Bagua Ordering and The Five Elements

The I-Ching outlines a fortune-telling system based on 64 hexgrams (or 64-gua).  Each individual hexgram is named individually in the I-Ching, but the particular hexgram is also referred to by the two trigrams forming the first three and last three lines of the hexgram respectively.

For example, the hexgram 益 is made up of the trigram of wind 風 (☴) and thunder 雷 (☳), representing the first three and last three lines of the hexgram for 益 respectively.

The I-Ching itself, strangely, avoids referring to trigrams, but only to hexgrams.  This is strange as future scholars all put heavy emphasis on the trigrams as the first level of categorization, almost as if the 64 hexgrams are mere extensions of the 8 trigrams -- a Cartesian product of the trigrams onto itself.  On the contrary, the I-Ching specifically avoids formally connecting the hexgrams with trigrams in this manner, and only seems to use trigrams as a convenient notation to refer to hexgrams.

In a future post, this issue will be investigated further with the concept that the Bagua trigrams form a group, and the 64 hexgrams also forming a group.

The I-Ching also does not specify a particular ordering or sequencing of the trigrams or hexgrams, and subsequently several popular orders have occurred -- simply because they must be ordered somehow to put into writing.

The Binary Order

One particular popular ordering of Bagua trigrams is the binary order, seemingly dated to the Suing (宋) dynasty close to 1000AD.  The ordering starts with the trigram of 乾 with all solid lines and essentially treat each broken line as a 1, and each solid line as a 0, starting with the top line as the least significant bit:

  乾 ☰, 兌 ☱, 離 ☲, 震 ☳, 巽 ☴, 坎 ☵, 艮 ☶, 坤 ☷

The Binary Order With The Five Elements

The Binary Order is interesting since it coincides with the Five Elements quite nicely:

 Notice that the I-Ching itself does not refer to the Five Elements, which is an independent development parallel to the I-Ching.  The Binary Order and the connection of the Bagua with the Five Elements appear to be the result of subsequent scholastic developments.

The Ordering of the Five Elements When Matched with the Bagua

Also notice that the ordering of the Five Elements is different from typical orders.  In typical Five Elements lists, an order following the "beget/benefit" (生) cycle or the "harm" (剋) cycle is used.  The matching of the Bagua Binary Order with the Five Elements yields a third ordering that is neither of those.

However, if one inspects this order closely, one can find that it matches other fortune-telling disciplines closely, namely the 12-Zhi (12 地支) categories in the Four Pillars system and the Nine-Squares Matrix (九宮) in fengshui.  Both of these will be discussed fully in future entries.

Similarities With 12-Zhi

The 12-Zhi categories can be modeled as in the following diagram:

Basically, the Five Elements are distributed in three axes, with metal (金) and wood/wind/air (木) forming one axis and opposing each other, fire (火) and water (水) forming another axis and opposing each other, and earth (土) being its own axis -- divided into wet earth (濕土) and dry earth (燥土), most likely the two poles of that axis.

Rotating anti-clockwise from metal and arriving at the earth axis last yields this particular ordering.

Similarities With Nine-Squares Matrix

The Nine-Squares Matrix used in fengshui has the following matching between the Bagua trigrams and the nine squares:

As can be seen, again going anti-clockwise from metal "sort of" yields this particular ordering of the Five Elements, with the exception that earth is squeezed in between metal and first, as well as between wood and water.

The difference treatments of earth in the Nine Squares Matrix and the 12-Zhi suggest that the Nine Squares Matrix may be a flat projection of a three-dimensional model (i.e. similar to 12-Zhi) with three axes.  This coincides with the fact that, historically, Chinese fengshui was preoccupied with two dimensions -- there being no high-rises in ancient China.

This almost indicates that the Nine Squares Matrix fengshui system of fortune-telling is a two-dimensional approximation of a three-dimensional model.

The Siblings Order

There is yet another popular ordering of Bagua trigrams -- the so-called Siblings Order.  This order is much more historical, dating almost back to the origins of the I-Ching itself:

Parent: 乾 ☰ (father) -- Yang
- Children: 艮 ☶ (youngest son), 坎 ☵ (middle son), 震 ☳ (eldest son)

Parent: 坤 ☷ (mother) -- Yin
- Children: 兌 ☱ (youngest daughter), 離 ☲ (middle daughter), 巽 ☴ (eldest daughter)

Definitions of the siblings are Siblings Order given in 說卦傳 (Documentaries on Trigrams):

乾，天也，故稱乎父；坤，地也，故稱乎母；震一索而得男，故謂之長男；巽一索而得女，故謂之長女；坎再索而得男，故謂之中男；離再索而得女，故謂之中女；艮三索而得男，故謂之少男；兌三索而得女，故謂之少女。
The 乾 trigram signifies the heavens and is thus the father. 坤 signifies the earth and is thus the mother. 震 contains the first solid line, making it the eldest son; 巽 contains the first broken line, making it the eldest daughter. 坎 contains the second solid line, making it the middle son. 離 contains the second solid line, making it the middle daughter. 艮 contains the third solid line, making it the youngest son. 兌 contains the third broken line, making it the youngest daughter.

Incidentally, according to unearthed ancient documents (帛書周易) written on cloth dating back to the Han Dynasty around 200BC, the order of the eight trigrams are also given as the Sibling Order instead of other more modern orders that put 乾 ☰ and 坤 ☷ at opposite ends.

Notice also that, in the Siblings Order, there are more broken lines in trigrams categorized as male (except for the father itself, which has no broken line), and more solid lines in trigrams categorized as female (except for the mother itself, which has no solid lines).  A popular explanation:

陽一君而二民，君子之道也；陰二君而一民，小人之道也。
Yang trigrams have one head and two subordinates  -- which is the way of the good.
Yin trigrams have two heads and one subordinate -- which is the way of the evil.

Another popular explanation, which actually seems to make more sense:

陽卦多陰，陰卦多陽，其故何也？
Yang trigrams have more yin lines, and yin trigrams have more yang lines.  Why?
陽卦奇, 陰卦偶。
Yang trigrams have odd numbers of yang lines.
Yin trigrams have even numbers of yang lines.

This explanation actually coincides with the Bagua Procedure outlined in a previous entry, when odd multiples of 4 yield a solid (or yang) line, and even multiples of 4 yield a broken (or yin) line.

The Siblings Order and Group Theory

The Siblings Order is apparently the oldest, most ancient, and most authoritative ordering, but one that makes very little practical sense. Unless one actually looks at the eight Bagua trigrams (and the 64 hexgrams) forming a group, which is the topic of the next entry.

The group structure of the trigrams/hexgrams also explains why ancient people should invent such a strange ordering, apparently without any practical reason, before inventing other more rational types of ordering. The Siblings Order arises naturally in certain groups of order eight.