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

Modeling Tzolk'in

Tzolk'in is fairly well-known as the sacred calendar of pre-Columbian Maya people.  Its significance to that civilization has been treated by several prominent authors (q.v. Arguelles, Calleman, and Jenkins) in recent years.  It comprises 13 x 20 = 260 days, and can be used to represent many cycles of time used by the historical Maya, and to date.  Several additional factorizations of the Tzolk'in harmonic remain, though it is unclear what terrestrial, astronomical, or physiological cycles (if any) correspond to those.  Its better-known cycles include:

• Five 52-day "seasons"
• Four 65-day "seasons"
• Thirteen 20-day periods called uinal
• Twenty 13-day periods called trecena

If one should desire to begin observing and studying Tzolk'in and related cycles and how they play-out in one's life, it would prove useful to have a model handy, and if we wish to model the Tzolk'in, which everyday objects might be used?  

In the West, the number thirteen is often counted among the casualties of the marginalization of the Divine Feminine.  This topic has been treated by other authors in greater detail and quality, and will not be repeated here at any length.  A few examples are here provided: "unlucky 13,"  Friday the 13^th  (day named for Norse goddess Freya), and the common absence of 13^th floors in buildings.  What this has done with respect to Tzolk'in, is to make objects embodying harmonics of thirteen fairly uncommon, compared to say, the number twelve.  


Returning to the opening topic, studying Tzolk'in with any serious intent might be made much more convenient if objects embodying harmonics of thirteen were more commonplace.  Casual investigation does produce a few familiar objects that loosely embody the 13:20 Tzolk'in harmonic.  For example, there are 4 x 13 = 52 weeks in the Julian calendar.  A regular deck of playing cards (sans jokers) contains 4 x 13 = 52 cards.  Tarot decks, from which playing cards are derived, comprise 6 x 13 = 78 cards.  Lastly, our very own Latin alphabet has 2 x 13 = 26 characters.  With adaptation and ingenuity, any of these objects (in theory) could be used as the basis for our model.

Tzolk'in model (closed)

A 3 x 3 x 3 arrangement of cubes is composed of 27 individual cubes.  Such an arrangement may be termed a 'hypercube' because it is a cubic shape formed from cubes.  This cube-within-cube fractal is akin to a dimension within (or beyond) the three  spatial dimensions. This formation is composed of three horizontal layers, each comprising three rows and three columns of cubes.  Removing the central cube yields a 3-dimensional formation of 26 cubes -- the sought Tzolk'in model.  Such a representation is arguably superior to one we might design from the objects listed above because it is markedly tangible, something we can touch and manipulate in various ways, a quality lacked by more conceptual representations.

Removal of the central cube is no arbitrary contrivance; it symbolizes the establishment of akasha, a concept intimately related to the notion of 'space', which is required for the existence of physical objects.  Consciousness, then, is fairly equivalent to space since consciousness is similarly required of the existence of mental objects.  


Of what use is any calendar without people to schedule their days and lives by means of it?  This 'empty' central position is thus required for the existence of the observer to 'mind' (attend to) the calendar and its cycles.  Additional support for this hypothesis is provided in Appendix I.

Tzolk'in is used in conjunction with the agricultural cycle. The Tzolk'in number 260 is alleged to be closely linked to human biology.  The number of discrete cell types in the human body is estimated at 260.  The average period of human gestation is estimated at 266 days.  The harmonic numbers 13 and 20 are said to correspond to the thirteen major joints of the human body (ankles, knees, hips, shoulders, elbows, wrists, and neck); and the twenty digits (fingers and toes).


The number 260 is also said to be related to prominent astronomical cycles, including the precessional cycle of ~26,000 years.  The pre-Columbian Maya are also said to have predicted eclipses by means of Tzolk'in.  The motion of planet Venus, well-known and highly-regarded by the Maya, was tracked by means of Tzolk'in.  

Physical construction of the Tzolk'in model quickly highlighted a practical issue: with a 'hole' at the center of the structure, the center cube of the crown layer lacks support to keep it in place.  A simple solution was to use adhesive to fix the cubes in place.  This solves the problem, but limits our ability to examine and manipulate the model, hence our model's utility is compromised.

The search for solutions to this problem revealed that subsets of the 26 cubes might be formed into fixed shapes that would not only provide a stable structure to surround the space at the center, it would also reduce the overall number of pieces required for assembly, thus simplifying the model.  It was also determined that while there are many ways to group the 26 cubes into fixed shapes to form a model that provides integral support for the space at its center, not every assembly is equally desirable.  The fixed shapes chosen for groupings should not be arbitrary; rather, they should meaningfully reflect sub-cycles of Tzolk'in.

Reasoning from the basis of 2 x 13 = 26 in composing our Tzolk'in model, it seems appropriate that the model would embody bilateral symmetry just as does the human body to which the it is said to relate. If we appeal to the use of Tzolk'in as a time-keeping device, it seems equally reasonable to consider the division of night and day as another basis for desiring symmetry in its formation.  Additionally, by employing symmetry in the design of the model, we effectively halve the amount of work required, since one half will mate the other.  
In sum, we are seeking to represent the number 13 with blocks in such a way that two such representations will produce a 3 x 3 x 3 formation with a space (symbolizing the observer) at the center. 

Since the base and crown layers of the Tzolk'in model each comprise nine cubes arranged in a square formation, nine cubes might likewise form the basis for each half of the Tzolk'in model.  Four additional cubes could then be placed atop these nine, while still allowing for the observer's position at center.  These restrictions on the assembly of the Tzolk'in model greatly limit the number of possible constructions.  For the sake of brevity and readability, this paper will not detail each the various means of constructing the model, but will instead concentrate on one particular construction that is presumed to obey each of the outlined restrictions while producing a useful model for studying Tzolk'in.

In a related paper, significant correspondences between features of I Ching (the Chinese Book of Changes) and the Mesoamerican Tzolk'in were detailed.  That theme is continued here.  One mentionable correspondence between the pre-Columbian Maya and the ancient Chinese regards veneration of the turtle or tortoise; in particular, the oracular use of tortoise shells.  References to the tortoise can be found in I Ching (hexagram 27, line 1; hexagram 41, line 5, and hexagram 42, line 2), reinforcing the claim of reverence paid to these creatures by the ancient Chinese.

In antiquity, tortoise shells were used to perform divinations.  The precise means by which this was done has apparently been lost to time, but historical records indicate that the later yarrow-stalk oracle was a great technological improvement over the elder tortoise-shell oracle.  We are given to know that the shells were prepared by first scribing them, then subjecting them to heat (as by placing them in fire).  The resulting cracks in the shell were then read by the diviner, who contextualized the reading through the question posed by the inquirer.  As with I Ching and Tzolk'in, the connection between tortoise shells and Tzolk'in is less than obvious.  Provided an illustration, however, we may begin to intuit the link.

Overhead view of live tortoise

The body of the tortoise shell (right) appears as a dome formed from thirteen fused scutes (plates), ringed by a number of smaller scutes.  Thirteen plates constitute the domed portion of the shell.  The importance of the number thirteen to the historical Maya has already been demonstrated here.  It was similarly described in a paper describing the connection between I Ching and Tzolk'in.

The correspondence continues: the thirteen plates of a tortoise's shell are arranged in a specific pattern: Five plates are centrally- and vertically-arrayed; these are braced on either side by a vertical array of four plates.  This 4-5-4 pattern is also present in our design of our Tzolk'in model. 

The halves of the Tzolk'in model are thus formed from three shapes; the complete model totals six pieces.  For each half, two shapes are composed of four blocks each, the remaining shape comprises five blocks, for a total of thirteen blocks per half of the model.  The halves of the Tzolk'in model are constructed symmetrically, but not identically; rather, they are anti-symmetric, or mirror-images, of one another. The following picture illustrates this.

Each five-block shape constitutes the majority of the base and crown of the model. To assemble the model, the two five-block shapes are laid flat and non-congruently, or with opposing “handedness.” The four-block pieces are then made to stand upright in the empty spaces of the former, within the 3x3 "footprint" established by the five-block shape.  The graphics in the appendix demonstrate the assembly in greater detail.

Lastly, support for the Tzolk'in model may be found in the original literature with which it is presumed to agree.  I Ching makes specific reference the tortoise shell in two closely-related hexagrams: line 5 of hexagram #41 ('Decrease') and line 2 of hexagram #42 ('Increase').  By 'closely-related' is meant that these two hexagrams comprise a pair in the traditional King Wen sequence.  Such pairs are figurative inversions; turn one on its head, and it is indistinguishable from its mate.  Thus, the pairs may also be thought as mirror-images.

Also of note is that line 5 of hexagram #41 and line 2 of hexagram #42 are each YIN and central to the YANG trigrams in which they appear (Mountain; outer trigram of , hexagram 41; and Thunder; lower trigram of hexagram 42). It is all the more fitting that even the names of these hexagrams ('Increase' and 'Decrease') denote anti-symmetric action or condition.  Anti-symmetry or complementarity are discussed elsewhere in more detail.


In each of the lines just mentioned, appears the phrase, “parties adding to the stores (of its subject) ten pairs of tortoise shells.”  A tortoise shell, as detailed above, characterizes the number thirteen.  “Ten pairs,” (i.e., twenty tortoise shells) constitute a precise description of the Tzolk'in harmonic 13:20, unambiguously indicating the number 260.  Q.E.D.

NB: It must be mentioned that this model currently suffers a deficit in that it does not presently account/incorporate the 7-day 6-night "rhythm," a feature that finds much support in current literature about the calendars of the Maya people, as well in original references.  It is the hope of the author to remedy this deficit in future treatments of this thesis.


References:

I Ching, ed. J. Legge; 
I Ching, ed. Karcher & Ritsema

C.J. Calleman; (2004), (2009)

http://www.13moon.com

http://www.lawoftime.org
I Ching Mandalas, Cleary (1989), Shambhala Publications Inc. (cover art)
"Consciousness and Calendars," Ian Xel Lundgold;  excerpted from "Mayan Calendar Comes North," June 22, 2004

Appendix I
Selected slides from a presentation given by Ian Xel Lundgold (2004)

Consciousness and Orientation

Appendix II 

Cross-cultural comparisons with the Tzolk'in model

Mayan Eight Division Sky Place

 Ba Gua (8 diagrams) of the I Ching Tradition

Cleary (1989), Shambhala Publications Inc.

Central section of Tzolk'in model

Appendix III
Several views of the Tzolk'in model using common wooden cubes

Closed model

Model with "upper half" removed

Interval view of separated halves

Halves separated and flattened

Exploded view of flattened halves

Quetzacoatl, the Winged Serpent