Tuesday, October 25, 2011

"Arrangements and Relations"

(Wen pairs presented on a square grid)
Wen's pairs are well-ordered, but they lack a well-defined spatial arrangement

4 homogenous pairs of hexagrams (formed from identical trigrams) related through the complementarity relation

28 pairs of heterogenous hexagrams (formed from dissimilar trigrams) related through the figurative inversion relation

Xiantian arrangements are spatially defined with a clear pairing/ordering syntax

xiantian magic square
traditional xiantian arrangement

4 homogenous pairs of hexagrams form a diameter

28 hexagrams gird either side of this diameter
internal view of 4 x 4 x 4 hypercube

The hypercubic hexagram arrangement is well-defined but lacks ordering or pairing syntax entirely.

4 pairs of hexagrams comprise a cubic "core"

28 pairs of hexagrams form a "hull" around the core

unit cube 
Here, we use the unit cube to order the tesseract (hypercube).  Each of the unit cube's eight vertices (representing the trigrams) has seven outbound paths to each of the other vertices, totaling 56 paths.  Half of these paths are duplicates, in the sense that the paths connect the same pair of vertices.  Thus, the 56 paths comprise 28 pairs of heterogenous* hexagrams.  An outbound path and its reversal constitute a Wen Pair.
Each vertex can also loop back to itself.  These 8 loops constitute 4 pairs of homogenous* hexagrams.  Loops at antipodal corners (e.g. (0,1,1) and (1,0,0) ) also designate Wen pairs.

1 comment:

  1. Arrangement of the 64 hexagrams on the ashtapada grid naturally partitions them into 4 groups with groups-sizes of [4|12|20|28]. As explained elsewhere, these numbers (expressed as fractions of sixty-four) express divinatory probabilities used in the oracular ritual. As such, they depict platonic symbols known as xiang. Xiang express four primal natures*: dynamic YIN, dynamic YANG, static YANG, and static YIN (respecting the four groups listed above).
    What we find is that the analysis presented on this page repeatedly produces depictions of dynamic YIN and static YIN. The author is intent on locating depictions of the other two xiang.

    *static & dynamic vs. active & structive