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?

18 Magic Spells

Having derived candidates for the 18 spells, it remains for us to choose or develop a framework for interpreting them.  The following are what came immediately to mind:

1. divination
2. DNA (genome)

Xian Tian magic square

The 18 collections or "spells" were derived from the xian tian magic square arrangement (discussed elsewhere) and are listed below.  The hyper-linked numbers correspond to the traditional hexagram ordering attributed to King Wen, while the numbers in the graphic at right are numerical values of the same.  The first few "spells" have been loosely interpreted using the divinatory language of the I Ching:



Spells from ROWS
01)     2,44,13,19,15,6,25,11
receptive/field, coupling/meeting, humbling, approaching, arguing, fidelity, peace 
02)     9,51,40,53,61,55,32,20
small harvest, thunder, deliverance, advancement, re-centering, fullness, endurance, viewing
03)     14,3,29,56,38,63,48,35
04)     45,18,22,58,31,4,27,43
05)     23,28,49,41,52,47,17,26
06)     5,21,64,39,60,30,50,8
07)     34,42,59,62,54,37,57,16
08)     12,46,36,10,33,7,24,1

Spells from COLUMNS
09)     2,9,14,45,23,5,34,12
10)     44,51,3,18,28,21,42,46
11)     13,40,29,22,49,64,59,36
12)     19,53,56,58,41,39,62,10
13)     15,61,38,31,52,60,54,33
14)     6,55,63,4,47,30,37,7
15)     25,32,48,27,17,50,57,24
16)     11,20,35,43,26,8,16,1

Spells from DIAGONALS
17)     2,51,29,58,52,30,57,1
18)     11,32,63,31,41,64,42,12

Magic Spells for Odin/Mercury

Ashtāpada: the uncheckered 8x8 board

I Ching ELEMENTS
64 six-line figures or "hexagrams" (e.g. ䷄), each with associated judgments, imagery, and texts.  May be written using line drawings: ⚊ ⚋ or with strings of binary digits; for example: 010111b is a binary expression of the hexagram figure above.


I Ching ARRANGEMENT



Ashtāpada, the uncheckered 8×8 board, of Hindu derivation.  The literal meaning of "ashtapadi" is "eight steps." This word is the origin of the word ashtāpada, an Indian board game, the forerunner of chess. The word now primarily refers to the board itself.

Vastu-Purusha mandala

• Related to the game Chaturanga: (Sanskrit: "four-limbed"), the ancestor of modern chess
• Related to Manduka/Chandita mandala 
• Populated by the 64 hexagrams, it represents 2080 
• Characterized as a "field of action [activity]," indicating the body, or even the Universe suggesting perhaps the identity relation between these 

• related to "kshetra," meaning 'field' or 'body' 
• related to Vastu-Purusha mandala
• q.v. Bhavagad Gita: ch. 13
• q.v. Ta Chuan p1:c11:v5-6 
• q.v. "Symbolism of Chess," Titus Burkhardt

I Ching SYNTAX
Xian Tian, a symmetrical arrangement of 32 pairs of binary complements over the ashtāpada. 

• Any given hexagram on the ashtāpada has a complement at 180 degrees of rotation 
• The binary values of each complementary hexagram pair sums to 65 
• The expression 64!! (double factorial), equal to [64*62*60...8*4*2] computes the xian tian arrangements that can exist

Xian Tian magic square

Our xian tian magic square (XMS) is an arrangement wherein all linear collections of eight numerical elements (rows, columns, and main diagonals) sum to 260. 

• XMS produces two proper partitions of the populated ashtāpada.
• The eight rows constitute a partition, the eight columns constitute another partition, and the diagonals comprise two additional octonary collections for a total of 18 linear octets. 
• The author is presently unclear how to calculate the number of xian tian arrangements that obey the magic square restriction, though it is obviously a subset of total number of xian tian arrangements . 

Explication:

Circular arrangement of the Armanen runes

The 18 linear collections of eight hexagrams within Mercury's magic square are suggestive the 18 spells (also charms, runes) gained by Odin as described in the Runatal and Ljooatal sections of the Hávamál.

The author proposes that the 18 linear octets of hexagrams, arranged on the XMS, are candidates for the eighteen spells of Odin. In the binary language of computing hardware, eight bits (binary digits) commonly constitute a 'word.' 
In this discussion, the sixty-four 6-bit hexagrams serve as the words or runes. Therefore each of the eighteen collections comprises a unique combination of eight 6-bit "words."  Each of our "spells" consists of eight words (alternatively, eight hexagrams or runes), having a numerical or numerological value of 260.


Words certainly imply 'spells' and 'spelling,' as to spell a word is to precisely enumerate its letters in correct sequence. The word 'gospel,' for example, literally means "good spell," or "good word."  


Words and runes are symbolic representations of ideas.  Runes are fairly distinct from letters, it seems, since individual letters seem devoid of semantic value viz. the word in which they appear; while runes preserve a measure of their individual meanings in isolation. Additionally, runes seem more akin to words in a phrase than to letters in a word.

I Ching Divination Ritual

Attribution: http://hermetic.com/sabazius/yijing.htm

To employ the yarrow stalk method, one begins with a bundle of 50 straight, dried stalks of the yarrow herb.  One stalk is set aside and not used again. 

• The bundle of the remaining 49 stalks is then divided into two bundles.
• One stalk is taken from the bundle on the right and placed between the ring finger and little finger of the left hand.
• The bundle on the left is then taken up in the left hand, and stalks are removed therefrom and set aside in groups of 4 using the right hand, until 4 or fewer remain. The remaining stalks are placed between the middle finger and ring finger of the left hand.
• The bundle on the right is then taken up in the left hand, and stalks are removed therefrom and set aside in groups of 4 using the right hand, until 4 or fewer remain. The remaining stalks are placed between the index finger and middle finger of the left hand.
• The total number of stalks between the fingers of the left hand are then counted and noted. The possibilities are 9 and 5. Nine stalks results in the value of 2, and five stalks results in the value of 3. Here are the possible outcomes of the first operation:

Between index and middle fingers:	4	3	1	2
Between middle and ring fingers:	4	1	3	2
Between ring and little fingers:	1	1	1	1
Total:	9	5	5	5
Assigned Value:	2	3	3	3

• These 9 or 5 stalks are then set aside and the remaining bundle of 40 or 44 stalks is divided and counted out in the same manner. This time, eight stalks results in the value of 2, and four stalks results in the value of 3. Here are the possible outcomes of the second operation:

Between index and middle fingers:	4	3	1	2
Between middle and ring fingers:	3	4	2	1
Between ring and little fingers:	1	1	1	1
Total:	8	8	4	4
Assigned Value:	2	2	3	3

• These 8 or 4 stalks are then set aside and the remaining bundle of 32, 36 or 40 stalks is divided and counted out in the same manner for a third operation, the possible outcomes of which are identical to those of the second operation.

Each operation has now produced a value of either 2 or 3, and these three values are now added together to produce a total of 6, 7, 8, or 9. The totals of 6 and 8 yield a (static) Yin line (-- --), the totals 7 and 9 yield a (static) Yang line (-----). A total of 6 is considered a moving Yin line (--x--), and a total of 9 is considered a moving Yang line (--o--).

This first line is placed at the bottom of the hexagram and the entire operation is repeated five times to produce the remaining five lines. The sixth line is placed at the top of the hexagram.

ed. note: after the third manipulation, the bundle contains one of (24|28|32|36) stalks, which when divided by four gives the corresponding ritual number (6,7,8,9)

Octopartition pt. II

This uncommon term is used to describe the act of dividing a thing into eight sections, called octants; colloquially, "pieces of eight."  I Ching lends itself well to illustrating the notion since it can be octo-partitioned twice in succession as the following graphic depicts.

The largest cube represents I Ching as a holism.  The mid-sized cube represents Ba Gua (one of the Eight Diagrams), a three-line figure or trigram, two of which, when arranged in superposition, produce an hexagram.  The smallest cube, its six faces standing for the six lines, represents an individual hexagram out of the 64 that comprise the I Ching.

Taoist Cosmogony pt. II

To recap, the two First Powers interact to produce the Four Symbols.  The Four Symbols intermingle with the two First Powers again to produce the eight diagrams, or Ba Gua.

arbitrary Ba Gua

Observe that the Ba Gua effectively combine to produce an octo-partitioned space in three dimensions.  The binary metric space allows for two values per dimension.  One of the Ba Gua is depicted here, with each of its corner given as a triplet of binary digits.

Three binary axes combine to yield 2 x 2 x 2 = 8 partitions in the metric space we've defined.  Each of the eight partitions may then be visualized as a cube.  The eight partitions also can represent the Ba Gua as eight cubes.

Eight Ba Gua comprise an hypercube, 

These 8 cubes come together at a common vertex (ORIGIN) to form a composite cube of a higher order, or hypercube.    Each of the eight Ba Gua cubes contributes one distinct corner to the higher-order composite cube.  A vector (path) drawn from the ORIGIN to each of the 8 corners of the hypercube defines direction in space and the bounds of the space.  If all eight paths measure equally, a spherical space results.

Hexagram, Trigram, I Ching

Movement between corners indicates motion, or CHANGE.  Note that all observable phenomena are intrinsically based on motion.  Consider that we can't really observe something that does not change, and if we were to try, what would be the point?  Most interesting matters involve CHANGE, and understanding CHANGE is the whole point of I Ching.

Complementary Pairs

When displayed on the xiantian field, complementary (antipodal) pairs appear separated by 180° of rotation.  Such pairs, insofar as their binary values are considered, are complementary pairs, meaning that one member always has YANG where the other has YIN.  Complementary pairs sum to 65.  In the following animation, xiantian is rotated 45° clockwise.

32 Complementary pairs

Yao Numbers and the Tree of Life

The following animation is a graphical depiction the nine species of "Yao Numbers," a concept introduced in Ta Chuan, the Great Treatise.  It displays the nine families in forward and reverse sequence.  It is the so-called Fu Xi (also, xiantian) diagram rotated 45 degrees CW.  We suggest that these nine families are analogous to the "Nine Worlds" surrounding Yggdrasil, the Teutonic Tree of Life.
  

The animation appears to depict a signal initiated at one pole (congruent with the realms called Hel and Asgard in the Teutonic tradition), propagated through the medium, then confirmed by the receiving pole.  Conceptually, it bears  resemblance to a common depiction of Yggdrasil, the Teutonic Tree of Life with its "Nine Homeworlds."  


From the poles  the signal passes through either group of three hexagrams with pairs formed by complementarity or "separated pairs".  These are congruent to the planes called Bright Alfheim & Dark Alfheim in the Teutonic scheme. The  purported power of these realms is to mediate between the worlds of Asgard and Midgard:

There is also a transitional world between Asgardhr and Midgardhr, where the energies of the former are transmitted to the latter world. This realm is called Alfheimr (world of the Elves), and is characterised by the higher aspects of light and air (equivalent perhaps to the occult conception of the Etheric planes). In this world, the highest energies of the human world mix with the lowest energies of the Gods. Beneath Midgardhr (the physical world) is a corresponding region intermediate to Midgardr and Hel, called Svart-Alfheim (world of the black elves). In this region are found the mysteries of earthly manifestation, represented by the dwarves.  

Teutonic Cosmology by M. Alan Kazlev

In most detailed depictions of the Tree of Life, a formation comprising a central node bounded by a group of four nodes (q.v. quincunx) is present.  In this presentation, the largest and central of the five families comprises twenty hexagrams, each with three YIN and three YANG lines, and each having yao-number <180>.  This node is congruent to Teutonic Midgard, the home of Men.


The four groups surrounding the central group have {sizes} and <yao numbers> as shown: ({6} <172>, {12} <176>, {12} <184>, {6} <188>).
The congruence of these groups to the Teutonic tradition is less clear than the others.  The candidates are: Jontunheim, Vanaheim, Niflheim, & Muspelheim.

The following crude table displays the 64 hexagrams and the nine yao-number "families" into which they fall, the quantity of hexagrams per family, as well as yao-numbers and line-analysis (#YIN lines and #YANG lines) for each hexagram.  Wen pairs are shown within parentheses where possible.


0 YANG,6 YIN <144> { 1} [2]*
2 YANG,4 YIN <168> { 3} [27 29 62]*
5 YANG,1 YIN <172> { 6} [(9 10) (13 14) (43 44)]
4 YANG,2 YIN <176> {12} [(5 6) (25 26) (33 34) (37 38) (49 50) (57 58)]
3 YANG,3 YIN <180> {20} [(11 12) (17 18) (21 22) (31 32) (41 42) (47 48) (53 54) (55 56) (59 60) (63 64)]
2 YANG,4 YIN <184> {12} [(3 4) (19 20) (35 36) (39 40) (45 46) (51 52)]
1 YANG,5 YIN <188> { 6} [(7 8) (15 16) (23 24)]
4 YANG,2 YIN <192> { 3} [28 30 61]*
6 YANG,0 YIN <216> { 1} [1]*


*indicates Wen pairs formed via complementary opposition; Wen pairs of this nature fall into separate families.

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.

Taoist Cosmogony pt. I

 Before all things, there is Tao.  Subjective and potential, limitless and empty, it is the source of all things knowable and unknowable.  Several early verses of Tao Teh Ching speak at length about Tao while conceding that words cannot fully describe it and that names are only words.

The Limitless produced the Great Limit/Great Axis -- 0-dimensional -- inside-outside: the first division

The Great Axis produced the two First Powers, Ch'ien and K'un (Heaven and Earth, YIN and YANG) -- 1-dimensional -- up-down: the second division

The two First Powers interacted to create the 4 xiang, [symbols / seasons / directions / elements / forces / nucleotides / experiential dimensions] -- 2-dimensional -- left-right & front-back: the third division:

3-dimensional -- these three divisions, or axes, define three entangled or looped "plane-spaces" corresponding to the Three Operations of Heaven, Humanity, and Earth.


4-dimensional -- the 4 Symbols each recombine with the Great Axis, expanding along it and extending through space toward the poles, thus creating 8-Partition Place.  The octants of 8-Partition Place are found to correlate mathematically to the 8 Diagrams (Ba Gua).


The above diagram depicts the Gates of Change, YANG and YIN at either pole of the vertical axis.  It also depicts the Four Symbols forming a Great Square (the plane-space shown above).  Observe that the Great Square extends upwards and downwards towards each pole, (visualize infinitely many "layers" of Great Squares congruently 'folded' or 'stacked' along a common vertical axis) thus creating three-dimensional space, or akasha.


These Great Squares provide domains for expression of the Four Symbols and provide the conditions for Consciousness to interact with the field of Experience.

The Canon of Change

King Wen sequence

 Our jaunt through the Taoist version of Creation had a goal: it teaches the basic structure of the Book of Change:
Sixty-four six-line figures, each composed of two three-line figures, positioned one atop the other.
The 8 three-line figures, Ba Gua, or trigrams, enumerate the eight partitions of space.  As they are responsible for creating and supporting space, they can be assumed capable of representing anything that space can contain, thus they are archetypal.  They combine in pairs to account for the time dimension as represented by Change.


Each line can 'hold' one of two possible values, yin or yang, '0' or '1', up or down.  The lines can be understood as representing the Three Operations that produced them: Earth, Mankind, and Heaven.  At top right are presented the 64 Changes in King Wen's ordering.  The Canon of Changes, as it is sometimes called, contains 6 x 64 = 384 lines in total, each bearing a decision on its significance in the context of the hexagram in which it appears and its position within that hexagram, and the trigrams of which hexagram is composed.  This framework provides for a flexible and fertile interpretative system.