Preface to "Evidence of Relation"

"Just because no one understands what you speak doesn't mean [what you say] is deep"  --Jessica Care Moore
“If you can't explain it simply, you don't understand it well enough"  --Einstein

1. One theory advanced by this weblog, that the Book of Changes bears close relation to the Maya calendar system, finds (in the author's view) considerable support from the entry here prefaced.
2. For the first time we have been able to demonstrate, more-or-less objectively, that the numerical foundations of two closely-coupled Maya calendars (tzolk'in and tun) are mathematically derivable from the essential elements and form of the Book of Changes.  In kind contrast to attempts by certain other authors (e.g. McKenna and Arguelles, from whose works the present author drew inspiration and direction), our thesis eschews jargon and complicated maths.

Change, in the common sense, surely involves space-time.  Barring quantum superposition and related phenomena, for a single object to exist in, say, two discrete states O and O', some interval must elapse wherein occurs the alternation from one state to the other; otherwise, one or more of our premises were violated.   This is so fundamental to experience that it resists further explanation.  The word 'event,' an happening, explicitly involves space-time.  Quantum physicists insist that the nature of physical Experience is essentially event-based, thus discontinuous.  This discontinuity manifests in the field of space-time, but we -- enveloped, as it were, within it -- largely fail to perceive this.


To assert, therefore, that the Book of Changes is related to space-time and its measurement is a reasonable proposition  already treated at some length here and here.  Returning to our theory, the numbers 260 and 360 are found to be inherent characteristics of the essential elements of the Book of Changes and its form.  This discovery involved two differing interpretative modes or "views."

Wen's pairs

One view is that presented by the legendary King Wen, who partitioned the 64 figures into 32 pairs. His method of pairing converts the raw data represented by the 64 figures into information. The majority of these pairs (in silver) are figurative inverses; that is, excepting 180 degrees of rotation, they are identical.  The remaining few pairs (in gold) not related through inversion are complementary opposites -- yin exchanged for yang and conversely.

3-D depiction of Wen and xiantian pairs

In the graphic at right, Wen pairs are represented by the shell, while xiantian pairs form the core.  Together they comprise the Changes depicted here in 3-dimensional form. The semantic content relevant to our discussion of Wen's pairings emerges from a simple transformation.


The second view involves a peculiar spatial arrangement of complementary opposites.  The semantic extracted by way of this view seems to require more than one dimension to permit its clear and concise expression.  Once the 32 pairs of complementary opposites are transformed -- this time by proper arrangement on a square matrix -- the latent information manifests.


As a side note, we observe that these two modalities interrelate by means of xiantian (complementary opposition).  In a related sense, tzolk'in and tun combine to form the Maya Long Count. 


We reiterate earlier assertions: the numbers 260 and 360 are derived simply from the 64 elements and form of the Book of Changes, not from tradition, although tradition* certainly informs and confirms our findings.  Moreover, we do not "massage" the numbers out of the Changes using elaborate calculations.  Finally, no specialized vocabulary is required to detail our findings.
*Tradition, in this context, refers to Ta Chuan, an appendix of the Book of Changes.

Evidence of a Relation from Chinese Book of Changes to the Maya Calendar

In our discussion of 'yao-numbers,' the existence to which is alluded in Ta Chuan Part 1 Chapter 9, we demonstrate that the Wen pairs embody a form of complementarity that is based on the number 360.


On performing a trivial transform on the Wen pairs, we obtained a distribution of ten (initially nine*) groups of hexagrams with the following properties:

• The ten groups exist in five matched pairs
• The same quantity of hexagrams is present in both halves of a given pair of groups
• All hexagrams in any given group have the same yao-number
• Yao-numbers of paired groups complement to 360
• Summing yao-numbers over all 32 pairs of hexagrams produces 11,520

The ten groups and their yao-numbers:

• 144/216 -- one member each 
• 168/192 -- three members each 
• 172/188 -- six members each 
• 180/180 -- ten members each 
• 176/184 -- twelve members each 

* Initially, groups presently numbered #4 and #7 were regarded as a single group of twenty.  Since both have yao-number 180, partitions of this group of twenty appear arbitrary.  However, Y yao-number, tends to increase monotonically from left to right across the groups.  A single group of twenty does find an appropriate location in the middle as it regards yao-numbers as index.  

In a separate discussion of hexagram arrangements, we observed that the xiantian arrangement (i.e. complementary opposites) over the ashtapada (i.e. 8x8 field) exhibits another kind of complementarity based on the number 65.  Hexagrams may be interpreted as binary expressions of the counting numbers [1..64].  When arranged on the ashtapada according to xiantian, any pair of hexagrams separated by 180 degrees of rotation will sum to 65.

Xiantian magic square

We later determined that a particular xiantian arrangement termed XMS (xiantian magic square), based on the 8th-order magic square, exhibits row-wise and columnar partitioning whereby each linear collection of eight hexagrams sums to the number 260.  The two main diagonals also exhibit this trait, producing a total of 18 linear octets  of hexagrams, each collection summing to 260.  Further investigation reveals that any symmetrical selection of eight hexagrams over the XMS will also account 260.  The 64 elements sum to 2080 of any proper 8th-order magic square.


These two modes of complementarity appear quite different in character.  The former is based on a simple transformation of a peculiar pairing relationship attributed to King Wen of Chinese antiquity.  The latter mode arises in large part from a distinct spatial arrangement of the same 64 elements interpreted as numbers.  Neither mode seems especially well-related to the other, yet both coexist in the same set of 64 hexagrams and express two different forms of complementarity.  It must be granted 1) that both modes are based on 'pairing,' but the pairings (xiantian vs. Wen) are fairly discrete; and 2) that complementarity figures prominently in both modes.


Databases "views" present varying modalities of the underlying object, yet the object remains one and the same.  Two calendars in prominent use by the pre-Columbian Maya, the tun and tzolk'in, are based on the numbers 260 and 360, respectively.  Is it mere coincidence that the mathematically- and astronomically-astute Maya would make calendric use of these numbers, now shown to be derivable from the Book of Changes dated some 2000+ years prior?  Or is it more logical and likely that the Maya calendar system and the Chinese Book of Changes are simply expressions of the same fundamental object?

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.

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

Fu Xi's Square: Tzolk'in Octo-partition

The diagram displayed at right presents the Book of Changes in the tabular form known as the Fu Xi sequence, widely held to be the original form of the sixty-four hexagrams presented by Fu Xi himself.  The Fu Xi diagram is also the earliest known depiction of sequenced binary integers, the discovery of which are commonly (and mistakenly) attributed to Gottfried Leibniz who reportedly learned of binary notation through a treatise written by Jesuit scholar Joachim Bouvet.

The white numbers at the upper-right corner of each cell of the diagram correspond to their ordering in the traditional King Wen sequence.  


The 64 hexagrams of the Fu Xi diagram are presented in an orderly structural arrangement:

Each hexagram (six-line figure) comprises a inner/lower trigram (three-line figure), and an outer/upper trigram.  In the diagram, columns are ordered by a hexagram's upper trigram; rows are ordered by the lower trigram.  Rows and columns cycle through the same trigram sequence:

☷ EARTH [1], ☶ MOUNTAIN [2], ☵ WATER [3], ☴ WIND [4], ☳ THUNDER [5], ☲ FIRE [6], ☱ LAKE [7], ☰ SKY [8]

The bracketed numbers following each trigram name above are associated with the Fu Xi  sequence of trigrams.  For example, the row with lower trigram MOUNTAIN is numbered [2].  The column numbered [3] has upper trigram WATER.  This row-column combination [2,3] locates hexagram #39 ䷦ (Obstruction).  This schema is similar to chess notation.

The following presentation of the Fu Xi diagram is doubly-indexed: once (in white) according to the King Wen's traditional ordering of the hexagrams (upper right corner); and as before, by Fu Xi's binary value (in black) for the hexagram (lower left).  The diagram is marked with eight color-coded pairs of hexagrams.

Arranged in this fashion, one easily observes that the binary values (in black) of the colored pairs sum to sixty-five.  In truth, this relation holds true over the entire 8 x 8 table; the sixteen colored figures presented are an arbitrary subset.
If we were to apply chessboard notation to the entire Fu Xi diagram: [1,1] at upper left  and [8,8] at its lower-right, any two figures with coordinates that combine to [9,9] are complementary antipodal pairs, having binary values which sum to 65.  


Our previous example of hexagram #39 (Biting Through) has hexagram #38 (Opposition) as its complementary antipodal pair.  


These thirty-two pairs of hexagrams are each complementary in the sense that each member of a pair has YANG lines where the other has YIN lines, and conversely.   


They are antipodal in the sense that they are separated by 180°of rotation, thus the pairs are maximally separated within the bounds of the square.

Finally, the binary values of these complementary antipodal pairs invariably sum to sixty-five.  In this context, the number 65 may be seen as suggestive of completeness or continuum.  Alternatively, as 1 querent + 64 hexagrams = 65, that number can symbolize divination, communion with the divine.


As each of the sixty-four hexagram figures has a discrete binary value ranging [1..64], they form thirty-two complementary antipodal pairs of hexagram figures.  Therefore, the Fu Xi diagram comprises a metric space of 65 * 32 = 2080,  also known to be the 64^th triangular number.

We also observe that the number 2080 factors into 8 x 13 x 20 which implies that even this representation of the Book of Changes may be octo-partitioned (divided by eight).  

Pieces of Eight

Elsewhere we suggested that the Book of Changes may also be represented as a 4 x 4 x 4 hypercube as in the diagram at right.
Observe that the 2 x 2 x 2 hypercube (at left in the picture) is an octant (one-eighth piece) of the 4 x 4 x 4 hypercube.  Therefore, the 13 x 20 metric space is an octonary partition of the Book of Changes.


Students of the pre-Columbian Mayan culture will recognize 13 x 20 as relating to the sacred 260-day tzolk'in calendar.  Since tzolk'in comprises 260 days and is analogous to one-eighth of the Book of Changes, eight tzolk'in account 2080 days.  Coincidentally, a year of full-time work (40 hours * 5 days * 52 weeks) comprises 2080 hours.

We can also use the 4 x 4 x 4 hypercube representation of the Book of Changes to model tzolk'in. Observe: 

This suggests that tzolk'in's 13 x 20 metric space (260 days) can be fractioned into 2080 units, each unit accounting for one-eighth of a standard day, or three hours.  Eight of those 3-hour units would, if modeled using cubes, form a 2 x 2 x 2 hypercube, representing a standard 24-hour day.



Tzolk'in's own octonary partition (represented, for example, by the 2 x 2 x 2 hypercube) is a half-season of 32.5 days (260/8).  More common divisions of tzolk'in include the four seasons of sixty-five days, five 52-day periods, and twenty 13-day trecenas.