Thursday, December 30, 2010

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 13th  (day named for Norse goddess Freya), and the common absence of 13th floors in buildingsWhat 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.

I Ching, ed. J. Legge; 
I Ching, ed. Karcher & Ritsema
C.J. Calleman; (2004), (2009)
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

Tuesday, October 19, 2010

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 64th 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.

Tuesday, October 12, 2010

Quantifying Change: Conjecture

Quantifying Change: Conjecture

One feature of the King Wen sequence particularly intrigues the author: Changes (expanded first-order differences) are defined over the ordered set of Wen pairs; this apparently causes the Changes to reproduce the King Wen sequence.  In the table of Changes following, as one reads across the vertical columns and down the rows in order, the Changes replay the King Wen sequence of the hexagrams with increased detail at the level of the individual hexagram lines.  The Change operation effectively recovers information about the internal statics and dynamics of the hexagram figures.  This recovered information allows us to derive the yao-numbers corresponding to the sixty-four hexagrams.  The yao-numbers, in turn, give us a means to quantify and categorize Change.
We term a pair 'unbalanced' when the yao-numbers of the pair-members are unequal.  Such unbalanced pairs appear four times in I Ching:  [1,2], [27,28], [29,30], and [61,62].  The hexagrams composing these pairs, incidentally, do not produce one another through fangua or hexagram inversion (reversing the order of the lines); they employ pantonggua (complementary opposition) to form a pair.  

King Wen pairs each seem to represent extremes of a continuum, since we can always determine the other half of a Wen pair provided we know the generative rules.  The Gates of Change, Chi’en and K’un, are the “father and mother” of the cosmos, but each King Wen pair can be similarly seen as constituting its own cosmos or continuum.  The self-similarity that permeates the actual world appears to be re-enacted via the relationships that obtain through the Change operation.

Now complete, how can we validate these findings?  What, if any, assurance have we that the Changes have a reflection in consensus reality?  Can we find corroboration of these findings there?

Quantifying Change: Uniting the Oracle with the Changes

Quantifying Change: Uniting the Oracle with the Changes

One troublesome aspect of I Ching for the author had been that there appeared to be no connection between the yarrow-stalk oracle (ostensibly presented by Confucius) and the text. To the point, the text of I Ching does not speak in terms of the four xiang, only in terms of greater YIN [6] or greater YANG [9].  Lesser YIN [8] and lesser YANG [7] are not referenced in the text.  

The divination ritual provided by Ta Chuan introduces the four digram figures denoting HEAVEN or greater YANG [⚌], FIRE or lesser YIN [⚍], WATER or lesser YANG [⚎], and EARTH or greater YIN [⚏] by the numbers [9,8,7,6] respectively.  The xiang, as the digrams are called, have a long heritage within I Ching tradition. The xiang constitute all permutations of the two basic lines [⚊], [⚋] taken in pairs (2 x 2 = 4).  Nine and six indicate changing conditions in a divination, but the reader of I Ching cannot possibly observe changing conditions in a hexagram figure that displays them statically.

Put differently, examining the 64 hexagram figures outside the context of divination provides little means to discern whether a given hexagram has moving lines, or the positions at which they occur – changing lines are essentially meaningless outside the divination context.  Without knowledge of where and how Change occurs, one attempting to systematize or quantify the Change would be left to broadly assign 9 (or 7) in place of YANG, and 6 (or 8) in place of YIN wherever YIN and YANG lines are encountered.

An initial step taken towards quantifying and measuring Change involved adopting a shift in perspective from which it follows that Ch'ien ("all nines") and K'un ("all sixes") refer to situations with all 6 positions are changing in parallel.  As situations, all hexagrams are thus dynamic contexts, not merely still images as they are appear in the text.  

In Taoist cosmogony, xiang are regarded as the primordial reality that emerged after YIN [⚋] and YANG [⚊] separate and emerge from the Taiji singularity, which differentiated itself from the Way (Tao) in order that Reality might be made manifest.  Thus, in the context of I Ching, Changes may also be understood as comprising three digrams which represent the Three Powers: EARTH, HUMANITY, and HEAVEN.  The relevance of this will shortly be made apparent.

Initially, the expanded first-order difference integers accounted only two types of line-changes: YIN changing to YANG, and conversely, at any of the six positions.  In the following depiction of Change #5, the white-spaces in the 2nd and 5th places betray flaws in the original version of the expansion procedure:

Such white-spaces invariably arose from the two non-change conditions: unchanging (or static) YANG, and unchanging (or static) YIN.  It became clear that for any Wen pair and at any of the six positions, four kinds of line-changes may manifest.  To omit or treat identically the two static conditions when they clearly denote different states is to commit the same errors McKenna committed with Timewave by only considering the number of line-changes while ignoring the places at which they occur and by ignoring non-changing positions.  

On learning of the xiang and their suitability for representing the different kinds of line-changes, the author refined the expanded first-order difference operation (renamed as Change) by using xiang bi-grams to denote the four discrete types of line-changes.  This Change procedure was iterated over the full set of 32 Wen pairs, producing thirty-two Changes.  However, yao-number 144 does not occur in these thirty-two Changes produced per the rules given in Ta Chuan.  Furthermore, the yao-numbers corresponding to these thirty-two Changes did not sum to 11,520.

It was determined that the only way to produce the yao-number 144 thus effecting "all sixes” changing at once is for K'un to transform into Ch'ien.  This occurs when one reverses the ordering of King Wen pair #1 to indicate #2 Earth changing to #1 Heaven.  This implies that the remaining Changes are produced by reverse passage through the King Wen sequence.

Initially, King Wen pairs were deemed properly-formed when an oddly-indexed hexagram is followed by an evenly-indexed hexagram; e.g., [1,2] or [29,30].  Subsequently, the rule for a properly-formed King Wen pair was expanded to include the reverses of the Wen pairs, and the procedure was amended to iterate over the additional thirty-two Wen anti-pairs (reversed Wen pairs) which completed the Canon of sixty-four Changes.  The yao-numbers (11,520) were then found to sum correctly, and  the yao-numbers corresponding to hexagrams #1 and #2 properly sum to 360.   

NB: While the yao-numbers of Wen pair [1,2] indeed sum to 360 (also true for anti-pair [2,1]) , we discovered that pairs do not always sum to 360.  The smallest yao-number sum of a King Wen pair was observed to be 344: [9,10], [13,14], [43,44]; the largest yao-number sum observed was 376: [7,8],[15,16], [23,24].

We now have a probability distribution for the King Wen Sequence.  The yao-numbers and their relative frequencies are:  144 (1), 168 (3), 172 (6), 180 (20), 184 (12), 188 (6), 192 (3), 216 (1).   A graphical representation of the “King Wen Distribution” (including distribution statistics) follows.
King Wen's Distribution

Quantifying Change: Number Magic

Quantifying Change: Number Magic
The Master said: "He who knows the method of change and transformation may be said to know what is done by that spiritual power."
This post specifically treats a section of Ta Chuan (the Great Treatise) chapters 52 through 58. This text may be found in James Legge's translation of I Ching, and in Stephen Karcher's (2000) translation of Ta Chuan, where the corresponding chapter is entitled "Number Magic and Consultation." The goal of this work is to solidify the means of quantifying and measuring change. Central to this approach is adopting a new look at the 64 familiar hexagram figures.

"Number Magic" begins by designating even numbers as YIN and odd numbers as YANG. To this point, the He Tu diagram depicts the counting numbers one through ten with light-colored (YANG) and dark-colored (YIN) dots.

The He Tu (diagram) is traditionally said to originate from the emergence of a dragon-headed horse with carp-like scales that emerged from the Yellow River. Its body was covered in strange markings which were noted by the sages of the time and studied in depth. Eventually, these markings were codified into a set of dots (see the diagram at right), called the He Tu, or River Map. Many observations about nature were derived from study of the He Tu, including the existence of the north-south axis of the Earth, and the idea that heat rises while cold descends. Gradually, associations with directional and Five Phase energies were incorporated into the He Tu.

NB: Note the enumeration of the four ritual numbers [6,7,8,9] on the periphery of the He Tu diagram.
(Excerpt attributed to, graphic courtesy of; referenced June 2010)

After a description of the yarrow-stalk oracle, also called "the operation by threes and fives" we are given:
The numbers (required) for Ch'ien (or the undivided line) amount to 216; those for K'un (or the divided line), to 144. Together they are 360, corresponding to the days of the year.The number produced by the lines in the two parts (of the Yî) amount to 11,520, corresponding to the number of all things. Therefore by means of the four operations is the Yî completed.
Legge's commentary on this passage:
"The actual number of the undivided and divided lines in the hexagrams is the same [192 of each]. But the representative number of an undivided line is 9, and of a divided line 6. Now 9 x 4 (the number of the emblematic figures) x 6 (the lines of each hexagram) = 216; and 6 x 4 x 6 = 144. The sum of these products is 360, which was assumed, for the purpose of working the intercalation, as the standard length of the year. But this was derived from observation, and other considerations;--it did not come out of the Yî."


Ch'ien appears in a divination when one casts nines (greater YANG) in all six places
K'un appears in a divination, when one casts sixes (greater YIN) in all six places

We propose the following derivation for the numbers
216 = 9 (symbolic for greater YANG) * 4 (xiang) * 6 (positions)
144 = 6 (symbolic for greater YIN) * 4 (xiang) * 6 (positions)
216 + 144 = 360 (days in a sacred year)

The number produced by the lines in the two parts (of the Yî) amount to 11,520, corresponding to the number of all things.

In the quoted passage above, “The number of all things,” 11,520, approximates the "ten thousand things," a common reference in the Tao Te Ching. The ten thousand things indicate the products of the interaction of Heaven and Earth, namely, all beings and phenomena between Heaven and Earth. The author admits to some uncertainty on this point, however, because no known references include Heaven and Earth among the ten thousand things. The term applies only to those things that are produced by the interaction of Heaven and Earth. The number of all things, 11,520, is shown by Legge to comprise the yao-numbers of Heaven and Earth, including the ten thousand things, the latter symbolized by the other sixty-two hexagrams.

Legge's notes:

The number in paragraph 53 (11,520) arises thus: 192 (the number of each series of lines in the sixty-four hexagrams) x 36 (obtained as above) = 6912, and 192 x 24 = 4608, the sum of which = 11,520. This is said to be 'the number of all things,' the meaning of which I do not know. The 'four operations' are those described in paragraph 31.

I Ching is composed of 384 (6 x 64) lines, half of which are YANG, half are YIN.  This means that 11,520 is the sum of:
6912 = 192 (YANG lines) x 4 (xiang) x 9 (symbolic for greater YANG)
4608 = 192 (YIN lines) x 4 (xiang) x 6 (symbolic for greater YIN)

We should note, however, the subtle clue to a construct inside the Changes: the yao-numbers of hexagrams #1 and #2 are 216 and 144, and the sum of the yao-numbers of all the hexagrams is 11,520 (derived above).  

If Heaven and Earth have yao-numbers resulting from divination ("operation by threes and fives"), we may confidently infer that each of the hexagrams has its own yao-number.  Our goal then, is to find an heuristic to produce an hexagram to yao-number assignment in accordance with the clues given in the text.

We find from the following exercises below that not just any substitution will work, summing yao-numbers over the entirety of I Ching must equal 11,520, and hexagrams #1 and #2 must have yao-numbers 216 and 144 respectively.  We observe that the mean value of any given yao is 7.5 (the average of [6,7,8,9]).  Therefore the lower limit of distributional variance would result from assigning 7.5 to all lines.  Thus, the sum of YANG yao-numbers would equal the sum of YIN yao-numbers.  

The median case of distributional variance results from example A in the table below.  The upper limit of distributional variance of results from example B following.

A: Substituting 7s for YANG and 8s for YIN:
All lines in the CHANGES: 6 x 64 = 384
Lines allotted to YANG and YIN: 192 + 192 = 384
4 * 7 * 192 = 5376 (7 representing YANG lines)
4 * 8 * 192 = 6144 (8 representing YIN lines)
5376 + 6144 = 11520 (
BSubstituting 9s for YANG and 6s for YIN:
All lines in the CHANGES: 6 x 64 = 384
Lines allotted to YANG and YIN: 192 + 192 = 384
4 * 9 * 192 = 6912 (9 representing YANG lines)
4 * 6 * 192 = 4608 (6 representing YIN lines)
6912 + 4608 = 11520 (Legge's solution)

Note that the Changes corresponding to hexagrams #1 and #2 would not have the correct stick-numbers under scheme A, though the sum of the numbers yielded is 360.  These explorations demonstrate that the means of representing YIN and YANG over the sixty-four Changes has consequences for the balance of YIN and YANG.  

From these results we confirm that the heuristic we sought for generating a yao-sum from a Change is equivalent to multiplying by four the appropriate ritual number at each of the six positions.

Returning to our example of Change #28 which emerges from hexagram #55 and hexagram #56, we enumerate the ritual numbers of each position, to yield [987786].  Multiplying each of these six ritual values by four produces [45 * 4 =] 180, which is the yao-sum for Change #28 (which we derived from the Change operation).

Conclusion: While Legge's method is mathematically correct, it alters the distribution of Change by limiting its consideration of ritual numbers to greater YIN [6] and greater YANG [9], while ignoring lesser YIN [8] and lesser YANG [7].