broken, or yielding line
solid, or firm lineIn this section I describe the structure of the hexagram and the combinatorics which determine the total number of hexagrams as well as the total number of judgments possible.
Each hexagram in the I Ching consists of six lines. Each line can be either a yielding (yin) line or a firm (yang) line. A yin line is represented by a broken line, and a yang line is represented by a solid line.
The lines are numbered from bottom to top.
If you have only a single line, there are only two possible situations which can be represented, yin or yang.
When two lines exist, there are four combinations of yin and yang, for example:
yin and yin
yin and yang
yang and yin
yang and yang
The order of yin and yang are significant here because each refers to a different line.
With three lines, there are eight possibilities:
yin and yin and yin
yin and yin and yang
yin and yang and yin
yin and yang and yang
yang and yin and yin
yang and yin and yang
yang and yang and yin
yang and yang and yang
This structure, which has three lines, is called a "trigram". The trigram is a fundamental part of the construction of the hexagram. See the references for details on the trigrams. You can see four of the trigrams on the Korean flag.
 |
heaven |
 |
earth |
 |
sun |
 |
moon |
If you look for a pattern in the numbers, you will see that the sequence of numbers is a power of two series. For example,
For 1 line, there are 2 possibilities = 21
For 2 lines, there are 4 possibilities = 22
For 3 lines, there are 8 possibilities = 23
This logic can be extended to calculate how many total hexagrams there are:
For 6 lines, there are 26 possibilities = 64
If you consult the references, you will see that this, indeed, is the case. There are exactly 64 hexagrams.
In addition to the yielding and firm characteristics, each line can be a static line or a moving line. A moving yin line is represented by:
moving yin line
and a moving yang line is represented by:
moving yang line
The moving lines are still the same yielding and firm lines as their static counterparts, however the moving lines are in transition. When a hexagram contains moving lines, it still defines the same hexagram which would result from only static lines. But more judgments are appended to the hexagram when it contains moving lines because it is in transition to a new state.
The transition of a moving line is to its complementary line. For example a moving yin line moves into a yang line. Conversely, a moving yang line moves into a yin line. This motion transforms the hexagram into a different hexagram.
For example, the hexagram
is hexagram 1, the Creative. However, due to the fact that the sixth line is a moving line, the hexagram is transforming into the hexagram:
which is hexagram 43, Breakthrough. This represents a transitional situation, hence judgments have been added to the movement of the sixth line. If a seeker had obtained this hexagram as the result of a query, he would take into account the judgment for hexagram 1, the Creative and also take into account the judgment for a "9 in the sixth place". The 9 is the value which determines a moving yang line. It will be discussed in more detail later. Each moving line in each place has a judgment appended to it by the Duke of Chou. All the judgments for all of the moving lines are taken into account when reading the hexagram.
The final judgment to be taken into account for this hexagram would be the hexagram to which it moves, hexagram 43, Breakthrough. Only the basic judgments apply when reading the second hexagram.
If a hexagram is obtained with only static lines, only the basic judgment of the hexagram is used in answering the query.
So, there are still 64 hexagrams. But each hexagram may contain from zero to eight moving lines in any combination. Since additional judgments are appended if the hexagram contains moving lines, there are more than 64 combinations of judgments.
A judgment is added to the hexagram for each moving line it contains. Therefore each hexagram can have appended to it any combination of an additional six judgments depending upon which lines move. The analysis goes as follows:
If we take movement and stasis into account with the lines, there are really four states possible on the determination of any single line. They are:
So, with only one line determined, there are only the four possible outcomes shown above.
When we have two lines, there are 16 possible outcomes. For example,
static yin + static yin
static yin + static yang
static yang + static yin
static yang + static yang
static yin + moving yin
static yin + moving yang
static yang + moving yin
static yang + moving yang
moving yin + static yin
moving yin + static yang
moving yang + static yin
moving yang + static yang
moving yin + moving yin
moving yin + moving yang
moving yang + moving yin
moving yang + moving yang
While the listing of all possible outcomes is becoming cumbersome, it brings out the fact that the number of combinations is finite and can be listed directly. However, these numbers can also be calculated as we did before. You may note the following pattern in the numbers:
1 line has 4 possibilities = 41
2 lines have 16 possibilities = 42
The pattern continues, and it becomes apparent that the number of judgments which can be determined with the system is a power of 4 series. Hence the total number of judgment outcomes is:
6 lines have 46 possibilities = 4096
Therefore, when taking the moving lines and static lines into account there are 4096 possible outcomes from a specific query.
This analysis has been done partly to help define the construction of the hexagrams and partly to introduce the combinatoric mathematics which will be used in the probability analysis for the oracles.
