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.

Teutonic Cosmology

Teutonic Cosmology
by M.Alan Kazlev




The pre-Christian Teutonic cosmology of the Yggdrasil Tree defines the various worlds (there being nine in all) on the basis of vertical, existential, and horizontal parameters
(the following account is based on Kvelduf Gundarsson, Teutonic Magic, (Llewellyn, 1990, pp. 1-6)).

The vertical dimension has the world of the Gods (Asgardhr - equivalent to the Kabbalistic Atzilut and Beriah) above, the underworld (Hel, from which presumably we get the Christian term) below, and the Earth plane (Midgardhr) in the middle.  

Copyright Robert N. St.Clair

University of Louisville

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.

In addition to this vertical continuum of being is the horizontal dimension.  This is defined by boundaries.  The ocean separates the human world (Midgardhr) from the outer worlds (Utgardhr) around it, while Midgardhr (literally the "middle enclosure") is itself distinguished from the inner world of the Gods (Asgardhr).  This cosmology is modelled on the neolithic farmstead (gardhr) which divides the inner social space (innangardhs) from the uncontrolled, unknown, dangerous outer space (utangardhs).  Obviously this is in turn symbolic of the Jungian model of the psyche; the safe secure ego or conscious self in the middle, and the vast, dark, unknown unconscious without.

In this cosmology there are four utangardhs (outer worlds) around and beyond the middle or earth world (midgardhr).  Each corresponds to a particular elemental nature, and is inhabited by particular races of beings.


To the east is the world of Jotunheimr, inhabited by various races of giants.  This is characterised by the element air, and so is a frosty rocky world lashed by violent winds and storms.


To the west is the more benevolent world of Vanaheimr, associated with the element water, and inhabited by the Vanir, gods of fertility, riches, and joy.


In the north is Niflheimr, the world of primal ice, and to the south is Muspellheimr, the world of primal fire.  The inhabitants of these worlds would be totally destructive, due to their pure elemental nature, if released into the innangards or inner worlds.

Tree of Life: Nested Octahedron

The model (left) employs the mythos of the Tree of Life. In the model, Mankind and its home planet locate within the octahedral space at the center of the star tetrahedron.

YIN & YANG: the Two First Powers at the poles of the Universal Axis.

4 Symbols (xiang): at the corners of the rectangle, symbolize  seasonality & temporality.

8 trigrams (ba gua): arranged on the star tetrahedron, symbolizing primal forces of Nature.


Hou Tian (familial) arrangement of the Ba Gua

Father                    Mother  
Heaven                   Earth
Sons                       Daughters

1.      Thunder        1.     Wind

2.      Water            2.     Fire

3.      Mountain       3.     Lake

The Two First Powers, YIN and YANG give rise to xiang. Xiang produce the ba gua. Ba gua yield Change; Change enfolds the Universe. This assertion is found in scripture:

Therefore there is in the Changes the Great Primal Beginning. This generates the two primary forces. The two primary forces generate the four images. The four images generate the eight trigrams. The eight trigrams determine good fortune and misfortune. Good fortune and misfortune create the great field of action." (Commentary on I Ching, tr. Wilhelm and Baynes 1967:318-9)

Derivation of the Natal Hexagram

Sample Natal hexagram construction:

	ZODIAC	ELEMENT	SYMBOL
SUN	SCORPIO	WATER	⚎
ASCENDANT	CAPRICORN	EARTH	⚏
MOON	LIBRA	AIR	⚌

An individual desiring to derive their own natal hexagram would simply use their birth horoscope to fill in the middle two columns of the chart.  The contents of the final column are determined by the following schema*, which maps the astrological elements to the Chinese xiang (symbols):
⚌: Dynamic YANG = AIR
⚏: Dynamic YIN = EARTH
⚎: Stable YANG = WATER
⚍: Stable YIN = FIRE


Once the final column is completed, the three symbols are simply copied line-for-line to form an hexagram.

Resulting Natal Hexagram (from sample data above)

⚎       ☶ (MOUNTAIN)
⚏ ==>   =============  ==>   ䷨ #41 (DECREASE)
⚌       ☱ (LAKE)


**Corresponding Tarot trump: XX (JUDGEMENT)

This presentation is derived (with minor modification) from the works of:
Charlie Higgins, “Mensionization Complementation: Construction of an I Ching/Astrological Natal Hexagram” (1997)

**Steve Krakowski's, “An Ancient and Occult Genetic Code” (1996) provides the mapping between Tarot trumps and I Ching hexagrams.

Xiang: The Yarrow Oracle Depicted Graphically

The yarrow oracle method as related through Ta Chuan (the Great Treatise) is attributed to Kung-fu Tze. The querent begins with 50 (fifty) yarrow stalks and manipulates them in a prescribed and probabilistic manner until few remain.  This remainder may take on one of exactly four possible symbolic values, though the actual number of stalks may vary according the the precise form of the ritual.  The symbolic values are sometimes referred to as 'xiang.'  Each manipulation of the stalks produces one line of a six-line figure commonly known as an hexagram.  Thus, six "randomly-generated" xiang produce an hexagram.  Once generated, the querent references the scripture according to that hexagram.

In recent years, capable persons have employed mathematics to reduce the yarrow-stalk oracle to the raw probabilities it produces for each of the four types of lines that may obtain from a divination operation.  The following table is a description of those lines and their relative probability of occurrence in divination:

dynamic yin,   represented by '6' and symbol ⚏ ( 4 in 64 = 0.0625)

dynamic yang, represented by '9' and symbol ⚌ (12 in 64 = 0.1875)

static yang,     represented by '7' and symbol ⚎ (20 in 64 = 0.3125)

static yin,       represented by '8' and symbol ⚍ (28 in 64 = 0.4375)

The graphic depicted here shows the xiantian arrangement overlaid on a tableau of concentric squares, each discretely colored. This presentation is remarkable in that it implicitly encodes the yarrow oracle probabilities; i.e.  each ring comprises a specific proportion of the hexagrams as shown in the table above.

Xiang: Relating Pranayama (breathing meditation) to I Ching

The four expressions below relate to the xiang of the I Ching/Taoist cultural tradition.  The meanings and breath counts are presented.  Circular breathing technique -- in through nose, out through mouth, or conversely -- is recommended.  The circular breathing and varying counts are presumed to assist in bridling mental activity by fixing it to the breath.  This exercise mimics the expansion/contraction cycle of the Universe at large as professed by the Vedic tradition.

(6) INHALATION -- DYNAMIC YIN
(8) (Hold to Fullness) -- STATIC YIN
(9) EXHALATION -- DYNAMIC YANG
(7) (Hold to Emptiness) -- STATIC YANG

The complete cycle is 30 counts, such that one may easily time or measure the duration of a meditation session using the fingers.  A complete minute of 60 seconds could consist of one cycle of circular breathing by inhaling through the nose and out through the mouth, and a reciprocal cycle of inhalation through the mouth and exhaling through the nose.  One may additionally employ mudra, asana, or mantra along with the breath meditation, thus making it a fully-fledged tantric exercise.  

The traditional associations of the first Two Powers are here presented

YIN: capaciousness, struction, form

YANG: creativity, action, function