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21st July 2008 | Draft
Sustainability through Magically Dancing Patterns8x8, 9x9, 19x19 -- I Ching, Tao Te Ching / T'ai Hsüan Ching, Wéiqí (Go)- / - This exploration has been stimulated by exchanges with Ye Zude, Maurice Yolles, Chris Lucas and Peter Collins, none of whom should be held responsible for its essentially speculative nature. Introduction Interweaving 8x8 and 9x9 within 19x19 Distinguishing the 8x8 patterns within the 19x19 pattern Consideration of any process of extension to a 9x9 "arrangement" Possibility of a dynamic pattern methodology "Dancing patterns" Formation dancing: a "body-centered" alternative? Cross, swastika variants and lauburu Some dynamic pattern questions Triangular representation of 64 I Ching hexagrams Double triangular representation of hexagrams: Star of David Patterns of possible interactions between linear and triangular configurations Dynamics, resonance hybrids and Pascal lines Mirroring, enantiodromia and the engine of change Patterning 60 within 19x19 Relevance of 19x19 to annual calendar and associated cycles Towards the specification of a pattern dancing applet Sustainability: sudoku, go, chess, yantras, sigils and bagua Sustainable "cognitive engine" and emergent "wisdom engine"? Conclusions References IntroductionThis is an exploration of the interrelationship between patterns fundamental in different ways to cultures of the East and the West. Specifically the focus is on the 8x8 pattern constituted by the 64 hexagrams of the I Ching (Yijing) and the 9x9 pattern constituted by the 81 tetragrams (or quadgrams) of the T'ai Hsüan Ching (Tai Xuan Jing). These two sets of patterns have as their root the 3x3 pattern of the BaGua and the 4x4 pattern. The 3-fold pattern is fundamental to thinking based on the enneagram. The 4-fold patterns has been extensively explored from a Western perspective by Carl Jung and thereafter in such patterns as the MBTI. The 3x4 pattern is of prime significance to thinking based on 12, notably that developed by Arthur Young on learning/action cycles. Much attention has been given to such patterns by the mathematically inclined through the challenge of so-called "magic squares" and the interesting arrangements of numbers which emerge as significant. Also of relevance is the manner in which such patterns have become fundamental to two distinct board games. The Eastern game of go (Wéiqí) is based on a board of 19x19. The Western game of chess is based on a board of 8x8. Such games have been the focus of a considerable amount of computer-enhanced thinking. This exploration follows from a series of earlier papers on the I Ching and the Tao Te Ching (9-fold Higher Order Patterning of Tao Te Ching Insights: possibilities in the mathematics of magic squares, cubes and hypercubes. 2003; 9-fold Magic Square Pattern of Tao Te Ching Insights experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2006). Early concern for the challenge was articulated in Representation, Comprehension and Communication of Sets: the role of number (1978). Interweaving 8x8 and 9x9 within 19x19Whilst such interweaving may have been highlighted as obvious in more specialized contexts, the possibility and significance of interrelating 8x8 and 9x9 only became apparent to this writer through consideration of the 19x19 pattern of the game of go. The patterns may be interwoven as indicated in the following table. Essentially 4 sets of 8x8 (the game of chess) may be set into the corners of a 19x19 table -- such as to make provision for extra rows/columns permitting each 8x8 set to be bordered on the outer sides, effectively creating 4 sets of 9x9 arrays. Within the 19x19 pattern this leaves a single central column/row. Given the widespread focus on 10-based patterns, however, it is interesting that this highlighted within this framework, but only provided that the 4 sets (of 8x8 expanded to 9x9) are understood as overlapping or "sharing" the central row/column -- by which they are thereby "held together". Table 1: 19x19 Wéiqí (Go) board divided into quadrants of 8x8
Distinguishing the 8x8 patterns within the 19x19 patternThere may well be many ways of considering interesting arrangements of the cells of an 8x8 pattern within the 19x19 pattern. It is however appropriate, and useful, to start with a commonly accepted pattern. That used for this exercise, as an illustration, is the one traditionally used in distributing the hexagrams of the I Ching. That is the pattern used in the lower right hand quadrant. This pattern can then be simply transformed into the distinct patterns of the other three corners -- by a process evident by inspection of the table as a whole. This sets aside -- or postpones -- any questions of other possible arrangements. Consideration of any process of extension to a 9x9 "arrangement"Various processes could be used to extend the 8x8 pattern into 9x9 -- whether starting from 8x8 or from 9x9. Any such 9x9 pattern can form one quadrant of the 19x19 arrangement above, rotating the arrangement from quadrant to quadrant as illustrated with the 8x8 arrangement. Of particular interest is the traditional focus on magic squares and the possibility of ensuring the emergence of interesting patterns -- aesthetically interesting to mathematicians. The writer is not competent to explore or evaluate the many such possibilities that have been highlighted in the literature. A number of pointers are explored separately (9-fold Magic Square Pattern of Tao Te Ching Insights: experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2006). One recently discovered valuable example of a 9x9 magic square, cited therein, is as follows..
The focus here is rather on the implications of the concern with whether a particular arrangement is "right", appropriate or "better" than another. One problematic consequence of this is that a preferred arrangement is then promoted in terms of its particular aesthetics and degree of perfection. The arrangement is given a degree of rigidity within which certain dynamic relationships between its elements are possible. It may acquire a name and be associated with its discoverer -- even to the point of being recognized to some degree as intellectual property. Methodologically the process may also be unsatisfactory given the range of possible arrangements of varying merit and potential significance. In terms of their wider significance, given the mathematical inclination desirable, the implication of the patterns may be obscured and subject to excessive mystification -- especially because of their challenge to comprehension when expressed numerically. Possibility of a dynamic pattern methodologyThe typical search for "arrangements" as with "magic squares" aims at discovery of the most integrative patterns according to various criteria. The focus is on the highest degree of order that can be embedded in the arrangement -- hence the use of "perfection" as a descriptor in the magic square literature. The challenge is that this perfection is only visible to those capable of decoding the pattern. Typically it is essentially incomprehensible to more than a few. Another approach may however be taken by recognizing the process of pattern selection -- effectively by recognizing and integrating the pattern of consideration of (aesthetic) alternatives. This implies a shift from a single target "best" static arrangement, whatever its justification, to the alternation dynamic between several (even many) distinct possible arrangements. It is this dynamic which is characteristic of any selection process. This dynamic is also characteristic of the interaction (or dialogue) between the variety of alternatives as they are embodied in methodologies, belief systems and institutions -- and even cultures. Hence, for example, the peculiar differences in stress variously placed on 8x8, 9x9, 10x10 and 19x19 in cultures of East and West. And, as a consequence, hence the challenge of interrelationship between those cultural frameworks -- as static rigid frameworks. "Dancing patterns"With such a shift to a dynamic perspective, how might the pattern of the dance be understood in the interplay between alternative possibilities? Note that the argument here is in favour of ensuring that the pattern of the "dance" becomes apparent by other means -- rather than being remarkably obscured to most by coding the squares in the table with numbers. With the advent of widespread access to computers, the kind of pattern dance could be made readily apparent (and widely comprehensible) by using shifting patterns of colour coding -- rather than numbers. Java applets are an obvious choice -- possibly enhanced by sonification. Their use in relation to colour charts provides the simplest example. The intuition for this exploration is the recognition that both in chess and go, those of significant expertise see the board as a whole in terms of shifting patterns of force (see Pavle Bidev. Chess: a mathematical model of the cosmos, 1979; Reuben Fine. The Psychology of the Chess Player, 1956). For those specializing in number theory, sets of numbers do indeed "dance". They have valencies that enable them to form various partnerships -- resulting in lines of relationship along columns, rows and diagonals. Few have real access to such understanding or its potential implications. The concern here with articulating particular patterns is therefore set in the context of understanding how any dance between them might be presented and comprehended. Some accessible metaphors may be variously helpful:
Such examples point to the possibility of using understanding of various possible arrangements of numbers within the table to drive colour coding of such numbers -- whether at the level of the cell, the row or column, or the diagonals -- such that transformations between the arrangements can be visually tracked as a morphing process. The question is what renders the dynamic aesthetically significant and memorable -- and inherently comprehensible? Formation dancing: a "body-centered" alternative?On the assumption that the singular 8x8 arrangement of the 6-fold hexagrams might be position centrally within the 19x19 array, this might be done as follows. Successive concentric outer rings might then be represented by 5-fold, 4-fold, 3-fold, 2-fold and 1-fold indicators. Note that the 1-fold might take the form of either a complete or broken line (yang and yin signs) that could then alternate in the positions of the outermost ring. Similarly the 4 forms of the 2-fold might then alternate in the next inner ring -- with a similar process for the 8 3-fold, the 16 4-fold and the 32 5-fold. Table 3: 19x19 board with indication of central set of 64 hexagrams
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