Yintelligence TM : The Mapping of the Pre-Heaven or FuXi Hexagrams to the Post-Heaven or King Wen Hexagrams

The Yijing易經 belongs to the famous group of the five classics. In the Ming and Qing dynasties, the Four Books and Five Classics 四書五經 were the subject of mandatory study by those Confucian scholars who wished to become government officials. The Yijing (Faure & Javary, 2002) consists of 64 figures called hexagrams 卦 formed by six Yin (broken) or Yang (unbroken) lines and commentaries on each of these hexagrams and their lines. The whole set of the hexagrams and their interdependencies make a dynamic model for the understanding or the anticipating of the different steps change goes through. The Yijing does not predict the future but gives a snap-shot of the actual situation and the holistic potentialities which may be deployed or not in order to better adapt to the on-going change. The different stages of a change are explained by the Yin-Yang polarity and the never-ending transformation of old-Yin 老陰 in young-Yang 少陽, old-Yang 老陽, young-Yin 少陰 and old-Yin again. The 64 hexagrams are built by pairing the 8 basic trigrams obtained by adding to each bigram a supplementary Yin or Yang line. Figure 1 depicts a 4-regions space (East/young Yang, South /old Yang, West/young Yin, North/old Yin) and a cyclic movement. A forthcoming article will explain the relation between this space organization and the whole set of 64 hexagrams as a network. The cyclic change is of major interest. Purpose: The 64 hexagrams of the Yijing are organized in 16 “first degree” nuclear families (Javary, 1997), which may be consolidated in 4 “second degree” nuclear families. There are another 16 families organizing the 64 hexagrams in a different structure. In the literature they are called the pre-heaven 先天 hexagrams (Schlumberger, 1987), and in this paper they are referred to as the FuXi 伏羲 hexagrams. They form 16 cyclic families, whose structure will be analyzed in this paper. This cyclicity induces a “predecessor-successor” relationship between the hexagrams belonging to the same FuXi family: H1H2H3H4H1. While the 4 members of each nuclear family are centered on a “master” hexagram that corresponds to a common inner lines’ structure, each FuXi family is cyclically structured and we call the hexagrams belonging to the same family FuXi related. Each Wen hexagram corresponds exactly to a FuXi hexagram. This correspondence reflects the respective trigrams’ arrangements of King Wen and FuXi. The trigrams of the FuXi hexagram and those of the corresponding Wen hexagram occupy the same locations in the FuXi and the King Wen arrangements displayed in the Figure 2. This paper explains the relation between the Wen hexagrams and their corresponding FuXi hexagrams based on structural and semantic properties. Summary: It has been demonstrated, that the pre-heaven and post-heaven hexagrams are pairwise related through specific relations defined by the opposition of the post-heaven hexagrams and 2 members of particular nuclear families which have opposite envelopes. Basically, the links between pre-heaven or FuXi hexagrams and post-heaven or Wen hexagrams are based on the hexagrams’ structure, especially on the opposition of 2 hexagrams. Two FuXi related hexagrams are connected to each other by a “predecessor-successor” like relation, the pre-heaven or FuXi hexagram somehow gives the necessary prerequisite which the post-heaven hexagram must own. In a pair of opposite hexagrams, the one possesses the complementary dimensions which are absent in the other. All these links may be represented using the trigrams deployed adequately on two circles, one with the pre-heaven trigrams, the FuXi arrangement, and the second with the post-heaven trigram or King Wen arrangement. Two corresponding trigrams are identically positioned in the two circular arrangements. The logical and sophisticated system applied to link the Wen to the FuXi hexagrams again reveals how subtly the whole system of 64 hexagrams has been conceptualized and evidences a new and original insight into the structure of the Yijing.


Introduction The Two Trigram's Arrangements form FuXi and King Wen
There are traditionally two basic organizations of the 8 tri-grams, the FuXi 伏羲 and King Wen 文王 arrangements (Figure 2).Each of these illustrates a specific dynamic and the interdependencies between the trigrams.
The implicate arrangement shows a perfect symmetry of the trigrams, which are diametrically opposed.Considering the explicate arrangement this symmetry only exits for the South/ North axis with the Li 離 and Kan 坎 trigrams.
The logic of these arrangements has been explained by different authors and we refer their studies to this literature.It is noticeable, that these two arrangements are always described independently from each other, as if they were disconnected.This article aims to demonstrate how closely they are linked together.

The Cyclic Nature of the Mapping and the 16 FuXi Families
The two arrangements correspond to two different permutations of the order of the trigrams.Two corresponding trigrams occupy the same place in the respective arrangements.As an example (Figure 3) Qian 乾 (in Pre-Heaven) corresponds to Li (in Post-heaven), Li to Zhen 震, Zhen to Gen 艮 and Gen goes back to Qian.

The Implicate Casting and the FuXi Hexagrams
As explained in the abstract, the explicate casting generates the situation hexagram with its changing old Yin and old Yang lines, defining the mutation hexagram.
The implicate casting results in transforming the situation and mutation hexagrams according to the King Wen and FuXi trigrams' arrangements.The building trigrams of the explicate hexagrams in the King Wen arrangement correspond to the implicate trigrams having the same position in the FuXi arrangement, thus delivering the FuXi hexagrams for the situation and mutation hexagrams.
It is mandatory to ascertain a precise analysis of the considered situation to clearly understand the specific meaning of the cast hexagram and its relations with the others.The whole set of 64 hexagrams acts as a network, and every hexagram plays a well differentiated role in this network.These roles are implicitly contained in the structure of the hexagram and in the different connections linking the hexagrams.Because each hexagram reflects a precise situation, a part of the interpretation consists in delimiting the semantic of each hexagram from the others.
The Yijing organizes all its hexagrams as 4-member families.
The 16 first-degree nuclear families feature 4 hexagrams which share common attributes.The 16 FuXi families, where the hexagrams are either preceding or succeeding in a circular chain.The nuclear structure is not always obvious but in fact less complicated to understand than the FuXi structure.In nuclear families the relations are somehow static, in a FuXi family there is a circular information flow which is continuously traversing the 4 members, feeding them with specifically needed inputs which trigger some actions or events.What unifies the 4 hexagrams in a nuclear family is the nuclear hexagram considered as a source of attributes for the other family's members.
The FuXi families' signification lies more in what differentiates two hexagrams.The pre-heaven and post-heaven hexagrams are pairwise related through specific relations defined by the opposition of the post-heaven hexagrams.This opposition is translated and expressed through the choice of 2 members of particular nuclear families having opposite envelopes; the latter hexagrams are then the related FuXi hexagrams.Thus indicating and delivering attributes, the post-heaven hexagrams have to integrate or realize some prerequisite conditions requested or recommended by its predecessor in the FuXi family.Two FuXi related hexagrams are bonded by a "predecessor-successor" like relation, the pre-heaven or FuXi hexagram somehow supplies the necessary qualifications which the post-heaven hexagram must own.In a pair of opposite hexagrams, one expresses complementary dimensions which are absent in the other.
A further article will propose an attempt to analyze the different FuXi families as distinct semantic objects and their importance in the interpretation of castings.

Definition of the Envelope Families
The First-and Second-Degree Nuclear Hexagrams and the Envelope Families According to the definition of a nuclear hexagram 1 , there are G.FELLEY 16 hexagrams corresponding to the structure of a first degree nuclear hexagram and each first degree nuclear hexagram builds a first degree nuclear family assembling 4 hexagrams which differ only by their specific envelopes.
Each of the 4 second degree nuclear hexagrams (Figure 4) generates its own 4-hexagrams family and the family's members are all first-degree nuclear hexagrams and differ only through the envelope configuration inside their own family.
The four second degree nuclear hexagrams condense some basic properties common to all the respective family members, which as first-degree nuclear hexagrams also condense some basic properties of their family members.All the nuclear hexagrams, first-and second degree may be considered as semantic consolidators.
The 4 envelopes act as differentiators in a family and are assemblers across the 4 considered second-degree nuclear families.Gathering together all the hexagrams having the same envelope generates 4 new families called envelope families (Figure 5).

The South, North, West and East Envelope Families
These families own the following properties: 1) Members of the same family differ from each other through the inner lines (2/3/4/5).They have the same envelope.
2) Opposite envelopes generate opposite families.The members of the South-family have their opposite in the North-family and the same relations hold for the members of the East-and West-families.
3) Mutating the envelopes' lines (one or two) maps one family into another family.
4) Each member of an envelope family is a first degree nuclear hexagram and therefore can be considered as a wildcard for 4 hexagrams differing from each other through their specific envelope but exhibiting the same lines configuration (2/3/4/5).

Definition of the FuXi Nuclear Pairs and FuXi Hexagrams
The FuXi Nuclear Pairs Each member of the 4 envelope families is representative for 4 hexagrams as a first-degree nuclear hexagram.
Pairing every member of the South-envelope family with one member belonging to the North-envelope family delivers 16 pairs of first-degree nuclear hexagrams, as Table 1 shows.The same occurs when pairing every member of the East-envelope family with members of the West-envelope family which delivers 16 pairs of first-degree nuclear hexagrams.These pairs of first-degree nuclear hexagrams are called the FuXi nuclear pairs.To facilitate the representation of these combinations, the hexagrams are replaced by their corresponding numbers, therefore the newly associated pairs look like this in Table 1.

The FuXi Hexagrams
The members of the envelope families are all first degree nuclear hexagrams, representing 4 hexagrams linked together by the same nuclear hexagram.Table 2 shows the logic of pairing the members of the South-envelope family belonging to hexagram 1 ("The Creative" 乾) with the members of the 4 Northenvelope families.To each member of the first-degree nuclear family generated by hexagram 1 exactly one member of one of the four first-degree nuclear families generated by the hexagrams 2/28/39/40 corresponds.The mapping of the South-envelope family corresponding to the hexagram 1 produces 4 pairs of hexagrams.Each of these pairs consists of one hexagram belonging to the first-degree nuclear family of hexagram 1 (either hexagram 1 or 44 or 28 or 43) and the second one is chosen out of the first-degree nuclear families of hexagram 2/28/39/40, thus delivering 4 pairs of hexagrams called the FuXi hexagrams.In this paper the FuXi hexagrams will be of gold-brown color like the paired fields in The same algorithm has to be repeated for the first-degree nuclear hexagrams 27/37/38, members of the South-envelope family.The left-side part of Table 2 changes accordingly to the first-degree nuclear families defined by the hexagrams 27/37/38 and the right-side part remains the same.

Conditions for the FuXi Families Principle1: The FuXi Sudoku
To each pair of FuXi hexagrams corresponds a pair of opposite Wen hexagrams.Table 3 illustrates this requirement and deserves a detailed explanation.In the left two cells there are two opposite Wen hexagrams XY, the two cells in the middle Table 3.
Opposite Wen hexagrams and their corresponding pair of FuXi hexagrams and FuXi nuclear hexagrams.(X * = opposite Hexagram of X).
Two opp.hex.FuXi pair of hex.FuXi nuclear pair of hex.X X * FuXi of X FuXi of X * Nuclear of FuXi X.
Nuclear of FuXi to X * .
contain a pair of FuXi hexagrams, each of them is a member of a nuclear family defined in the cells to the right, which contain a pair of FuXi nuclear hexagrams.The two colors indicate the relations existing between the 6 hexagrams.Table 3 shows how the opposition of two Wen hexagrams (explicate casting) is translated in the implicate casting.The property of opposition between the hexagrams in the explicate casting is "smoothed or nuanced" in the implicate casting.A FuXi nuclear pair is a pair of first-degree nuclear hexagrams with opposite envelopes and each of the corresponding FuXi hexagrams belong to one of these FuXi nuclear families.
Principle 2: Inverse Free A FuXi family never contains two opposite hexagrams.
Principle 3: FuXi Invariance Mutating all the corresponding lines2 of all the hexagrams of a FuXi family is still a FuXi family.There are families mapped onto other families or onto themselves, especially FuXi families whose hexagrams do not have corresponding lines are considered as self-reflecting.
Hexagram H1 is the FuXi of H2 this is the FuXi of H3 which is the FuXi of H4

Special Groups of Hexagrams Two FuXi Families without Corresponding Lines
All the 8 hexagrams, obtained by doubling the 8 trigrams (Figure 6) do not have corresponding lines.Therefore (Principle 3) they compose two self-reflecting FuXi families.
While it is clear that these eight hexagrams form two FuXi families, there are still many different ways to select the members of each family and to arrange the family in the proper order.

Two Further FuXi Families Whose Reverse and Opposite Hexagrams are Identical
There are exactly 8 hexagrams fulfilling this requirement (Figure 7).
Applying the Principles 2 and 3 to this group of hexagrams (Felley, 2013), it is clear that opposite hexagrams do not belong to the same family and the mutation of the corresponding lines switch from one family to the other.Using these properties, it is possible to pair these 8 hexagrams so that each member of a pair belongs to a different FuXi family.The hexagrams 11/17/53/63 build one FuXi family and the hexagrams 12/18/54/64 the second one.Each hexagram in the first group has its opposite in the second group and the mutation of all the corresponding lines of the hexagrams in the first group just  map into the second group.The other principles will set the hexagrams into the correct order.This is defining the mapping of the eight trigrams to themselves and therefore recovering the King Wen and FuXi trigrams' arrangements.
Next, all these hexagrams will be consistently organized in families using the Principles 1 to 4 defined in "Conditions for the FuXi Families".
Applying3 carefully the Principles 1 to 4 defined above to the FuXi nuclear pairs 1/2 and 53/54, links the hexagrams displayed in the next Table 4.
Extending the same principles to the FuXi nuclear pairs 53/63, 54/64, 63/64 establishes the remaining correspondences between the trigrams, as illustrated in the Table 5.
Comparing the hexagrams in the first two left columns with the ones in the third and fourth columns establishes the correspondences between the King Wen and FuXi trigrams' arrangements and the two tables can be summarized in the two trigrams arrangements from King Wen and FuXi (Figure 8).

Conclusion and Further Investigations Practical Significance of the Established Result
Usually a Yijing's casting consists of two principal hexagrams: a situation and a mutation if there are either old-Yin or old-Yang lines.To interpret the casting correctly it is mandatory to thoroughly understand what differentiates the significant situation and mutation hexagrams from the other 62 remaining hexagrams.To do this, considering the nuclear, opposite, and the lines hexagrams, pre-heaven or FuXi, is very helpful.If the roles of the nuclear, opposite and line hexagrams are relatively known and clear, the importance of the pre-heaven or FuXi hexagram is generally not sufficiently recognized.The under-standing of how the pre-heaven or FuXi hexagrams connect to  the others will, no doubt, substantially strengthen the statements of a casting, and provide them with a better traceability and comprehension.
The whole set of post-heaven hexagrams is called the explicate casting.The FuXi hexagrams are used to enrich its interpretation by evidencing the implicate casting related to the corresponding situation and mutation hexagrams.By linking the explicate hexagrams to their related FuXi hexagrams the interpretation of the situation unfolds and reveals correlations between the different parameters which influence the analyzed change process.
It also contributes to substantially enforce: 1) The specific understanding of the 64 hexagrams and their interdependencies, especially the symmetry existing between the hexagrams' geometry and the carried semantic elements.

G. FELLEY
2) The integration of the FuXi hexagrams in a casting's interpretation considerably helps to demonstrate the relevance and consistency of the casting by revealing hidden connections between the selected hexagrams.
3) The perception of the Yijing internal logic as a 4-valued logical system and therefore the support of a thin granular conceptual framework for decision making.
4) The assumption that the trigrams play a significant role in organizing the hexagrams especially to describe the mapping of the pre-heaven to the post-heaven hexagrams.
These sequences may be considered as a line of growing complexity in the hexagrams' specific logic.Starting with hexagrams 1 and 2, equipped with a uniform, erratic Yang and Yin logic, this sequence goes through different states to end with hexagrams 63 and 64 both displaying a sophisticated intricacy.
The process is launched with the re-arrangement of the trigrams Qian and Kun 坤, creating six lines, which generate the two-cyclic dynamic of hexagrams 11/12 (Figure 10).This dynamic culminates in the delicate interplay of the hexagrams 63 and 64, the last closes the Book of Changes igniting a new organization.
This entanglement of the hexagrams is deployed by means of mutating the corresponding lines of the pair of hexagrams 11/12.Mutating the lines 1 and 4 produces hexagrams 32/42, mutating the lines 3 and 6 produces hexagrams 41/31 and mutating the inner lines 2 and 5 results in the last two hexagrams 63 and 64.While the hexagrams 31, 32, 41, 42 are related to each other being either reverse or opposite and the hexagrams 11 and 12 are reverse and opposite 4 , the hexagrams 63/64 are the only two consolidating all these attributes with the fact that they are the reciprocal first-degree nuclear hexagrams and identical to their second-degree nuclear hexagrams, the last property being also shared by the hexagrams 1 and 2. Thus the four hexagrams 1, 2, 63, 64 play an outstanding role in the Yijing's inherent logical structure.This is based on a four-valued logic defined by the very deep characteristics of each of these four special hexagrams.These characteristics will be analyzed more in detail in a forthcoming article at a later date.

Principle of Inheritance or Seeding
Within a given FuXi family, the cyclic chain of the four hexagrams defines a predecessor and a successor for each hexagram, thus creating a kind of "inheritance" of hexagram specific attributes.The impact of these influences deserves more attention and will be analyzed in a forthcoming article.
As an example, hexagrams 3 and 50 are considered in more detail (Table 6).These two hexagrams have a special meaning, because they also divide the Book of Changes in two parts.All the hexagrams between these two also have their opposite between them and consequently all hexagrams between the hexagrams 51 and 64 have their opposite in that range 5 .
The opposition of the Wen hexagrams 3 and 50 corresponds to the opposition of the FuXi nuclear hexagrams 40 and 37. Noticeable is the fact that the two FuXi hexagrams 36 and 10 are almost opposite each other.Only the line 1 does not fulfill the right conditions.
The hexagrams 3 and 50 both describe situations of qualitatively different renewals, the hexagrams 36 and 10 explain qualitatively different behavioral attitudes rooted in the opposite nuclear hexagrams 40 and 37, most suitable to achieve the respective goals of the hexagrams 3 and 50.
To investigate all these interdependencies between the hexagrams belonging to the same FuXi family and their assigned FuXi nuclear pairs will be the central theme of forthcoming publications.This example aims to demonstrate the possible benefits derived from this consideration of the generation process for the FuXi hexagrams.

The FuXi Hexagrams as Marks for the Book of Changes
It is usual to consider the hexagrams 1 to 30 as the first part of the Book of Changes and the remaining hexagrams 31 to 64 as its second part.Each part contains 18 different hexagram figures, some of them can be reversed; some cannot.
The hexagrams 1 and 2 are respectively the FuXi hexagrams of the hexagrams "Fire" 離 and "Abysmal" 坎6 , and they close the first part of the Book of changes.
Further, as illustrated by Table 7, hexagram 3, "The difficulty of the Beginning" 屯, is the FuXi of hexagram 31, "Influence" 咸, the hexagram initiating the second part of the Book of Changes.This correspondence between the implicate or pre-heaven order and the partitions of the Book of Changes demonstrates again how smartly, precisely and differentiatedly the Yijing methodology was conceptualized with the support of the hexagram's structure!How far may the FuXi arrangement help to understand the organization of the Book of Changes?
The network of the FuXi families interferes strongly with the nuclear families.The perspective of looking at the structural roots of the pre-heaven or implicate hexagrams developed in this paper opens new ways to better understand the links between the hexagrams and to more appreciate the semantic identity of each of them.This results in an enhanced analysis and better groundwork to validate the strategy suggested by the casting and therefore to help promote the Yijing as a valuable tool for decision making in a managerial environment.

FuXi Hexagrams and Nuclear Families
A further prospective field deserving more attention is to consider identifiable relations between the hexagrams coupled with FuXi hexagrams rooted in the same nuclear family.As an example see hexagram 64 (Table 8).
Since 2008 I have been attending the monthly seminar series given by Pierre Faure in Paris.In my opinion Pierre Faure is one of the best Western Yijing experts.His profound and holistic knowledge of the Book of Changes and his enthusiasm to share it with other Yijing practitioners and Yijing addicted persons has substantially helped me to better understand the fantastic entanglement of the hexagrams and to develop the ideas used here to establish the connection between these two essential trigrams' arrangements.
What kinds of characteristics have the hexagrams 11, 17, 24, 43 in common?What happens when hexagram 63 substitutes hexagram 64?This is a large set of questions, whose answers will help to enhance the understanding of the rules of change.

To Summarize
A hexagram consists of a structured figure of six-lines and specific semantic contents.Moving from one hexagram's geometry to another one moves in a "parallel manner" the semantic contents.This is no doubt one of the most interesting characteristic features of the Yijing.Also many thanks to Dr. Barbara Davies who guided me many years ago to the igniting discovery of Yijing, to Evelyn Frisch, my life partner, for having supported for so long all the Yijing floodings in our everyday life, to Christine Lorgé for reviewing this manuscript, to Prof. Dr. Rolf Dornberger and Prof. Dr. Arie Verkuil, from the UNW, who always encouraged me and gave me their support to deploy Yijing-centered activities in the UNW, and to all my friends and colleagues not explicitly mentioned here for having helped me in some way or other.

Figure 3 .
Figure 3. Illustrating the 4-cyclicity of the FuXi and King Wen arrangements with the trigram Qian.
Figure 4.The 4 second-degree nuclear families.South North East West Detailed pairing for the South-envelope family of nuclear hexagram 1 with the sequence of North-envelope families, represented by 4 nuclear hexagrams 2, 28, 39, 40.

Figure 10 .
Figure 9.A special sequence of pairs of hexagrams.

Table 1 .
Pairings of the envelope-families.