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 64^th 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.

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].

Quantifying Change: Defining the Chan...

Quantifying Change: Defining the Change operation

Considering any King Wen pair of hexagrams, making comparisons across the members of the pair at each of the six positions, observing and denoting where line-changes occur, the type of line-changes, and where line-changes do not occur, describes the Change operation.  The product of this operation constitutes an individual Change.  Fully-iterated* over the King Wen sequence, the Change operation produces sixty-four discrete Changes.  These sixty-four Changes bear a close semantic relationship to the sixty-four hexagrams.

NB: *Full-iteration means to apply the Change operation as one traverses the pairs of the King Wen sequence from beginning to end and back to the beginning.

Example: Change #28 is composed of the pair [55,56].  Change #28 describes a situation where two positions, the bottom and top, are changing in different directions:

Position 6 shows YANG changing to YIN

Positions 5 and 2 show static YIN

Positions 4 and 3 show static YANG

Position 1 shows YANG alternating to YIN.

By contrast, its anti-pair, Change #37 (composed of the hexagram pair [56,55] also has two changing lines and four stable lines, but the polarity of each changing line is the opposite of Change #28.  Static lines, by nature, do not change polarity.

Position 6 shows YIN changing to YANG

Positions 5 and 2 show static YIN

Positions 4 and 3 show static YANG

Position 1 shows YIN alternating to YANG.

While a Change has the same shape as a hexagram, it must be emphasized that each Change captures two qualities at each of the six positions which describe what transpired within the pair resulting from the Change:

• polarity of the position (in terms of YIN versus YANG)
• state of that position's polarity (in terms of static versus dynamic).

Polarity and state are terms used to describe qualities that describe mutually-exclusive conditions of a line.  Representing them simultaneously requires (2 x 2 = 4) four values.  Fortunately, we are not required decide how to represent these conditions.  The appendices of I Ching include a section entitled Ta Chuan, the Great Treatise.  Ta Chuan is an invaluable resource for accessing the philosophical and cosmogonical context of the Changes.  Ta Chuan provides a description of the “method of threes and fives” (i.e., the yarrow stalk oracle), a divination ritual commonly used at the time of its writing.  The description of the yarrow-stalk oracle includes a discussion of the ritual numbers [6,7,8,9].  These ritual numbers will serve to track polarity and state as described above.

The first thirty-two King Wen pairs (H,H') are as follows: [1,2], [3,4], ... [63,64]. The remaining thirty-two King Wen pairs are the reversals, pair-wise and sequential, of the first thirty-two pairs: [64,63] [62,61], ... [2,1]. Performing the Change operation (taking the expanded first-order difference within the sixty-four King Wen pairs), produces a sequence of sixty-four figures which record the positional state-changes that occurred within each pair as the Wen sequence is traversed to its end and back.  The pair-wise Change operation is continuous over the sequence of sixty-four hexagrams.  

The ritual numbers six, seven, eight, and nine are central to the divination ritual and are described as xiang, or symbols, which we employ for representing the four kinds of Change.   The cosmogonical origin of the four xiang appears to stem from the initial intermingling of the primordial YIN and YANG (the Gates of Change, Ch'ien and K'un, represented by Changes #1 and #2).  Indeed, this can shown geometrically:

The Cartesian plane is composed of two axes which we may use these represent Ch'ien and K'un. We may use resulting quadrants to stand for the xiang:

⚏  [6] = greater/dynamic YIN (alternates to YANG)
⚎  [7] = lesser/static YANG
⚍  [8] = lesser/static YIN
⚌  [9] = greater/dynamic YANG (alternates to YIN)

The positional changes in polarity (YIN or YANG) may be combined to produce a notion of the quantity of Change thus represented.  Such a quantity may take on, for example, the values [2,4,6].  These values reflect the total number of positions undergoing change within a given pair of hexagrams when ordered in the King Wen sequence.  There are other ways of interpreting the quantity of Change as well. For instance, one might count greater YIN and greater YANG individually, which would allow for values [1,2,3].

We also are able to infer some notion of direction as it regards traversal through the King Wen sequence.  Forward motion through the King Wen sequence is represented by King Wen pairs #1 to #32, return motion is represented by King Wen pairs #33 to #64, which reverse the order of the pairs. Because the Change operation preserves the direction of traversal through the King Wen sequence, the Change operation appears to violate commutativity, though superficially.  

The following table presents the sixty-four King Wen pairs and anti-pairs in a color-coded tabular format.  Change indices, hexagram indices and yao-sums are in the various columns. The thirty-two King Wen pairs are read from upper left to lower right.  The thirty-two anti-pairs are read from lower-right to upper left.

Quantifying Change: Introduction

Quantifying Change: Introduction

In this series of posts, the author purposes to develop a mathematical framework for studying Change, the constant, unvarying.  This paper assumes a practical familiarity with I Ching and its internal structures.
In presenting this treatment of Quantifying Change, the author has spared little effort in making it accessible to the casual student of I Ching, or even to the interested layperson by reducing unnecessary jargon.  Regretfully, everyday language often falls short of providing what is required for this endeavor.  This is especially true where mathematics are concerned.  Clear expression of even straightforward math concepts requires verbal precision that everyday language seldom provides.  The author, where specialized terms have been employed, has attempted to make the meanings clear and relevant to the application of the terms.

Graphic descriptions (far more useful in this regard) have been employed where words proved inadequate.  Future treatments of this topic will make broader use of graphical explanations.

---

The philosophy espoused by commentators is that study of the Changes can produce a penetrating understanding of the processes of Change as they manifest in our personal lives, in the cosmos, and at the various scales of dimension.  It is reputed that mastery of this discipline enables passage through the world essentially without misstep since one will have developed the ability to reliably discern the appropriate course of action in any situation.

'Change' is capitalized, above and elsewhere, to distinguish common usage from our use in the sense of the philosophy of I Ching.  Generally speaking, reference to 'Change' indicates the eternal process of transformation, creation and destruction, formation and disintegration.  

Herein, reference to 'Changes' or the 'Change operation' will also indicate what were previously called “expanded first-order difference integers*.”  The Changes are a product of the King Wen sequence of hexagrams.  The King Wen sequence is the likeliest form in which one will encounter the Changes. The King Wen sequence is arranged in pairs of hexagrams, beginning with Ch’ien and K’un, the  Gates of Change, represented by hexagrams #1 Heaven and #2 Earth.  It ends with hexagrams #63 After Completion and #64 Before Completion.

NB: *First-order differences is a term coined by the late Terrence McKenna in his presentation of Timewave (elsewhere discussed).  Our treatment of the Changes is a fuller development of McKenna's nascent concept.

It is unclear (and dubious) that any comprehensive explanation of the ordering principle behind the Wen sequence has been published.  This is to say that the sequence has not, to the knowledge of the author, been proven computable or 'deterministic' in the sense that the sequence may be generated from a reasonably small set of inputs and rules.  Compare this to say, the first 64 digits of π, or the of any common mathematical function or value.

---

<Terminology>
Before Change and the nature of the King Wen pairs are discussed, some terms:

• xiang: literal meaning: symbol; in this context, indicates the primordial reality (represented by HEAVEN, EARTH, FIRE, and WATER) that is believed to have emerged after the “separation” of YIN and YANG
• line:  (also: , place, position, or yao) any of the six steps or stages of a situation or process as given by I Ching, ordered from bottom to top.  Lines take on two basic forms: YIN (open, divided), and YANG (closed, undivided).  Through Change, a line may move to a different position within a hexagram, or transform into its opposite; that is, YIN may alternate to YANG, and conversely.
• ritual number: (closely-related to xiang) a product of the yarrow-stalk oracle which determines the specific type of each line in a hexagram. Ritual numbers may have the following values: [6, 7, 8, 9].
• digram (also bi-gram): (related to xiang) any of four combinations of two lines (places, positions, or yao) taken as a group.  Digrams are figurative representations of xiang, and have traditional ritual number assignments.
• trigram: eight discrete trigrams exist; each is a ordered collection of three contiguous lines that compose half of a hexagram.  Two trigrams are traditionally oriented in superposition.  They are named HEAVEN, EARTH, THUNDER, WIND, WATER, FIRE, MOUNTAIN, LAKE.
• hexagram: any of sixty-four distinct figures composed of six places, lines, positions, or yao. Hexagrams may also be interpreted as comprising three digrams or two trigrams.
• trigram exchange: an operation where the inner/lower trigram exchanges positioning with the outer/upper trigram such that the inner state becomes the outer state, and conversely.
• pantonggua: place-wise or positional negation (to/from YIN from/to YANG), applied to an entire hexagram.  YIN lines will alternate to YANG, and conversely.
• fangua: figure-wise inversion of a six-place hexagram that re-orders it from topmost place to lowest); applied to an entire hexagram.  Line 6 changes places with  position 1, line 5 changes places with line 2, line 4 changes places with line 3, and so on.
• King Wen pairs: (also: anti-pairs) sixty-four of these exist.  They individually comprise a unique hexagram (H) and its mate (H’).  In general, the mate (H’) is defined by trigram exchange.  Exception: in eight cases this operation produces the self-same hexagram; for those eight cases, the mate is defined as the pantonggua (the positional negation).
• Changes: (similar to hexagram) An ordered sequence of sixty-four six-line figures reflecting the state-change occurring within King Wen pairs as the sequence is fully-traversed (from beginning to end and back).