shaRQ: Math & Myth: 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)