THE two diagrams shown in Picture 1 were once presented to German mathematician Gottfried Wilhelm Leibniz by Joachim Bouvet, a missionary working in China. The diagrams came into existence in the mid-11th century, although the gua and the hexagrams date back at least another 1,400 years.

We’ve already known that each 6-yao gua (hexagram) consists of two 3-yao gua (trigrams). For example, Tai Gua
( ) is Sky (☰) underneath Earth (☷) while Pi Gua () is to the contrary; Weiji Gua () is Water (☵) underneath Fire (☲) while Jiji Gua () is to the contrary, Water above Fire. When two Waters (☵) are stacked together, it becomes Kan Gua (). Therefore, we just need to remember the eight 3-yao gua (Bagua), since any I Ching hexagram and their order are a combination of two Bagua.

Let’s take a look at the square diagram in the Picture 1. The eight 3-yao gua are laid in the same order both horizontally and vertically — the eight columns from left to right and the eight rows from top to bottom are arranged in the same order of “0-Earth, 1-Mountain, 2-Water, 3-Wind, 4-Thunder, 5-Fire, 6-Valley, and 7-Sky.” Of all the 64 I Ching gua, the most important eight are called “Ching Gua.” Each of them is consisting of two same 3-yao gua, namely, Kun Gua of two Earths, Gen Gua () of two Mountains, Kan Gua of two Waters, Xun Gua ()
of two Winds, Zhen Gua () of two Thunders, Li Gua () of two Fires, Dui Gua () of two Valleys and Qian Gua of two Skys. Eight Ching gua are positioned in the primary diagonal line from northwest to southeast in the square diagram. You can calculate the ID number for 64 6-yao gua in two ways. One is to calculate the ID of each gua according to the formula “ID number = a+b*8,” where b standing for the row number and a for the column number of the location of the gua in the square diagram. For example, the coordinate of Kan Gua ()
is (2,2), thus, its ID number is 2+2*8=18.

An alternate way to calculate the ID number is using binary math — the 0-1 system — to obtain the gua’s ID number, which is equivalent to Leibniz’s method. For example, Kan Gua () is 010010 according to the rules of the binary method. When a yang yao (­—) is defined as 1 and a yin yao (--) as 0, the Shao Yong sequence of Kan Gua is 0*25 + 1*24 + 0*23+0*22 + 1*21 + 0*20=18. Inspired by Shao Yong, Leibniz’s binary ID numbers of I Ching gua are fairly easy to verify, as shown in the Picture 2.

Now, take a look at the circular diagram. Arranged in the Shao Yong sequence, the first 32 gua from Kun Gua (0) to Gou Gua (31) form the right half of the circle, counter-clockwise. The remaining 32 gua are arranged reversely, from Qian Gua (63) to Fu Gua (32) , to form the left half of the circle. So we will get an S-shaped line if we link all the gua from Kun Gua (0) to Bo Bua (1) to .... Gou Gua (31) to Fu Gua (32) to ... to Guai Gua (62) to Qian Gua (63). Actually, the line implies the Taijitu of two fish nestling head to tail against each other. We can also find it in the square diagram. The first 32 gua are arranged in order from the first to the fourth rows from left to right, which form the right half of the circle. The remaining 32 gua are positioned in rows from row 8 to 5 from right to left in an inverse order, or from Qian Gua (63) to Fu Gua (32) to form the left half circle.

We have mentioned in the last issue that Leibniz made a big comprise in his calculation of the sequence of I Ching gua as he wanted to show the perfect symmetry of Shao Yong’s diagrams. In Shao Yong’s method, the top yao was defined as the first yao with the weight of 1. From top to bottom, the weights of each consecutive yao is hence 2, 4, 8, 16, and 32, respectively. So the binary number of Pi Gua
( , 111000) is 7 as its ID number: 0*25 + 0*24 + 0*23 + 1*22 + 1*21 + 1*20 = 7. However, according to the rules of the binary method, the weight of bottom yao should be 1, and the weight should be doubled each consecutive yao up to the top yao 32, in an order from bottom to top. As a result, Pi Gua’s ID number should be 56 as 1*25 + 1*24 + 1*23 + 0*22 + 0*21 + 0*20. So the ID number of Tai Gua (000111) is, indeed, 7.

So the authentic binary method is actually Shao Yong’s method, calculated from the bottom up. So the Bagua sequence should instead be 0-Earth, 1-Thunder, 2-Water, 3-Valley, 4-Mountain, 5-Fire, 6-Wind, and 7-Sky. Sky(7)-Earth(0), Thunder(1)-Wind(6), Water(2)-Fire(5), and Mountain(4)-Valley(3) make up four pairs, with each pair’s ID summing up to 7. The front gua of each pair belongs to the yang sphere and the rear gua belongs to the yin sphere. In order to separate the gua in the yang sphere from the yin sphere and to keep them in a symmetrical manner, it is better to exchange the positions of Mountain (☶) and Valley (☱) as shown in the following:

Therefore, we have a square diagram (Picture 3) arranged in the sequence of ID numbers. As explained before, there are two methods to get the ID number for gua. Besides the binary method shown above, we can use the coordinate (a, b) of a gua in the diagram to calculate its ID number with the formula “ID number = a*8+b.” For example, the coordinate of Jiji Gua is (2,5). So its ID is 2*8+5=21. Tai Gua located at (0,7), so its ID is 7. Dui Gua’s ID is 36 (4*8+4), as its location coordinate is 4,4, and so on.

Although the I Ching hexagrams are drawn flat, usually on a piece of paper, each of the 6 yao of a hexagram represents one dimension in space. As a matter of fact, the 64 hexagrams represent 64 gua vary, and flow in a six-dimensional space. However, it is impossible to illustrate movement in a six dimensional space.

Fortunately, Fu Xi, the legendary framer of the I Ching system, was wise enough to realize the limitations of human race, so he combined two 3-yao gua to make up a 6-yao gua in I Ching. So, to easily understand the location of Bagua in a three-dimensional space, we are able to perceive the location of a 6-yao gua, in a 3-dimensional space within this space.

Picture 4 (a) indicates the location of Bagua in a cube. We can imagine it as a Rubik’s cube of 2x2x2. The four gua shown in Picture 4 (b) are located on the top in the cube while the rest four gua in Picture 4 (c) are situated on the bottom. In I Ching, we call them “Eight Gong” (or Eight Palaces). Within each of the Eight Gong there are eight gua. We may imagine that there is a mini Rubik’s cube of the same arrangement located in each Gong or palace.