I Ching in 3-D

The presented diagram depicts the 8 x 8 field known as ashtapada. The colored bands represent the oracular probabilities corresponding to the four xiang (symbols). In this entry, the author extends the notions implied by this diagram into three-dimensions through the use of 64 colored cubes.

The idea is simple: a cube has six faces; while an hexagram has six places, commonly denoted as bottom, 2nd, 3rd, 4th, 5th, and top.  Each cube face, therefore, may represent one place of an hexagram and its associated text.

representation of xiang

Tradition asserts that bottom place is nascent, and top place is retiring.  It follows, therefore, that the bulk of a hexagram's meaning or action is found in the middle of the hexagram, sometimes called a 'tetragram'.  These four middle places of the hexagram, akin the four walls of a room, correspond to the four sides of a cube; corresponding furthermore, to the four seasons, four cardinal directions (xiang), four elements, four tarot suits, or any other quaternity.


Since the oracular probabilities are depicted by the colors, all that remains is to devise unambiguous means of associating a particular cube face with an hexagram place.  
We may appropriately mark the six faces, 1) to indicate the relative positions of the faces, and 2) to determine their yin or yang character, and perhaps, 3) with its index (for easy reference).  This permits us to determine easily the quality of the upturned face when a cube is selected.


Proceeding to appropriately label each of the 384 cube faces, we now have a device whereby the identity of an hexagram is determined by the unique configuration of its faces. Simultaneously, that same cube indicates by its color if the upturned face is a changing line or static.  Alternately, six cubes may be selected blindly with replacement to generate an hexagram line-by-line with color alone as an the line-type indicator . 

Addendum (26April12):

Yarrow Oracle

The author has fashioned for himself a compact set of divination cubes that are functionally equivalent to the yarrow oracle.   Since the yarrow probabilities reduce to multiples of 1/16, four colors in the proper proportions reproduce the oracle.  Six cubes are selected individually and randomly with replacement.  Further, the author employs a webcam to film a blind-selection procedure to reduce the possibility of inadvertently influencing the outcome while divining.

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.

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.

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.

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 http://www.maelearning.com/onlinecourses/store/details.asp?id=4, graphic courtesy of http://www.8whitestarfengshui.com/The_Luo_Shu_and_He_Tu.html; 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."

Explication:

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.

Explication:

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 (

B: Substituting 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].