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The categorization of the gua and its underlying meaning
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TO briefly reassess what we have learned about the I Ching, let’s remember that a full set consists of 64 gua, which belong to either the yin sphere or the yang sphere, with exactly 32 gua in each.
There are 32 pairs of gua, forming 32 micro circulations. Each pair has one gua from the yin sphere, and another from the yang sphere.
According to the binary algorithm (similar to the Shao Yong’s “doubling method”), each gua is assigned an ID number, which results in two sequences: one is the “Shao Yong-Leibniz Sequence” with the weight doubling from the top yao down to the base yao, and the other is the “Digital Sequence”, which calculates the ID number with the weight doubling from the base yao up to the top.
Based on this, we develop the “Yangdu Sequence”, which ranks the gua in terms of the “Rule of Yangdu” (i.e., the intensity of sunshine). Those gua with higher yangdu should be ranked before the gua with a lower yangdu. The yangdu are defined by the number of yang yao and their positions: the more yang yao, the higher the yangdu, and if the number of yang yao is the same, the yangdu is higher if the yang yao’s position is, too.
As such, the macro and micro circulations of I Ching gua are derived from the Yangdu Sequence of Bagua (3-yao), which is: Earth 0, Thunder 1, Water 2, Mountain 3, Valley 4, Fire 5, Wind 6, Sky 7. Thunder, Water, Mountain and Sky are yang gua while Wind, Fire, Valley and Earth are yin gua. The pairing of Bagua are thus formed (see Picture 1): Earth-Sky, Thunder-Wind, Water-Fire, and Mountain-Valley. Each pair’s ID number totals 7.
Inversely and reversely pairing
Picture 1 displays the arrangement of gua according to the Rule of Yangdu, the primary axis (northwest-southeast) divides the diagram into two triangles. The 32 gua in the bottom left triangle including Zhen, Kan, Gen and Qian on the axis are in the yang sphere. Their ID is between 1 and 32. The remaining 32 gua in the top right triangle belong to the yin sphere, including Dui, Li, Xun, and Kun on the axis, with IDs from 33 to 0.
There are seven different types of gua, classified by the number of yang yao (or oppositely in the number of yin yao). 1 Zero Yang Gua; 1 Six Yang Gua; 6 One Yang Gua (3 of them are in fact Five Yin Gua); 15 Two Yang Gua (6 of them are in fact Four Yin Gua); 20 Three Yang Gua (10 of them are in fact Three Yin Gua); 15 Four Yang Gua (9 of them are in fact Two Yin Gua); 6 Five Yang Gua (3 of them are in fact One Yin Gua). They are highlighted in distinctive colors in Picture 1.
Two gua in any pair in Picture 1 are symmetric to the center, or Tengen (天元, a symbol of blue star). The difference of the ID number, or yangdu, of any paired gua must be 32, which means each pair has a total of six yang yao, indicating the sum of their yangdu as 32.
It is therefore clear from the hexagram that there are two pairing principles to all the gua: the inversely pairing principle (错卦) and the reversely pairing principle (综卦). The former means the yao at the same position of a paired gua is opposite to each other, yin vs yang. The latter means the hexagram of paired gua are mutually upside down. In Zhouyi, there are 24 reversely paired gua and eight inversely paired gua. However, according to the binary algorithm, all the 32 paired gua are all inversely paired regardless of whichever the sequence they are paired from.
Consequently, we have 20 “groups” of I Ching gua: Eight of them are Two Gua Groups that are made up of an inversely paired gua in Zhouyi. The rest are 12 Four Gua Groups, which are made up of two inversely paired gua, which can be further recombined into two reversely paired gua in Zhouyi, and two re-inversely paired gua（错综卦).
When we “synthesize” different sequences into one square diagram as in Picture 2 and put the columns in Shao Yong Sequence and rows in Digital Sequence, with an adjustment in yangdu, the 64 gua are shown to be perfectly symmetric. Please pay attention to the 32 “primary gua” (in color).
All inversely paired gua are symmetric to Tengen. All reversely paired gua (original pairs in Zhouyi) are symmetric to the primary axis. And all re-inversely paired gua are symmetric to the secondary axis (northeast-southwest).
To make the relationships in Picture 3 stand out, we rotate the square diagram clockwise, and the paired gua become very prominent. For illustration, the 8 gua on the primary axis (now the vertical axis) are four inverse pairs. They have no way to form a reverse pair. In other words, they themselves are reversely paired gua. The 8 gua on the secondary axis (now the horizontal axis) form 4 pairs of inversely and reversely paired ones. So they themselves are re-inversely paired gua.
The remaining 48 gua form 12 groups. For examples, in the group of Zhen Gua(51), Gen Gua(52), Xun Gua(57) and Dui Gua(58), Zhen Gua-Gen Gua and Xun Gua-Dui Gua are two reversely paired (original in Zhouyi); Zhen Gua-Xun Gua and Gen Gua-Dui Gua are two inversely paired; and Zhen Gua-Dui Gua and Gen Gua-Xun Gua are two re-inversely paired! For the same token, in the group of Tun Gua(3), Meng Gua(4), Ge Gua(49) and Ding Gua(50), Tun Gua-Meng Gua and Ge Gua-Ding Gua are two reversely paired (as originally in Zhouyi); Tun Gua-Ding Gua and Meng Gua-Ge Gua are two inversely paired; and Tun Gua- Ge Gua and Meng Gua-Ding Gua are two re-inversely paired.
Having acquired some understanding of I Ching, you may now be able to find some groups of gua and discover their pairing relationships, reversely paired, inversely paired, or re-inversely paired.
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