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.

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