Hexagram Navigator | ||
Canonical Taiji Contrast Cluster 1This cluster conforms to the equation o(P) = e(P), but, more importantly: In this table, the hexagrams are related to Taiji diagrams. Each hexagram represents one half of a diagram. Thus, A: is the yin top half of the top diagram and B: is the yang bottom half of the same diagram. Similarly for C and D and the second diagram. The visual similarity between the hexagrams and the Taiji diagrams is quite compelling. However, the algebraic relationships between the symbols are the important aspect. Which leads us to the Taiji Equation: Taiji Dengshi 太極等式 X = e(~Y) So, as shown by the equations to left of the Taiji diagrams, the relation between one half of any given Taiji diagram and the other is X = e(~Y). That is, two aspects change within the diagram: both the polarity and the position. The polarity change is represented by the Boolean complement ~(.) - because this exchanges yin and yang; and the position change by trigram exchange e(.) - this because e(.) is equivalent to a rotation by half, three lines in the case of hexagrams e(P) = P >> 3. Note that although the Taiji diagrams could be rotated to any particular orientation, here I am only interested in the fact that for any given rotation, there is a corresponding rotation 180^{o} away. The relationship between segments in two different Taiji diagrams is described in the equations to the right of the diagrams. Here, only one of polarity or position changes. That is, if position is retained, then polarity changes; however, if polarity is retained, then position must change. For example, A = ~C, showing that if the top part of the first Taiji diagram is yin, the the top part of the second diagram is yang. Similarly, A = e(D) shows that the top part of the first diagram is rotated to the bottom part of the second diagram. So, it is important for the full meaning of the Taiji equation X = e(~Y), that all of the component expressions in the equation have distinct values. That is, e(X) ≠ ~X and X ≠ e(X). The colour coding of the table cells should help make the relationships clearer as we use the various pairs to develop contrasts. So, we can say that, for example:
These examples show how the meanings of a symbol can shift as the point of contrast switches from one symbol to another. Notice that both of the Taiji diagrams presented above are both rotating in the same direction: clockwise. Contrast this with the Taiji diagram shown on the left, which is rotating anticlockwise. Because of the symmetry in the hexagram symbols of this cluster (i.e. P = o(P)), we cannot represent a change in direction of rotation using the algebraic relationships. To discuss this degree of variation fully we need a contrast cluster with eight members. A good example might be this. |