Tuesday, September 18, 2012

Axes and Reflections on the 8 x 8

16 axial figures
Ashtapada, the 8 x 8 field, is here populated with xian tian (complementary opposition) hexagram arrangement.  We observe that two pairs of orthogonal axes (opposed at 90 degrees) cross the field.  These axes include HORIZONTAL with VERTICAL, and DIAGONAL with SLANT.  
The 48 remaining cells of the field are effectively partitioned into four groups of twelve contiguous cells by the SLANT and DIAGONAL axes.  The VERTICAL and HORIZONTAL axes further bisect these four groups of 12 cells.
  • HORIZONTAL axis (unmarked) divides the field into an upper half and a lower half
  • VERTICAL axis (unmarked) divides the field into left and right halves
  • DIAGONAL axis is defined by the eight cells that span the corners lower left to upper right, or conversely.  
  • SLANT axis is defined by the eight cells that span the corners lower right to upper left, or conversely.
It should be noted that the figures located on the DIAGONAL and SLANT axes exhibit the associated relationships when they are folded or reflected across each other.  Example: figures (#52,#53) located on the SLANT axis (which crosses the DIAGONAL), are related through transposing and complementing both trigrams.  

48 non-axial figures

Imagine our grid printed on a square piece of paper which is then folded across length and width, and across both diagonals.  An axis is equivalent to any singular fold-line; reflection across an axis results in two cells that "mirror" each other on either side of the fold-line.  If the paper were actually folded as described, the mirrored cells would overlap perfectly.  Now with a working definition of axial reflection over our grid, let's look at relationships among the reflected figures.

  1. Reflection across the HORIZONTAL axis complements a figure's lower trigram, leaving the upper trigram unchanged; e.g.: (#46,#24)**
  2. Reflection across the VERTICAL axis complements a figure's  upper trigram, leaving the lower trigram unchanged; e.g.: (#59,#40)
  3. Reflection across the SLANT axis transposes OR complements both trigrams of a figure; e.g.: (#28,#61)
  4. Reflection across the DIAGONAL axis transposes AND complements both trigrams of a figure; e.g.: (#4,#38)  ** parenthesized numbers indicate the traditional hexagram ordering, located at upper-right corner of cells in white text.
Likewise, paired figures (#11,#12) located on the DIAGONAL axis (which crosses the SLANT), are related through transposing trigrams; alternatively, through complementing trigrams.
Reflection across a single axis is equivalent to rotating the grid in place by 90 degrees; therefore, reflection across two orthogonal axes is equivalent to rotating the grid in place by 180 degrees.  

For any figure selected, crossing two orthogonal axes results in complementing the entire figure.  This fact is intrinsic to the xian tian arrangement whereby pairs of complementary hexagrams are separated on the field by 180 degrees of rotation, or two orthogonal axes.   Since DIAGONAL and SLANT axes cross both HORIZONTAL and DIAGONAL axes, the reflections of the figures on those axes are complementary.

No matter the combination of axial reflections, any selected figure and its several reflections remain confined within the same concentric band of figures on the field.  These are termed xiang probability bands because of their relation to the divinatory probabilities for generating each of the four kinds of lines (xiang) that may appear when consulting the oracle.

XMS (XIANTIAN MAGIC SQUAREelsewhere described in detail) is a magic square that is also a xian tian or complementary opposition arrangement; therefore, the relationships described above likely also hold for XMS.

This diagram embodies many of the same qualities as the 8 x 8 xiantian diagram

xiang probability bands

NB: The two outer bands of the xiang probability square representing static YIN and static YANG contain 28 + 20 = 48 cells, the same number as the non-axial cells Therefore, these cells may stand for static YIN with static YANG, or a stable condition overall.  The same logic represents the axial cells as 12 + 4 = 16, dynamic YANG with dynamic YIN, a changing condition.


  1. This post marks the first time that the notion of the trigram has been observed as a functional construct as opposed to a construct used to analyze or assemble hexagrams.
    To this point, we have made mention of trigrams and their function in the context of Change. Investigating axes of reflection upon the XMS (xian tian magic square) revealed that reflection across the top-bottom or left-right axes of the XMS resulted in a pair of hexagrams that differed only by complementing the top or bottom trigram.

  2. On reflection, it seems clear that the role of the axes in producing the reflections on the 8x8 is utterly dependent on the strict ordering imposed by the xiantian arrangement, since it seems clear that random orderings on the 8x8 could not produce the same results.
    Though we have seen the axes produce trigrams from the hexagrams, the trigrams cannot, on this basis, be considered intrinsic to Change