua group. This set is called the Basis set.

Each of the eight hexgrams is thus modified in a predictable manner through seven generations (or variations).  The logic of each generation is easy to discern by inspecting the lines of the hexgrams, and they are:

First generation: The lowest line of the base hexgram flipped
Second generation: The two lowest lines of the base hexgram flipped
Third generation: The three lowest lines (i.e. the lower trigram) of the base hexgram flipped
Fourth generation: All but the top two lines of the base hexgram flipped
Fifth generation: All but the top line of the base hexgram flipped
Sixth generation: The fifth line (from the bottom) and the lower trigram of the base hexgram flipped
Seventh generation: The fifth line (from the bottom) of the base hexgram flipped

Notice the sixth and the seventh generations (variations) as they have special names (遊魂 and 歸魂,, the Wandering Spirit and Returning Spirit respectively) and special significance in Jing Fang's fortune telling technique.

The Seven Generations as a Change Group

It is immediately obvious that the seven generations (variations), plus the identity, may form a group representing change of the hexgram lines. In particular, each generation modifies the base hexgram such that, when the top trigram is compared with the bottom trigram (which should be identical in the base hextram), the two trigrams have different lines in different positions based on the generation in question.

As there are three lines in each of the top/bottom trigrams of the base hexgram, there are eight possible combinations of differences of the three lines (counting identity as one).  It can be seen that each of the seven generations, plus the identity, map directly to one combination.

For example, the first generation has the lower line different between the top and bottom trigrams.  The second generation the lower and middle lines. The third generation with all three lines different. The fourth generation, has only the top two lines different, whilst the fifth generation has only the top line different. The sixth and seventh generations are have the middle line different and the top/bottom lines different respectively.

To illustrate with a 3D model:

Notice that, in the 3D representation of this change group, the two most important generations (sixth and seventh) reside on opposite ends of the cube and form the two end-points of the diagonal traversal. Otherwise, the group elements traverse the cube one edge at a time.

Note: This begs the question that whether the seven generations should have been reordered such that all traversals of the cube occur one edge at a time, such as 0 -> 1 -> 2 -> 3 -> 6 -> 5 -> 4 -> 7.

Reconciling The Houses With The 雜卦傳

Unique among the various commentaries of the I-Ching is the 雜卦傳 (Ad Hoc Commentaries on the Hexgrams), an ancient document that focuses on the differences between hexgrams, and their symmetries, both in structure and in meaning.  This document is most crucial in discerning the meaning of each hexgram (other than the hexgram's name) and is frequently quoted, as the I-Ching itself does not make the meaning of each hexgram clear.

The 雜卦傳 matches many hexgrams into pairs, most of them with opposite meanings (but not all). The pairs are constructed by flipping each hexgram upside-down to find its inverse -- thus immediately suggesting an abelian group in operation.  Hexgrams which look the same when flipped upside-down are paired with their mirrored images (i.e. the hexgram formed by flipping each line of the original hexgram from solid to broken and vice versa).

One important feature of the 雜卦傳 is that it does not pair all the hexgrams in a similar manner -- for example, eight hexgrams (姤, 既濟, 未濟, 夬, 大過, 頤, 漸, 歸妹) are not paired for unknown reasons.  Another remarkable feature is that, although many of the meanings cited by the document are opposite in pairs, some pairs are described by meanings that are not clear opposites of each other, some may even be unrelated.

These features suggest that the 雜卦傳 was merely an ancient attempt to make sense of the group structure of the hexgrams by using two inverse-finding techniques -- i.e. flipping upside-down and mirrored image. These techniques failed for those hexgrams that do not follow these two patterns (as seen later), and the author was not able to reconcile them.

The Basis Set is Z8

If one looks at the change group used to crate the seven generations, another characteristic immediately pops out -- the first generation is related to the fifth generation, whilst the second generation is related to the fourth generation. The third generation relates to itself. The relation is that of flipping the lines changed upside-down (thus changing the lowest line becomes changing the topmost line) and taking a mirrored image (thus a line changing becomes non-changing and vice versa).

For example:

It is thus possible to reconcile the methods used by the 雜卦傳 with Jing Fang's houses if we choose a group for the basis set that maps a trigram to its inverse which is the mirrored image or its flipped upside-down version -- or Z8 as discussed in the previous entry!

If Z8 is chosen as the group for the basis set, then hexgrams the first 5 generations (plus the identity) are all automatically paired with hexgrams that are upside-down versions of themselves, except for the houses 乾 and 坤, which are paired with hexgrams within their own houses (since 乾 is the identity and 坤 is -1, self-inversed, in Z8). This discrepancy is perhaps one of the reasons for the difficulties encountered by the author of 雜卦傳, together with another discrepancy which concerns the last two generations.

The Change Group is Z2 x Z4

Judging from the first five generations and their inverses, it is obvious that the change group can only be either Z8 or Z2 x Z4.

The last two generations are more difficult to map, as they do not follow the pattern of the first five generations, simply because they concern trigrams with either the middle line or the top/bottom lines different. In either case, flipping the changes upside-down yields the exact same item, and thus it is not possible to form an inverse-pair in the same manner as the first five generations, since both would have the same lines that are different.  Because of this, it is strongly likely that the last two generations are self-inverses, similar to the third generation.

Another strong support for the last two generations being self-inverses has to do with a few hexgrams that are not in the basis set but still are the same when flipped upside-down -- i.e. 中孚, 頤, 大過, 小過. All such trigrams occur in the sixth generation. In particular, the names of the two hexgrams 大過 and 小過 (literally, "over-large" and "over-small") suggest that they are related, and if treating the last two generations are self-inverses, these hexgrams are paired with each other as inverse-pairs: 中孚 with 頤, 大過 with 小過 (which can also be interpreted as "plus" and "minus").

Therefore, it is concluded that the most likely group for the change group is Z2 x Z4.

I-Ching is Z8 x Z2 x Z4

It is now possible to reconcile the group structure of the 64 hexgrams with interpretations given in the 雜卦傳:

In this diagram:

Blue = Hexgrams not paired by the 雜卦傳

Yellow = Hexgrams paired (correctly or incorrectly) by the 雜卦傳 as mirrored images and with opposite meanings

Light Green = Hexgrams correctly paired by the 雜卦傳 with opposite meanings
Dark Green = Hexgrams correctly paired by the 雜卦傳 but with meanings that are not opposites

Light Red = Hexgrams incorrectly paired by the 雜卦傳 with opposite meanings
Dark Red = Hexgrams incorrectly paired by the 雜卦傳 and with meanings that are not opposites

As can be seen clearly, the author of the 雜卦傳 appeared to get it right on most of the hexgrams, except for the ones in the sixth/seventh generations, as well as those in the 乾 and 坤 houses (which map to inverses within their own houses). Incidentally, six out of the eight blue hexgrams (those not paired by the author) reside in these special-case zones.

A quick look at the meanings of the discrepancies (i.e. red and blue ones) suggests that the "correct" pairings may actually make better sense:

剝 (separate) <--> 逅 (meet/converge)
觀 (observe) <--> 遯/遁 (hide)
夬 (break) <--> 復 (restore)

大過 (delta plus) <--> 小過 (delta minus)
中孚 (reliable) <--> 頤 (middle ground)
隨 (follow, to wed?) <--> 歸妹 (receive/return bride)
師 (make war) <--> 同人 (harmony)

Ones under the "correct" pairings but with meanings less obvious are:

大壯 (strong) <--> 臨 (arrive)
明夷 (harm) <--> 訟 (argument)
漸 (gradual, improve) <--> 蠱 (rot)

Remaining Issue: The Last (Seventh) Generation

The sixth and seventh generators do not follow the same basic formula as the first five generations (six if counting the basis/identity). This is primarily due to the fact that the difference pattern between their upper and lower trigrams are identical when flipped upside-down.

Still, it would appear that the last (seventh) generation could be rendered differently -- for example, having the fifth line (counting from the top) flipped instead of the current second line, or having the forth and last lines (counting from the top) flipped, etc. The hexgrams residing in the seventh generation will be shuffled into different positions, but the group structure will stay valid.

As there are numerous ways to generate seven variations from a basis set which cover all possible difference patterns between the upper and lower trigrams, the rationale behind this particular choice is still unknown, except that it may lead to a consistent procedure (i.e. a calculus) of manipulating the hexgrams that parallel their group-theoretical behaviors.