Treatment of Hexagrams as Trigram Superpositions

Legge, 1879

Trigrams are composed of three "places" or "positions", each "containing" YIN (0) or YANG (1).  The trigrams will be denoted by triplets read left to right as in the diagram below.  In the wireframe diagram (below right), the numbered vertices stand for the trigrams.

Hexagrams are composed of a trigram pair arranged in superposition: lower and upper, or inner and outer. The places are numbered 1 through 6 from bottom to top.  Therefore, we can depict an hexagram using our unit square wireframe diagram by connecting any two vertices with directed paths.  The syntax for this poses the original vertex as lower trigram, the terminal vertex as upper.

For example, the directed path leading from (1,1,0) to (0,0,1) gives ䷨ hexagram #41, Reduction. Reversing the direction of the path produces ䷩ hexagram #42, Increase.

Each of the eight vertices has seven outbound paths to each of the other vertices, totaling 56 paths.  Half of these paths are duplicates, in the sense that the paths connect the same pair of trigrams; thus, the 56 paths comprise 28 pairs of heterogenous* hexagrams.  Each vertex can also loop back to itself.  These 8 loops constitute 4 pairs of homogenous* hexagrams.  An outbound path and its reversal constitute a Wen Pair; loops at antipodal corners (e.g. the example given above) also designate Wen pairs.
*The terms heterogenous and homogenous (applied to hexagrams) refer to the constituent trigrams

Given the yao-numbers derived from Wen's hexagram pairs, we are enabled to assign a weighting to each path, and treat the paths as proper vectors.

"Arrangements and Relations"

(Wen pairs presented on a square grid)

Wen's pairs are well-ordered, but they lack a well-defined spatial arrangement


4 homogenous pairs of hexagrams (formed from identical trigrams) related through the complementarity relation


28 pairs of heterogenous hexagrams (formed from dissimilar trigrams) related through the figurative inversion relation


Xiantian arrangements are spatially defined with a clear pairing/ordering syntax

xiantian magic square

traditional xiantian arrangement

4 homogenous pairs of hexagrams form a diameter


28 hexagrams gird either side of this diameter

internal view of 4 x 4 x 4 hypercube

The hypercubic hexagram arrangement is well-defined but lacks ordering or pairing syntax entirely.


4 pairs of hexagrams comprise a cubic "core"


28 pairs of hexagrams form a "hull" around the core

unit cube 

Here, we use the unit cube to order the tesseract (hypercube).  Each of the unit cube's eight vertices (representing the trigrams) has seven outbound paths to each of the other vertices, totaling 56 paths.  Half of these paths are duplicates, in the sense that the paths connect the same pair of vertices.  Thus, the 56 paths comprise 28 pairs of heterogenous* hexagrams.  An outbound path and its reversal constitute a Wen Pair.
Each vertex can also loop back to itself.  These 8 loops constitute 4 pairs of homogenous* hexagrams.  Loops at antipodal corners (e.g. (0,1,1) and (1,0,0) ) also designate Wen pairs.

Preface to "Evidence of Relation"

"Just because no one understands what you speak doesn't mean [what you say] is deep"  --Jessica Care Moore
“If you can't explain it simply, you don't understand it well enough"  --Einstein

1. One theory advanced by this weblog, that the Book of Changes bears close relation to the Maya calendar system, finds (in the author's view) considerable support from the entry here prefaced.
2. For the first time we have been able to demonstrate, more-or-less objectively, that the numerical foundations of two closely-coupled Maya calendars (tzolk'in and tun) are mathematically derivable from the essential elements and form of the Book of Changes.  In kind contrast to attempts by certain other authors (e.g. McKenna and Arguelles, from whose works the present author drew inspiration and direction), our thesis eschews jargon and complicated maths.

Change, in the common sense, surely involves space-time.  Barring quantum superposition and related phenomena, for a single object to exist in, say, two discrete states O and O', some interval must elapse wherein occurs the alternation from one state to the other; otherwise, one or more of our premises were violated.   This is so fundamental to experience that it resists further explanation.  The word 'event,' an happening, explicitly involves space-time.  Quantum physicists insist that the nature of physical Experience is essentially event-based, thus discontinuous.  This discontinuity manifests in the field of space-time, but we -- enveloped, as it were, within it -- largely fail to perceive this.


To assert, therefore, that the Book of Changes is related to space-time and its measurement is a reasonable proposition  already treated at some length here and here.  Returning to our theory, the numbers 260 and 360 are found to be inherent characteristics of the essential elements of the Book of Changes and its form.  This discovery involved two differing interpretative modes or "views."

Wen's pairs

One view is that presented by the legendary King Wen, who partitioned the 64 figures into 32 pairs. His method of pairing converts the raw data represented by the 64 figures into information. The majority of these pairs (in silver) are figurative inverses; that is, excepting 180 degrees of rotation, they are identical.  The remaining few pairs (in gold) not related through inversion are complementary opposites -- yin exchanged for yang and conversely.

3-D depiction of Wen and xiantian pairs

In the graphic at right, Wen pairs are represented by the shell, while xiantian pairs form the core.  Together they comprise the Changes depicted here in 3-dimensional form. The semantic content relevant to our discussion of Wen's pairings emerges from a simple transformation.


The second view involves a peculiar spatial arrangement of complementary opposites.  The semantic extracted by way of this view seems to require more than one dimension to permit its clear and concise expression.  Once the 32 pairs of complementary opposites are transformed -- this time by proper arrangement on a square matrix -- the latent information manifests.


As a side note, we observe that these two modalities interrelate by means of xiantian (complementary opposition).  In a related sense, tzolk'in and tun combine to form the Maya Long Count. 


We reiterate earlier assertions: the numbers 260 and 360 are derived simply from the 64 elements and form of the Book of Changes, not from tradition, although tradition* certainly informs and confirms our findings.  Moreover, we do not "massage" the numbers out of the Changes using elaborate calculations.  Finally, no specialized vocabulary is required to detail our findings.
*Tradition, in this context, refers to Ta Chuan, an appendix of the Book of Changes.

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.