7th December 2006 | Draft
9-fold Magic Square Pattern of Tao Te Ching Insights
experimentally associated with the 81 insights of the T'ai Hsüan Ching
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Experimental development of 9-fold Higher Order Patterning of Tao Te Ching Insights
Magic square presentation
Pan-magic square presentation
Most-perfect magic squares
As discussed in the related papers (Commentary on Tao Te Ching Interpretation: and the possibility of higher order patterning 2003; Hyperspace Clues to the Psychology of the Pattern that Connects, 2003; Musical Articulation of Pattern of Tao Te Ching Insights: experimental sonification based on magic square organization, 2003) the prime concern in this experiment is with the possibility of configuring fundamental insights in ways which facilitate their comprehension. A particular concern is with their interrelationship as a pattern, or system of checks and balances, rather than on the significance of any particular insight.
Potentially the sets of insights developed in ancient Chinese culture, and fundamental to the philosophy of life and governance, is therefore especially relevant at this time. Great importance was then attached to both the coherence of the pattern as a whole and its capacity to model the changes experienced and to be anticipated. The use of three of them for purposes of divination [review], now considered inappropriate by some [discussion], should not distract from their insights into the operations of complex psychosocial systems.
This is especially the case in the light of the increasing emphasis now placed on "values" and "wisdom" in relation to global governance, whether "faith-based" or not, and on elaboration of more appropriate strategies. If the sets of insights effectively functioned, or were used, as what would now be termed "global models" or "world models", then the adequacy of their predictive capacity should be distinguished from the adequacy of their descriptive/explanatory capacity -- especially given the requirements of them by their culture. The most comprehensive of these present day models of the world system face a particular challenge in that the "predictions" they offer fail to engage the individual as coherent and credible -- and therefore fail to engender political support for appropriate concerted action (cf Donnella Meadows, et al. Limits to Growth: the 30-Year Update, 2004).
These concerns also follow from an earlier exploration of the possibility of higher orders of cognitive engagement in strategic issues (Governance through Patterning Language: creative cognitive engagement contrasted with abdication of responsibility, 2006; Creative Cognitive Engagement: beyond the limitations of descriptive patterning, 2006).
This is an exercise in building on the tables in the separate paper 9-fold Higher Order Patterning of Tao Te Ching Insights: possibilities in the mathematics of magic squares, cubes and hypercubes (2003). There the 81 insights of the Tao Te Ching (Book of the Way and its Virtue) were presented experimentally in cells in a 9 x 9 square to explore the possible existence of higher order patterns of significance. Titles have here been added to the insights in the cells -- but these titles are derived from the 81 insights of the T'ai Hsüan Ching (Tai Xuan Jing / Canon of Supreme Mystery /The Great Dark Mystery) of Yang Hsiung (Yang Xiong), thus establishing a relationship between the two sets of insights. This paper is therefore a development of the previous one only through the addition of points relating to the T'ai Hsüan Ching in order to faciliate comprehension of any possible relationship with the Tao Te Ching.
Although the T'ai Hsüan Ching is a different publication, it is of the same era in Chinese culture. It has been described as one of the world's great philosophic poems comparable in scale and grandeur to the De Rerum Natura of Lucretius. As noted in the valuable clarificatory commentary by Michael Nylan and Nathan Sivin (The First Neo-Confucianism: an introduction to Yang Hsiung's "Canon Of Supreme Mystery", 1995) in distinguishing the T'ai Hsüan Ching ("the Mystery") from the I Ching ("the Changes") :
Although Nylan and Sirvin make no explicit mention of any relation between the Mystery and the Tao Te Ching, as purportedly articulated by Lao-Tzu, they do, like others, refer to the Tao Te Ching as "the Lao-Tzu" and note Yang Hsiung's recognition of such a connection in the following terms:
As with the I Ching, the T'ai Hsüan Ching was orginally one of several works that formed the Ta Pu or Grand Oracle. It is considered to be a companion volume to the I Ching -- which is far better known. Like the Tao Te Ching, the T'ai Hsüan Ching has 81 insights known in this case as Shou. Like the I Ching these are associated with a diagram of broken and unbroken lines. In the case of the 64 insights of the I Ching, each is represented by six such lines (a hexagram), each of which may be unbroken (yang), or broken once only (yin). In the case of the T'ai Hsüan Ching, these are represented by four such lines (a tetragram or quadgram), each of which may be unbroken, or broken once or twice. The sequence of numbers in the Tai Hsuan Ching is conventionally arranged into three groups of three called T'ien (1-27), Jen (28-54) and Ti (55-81) as discussed below.
Each of the 64 hexagrams of the I Ching is traditionally associated with a descriptive name for the explanatory details associated with it. There is no such descriptive name associated with the 81 insights of the Tao Te Ching , which are each presented through a set of poetic verses. These were reduced, experimentally, to a single phrase in an earlier phase of this experiment (Tao Te Ching Interpreted Succinctly: a 9-fold pattern of 81 insights presented as phrases, 2003). In the case of the T'ai Hsüan Ching, each of the 81 insights ("Heads"), has a title and is explicated through 9 very short philosophical verses (or "Appraisals", known as Tsan), typically presented in allegorical form -- and totalling 729 (as analyzed by Walters) or 731 (as analyzed by Nyland and Sivin). Although not immediately relevant to the following experiment, it is appropriate to note that each of the 81 insights is linked to one of the 64 hexagrams of the I Ching (with some duplication, of course) to evoke the old meanings and associations.
For Nylan and Sivin:
The sequence of 81 insights of the Tao Te Ching are however typically not clustered in any way. In the previous experiment, the conventional order was used to cluster the 81 insights into a 9 x 9 table to derive groups of 9 insights by row and, separately, by column. The question was whether a magic square clustering, much favoured in that period in China, would enable new insights to be elicited from the resultant pattern. It is important to note that the mathematical properties of magic squares continue to be of great interest to mathematicians -- but very little attention is paid to their potential role in ordering systems of concepts. In that period however the 3 x 3 Lo Shu magic square was essential to the ordering of the most fundamental Chinese insights.
The 81 insights of the T'ai Hsüan Ching were much more closely associated with magic square orderings than the Tao Te Ching. The following experiment is based on the work of Derek Walters (The T'ai Hsüan Ching: the hidden classic -- a lost companion of the I Ching, 1983, subsequently titled The Alternative I Ching, 1987) who reconstructed and translated it. Walters notes the relationship of the order of the T'ai Hsüan Ching to the arrangement of the classic Magic Square of Master Tsan -- using the first modern numbering system of the Han dynasty (a base ten system like that of the Romans) [more]. Walters explores the use of magic squares as a means of ordering the sets of philosophical verses (tsan) clarifying each of the 81 insights.
Prior to Nylan, Walters is emphatic at the contrast between the T'ai Hsüan Ching and the earlier I Ching it was designed to improve upon. He notes that in the I Ching:
The principal difference for Walters is that:
Part of the study by Walters is on the relevance of the tri-partite focus of the T'ai Hsüan Ching to current scientific thought. Understanding of such a third force could be usefully compared with the concept of morphogenesis as developed by Magoroh Maruyama (The Second Cybernetics: deviation-amplifying mutual causal processes, 1963; Morphogenesis and Morphostasis, Methodos, 12, 1960, pp. 251-296).
However as a reviewer of Nylan's translation, Richard Hunn strongly argues:
Part of the challenge of this experiment is to render comprehensible a dynamic framework through which such differences of perspective, so typical of academic dialogue, are understood as intrinsic to psychosocial dynamics rather than in some way external to them. The widespread emphasis on correct and incorrect views too readily reinforces the style of binary discourse that justifies bloody conflicts -- as common today as at the time at which such works originated -- and a reason for their elaboration.
This following very simple experiment associates the titles (as translated by Walters in one or two words) of the 81 insights of the T'ai Hsüan Ching to the single-phrase presentation of the Tao Te Ching -- in the tabular form explored previously. A choice was however made to convert the terms chosen by Walters into gerund form (of a synonym, if necessary), where this was not already the case. For example, Walters has #1 as The Center, converted here to Centering; he has #2 as Surrounding, not converted here, etc. The reason for this is an interest here in the dynamic associated with the insight rather than reinforcing any static sense of a description. This adaptation may indeed be misleading given that the version presented by Walters typically offers only a metaphoric allusion to the sense of the insight.
Note that, following Walters, Michael Nylan (The Canon of Supreme Mystery by Yang Hsiung. 1993) produced a new translation of the T'ai Hsüan Ching using other English title variants in some cases. The representation of the Tai Xuan Jing tetragram symbols according to the web Unicode 5.0 standard (range 1D300-1D35F) uses another set of titles but specifically notes that these are not correct translations of the usual Chinese terminology.
The question raised by this experiment is whether the titles (from the T'ai Hsüan Ching) and the phrase (from the Tao Te Ching) then have any relationship -- whether inherently meaningful, intuitive, or aesthetically suggestive -- despite the complex pathways through which the English texts were derived, and the criticism that could be validly made of this approach. Minimally however it permits an inspection of the juxtaposed elements from quite different sources. It is possible that any relationship that may exist is necessarily to be understood as a paradoxical challenge to comprehension like a Zen koan (gōng-àn in Chinese).
The configurations of insights in the following tables point to the possibility that, like stringed instruments, they in all probability each require a form of semantic or memetic "tuning" to be able to communicate the interplay of insights whose totality is named by the author as "mystery". To what extent do the interlocking numbers in any magic square patterning enable such comprehension?
The magic square configuration, of significance at that time, may help to determine whether the relatiuonship between the insights of the Tao Te Ching and the T'ai Hsüan Ching is very significant or fortuitous at best.
Table 1: The basis for the following table of the 81 insights of the Tao Te Ching is discussed in a separate commentary. The rows of the table provide 9 groups in terms of the conventional ordering in the Tao Te Ching. The columns of the table provide 9 different groups in terms of the alternative ordering represented by those columns. The title from the Tai Hsüan Ching (or adapted therefrom) has been added before each phrase, in italics, respecting the number order.
Table 2: The basis for the following table of the 81 insights of the Tao Te Ching is discussed in a separate commentary. It is an experiment in the organization of these insights into clusters. The table is made up of 9 nested tables (each of 9 cells). Each nested table corresponds to one of the rows from Table 1 above -- each row above being transformed into a nested table of 3x3 cells below. Note that the insight numbers in each row total to 369, as do the insight numbers in each column.
As a further experiment in organization, the insights were clustered according to the mathematical principle of the magic square (see Table 2). The structure of Table 2 is best understood by considering the first row of 9 insights (1 to 9) in Table 1. These 9 appear as the central nested table in the top row of 3 nested tables in Table 2. The 9 in that nested table are however presented in an order based on the structure of what is known in mathematics as a magic square -- -- namely the numbers of the insights (of the conventional ordering in the Tao Te Ching), whatever the direction of addition, whether vertically (8+3+4; 1+5+9; 6+7+2), horizontally (8+1+6; 3+5+7; 4+9+2), or diagonally (8+5+2; 4+5+6), total in each case to 15 (as indicated there as 1:15). Similarly if the numbers of each row are multiplied (8x1x6; 3x5x7; 4x9x2) they together total to 225 -- as do those of the columns (8x3x4; 1x5x9; 6x7x2).
In such a square the numbers of the first 9 insights (1 to 9) (of the conventional ordering in the Tao Te Ching), whatever the direction of addition, whether vertically (8+3+4; 1+5+9; 6+7+2), horizontally (8+1+6; 3+5+7; 4+9+2), or diagonally (8+5+2; 4+5+6), total in each case to 15 (as indicated there as 1:15). Similarly if the numbers of each row are multiplied (8x1x6; 3x5x7; 4x9x2) they together total to 225 -- as do those of the columns (8x3x4; 1x5x9; 6x7x2).
This is an adaptation of the Lo-Shu order known in classical China. In the table as a whole, the 9 nested tables have been positioned in a manner corresponding to this same order. Thus the first row of nested tables in Table 2 (above) groups the contents of rows 8, 1 and 6 respectively from Table 1 (namely rows marked there as VIII, I, and VI), the second groups 3, 5 and 7, with the third grouping 4, 9 and 2. The principle of the magic square is discussed elsewhere (notably by Alan Grogono), together with its long history dating back to 2800 BC [more | more | more | more].
The Lo Shu is the only magic square of order 3. Namely there is just one 3x3 magic square -- although with rotations and reflections, there are eight variations of what is essentially the same square. An associative magic square of order n is one for which every pair of numbers symmetrically opposite the center sum to n2+1. The Lo Shu square is associative -- but is not a panmagic square for which all the diagonals --including the broken diagonals obtained by "wrapping around" the edges -- total like the rows and columns.
Just as the magic square total for the first 3x3 nested table is 15 (indicated above in Table 2 as 1:15), each other 3x3 nested table gives rise to its own total (indicated beneath it, eg 4:96, 9:231, and 2:42). The 9 such totals from each nested table also constitute a magic square -- with a total figure of 369. As might be expected, if the table as a whole is treated as a 9x9 magic square, the total is also 369.
Interesting patterns can be generated from magic squares when the numbers of the squares are replaced by symmetric symbols.
Mathematically a "continuous" ("pan-magic", pan-diagonal, Nasik or Jaina) square has the additional property that even the broken diagonals add to the same total as those of the magic square. It was long supposed that a 9x9 pan-magic square did not exist, but one such based on the 81 numbers 0 to 80 is reported by Alan Grogono [more]. He explains this early belief as probably due to the absence of any obvious pattern to use to create a regular 9x9 square. Constructing a square by expanding a 3x3 square indeed produces a magic square as in Table 2 but not a pan-magic one. In addition, amongst odd-order pan-magic squares, most interest has been focused on the regular prime number squares. These lent themselves to analysis more readily and to calculation of the number of regular pan-magic squares which could be constructed with an underlying pattern.
Grogono argues that the analysis (and construction) of magic squares is more logical, and the results make more sense, when the smallest number is 0 -- instead of 1. This would imply that a 9x9 square of the Tao Te Ching insights should run from 0 to 80 instead of from 1 to 81. This would not affect the pattern of Table 2, provided that the rows from which it was derived in Table 1 were then renumbered from 0 to 8 (instead of from I to IX).
Of further interest, however, is to use the 9x9 pan-magic square order discovered by Grogono to redistribute the 81 insights. There is an interesting clue to the relevance of renumbering the first insight from 1 to 0 -- in the text of that first insight itself.
Given the properties of the pan-magic square, in this case the row containing 0 (the insight traditionally numbered 1) in his case was shifted to the central position (and checked in the online facility he provides to ensure that it remained a pan-magic square). This gives the following (Table 3a) from which the ordering in Table 3b was then produced -- retaining the numbering of the insights in Table 1 (namely 0 in Table 3a is 1 in Table 3b, in order to correspond to Table 1).
In 1999 Dan Washburn made the point that "The vastu-purusha-mandala is a square of 81 subsquares with 9 subsquares on each side. Take a Lo Shu magic sqaure of 3 and place a Lo Shu magic square of 3 in each of its 9 subsquares and you have a 9 x 9 square of 81 subsquares. So the vastu-purusha-mandala is the Lo Shu square squared, or seen in more detail." According to Vini Nathan (Vastu Purusha Mandala: Beyond Building Codes, Nexus Network Journal, vol. 4, no. 3, Summer 2002), The Vastu purusha mandala has been defined as "a collection of rules which attempt to facilitate the translation of theological concepts into architectural form." This law of proportions and rhythmic ordering of elements not only found full expression in temples, but extended to residential and urban planning as well. He argues that the influence of the Vastu purusha mandala extended beyond building activity to encompass the cultural milieu as well.
Note that the insight numbers in each row now total to 360 (instead of 369, as in Table 2), as do the insight numbers in each column.
Since the ternary numbers go from 0 to 80 (as indicated by Grogono in Table 3a) instead of from 1 to 81, their equivalent in the Tai Hsuan Ching is obtained by adding "plus 1" to the numbers in Table 3a, conserving the order.
Note that the insight numbers in each row now total to 369 (as in Table 2, and in contrast to the 360 of Table 3a), as do the insight numbers in each column). In addition the total of the insight numbers in any 3x3 nested square (even across highlighting) also total to 369 -- whereas those of the 3x3 nested squares (even those highlighted) in Table 2 are not equal (although those of the central 3x3 square only do indeed total to 369). Note that the difference of 9 between 360 and 369 derives from the difference in insight numbering from 0-80 against 1-81 (giving a difference of 9, whether in row or column totals).
The sequence of numbers in the Tai Hsuan Ching is conventionally arranged into three groups of three called T'ien (1-27), Jen (28-54) and Ti (55-81) as in the table below (which is not a magic square arrangement as explored here).
The property of pan-magic squares whereby there is a continuity of the "magic" property even along the broken diagonals may be consistent with the phenomenon expressed poetically in the much-quoted stanza of T S Eliot (in Little Gidding, 1942):
We shall not cease from exploration
Mathematically a magic square is bimagic (or 2-multimagic) if it remains "magic" after each of its numbers have been squared -- a bimagic square thus has the additional property that if each number in the square is multiplied by itself (squared, or raised to the second power) the resulting row, column, and diagonal sums are also magic. Bimagic squares are a subset of the class of multimagic squares; it is believed that no bimagic squares of order less than 8 exists (Benson and Jacoby 1976). The original 3x3 Lo Shu square is far from being bimagic, since the sums of the squared numbers (of the rows or columns) vary between 77 and 107. The discoverer of the first bimagic square, G. Pfeffermann later published in Les Tablettes du Chercheur (15 July 1891) the first 9th-order bimagic square. In the case of the examples of bimagic squares based on 9x9 in Table 4 (below), the rows and columns sum to 369 as before. But if each number is squared, the sum is then 20,049.
A special type of pan-diagonal magic square is characterized as most-perfect [more]. An example of a 12x12 most-perfect magic square is provided by Ian Stewart [more]. The numbers in every 2x2 square sum to 286. More generally every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= n2 + 1). Any pair of integers distant ** along a diagonal sum to T.
There are extensive resources on magic cubes and hypercubes [notably Harvey Heinz and Marián Trenkler] that may offer even more powerful ways of organizing the 81 insights. A magic cube is a three-dimensional version of the magic square in which the rows, columns, pillars (or "files"), and four space diagonals each sum to a single number known as the magic constant. If the cross section diagonals also sum to that constant, the magic cube is called a perfect magic cube; if they do not, the cube is called a semiperfect magic cube, or sometimes an Andrews cube (Gardner 1988). A pandiagonal cube is a perfect or semiperfect magic cube which is magic not only along the main space diagonals, but also on the broken space diagonals [more]. In a panmagic square, in addition to the main diagonals, the broken diagonals also sum to the magic constant.
Harvey Heinz (Magic Cubes - Introduction, 2003) has reviewed the variety of, often confusing, definitions and features of "magic cubes" (see also his Magic Cubes Definitions, which includes a discussion of cube features) and has allocated them to distinct classes according to the types of parts that must sum correctly for the more advanced cubes. His classes may be summarized here as:
Heinz notes that a magic cube is called normal if it consists of the numbers 1 to m3 (or 0 to m3 - 1). A magic cube is called associated if all pairs of two numbers diametrically equidistant from the center of the cube equal the sum of the first and last number in the series. If the associated cube (or other dimension of hypercube) is an odd order, then the center of the cube is a cell containing one half the sum of the first and last number in the series.
Heinz provides a generalized definition as follows: A hypercube of dimension n is perfect if all pan-n-agonals sum correctly, and all lower dimension hypercubes contained in it are perfect! He also provides spreadsheets for testing them. Heinz has collaborated with J. R. Hendricks to produce a A Unified Classification system for Magic Cubes (Journal of Recreational Mathematics, 2002).
The relationship of the 81 tetragrams of the Taoist classic Tai Hsuan Ching (or Tài Xuán Jïng) and the Tao Te Ching has most recently been explored in relationship to modern physics by Tony Smith (I Ching (Ho Tu and Lo Shu), Genetic Code, Tai Hsuan Ching, and the D4-D5-E6-E7-E8 VoDou Physics Model ). According to Smith:
A magic tesseract is a four-dimensional generalization of the two-dimensional magic square and the three-dimensional magic cube. Harvey Heinz defines a 4-dimensional hypercube (or tesseract) as perfect if all pan-quadragonals are correct, and all the magic squares and magic cubes within it are perfect. This means that the magic squares are all pandiagonal and the magic cubes are all pantriagonal and pandiagonal. There are 40m2 lines that sum correctly. They are m3 rows, m3 columns, m3 pillars, m3 files, 8m3 quadragonals, 16m3 triagonals, and 12m3 diagonals. Furthermore, a magic hypercube of any dimension n is perfect if all pan-n-agonals sum correctly, and all lower dimension hypercubes contained in it are perfect!
Natural appropriateness: Yang Hsiung is understood to have shared the vision of the I Ching of an order that united the cosmos, the sphere of action, and the individual. The concern of all such works, notably the Tao Te Ching, might be said to be with appropriateness in terms of the patterns and dynamics of nature -- the Way of Nature, exemplified by the insights of the Tao Te Ching. Any explorations of such matters merit careful consideration in the light of contemporary challenges of global governance of an endangered planet -- exemplified by the paradox of "sustainable development".
With respect to the title of the work, T'ai Hsüan Ching, as noted by Michael Nylan and Nathan Sivin (The First Neo-Confucianism: an introduction to Yang Hsiung's "Canon Of Supreme Mystery", 1995):
Complementarity: For its author, Yang Hsiung, the intent was to evoke a sense of the inseparability and complementarity between mystery and rational pattern. Again for Nylan and Sevin:
Degree of relationship: With respect to the significance of the pattern of relationships between the T'ai Hsüan Ching and the Tao Te Ching -- both pointing to a subtle understanding -- the tables above could be usefully seen as raising questions as to the nature or degree of that relationship, or of any ability to recognize it. It is of course highly unsatisfactory to assess such a relationship through a matching of single-word insight titles from the T'ai Hsüan Ching to succinct representations of insights from the Tao Te Ching. Each such "insight" is a memetic complex at the nexus of the pattern of associations it evokes. It is severely distorted by reducing its dimensionality in this way through the use of a somewhat arbitrary choice of term in English.
A simpler variant of this challenge was central to the methodology of isolating "human values" from the maze of relevant words and connotations in English -- which identified 2,200 "values" and 14,500 associative links between them in the Human Values Project of the Encyclopedia of World Problems and Human Potential. The 9 "Appraisals" associated with each insight in the T'ai Hsüan Ching might be understood as a systemization of the pattern of possible connotations. In addition to the possibility of a fruitful koan-like relationship (suggested above), there is of course the interesting implication of the so-called small world hypothesis that everyone in the world could be reached through a short chain of social acquaintances. This gave rise to the concept of six degrees of separation with generic implications (cf Duncan J Watts, Six Degrees: the science of a connected age, 2004), notably for knowledge diffusion (cf Sara Grant, Caves, Clusters, and Weak Ties: the six degrees world of inventors, Working Knowledge for Business Leaders, November 2004) and web-related products (Dan Farber, Less than six degrees of social networking and Web 2.0 goodness, June 2006).
Might there therefore be an analogous phenomenon in the knowledge universe whereby even the most seemingly disparate concepts could be linked through a short chain of semantically relevant associations?
Mnemonic interlocking: The focus here on the potential of magic squares can be usefully understood as a mnemonic device for comprehending a complex pattern of associations. It might possibly be compared with the everyday challenge of comprehending a subway network map as a whole -- or any systems diagram, and perhaps most notably those associated with global modelling exercises (cf Limits to Growth, as first commissioned by the Club of Rome in 1972). These were initally based on the world dynamics models of Jay Wright Forrester (see also an early effort at relating these to the individual World Dynamics and Psychodynamics: a step towards making abstract "world system" dynamic limitations meaningful to the individual, 1971).
As with a song or poem, it is the recurring elements and associations that ensure the integrity and memorability of the whole -- arguments developed in an associated proposal (A Singable Earth Charter, EU Constitution or Global Ethic?, 2006). A sense of the challenge, and willingness to address it, may be seen in current widespread enthusiam for sudoku -- which have their origin in magic squares. The objective is to achieve interlocking of symbols (of any kind) in an array -- an intellectually "satisfying" design "fit". The 81 memetic complexes of the T'ai Hsüan Ching and the Tao Te Ching pose the challenge of how to identify, understand, remember and communicate an appropriately interlocked pattern -- of relevance to an appropriate lifestyle and style of governance. As with sudoku, they point to the possibility of more challenging complex patterns of interconnectedness (including bimagic squares, cubes and hypercubes).
Individual engagement: The additional dimension introduced by the preoccupations of the two works linked here, notably through the 9x9 pan-magic square presentation (centered on #1: Centering), is the sense of a central focus for order with which the individual may fruitfully identify. This can be contrasted with conventional systems diagrams that have no such centre, singificantly failing to design in the individual observer, as a focus for action within the system. The closest to this is ironically the arrow on a subway may indicating the user's current location. However the "centering" in the pan-magic square is an extremely subtle notion that, paradoxically and provocatively, calls into question the naming and condition of the centre itself -- as often quoted from the Tao Te Ching in many variations :
The way that can be told is not the common way
Readers of any systems diagram, like the magic squares above, are therefore challenged as much to ask "who they think they are" and "where they think they are going", rather than simply to expect a simple answer that in no way challenges their sense of identity or the validity of their quest. They offer a puzzle that merits being taken seriously.
Focus of cognitive engagement: It is from such a perspective that it is worth recalling that magic squares were traditionally placed at the centre of mandalas as a focus for meditation. These mandalas might then be viewed as a form of cognitive "gearbox" through which "power is transmitted to an output device, normally rotary in form, and generally at a reduced rate of angular speed but at a higher motive torque". The "gears" presented in such mandalas typically involve concentric sets of (3, 4, 5, 6, 7, 8, 9, 10 or 12) archetypal symbols as conduits for particular modes of understanding. From such a perspective, the challenge of appropriateness for an individual lifestyle, and for collective governance, is to identify such "gears" and to ensure that they intermesh fruitfully.
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