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]

Progressing Toward a Spherical Model of Change

Yao-numbers depicted as radial vectors

Having determined a set of coordinates for each hexagram allows us to  generate 32 pairs of complementary vectors with which vector-analysis may be performed.  
The spherical model also enables us to derive insight from quantum mechanical systems by using the Bloch/Poincare model of the qubit as an entry-point.
The basis vectors in our model are hexagrams #1 and #2, found at the "poles" of our model; the remaining 31 pairs of  hexagram figures are said to derive from these.  Prior to performing an instance of divination, the answer to the posed question is like a superposition of the basis vectors, entailing 64 x 64 = 4,096 possible results.  Once the oracle is consulted, the superposition collapses to a single result.  



Future entries on this topic intend to determine the meridian lines in order to produce a model that fits the 64 hexagrams onto the unit sphere in a non-arbitrary fashion.  That is, the particular assignment of hexagrams to points on the surface of the sphere will be based a scheme that preserves the integrity of known relationships.

Subspatial Scaffolding, Aetheric Architecture

This graphic attempt to convey the notions of subspace, or the architecture of volumetric space.  
It is easy to take space for granted, like we assume a fish takes water for granted.  Indeed, were there no water, there'd be no fish -- likewise for space and ourselves.  
It's not so easy to wrap one's mind around non-volumetric reality, though some are familiar with 'Flatland,' a fictional 2-D world.  Space normally provides an habitat for objects to occupy, but under certain conditions, intense gravitation does not allow space to exist as we understand it.  Under those conditions, space is presumed to collapse.  The Big Bang cosmological theory proposes that the universe as we know it evolved from a singularity; otherwise there was no space nor time.  From this we may infer that some 'thing' or force exists that gives space the rigidity to resist gravitationally-induced collapse.  Some physicists have proposed that hypothesized (but undetected) "dark energy" fill this function.  To this point, physicist David Bohm once remarked in The Holographic Universe that "every cubic centimeter of empty space contains more energy than the total energy of all the matter in the known universe."   We propose that this role is fulfilled by an architecture, hypothesized in the above graphic.

In the diagram, mutually complementary elements intersect on orthogonal axes.  This arrangement is presumed to produce a framework that 'erects' space and maintains its volume, similar to the way that a balloon maintains its shape from the force exerted by the gas within.
Complementarity is depicted via the axes of 'opposing' color pairs: red and blue, yellow and green.  These opposing pairs span opposite corners of the bounding box. 
The nexus of the four axes at the center of the figure suggests a combination of the four axes that intersect there.  This is perhaps where the proposed structive force emerges.
We note that the central nexus is adjoined by the apexes of six square pyramids, the bases of which comprise the  faces of the bounding box and its enclosed cube.

11,520: "The Number of All Things"

This entry treats the connection between the 8x8 grid and 11,520, the "number of all things," attested in Ta Chuan (the "Great Treatise").  The 64 hexagram are commonly depicted on an 8 x 8 grid, known in Hindu culture as ashtapada. Its metric (generic term for measure) is 2080, and is thus related to 11,520:

1) Begin by enumerating its cells beginning with 1, through 64, noting that the cells sum to 2080.

2) Arrange the numbered cells such that the numbers within each row, column, and major diagonal sum to 260. 
Under this arrangement, the 18 columns, rows, and diagonals sum to 4680 = 18 * 260 = 13 * 360 = 18 * 13 * 20.  This number seems to link the Mayan 360-day tun calendar to the Mayan 260-day tzolkin calendar.  

3) Convert the numbers into hexagrams (6-line binary figures) , the least-valued hexagram valued having the value one (1).

4) Transform each figure into a yao-number by substituting its lines with the corresponding divination ritual numbers: '6' for YIN lines and '9' for YANG lines, summing the substitutions, then scaling each sum by 4.  This step is derived from instructions given in Ta Chuan, and has the effect of:

• "flattening" the hexagram (like a logarithm), which may be regarded as a "stack" of binary exponents 
• shifting the range of values from a continuum of [1..64] to [144..216] in discrete intervals of 12 

The 18 rows, columns, and diagonals noted above now sum to 1440, and produce a grand sum of 18 * 1440 = 25,920, equal to the duration of a precession cycle.

5) Finally, the hexagrams and their associated yao-numbers from steps #3 and #4 are sorted to produce seven groups with the following membership and distribution:

The 64 resulting yao-numbers are summed to produce 11,520, the "number of all things."

XMS and Genetic Code

German medical scientist Dr. Martin Schönberger is attributed with the initial observation that the 64 hexagrams of the genetic code are analogous to the 64 codons of the DNA genetic code.  His book, I Ching and the Genetic Code: The Hidden Key to Life, is regarded as the inaugural work on this topic.
 Other authors have convincingly discussed the similarity between the I Ching and the genetic code, including Steve Krakowski, whose work borrowed from Schönberger's, as well as Mark White M.D. Both these authors' work is cited by the current author.
Krakowski's work seeks to integrate Hebrew language, 22 Tarot trumps, genetic code, and I Ching.  Dr. White's introduction of "dodecahedral language" brings the discussion squarely into the realm of tangible objects by mapping the 64 hexagrams onto the twelve-faceted, twenty-verticed Platonic solid lovingly-called a "12-tope." 

xiantian magic square

The figure at right poses a comparison between the XMS (xiantian magic square, at right), and the tzolkin calendar (lower right) of the Maya. The author holds that there is a deep connection between these two objects, and uses math and art in his work to promote said thesis.

The XMS is a particular arrangement of the 64 hexagrams on the 8x8 grid whereby any hexagram is mated to its complementary opposite located at 180 degrees of rotation.  Each hexagram on the grid is both indexed, and assigned a scalar value; the scalar values of complementary pairs sum to 65.  This relationship is also proven by the fact that when the hexagrams are rendered into their numerical equivalents, all rows, columns, and main diagonals sum to 4 * 65 = 260.  The latter number is emblematic of the tzolkin calendar of the Maya people.

Arguelles and "His" Tzolkin

The 260-day tzolkin (at right) is displayed as a 13x20 grid with a highlighted subset of 52 days identified by Tony Shearer as galactic "portal days" or by Jose Arguelles as the "Loom of the Maya."  [The author is seeking/awaiting an detailed account of their derivation].  Arguelles' greatest achievement, arguably, was the popularization of Mayan calendrics.  This was also his greatest folly as well, since his efforts -- even post-mortem -- have taken on the trappings of new-age religion.  Nonetheless, Arguelles was possessed of a peculiar spiritual intuition which makes the current author reluctant to throw out the bathwater for fear that the baby may still be in it.  References to the Loom of the Maya viz. tzolk'in often involves references to DNA or the genetic code.  The pattern formed by the portal days is held by some to resemble the helical shape of DNA.  The following passage is from a site that promotes Arguelles' work:

This form of the Tzolkin as the "Harmonic Module," (shown above), inclusive of the 52 shaded squares which form what is called the "Loom of the Maya," is based on Dr. Jose Arguelles' presentation in The Mayan Factor, and is distinct from the form of the Tzolkin as taught and followed by the Quiche Maya of Guatemala. For instance, the 52-unit loom is a bi-lateral symmetry pattern which reflects the basic pattern of our DNA double helix, and was passed down from a secret lineage of Yucatec Mayan shamans, received and revealed by the works of Dr. Jose Arguelles. Integrating the galactic code of light into the genetic code of life, this "portal" formation is a resonant structure linked to the activation of our full DNA potential. Find out when these galactic activation portal days occur, and receive indepth descriptions of the 13 Tones of Creation and the 20 Solar Seals by utilizing the 13-Moon Natural Time Calendar.

So, while the Arguelles camp appears to recognize a connection between time and inner space (body its subtler structures), no evidences are offered to ground its bold assertions.

DETAILING the XMS and SUBSTANTIATING its RELATION to GENETIC CODE

The colored partitions of the XMS grid originate from a simple mathematical transform of the hexagrams that assigns a yao-number to each.  [The origin of the concept of yao-numbers is discussed in greater detail elsewhere; suffices it to say, they were not invented by the author.]  

Yao-groups and Yao-numbers

Yao-numbers range from 144 (hexagram #2, all YIN) to 216 (hexagram #1, all YANG), with intervals of 12 between the seven groupings.  The midpoint of the yao-number scale is found at 180.    
Yao-numbers measure the YIN- or YANG-ness of an hexagram on a continuum.  Thus, the 22 YIN-dominant hexagrams (having yao-numbers less than 180) are red-colored, while the 22 YANG-dominant hexagrams (having yao-numbers greater than 180) are blue-colored.  The remaining 20 hexagrams that are balanced in terms of YIN and YANG (yao-number 180) are white-colored; they also divide the grid in half.

While the similarity of the figures formed by the 20 white-colored cells and that of the 52 portal days forming the Loom of the Maya is superficial and debatable, what is less-debatable is that the 20 white-colored cells partition the grid into 22 pairs, curiously identical to the number of human autosomes, which are indistinguishable by sex.  The uncounted (23rd) pair of human chromosomes is responsible for sex-differentiation, and is necessarily different between males and females.  
The white-colored group of 20 hexagrams is fairly representative of the 20 essential amino acids of the human genome.

Thus, in a single diagram we have represented salient characteristics of the genetic code:

22 yang hexagrams <==> 22 chromosomes contributed by the male
22 yin hexagrams <==> 22 chromosomes contributed by the female
20 neutral hexagrams <==> 20 essential amino acids
An admittedly-vague helical or chromosomal shape

The analogy seems fairly solid so far; Argulles, we're not done with you just yet.