Yao Numbers and the Tree of Life

The following animation is a graphical depiction the nine species of "Yao Numbers," a concept introduced in Ta Chuan, the Great Treatise.  It displays the nine families in forward and reverse sequence.  It is the so-called Fu Xi (also, xiantian) diagram rotated 45 degrees CW.  We suggest that these nine families are analogous to the "Nine Worlds" surrounding Yggdrasil, the Teutonic Tree of Life.
  

The animation appears to depict a signal initiated at one pole (congruent with the realms called Hel and Asgard in the Teutonic tradition), propagated through the medium, then confirmed by the receiving pole.  Conceptually, it bears  resemblance to a common depiction of Yggdrasil, the Teutonic Tree of Life with its "Nine Homeworlds."  


From the poles  the signal passes through either group of three hexagrams with pairs formed by complementarity or "separated pairs".  These are congruent to the planes called Bright Alfheim & Dark Alfheim in the Teutonic scheme. The  purported power of these realms is to mediate between the worlds of Asgard and Midgard:

There is also a transitional world between Asgardhr and Midgardhr, where the energies of the former are transmitted to the latter world. This realm is called Alfheimr (world of the Elves), and is characterised by the higher aspects of light and air (equivalent perhaps to the occult conception of the Etheric planes). In this world, the highest energies of the human world mix with the lowest energies of the Gods. Beneath Midgardhr (the physical world) is a corresponding region intermediate to Midgardr and Hel, called Svart-Alfheim (world of the black elves). In this region are found the mysteries of earthly manifestation, represented by the dwarves.  

Teutonic Cosmology by M. Alan Kazlev

In most detailed depictions of the Tree of Life, a formation comprising a central node bounded by a group of four nodes (q.v. quincunx) is present.  In this presentation, the largest and central of the five families comprises twenty hexagrams, each with three YIN and three YANG lines, and each having yao-number <180>.  This node is congruent to Teutonic Midgard, the home of Men.


The four groups surrounding the central group have {sizes} and <yao numbers> as shown: ({6} <172>, {12} <176>, {12} <184>, {6} <188>).
The congruence of these groups to the Teutonic tradition is less clear than the others.  The candidates are: Jontunheim, Vanaheim, Niflheim, & Muspelheim.

The following crude table displays the 64 hexagrams and the nine yao-number "families" into which they fall, the quantity of hexagrams per family, as well as yao-numbers and line-analysis (#YIN lines and #YANG lines) for each hexagram.  Wen pairs are shown within parentheses where possible.


0 YANG,6 YIN <144> { 1} [2]*
2 YANG,4 YIN <168> { 3} [27 29 62]*
5 YANG,1 YIN <172> { 6} [(9 10) (13 14) (43 44)]
4 YANG,2 YIN <176> {12} [(5 6) (25 26) (33 34) (37 38) (49 50) (57 58)]
3 YANG,3 YIN <180> {20} [(11 12) (17 18) (21 22) (31 32) (41 42) (47 48) (53 54) (55 56) (59 60) (63 64)]
2 YANG,4 YIN <184> {12} [(3 4) (19 20) (35 36) (39 40) (45 46) (51 52)]
1 YANG,5 YIN <188> { 6} [(7 8) (15 16) (23 24)]
4 YANG,2 YIN <192> { 3} [28 30 61]*
6 YANG,0 YIN <216> { 1} [1]*


*indicates Wen pairs formed via complementary opposition; Wen pairs of this nature fall into separate families.

Mercury's Arrangement Hosts the 64 Gua

Mercury's magic square

Mercury's arrangement of the 64 Hexagrams

The numbers associated with Mercury are 8, 64, 260, and 2080. This is because:

• Each row and column and major diagonal of the magic square contains eight numbers.
• The square contains 64 numbers total, ranging from 1 to 64.
• Each row, column and diagonal adds up to 260.
• All of the numbers in the square add up to 2080

The image at right is a depiction of the 64 gua or hexagrams overlaid on Mercury's magic square.  The individual cells are indexed by their numerical equivalents at lower left, and by traditional King Wen indices at upper-right.  

As consequence of the above constraints, pairs separated by 180 degrees of rotation (so-called antipodal pairs) of the Mercury arrangement are complementary opposites.  That is, the numerical equivalents of the pair sum to 65).  

Mercury's magic square is a form of xiantian (complementary opposition) arrangement, but the constraints on Mercury's xiantian are stronger.  The binary values of its rows, columns, and major diagonals  invariably equal sum to 260 where those of the traditional xiantian do not.  This fact provides an common interface between the Chinese I Ching and the Mayan Tzolk'in.

Introduction to Yao-numbers

ABSTRACT
Yao-numbers or "stick-numbers" are the product of a mathematical transformation of the I Ching's 32 hexagram pairs as presented by King Wen.  In the case of twenty-eight pairs of hexagrams, the mates of a pair differ only by 180 degrees of rotation.  The last four pairs are complementary opposites per Fu Xi's hexagram and trigram arrangements.  


Yao-numbers seem to call attention to the four pairs of complementary opposites: pair-mates generally have identical yao-numbers except in four cases: (01 02) (27 28) (29 30) (61 62), where the mates have "unbalanced" yao-numbers.  Again, these four pairs bear the distinction of having been formed from complementary opposition, as opposed to pair formation from figurative inversion (180 degrees of rotation) in the remaining 28 pairs of hexagrams.


The transformation procedure (described below) produces nine discrete yao-numbers from the 32 Wen pairs of hexagrams. The following crude table displays the 64 hexagrams and the yao-number "families" into which they fall, the quantity of hexagrams per family, as well as yao-numbers and algebraic line-analysis (#YIN lines and #YANG lines) for each hexagram.


0 YANG,6 YIN (144) { 1} [2]
2 YANG,4 YIN (168) { 3} [27 29 62]
5 YANG,1 YIN (172) { 6} [(9 10) (13 14) (43 44)]
4 YANG,2 YIN (176) {12} [(5 6) (25 26) (33 34) (37 38) (49 50) (57 58)]
3 YANG,3 YIN (180) {20} [(11 12) (17 18) (21 22) (31 32) (41 42) (47 48) (53 54) (55 56) (59 60) (63 64)]
2 YANG,4 YIN (184) {12} [(3 4) (19 20) (35 36) (39 40) (45 46) (51 52)]
1 YANG,5 YIN (188) { 6} [(7 8) (15 16) (23 24)]
4 YANG,2 YIN (192) { 3} [28 30 61]
6 YANG,0 YIN (216) { 1} [1]

INTRODUCTION
In Taoist cosmology, the SINGULARITY splits into YIN and YANG; YIN and YANG then combine to produce Four Images, xiang: Heaven (9), Earth (6), Fire (8), and Water (7).  For use in divination ritual, the Taoist tradition assigned "ritual numbers" to the xiang as detailed in parentheses.  Also notable is the traditional association between xiang and the four cardinal directions.


Note: xiang are subtly distinct from the trigrams with the same name; the xiang emerge into being before the bagua (trigrams).  


The transformation that produces yao-numbers is described as follows:
Given the King Wen sequence of thirty-two hexagram pairs as input, employ the xiang bi-grams to measure any changes that occur within a given Wen pair at any of the six line positions.  Four possible measurement outcomes exist:


⚌ EARTH: YIN alternating to YANG (6), 
⚏ SKY: YANG alternating to YIN (9), 
⚍ FIRE: stable YIN (8), or 
⚎ WATER: stable YANG (7)


Note: the xiang bi-grams are represented using Unicode and may not render properly on browsers or systems using UTF-8 or other character encoding


We may contextualize and clarify the transformation procedure by interpreting Wen pairs as though they were products of ritual divination.  For example, We can pretend that, using yarrow stalks, we cast hexagram #1 with six moving lines; this normally indicates a situation initially characterized by hexagram #1 subsequently changing to one represented by hexagram #2.   We now encode the differences between #1 and #2 using xiang.  


Line-by-line comparison of hexagram #2 to #1, beginning at position 1 through position 6 shows YANG changing to YIN at all six places.

䷀ ==> ䷁

This implies moving YANG (ritual number 9).  We therefore encode Wen pair (1 2) as [999999].  


If instead we had cast #2 changing at all six places to #1, we would use moving YIN (ritual number 6), to encode Wen anti-pair (2 1) as [666666].  

䷁ ==> ䷀  

The significance of anti-pairs is discussed elsewhere, but as change is represented by movement, it seems to suggest reversal of direction or perhaps reversal of time.


The encoding for Wen pair (3 4) is [968896]; its anti-pair encodes as [698869].  

䷂ ==> ䷃

At positions 3 and 4 both hexagrams indicate YIN -- a non-changing condition -- encoded by stable YIN (8).


The encoding for Wen pair (5 6) is [979676]; [676979] is the encoding for its anti-pair (6 5).  

䷄ ==> ䷅

At positions 2 and 5, both hexagrams indicate YANG.  This, again, is a non-changing condition encoded by stable YANG (7).


Iterating this transformation procedure over the 32 Wen pairs produces 32 vectors of six xiang.  We call them 'vectors' because they depict ordered arrays of data.  The relevant section of Ta Chuan, the Great Treatise (Pt.1, Ch. 9, Sec. 4) provides three clues for interpreting these vectors: 

• Hexagram #1 has stick-number 216, 
• Hexagram #2 has stick number 144, 
• All 64 stick-numbers summed yield 11520, "the number of the myriad things."  

The rule for obtaining the correct stick-numbers from the vectors involves summing the ritual numbers of the xiang for a given vector and multiplying by four.  The derivation of this rule is discussed in the translators' notes to the Great Treatise.  


The divination ritual involves manipulating 49 yarrow stalks to produce a remainder of stalks, then dividing that remainder by four to produce a ritual number (6|7|8|9). Each ritual number corresponds to one of the four xiang (as described in the INTRODUCTION).  Each xiang produced by ritual divination represents a single hexagram line, which may be dynamic or static.  This procedure is conducted six times to produce a complete hexagram.  


We may infer from this that the rule for obtaining the correct yao-numbers involves the inverse operation to dividing by four, i.e., multiplying by four.


As shown in the following illustration, when all 32 pairs of vectors are thus treated, the combined stick-number sum calculates to 11,520 as stipulated in the Great Treatise.

So, what to do with this information?  What is its use?  What does it tell us?  A simple categorization of the 64 stick-numbers shows them falling into 9 discrete "families" (numbered 1 through 9, left to right) with all members of a family having a particular stick-number.  Additionally, Wen pairs tend to fall together within a given family -- excepting the two triplet groups of hexagrams (#2 & #8), and the two singleton hexagram groups (#1 & #9), where the four Wen pairs are separated from each other.

We find there is very strong correlation/correspondence between these nine families and the seven groups (below) produced by algebraically sorting the hexagrams; that is, according to the quantity of YIN and YANG lines in a given hexagram. Alternating between views of the two schema, we observe that the triplet groups move from their positions next to the poles to a position adjacent to the central group of twenty figures.  Both groups of twelve temporarily gain three, bringing their totals to fifteen.

We also observe that the four "unbalanced" hexagram pairs: (01,02) (27,28) (29,30) (61,62) [families #1, #2, #8, & #9]  all share a peculiar trait -- none of them has a proper Wen inverse as do the other twenty-eight pairs of hexagrams.   The pairs formed from these eight hexagrams are not figurative inverses, but complementary opposites.  Complementary opposition is the basis of the xiantian (Earlier Heaven) trigram arrangement, and the square (8x8) hexagram arrangement, both attributed to Fu Xi.


CONCLUSION
To recap, twenty-eight pairs of hexagrams are formed by 180 degrees of rotation per King Wen's ordering.  The remaining four pairs are complementary opposites as per Fu Xi's arrangement.  Pair-mates generally have identical stick-numbers excepting four cases: (01,02) (27,28) (29,30) (61,62). 


The stick-numbers (calculated above) appear to highlight the complementary opposition of these four pairs against a background of twenty-eight pairs formed from 180 degrees of rotation ("figure inversion").

Although we observe that pairings based on complementary opposition, it is currently unclear how output of the transformation procedure significantly correlates to output of the algebraic sort.

Lexicon:

Position refers to lines of an hexagram, numbered 1 (bottom-most), to 6 (top-most).  

Movement refers to change between YIN and YANG at any of the positions.  

Movement can be dynamic (or alternating) when members of a Wen pair disagree at any of the six positions.  Dynamic also refers to moving lines produced from ritual divination (YANG <==> YIN).

Movement can also be static, as when both pair-members agree at a given position, or when no moving lines appear in a divination.

Wheel Cross: Navigation Tool of Antiquity

Its overall appearance is reminiscent of the comic book character "X-Men" insignia (a cross within a circle); less recently, of a religious symbol.  Wikipedia's article on religious symbols provided the first pointer (below right) but symbols.com provided clear confirmation of this symbol's place within our overall field of investigation:

The wheel cross, sun cross, Odin's cross or Woden's cross. Nordic Odin and Teutonic Wotan or Woden was the supreme god of the Nordic religion before Christianity. Odin was the god of art, culture, warfare, and the dead; depicted as an old, one-eyed man with two ravens as his intelligence agents and messengers.  

Sun Cross 

 The structure  is one of the first non-pictorial graphs to appear when humankind was on the threshold of the Bronze Age. It is common on rock carvings. It appears in ancient Egypt, China, pre-Columbian America, and the Near East. From the facts available it seems as if  is associated with the wheel, not so much with its invention as with its revolutionary effect on the existing society. In ancient China this sign was associated with thunder, power, energy, head, and respect. 

This figure is also known as the gamma cross, as the Greek letter 'gamma' appears four times within the circle, radially rotated.  It is shown to play a prominent role in the Norse, Egyptian, Chinese, and pre-Columbian culture.  Perhaps excepting the Egyptian, this observation is consistent with our thesis that the religion of these cultures is based on representations the Tree of Life. 

The sun cross embodies several notable features: the circle, the right angle, the line segment, and the point.  These figures are all associated with measurement, surveying, mathematics, Platonic geometry, and Freemasonry.  Another feature is the fact that the sun cross can be reassembled into the Ankh cross, by remaking the Greek cross into a Tau cross, and placing the circle atop.

Crichton Miller's research on the Celtic cross convincingly demonstrates its use in antiquity as an instrument of spherical/nautical navigation based on the impeccably-supported theory that working models of the true Celtic cross embody a weighted rotating wheel.  as the circular portion. Moreover, the Celtic cross was used as a tool of determining longitude and latitude.  That Wotan's wheel cross and the Celtic cross bear both a figurative and literal resemblance is worthy of remark.