The Spherical Model of Change

As promised at the end of the previous entry,  here we present a non-arbitrary arrangement of the hexagrams on the sphere.  Our spherical arrangement is based on a construct called "yao-numbers", the existence of which is attested in an appendix of the Book of Changes known as Ta Chuan (The Great Treatise).

Spherical Model
(equatorial view)

Once the hexagrams were sorted by the size of the yao-group in which they fall [1, 6, 15, or 20], it seemed natural to sort them again by their scalar (xiantian) index. Recall that the xiantian or scalar index of an hexagram is the decimal value of an hexagram figure rendered into binary.  Scalar value is a reasonable indicator of an hexagram's magnitude; thus it appears a reasonable choice for placing the hexagrams of a given yao-group (which all lay on a common latitude) at specific meridians.

Spherical model
(depicted with vectors)

Expressed another way, the problem was to determine how to arrange the hexagrams on each latitude of the sphere in such a way that the ordering remained consistent with the complementarity of the 32 antipodal pairs.  Scalar value was used as a proxy for angular displacement (PHI) on a latitude, with larger values corresponding to larger angular measures.

Spherical Model
(polar view)

In the table following, each hexagram is assigned a coordinate pair (theta,phi) on the sphere: THETA is the measure of latitude (declination from the positive vertical) while PHI measures longitude (angular displacement from a given meridian) on a latitude.  The hexagrams are grouped, generally speaking, by yao-number and ordered within a yao-group by scalar value (XT).

As detailed in other entries, the yao-numbers of the paired hexagrams sum to 360.  Additionally, these pairs have XT numbers summing to 65, thus they are complementary pairs.  The spherical model presented here preserves those relationships.
One unexpected outcome of this effort was the discovery of a natural partition of yao-group[20] which, for lack of a self-consistent way to divide it, had been treated only as a unit.  With the aid of the XT index, however, this group of 20 hexagrams falls naturally into halves: one half with XT less than 32; the XT of the other ten hexagrams exceeding 32.  Only yao-group of 20 features this relationship.  Indeed, only this group requires additional means to distinguish pair-mates.  


Whereas they were previously presented as seven groups on discrete latitudes, the 64 hexagrams can now be portrayed as four groups of varying sizes, each group with equal-sized halves as shown in the table.  These eight sections comprise an octo-partition of spherical space, bringing us full-circle (pardon the pun).  

The model is expected to serve as an anti-stereographic projection of the xiantian magic square onto the unit sphere and may prove useful for visualizing projections of XMS subsets onto 3-space.  For example, how do the XMS main diagonals -- or any of the "18 spells" for that matter -- appear when displayed on the sphere?  The projection may also help with transforming the XMS field into a magic 4x4x4 cube (with magic constant 130).


[NB: Consequently, the yao-groups should perhaps be redefined for the sake of clarity in designation:
[2 12 30 20] with modifiers '+' or '-' to indicate latitudes above or below the equator, respectively.  Again, this coincides with the specification of a dodecahedron]