Complementary Pairs

When displayed on the xiantian field, complementary (antipodal) pairs appear separated by 180° of rotation.  Such pairs, insofar as their binary values are considered, are complementary pairs, meaning that one member always has YANG where the other has YIN.  Complementary pairs sum to 65.  In the following animation, xiantian is rotated 45° clockwise.

32 Complementary pairs

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.

Taoist Cosmogony pt. I

 Before all things, there is Tao.  Subjective and potential, limitless and empty, it is the source of all things knowable and unknowable.  Several early verses of Tao Teh Ching speak at length about Tao while conceding that words cannot fully describe it and that names are only words.

The Limitless produced the Great Limit/Great Axis -- 0-dimensional -- inside-outside: the first division

The Great Axis produced the two First Powers, Ch'ien and K'un (Heaven and Earth, YIN and YANG) -- 1-dimensional -- up-down: the second division

The two First Powers interacted to create the 4 xiang, [symbols / seasons / directions / elements / forces / nucleotides / experiential dimensions] -- 2-dimensional -- left-right & front-back: the third division:

3-dimensional -- these three divisions, or axes, define three entangled or looped "plane-spaces" corresponding to the Three Operations of Heaven, Humanity, and Earth.


4-dimensional -- the 4 Symbols each recombine with the Great Axis, expanding along it and extending through space toward the poles, thus creating 8-Partition Place.  The octants of 8-Partition Place are found to correlate mathematically to the 8 Diagrams (Ba Gua).


The above diagram depicts the Gates of Change, YANG and YIN at either pole of the vertical axis.  It also depicts the Four Symbols forming a Great Square (the plane-space shown above).  Observe that the Great Square extends upwards and downwards towards each pole, (visualize infinitely many "layers" of Great Squares congruently 'folded' or 'stacked' along a common vertical axis) thus creating three-dimensional space, or akasha.


These Great Squares provide domains for expression of the Four Symbols and provide the conditions for Consciousness to interact with the field of Experience.

The Canon of Change

King Wen sequence

 Our jaunt through the Taoist version of Creation had a goal: it teaches the basic structure of the Book of Change:
Sixty-four six-line figures, each composed of two three-line figures, positioned one atop the other.
The 8 three-line figures, Ba Gua, or trigrams, enumerate the eight partitions of space.  As they are responsible for creating and supporting space, they can be assumed capable of representing anything that space can contain, thus they are archetypal.  They combine in pairs to account for the time dimension as represented by Change.


Each line can 'hold' one of two possible values, yin or yang, '0' or '1', up or down.  The lines can be understood as representing the Three Operations that produced them: Earth, Mankind, and Heaven.  At top right are presented the 64 Changes in King Wen's ordering.  The Canon of Changes, as it is sometimes called, contains 6 x 64 = 384 lines in total, each bearing a decision on its significance in the context of the hexagram in which it appears and its position within that hexagram, and the trigrams of which hexagram is composed.  This framework provides for a flexible and fertile interpretative system.

Taoist Cosmogony pt III: Juxtaposition

The version of Chinese cosmogony we've been discussing comes from Ta Chuan, the Great Treatise; it is an appendix to the Chinese Book of Changes.


After the Four Symbols (xiang) produce the Eight Diagrams (Ba Gua), the Eight Diagrams interact to produce the Canon of 8 x 8 = 64 Changes (I Ching).  By crossing or combining each of the Eight Diagrams (Ba Gua) with itself and each of the others, all possible combinations of Change are represented.


Recall that Three Operations produce akasha, upon which all matter and phenomena depend.  Given akasha, a world and everything in it can be modeled, but we are left with just a static model -- a dead world.  To create a real world, we require Change.  To account for Change, we require a second model to represent the outcome of the Change.

In another discussion of Change, we suggested that vectors or paths between corners of the Ba Gua hypercube could be used to represent Change.  The option to juxtapose two individual Gua (trigrams: three-line figures) is simply an explicit presentation of the finished state, rather than using a vector to suggest the approached state.  Given the notation in the figure at top, we can represent movement from (1,1,1) --> (0,1,1) with an arrow pointing from the first corner to the second as just shown, or simply enumerate it as (111,011).  This interaction between two of the Eight Diagrams produces Change #9, Small Harvest.  The remaining sixty-three Changes may be likewise produced.


By opposing an inner or initial situation with a second final or outer situation, a Change is produced.  We have now accounted for the three dimensions of space and the fourth temporal dimension (as represented by Change).

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.