Saturday, May 5, 2012

11,520: "The Number of All Things"

This entry treats the connection between the 8x8 grid and 11,520, the "number of all things," attested in Ta Chuan (the "Great Treatise").  The 64 hexagram are commonly depicted on an 8 x 8 grid, known in Hindu culture as ashtapada. Its metric (generic term for measure) is 2080, and is thus related to 11,520:

1) Begin by enumerating its cells beginning with 1, through 64, noting that the cells sum to 2080.
2) Arrange the numbered cells such that the numbers within each row, column, and major diagonal sum to 260. 
Under this arrangement, the 18 columns, rows, and diagonals sum to 4680 = 18 * 260 = 13 * 360 = 18 * 13 * 20.  This number seems to link the Mayan 360-day tun calendar to the Mayan 260-day tzolkin calendar.  
3) Convert the numbers into hexagrams (6-line binary figures) , the least-valued hexagram valued having the value one (1).
4) Transform each figure into a yao-number by substituting its lines with the corresponding divination ritual numbers: '6' for YIN lines and '9' for YANG lines, summing the substitutions, then scaling each sum by 4.  This step is derived from instructions given in Ta Chuan, and has the effect of:

  • "flattening" the hexagram (like a logarithm), which may be regarded as a "stack" of binary exponents 
  • shifting the range of values from a continuum of [1..64] to [144..216] in discrete intervals of 12 
The 18 rows, columns, and diagonals noted above now sum to 1440, and produce a grand sum of 18 * 1440 = 25,920, equal to the duration of a precession cycle.
5) Finally, the hexagrams and their associated yao-numbers from steps #3 and #4 are sorted to produce seven groups with the following membership and distribution:
The 64 resulting yao-numbers are summed to produce 11,520, the "number of all things."


  1. The legend associated with the Hindu tradition of Vaastu Purusha associates the ashtapada with the theme of creation of the world through ritual sacrifice. This legend finds a cognate in the Norse legend of Ymir.

    Odin's self-sacrifice on the ash tree Yggdrasil, the 'Tree of Life' or 'World Tree,' gained him the knowledge of eighteen (or nine) magic spells. We infer a parallel between those spells and the 18 rows, columns, and major diagonals of the ashtapada. The highly-ordered numerical arrangement that is imposed by Mercury's magic square produces the numbers associated with precession of the equinoxes.

    The association of these legends with precession is confirmed by Norse legend of the Yuletide tree, where the tree is to be understood as metaphoric of the Earth's polar axis. The star placed at the top of the tree represents the pole star. (q.v.

  2. We also note a mathematical link between 11,520 and 25,920:

    The arithmetic progression 1 : 2 : 3 when scaled by 720 gives 720 : 1440 : 2160
    The geometric progression 1 : 4 : 9 when scaled by 2880 gives 2880 : 11520 : 25920

    We can offer an account for the differing number of spells, nine versus eighteen, gained by Odin:
    As graphically presented in the associated blog entry above, 18 * 1440 = 25,920; similarly, 9 * 2880 = 25,920.

  3. Great Treatise, pt.1, ch.9, v.4-6; Karcher (2000)

    The extreme of Ch'ien's sticks is 216. The extreme of K'un's sticks is 144. This gives 360. It corresponds to the days of the year.

    The total number of sticks in the two parts of Change is 11,520. This corresponds to the the ten thousand things.

    Therefore operations by four establish change. Eighteen transformations complete the diagram.

  4. Again, Ta Chuan gives us that 11,520 is the "number of all things corresponding to the "myriad (ten thousand) things" attested in Tao Teh Ching.

    The prime factorization of 11,520 is 2^8 * 3^2 * 5 = 64 * 180. This perhaps indicates sixty-four objects, each with a mean value of 180.
    Possible interpretations:
    64 cubes each with a value from the range [144..216] with a strictly-defined distribution for those values.
    6 faces per cube, corresponding to the six positions of an hexagram.
    4 edges per face perhaps indicating that the value of each position/face is to be multiplied by four to derive its corresponding yao-number.
    the average yao-number of 180 per cube perhaps indicating an average value of 30 yao per cube-face, corresponding with the sum of the four divination ritual numbers, [6 7 8 9].