Sung: Part II - Their Exibits

Chapter II - Mathematical Exhibits - Sixtyfour Hexagrams

This chapter extends the representations developed for the trigrams in Chapter I of Sung to hexagrams. The first extension is to take the polynomial given as Equation 1 and extend it to cover the space representing hexagrams (p28):
Equation 2: The Hexagram Space
(A+B)6 = A6+6A5B+15A4B2+20A3B3+15A2B4+6AB5+B6
Again "A" represents yang and "B" yin. In a similar manner to Equation 1 we can read this as saying (from left to right) that there is one hexagram with six yang lines, six hexagrams with five yang lines and one yin line, fifteen hexagrams with four yang lines and two yin lines, and so on.

As we might expect from the previous development from equations to graphical exhibits (see the Figure from p18) there is also a graphical representation of this equation. Sung presents this on p30 and it is reproduced below.

From Z.D. Sung, p30

This figure again clearly shows the stratification of the gua according to the amount of yang energy that they contain.

Then, on pp36-47 the 3-dimensional octant model for trigrams (see the Figure from p26) is extended to the hexagrams. The author does this by using a subdivided cube: the initial division of the large cube into octants gives the inner trigram, then each octant is itself divided into octants to give the outer trigram. This gives a cube with four layers, with each layer holding a four by four array of hexagrams.

Sung presents a number of different diagrams, trying to clarify the construction; for example, he gives one figure for each of the major octants of the cube. Here I shall give a single example, for the octant with the inner trigram of the Creative.

From Z.D. Sung, p40

This is an interesting model but there is no discussion - just a bare presentation. In order to clarify the exact structure of Sung's cubical arrangement I shall present the layers of the cube in a tabular format. Firstly note that the cube is composed of four layers, with each layer holding a four by four array of hexagrams.

Bottom Layer       First Layer
Second Layer Top Layer
Sung's cubical arrangement of the hexagrams

Consider the following points:

  • Each of the corners of the cube carries a hexagram composed of a doubled trigram. These are the hexagrams in blue squares in the table.
  • Note also that hexagrams on diagonally opposite corners are binary opposites.
  • In general, to find the binary opposite of a hexagram, look to its diagonal partner within the structure.
  • The eight hexagrams that form the heart of the cube (coloured purple in the table) are all of the hexagrams that can be formed by combining complementary trigrams.

No doubt there are many more relationships encoded in this structure. However, although Sung later mentions the first of the relationships listed above, these interesting structural properties are not explored at all.

There are three further figures in this chapter, one called "the cube of three quadrinomial dimensions" and two called "the product of a binomial cube by itself" (pp48-50) that I cannot really interpret and Sung's description is too terse to enable me (so far) to unravel his intention.

Finally, it is interesting to note that Russell [Rus67, pp26-27] seems to be describing exactly the same construction as this cube.