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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

- / -


Experimental development of 9-fold Higher Order Patterning of Tao Te Ching Insights
This is part of a commentary on the Tao Te Ching Interpreted Succinctly (original order) and (alternative order)
See also Commentary on Tao Te Ching Interpretation: and the possibility of higher order patterning
Navigational implications are explored in Hyperspace Clues to the Psychology of the Pattern that Connects.


Introduction
Experimental context
Experiment
Experimental presentation
Magic square presentation
Pan-magic square presentation
Bimagic squares
Most-perfect magic squares
Magic cubes
Magic hypercubes
Conclusion
References

Introduction

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).

Experimental context

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") :

The Mystery was the most influential among the many meant to remedy inconsistencies in the Changes and to add to the old discourse current ideas about the cosmic order, the sagely life, and the beauty and precision that can be drawn from words. Until the thirteenth century Yang Hsiung's writings were considered central to the orthodox search for universal pattern, and thereafter were forgotten...The Mystery made considerable demands on its readers. The clarity of its structure was intentionally balanced by the complexity of language that strives above all for allusiveness....The Mystery, like the Changes, was said to be hopelessly abstruse and of no practical benefit....Yang's aim in writing the Mystery [was] to instigate and guide the personal striving for integrity that is the only possible basis for a sound polity. This virtue is more than a matter of moral and psychic integration; it involves union with the Way of Nature and its Mystery....It does not offer magical power over nature. It simply aids reflection on the eternal patterns that underlie every aspect of experience and action. Assimilating those patterns, Yang was convinced, could guide the renewal of human creativity and the eventual recovery of order....His book applies rigorously and reflects, in its texts and guides to interpretation, the basic seasonal rhythms, the fundamental social relationships, and the functions of yin-yang and the Five Phases that pervade the natural and human worlds.

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:

It is from Lao-tzu's Mystery that that of Yang derives, although his moral stance differs: "As for Lao-tzu's discussion of the Way and its power, I have drawn upon it; but from his rejection of Good (jen) and Right (i), his elimination of ritual and study, I have taken nothing."

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 philosophic interest of Yang's Head texts lies in their intricate, nuanced picture of a grand cycle of change, his recognition of complexity within regular order. Conversely, his Appraisals are remarkable because, through highly figurative language and the interplay of cycles within cycles, Yang suggests regular patterns emerging from the inexhaustible variety and ambiguity of moral circumstances.

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:

the T'ai Hsüan Ching holds that there are three forces at work in nature; two of these Yin and Yang, represent the positive and negative fluxes of electro-magnetism; but there is also a third force which accounts for the creation of the truly novel. With Yin and Yang [alone] there is nothing new under the Sun. While everything can be classified as belonging to Yin and Yang, the dual philosophy can only account for what exists already; no matter what resultant products or ideas are spawned by the action of Yin and Yang, there is no entirely new element created...But new ideas, and new species, only arise from the third creative force... The third force, according to the T'ai Hsüan Ching, is the Jen (Mankind) force... the philosophy of the triad is totally at variance with the duality of the I Ching, and yet, paradoxically as though it might seem the T'ai Hsüan Ching is a work of tremendous originality (pp 8-9).

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:

Like Walters, Nylan claims special advantages for the 'tri-partite' division of the Tai Hsuan Ching. Walters had argued that the Tai Hsuan Ching accorded a more complete, active role to 'man' -- as against the Yi-Ching's allegedly 'fixed' dualistic system. All of this shows a poor grasp of what the Yi Ching (and Tai Hsuan Ching) actually teach. The Tso chuan section of the Yi-Ching stresses that 'Heaven, Earth and Man' are what comprise the Tao. Every trigram (and hexagram) in the Yi Ching reflects this threefold unity ('the 'three powers' or san-tsai) - and the Yi-Ching makes this clear on every count. Human 'agency' is therefore vital to the Yi-Ching. It was not a 'new' idea with the Tai Hsuan Ching. The antiquity of this intuition is evident in the formation of the Chinese script, the old Ku-wen forms, giving the character for 'king' or 'kingship' as a representation of the 'san-tsai' or 'three powers' -- linked by a vertical stroke, anciently, the kingly-priest in whom the san-tsai were united or focused. It is a basically a 'trigram' -- crossed by a vertical line. Like Walters, Nylan makes some rather bold claims for the Tai Hsuan Ching. But trying to place it in 'competition' against the Yi-Ching -- is naive. [more]

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.

Experiment

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.

Experimental presentation

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.

  a b c d e f g h i
I 1:Centering: Journeying through unnaming the myriad patterns of the past 2:Surrounding: Engaging without engaging 3: Slowing: Cultivating non-engagement 4: Obstructing: A Barrier: Having been there; having done that 5: Lessening: Engendering through complementarity 6: Suffering: Completing 7: Ascending: Enduring 8: Opposing: Easing forward, going wherever, without competition 9: Penetrating: Avoiding excess
II 10: Praising: Centering through learning 11: Mistaking: Benefiting from what is not 12: Rejuvenating: Sensing the inner 13: Increasing: Governing others appropriately 14: Focusing: Living the present 15: Perceiving: Subtly stilling to clarify the troubled 16: Exchanging: Accepting wisely the enduring cycle of beginning and ending 17: Softening: Acting simply, enabling others to value their own initiatives 18: Waiting: Failing to exalt merit
III 19:Conforming: Being untroubled through needing little and wanting less 20: Advancing: Living uncertainty, confusion and strangeness 21: Releasing: Knowing the strange uncertainties offered in the moment 22: Patterning: Acting contrarily 23: Harming: Being in the moment 24: Enjoying: Avoiding disproportion and self-satisfaction 25: Contending: Following the unnamable 26: Scrutinizing: Becoming insightful through assiduous handling of obligations 27:Transacting: Educating the challenged as the inspiration of the wise
IV 28: Changing: Knowing the other, and retaining one's identity and quality 29: Severing: Doing "nothing" to the world 30: Enduring: Leading through inspiration that does not seek to win 31: Pretending: Using weapons, when there is no choice, with a calm, still mind 32: Multiplying: Knowing when to cease making essential distinctions 33: Internalizing: Applying to oneself the skills developed successfully to deal with others 34: Relating: Achieving greatness without great doings 35: Accumulating: Holding fast to the eternal process through the very ordinary 36: Enclosing: Prevailing through weakness
V 37:Encycling: Self-organizing of myriad things 38: Abounding: Abiding in letting go and doing nothing 39: Dwelling: Enwholing to sustain the integrity of the subtle 40: Regulating: Returning from weakness 41: Befitting: Understanding appropriateness 42: Greeting: Losing as the key to the cycle of winning and losing 43: Meeting: Ensubtling to enliven the impenetrable 44: Cooking: Self-constraining fruitfully 45: Appreciating: Remaining calm and clear to ensure that the capacity for appropriateness is renewed
VI 46:Extending: Knowing that enough is enough 47: Comprehending: Understanding the truth and opportunity of the moment 48: Fitting: Unlearning 49: Escaping: Enminding the world to see the ordinary through the eyes of children 50: Doddering: Living in recognition that this implies dying 51: Ruling: Nurturing life according to natural processes 52: Placing: Understanding insignificant beginnings 53: Perpetuating: Ensuring modesty 54: Ensuring: Ensuring that rules for oneself are consistent with those for the world
VII 55:Diminishing: Knowing harmony as knowing the eternal 56: Humming: Knowing that discourages talking 57: Protecting: Avoiding instrumental thinking, initiation of innovative change and regulation of action 58: Re-uniting: Bumbling on without forcing 59: Assembling: Gathering insight to ensure staying power 60: Storing: Allowing potentially disruptive forces to have their place 61: Adorning: Lying low to ensure integrity and continuity 62: Doubting: Honoring the appropriate as a gift 63: Inspecting: Focusing on the challenge of beginnings
VIII 64: Sinking: Attending to what may have been neglected in the achievement of undertakings 65: Questioning: Being in ignorance of appropriate action 66: Departing: Following rather than leading 67: Obscuring: Leading the mightiest by not presuming to do so 68: Dimming: Avoiding competition 69: Lacking: Yielding to antagonism 70: Depriving: Being obscure 71: Stopping: Knowing without knowing 72: Persevering: Fearing the dangers of acting inappropriately
IX 73: Completing: Acting silently, non-competitively, and non-directively 74: Willing: Avoiding the presumptuousness of usurping the judgement on others 75: Erring: Living for more than the pursuit of wealth 76: Distressing: Bending in response to pressure 77: Domesticating: Redistributing from those who have to those who have not 78: Taking: Recognizing the counter-intuitive, paradoxical nature of appropriate action 79: Troubling: Fulfilling obligations 80: Attending: Enjoying the freedom of movement in relation to what is to hand 81: Nourishing: Doing without outdoing

Magic square presentation

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.

71: Stopping: Knowing without knowing 64: Sinking: Attending to what may have been neglected in the achievement of undertakings 69: Lacking: Yielding to antagonism   8: Opposing: Easing forward, going wherever, without competition 1:Centering: Journeying through unnaming the myriad patterns of the past 6: Suffering: Completing   53: Perpetuating: Ensuring modesty 46:Extending: Knowing that enough is enough 51: Ruling: Nurturing life according to natural processes
66: Departing: Following rather than leading 68: Dimming: Avoiding competition 70: Depriving: Being obscure   3: Slowing: Cultivating non-engagement 5: Lessening: Engendering through complementarity 7: Ascending: Enduring   48: Fitting: Unlearning 50: Doddering: Living in recognition that this implies dying 52: Placing: Understanding insignificant beginnings
67: Obscuring: Leading the mightiest by not presuming to do so 72: Persevering: Fearing the dangers of acting inappropriately 65: Questioning: Being in ignorance of appropriate action   4: Obstructing: A Barrier: Having been there; having done that 9: Penetrating: Avoiding excess 2:Surrounding: Engaging without engaging   49: Escaping: Enminding the world to see the ordinary through the eyes of children 54: Ensuring: Ensuring that rules for oneself are consistent with those for the world 47: Comprehending: Understanding the truth and opportunity of the moment
 
8:204
     
1:15
     
6:150
 
26: Scrutinizing: Becoming insightful through assiduous handling of obligations 19:Conforming: Being untroubled through needing little and wanting less 24: Enjoying: Avoiding disproportion and self-satisfaction   44: Cooking: Self-constraining fruitfully 37:Encycling: Self-organizing of myriad things 42: Greeting: Losing as the key to the cycle of winning and losing   62: Doubting: Honoring the appropriate as a gift 55:Diminishing: Knowing harmony as knowing the eternal 60: Storing: Allowing potentially disruptive forces to have their place
21: Releasing: Knowing the strange uncertainties offered in the moment 23: Harming: Being in the moment 25: Contending: Following the unnamable   39: Dwelling: Enwholing to sustain the integrity of the subtle 41: Befitting: Understanding appropriateness 43: Meeting: Ensubtling to enliven the impenetrable   57: Protecting: Avoiding instrumental thinking, initiation of innovative change and regulation of action 59: Assembling: Gathering insight to ensure staying power 61: Adorning: Lying low to ensure integrity and continuity
22: Patterning: Acting contrarily 27:Transacting: Educating the challenged as the inspiration of the wise 20: Advancing: Living uncertainty, confusion and strangeness   40: Regulating: Returning from weakness 45: Appreciating: Remaining calm and clear to ensure that the capacity for appropriateness is renewed 38: Abounding: Abiding in letting go and doing nothing   58: Re-uniting: Bumbling on without forcing 63: Inspecting: Focusing on the challenge of beginnings 56: Humming: Knowing that discourages talking
 
3:69
     
5:123
     
7:177
 
35: Accumulating: Holding fast to the eternal process through the very ordinary 28: Changing: Knowing the other, and retaining one's identity and quality 33: Internalizing: Applying to oneself the skills developed successfully to deal with others   80: Attending: Enjoying the freedom of movement in relation to what is to hand 73: Completing: Acting silently, non-competitively, and non-directively 78: Taking: Recognizing the counter-intuitive, paradoxical nature of appropriate action   17: Softening: Acting simply, enabling others to value their own initiatives

10: Praising: Centering through learning

15: Perceiving: Subtly stilling to clarify the troubled
30: Enduring: Leading through inspiration that does not seek to win 32: Multiplying: Knowing when to cease making essential distinctions 34: Relating: Achieving greatness without great doings   75: Erring: Living for more than the pursuit of wealth 77: Domesticating: Redistributing from those who have to those who have not 79: Troubling: Fulfilling obligations   12: Rejuvenating: Sensing the inner 14: Focusing: Living the present 16: Exchanging: Accepting wisely the enduring cycle of beginning and ending
31: Pretending: Using weapons, when there is no choice, with a calm, still mind 36: Enclosing: Prevailing through weakness 29: Severing: Doing "nothing" to the world   76: Distressing: Bending in response to pressure 81: Nourishing: Doing without outdoing 74: Willing: Avoiding the presumptuousness of usurping the judgement on others   13: Increasing: Governing others appropriately 18: Waiting: Failing to exalt merit 11: Mistaking: Benefiting from what is not
 
4:96
     
9:231
     
2:42
 

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.

Pan-magic square presentation

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.

Table 3a: Distribution of 81 numbers according to 9x9 pan-magic pattern (as discovered by Alan Grogono)

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.

36 51 30 65 80 59 10 25 4
64 79 58 9 24 3 38 53 32
23 2 17 49 28 43 75 54 69
48 27 42 77 56 71 22 1 16
76 55 70 21 0 15 50 29 44
8 14 20 34 40 46 60 66 72
33 39 45 62 68 74 7 13 19
61 67 73 6 12 18 35 41 47
11 26 5 37 52 31 63 78 57


Table 3b: Application of pan-magic pattern to order 81 insights of Tao Te Ching and T'ai Hsüan Ching

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).

37:Encycling: Self-organizing of myriad things 52: Placing: Understanding insignificant beginnings 31: Pretending: Using weapons, when there is no choice, with a calm, still mind 66: Departing: Following rather than leading 81: Nourishing: Doing without outdoing 60: Storing: Allowing potentially disruptive forces to have their place 11: Mistaking: Benefiting from what is not 26: Scrutinizing: Becoming insightful through assiduous handling of obligations 5: Lessening: Engendering through complementarity
65: Questioning: Being in ignorance of appropriate action 80: Attending: Enjoying the freedom of movement in relation to what is to hand 59: Assembling: Gathering insight to ensure staying power 10: Praising: Centering through learning 25: Contending: Following the unnamable 4: Obstructing: A Barrier: Having been there; having done that 39: Dwelling: Enwholing to sustain the integrity of the subtle 54: Ensuring: Ensuring that rules for oneself are consistent with those for the world 33: Internalizing: Applying to oneself the skills developed successfully to deal with others
24: Enjoying: Avoiding disproportion and self-satisfaction 3: Slowing: Cultivating non-engagement 18: Waiting: Failing to exalt merit 50: Doddering: Living in recognition that this implies dying 29: Severing: Doing "nothing" to the world 44: Cooking: Self-constraining fruitfully 76: Distressing: Bending in response to pressure 55:Diminishing: Knowing harmony as knowing the eternal 70: Depriving: Being obscure
49: Escaping: Enminding the world to see the ordinary through the eyes of children 28: Changing: Knowing the other, and retaining one's identity and quality 43: Meeting: Ensubtling to enliven the impenetrable 78: Taking: Recognizing the counter-intuitive, paradoxical nature of appropriate action 57: Protecting: Avoiding instrumental thinking, initiation of innovative change and regulation of action 72: Persevering: Fearing the dangers of acting inappropriately 23: Harming: Being in the moment 2:Surrounding: Engaging without engaging 17: Softening: Acting simply, enabling others to value their own initiatives
77: Domesticating: Redistributing from those who have to those who have not 56: Humming: Knowing that discourages talking 71: Stopping: Knowing without knowing 22: Patterning: Acting contrarily 1:Centering: Journeying through unnaming the myriad patterns of the past 16: Exchanging: Accepting wisely the enduring cycle of beginning and ending 51: Ruling: Nurturing life according to natural processes 30: Enduring: Leading through inspiration that does not seek to win 45: Appreciating: Remaining calm and clear to ensure that the capacity for appropriateness is renewed
9: Penetrating: Avoiding excess 15: Perceiving: Subtly stilling to clarify the troubled 21: Releasing: Knowing the strange uncertainties offered in the moment 35: Accumulating: Holding fast to the eternal process through the very ordinary 41: Befitting: Understanding appropriateness 47: Comprehending: Understanding the truth and opportunity of the moment 61: Adorning: Lying low to ensure integrity and continuity 67: Obscuring: Leading the mightiest by not presuming to do so 73: Completing: Acting silently, non-competitively, and non-directively
34: Relating: Achieving greatness without great doings 40: Regulating: Returning from weakness 46:Extending: Knowing that enough is enough 63: Inspecting: Focusing on the challenge of beginnings 69: Lacking: Yielding to antagonism 75: Erring: Living for more than the pursuit of wealth 8: Opposing: Easing forward, going wherever, without competition 14: Focusing: Living the present 20: Advancing: Living uncertainty, confusion and strangeness
62: Doubting: Honoring the appropriate as a gift 68: Dimming: Avoiding competition 74: Willing: Avoiding the presumptuousness of usurping the judgement on others 7: Ascending: Enduring 13: Increasing: Governing others appropriately 19:Conforming: Being untroubled through needing little and wanting less 36: Enclosing: Prevailing through weakness 42: Greeting: Losing as the key to the cycle of winning and losing 48: Fitting: Unlearning
12: Rejuvenating: Sensing the inner 27:Transacting: Educating the challenged as the inspiration of the wise 6: Suffering: Completing 38: Abounding: Abiding in letting go and doing nothing 53: Perpetuating: Ensuring modesty 32: Multiplying: Knowing when to cease making essential distinctions 64: Sinking: Attending to what may have been neglected in the achievement of undertakings 79: Troubling: Fulfilling obligations 58: Re-uniting: Bumbling on without forcing

Table 3c: Distinguishing Ti, Jen and T'ien from T'ai Hsüan Ching in Table 3b

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).

Ti
(yin)
Jen
(yin-yang)
T'ien
(yang)
73 64 55 46 37 28 19 10 1
74 65 56 47 38 29 20 11 2
75 66 57 48 39 30 21 12 3
76 67 58 49 40 31 22 13 4
77 68 59 50 41 32 23 14 5
78 69 60 51 42 33 24 15 6
79 70 61 52 43 34 25 16 7
80 71 62 53 44 35 26 17 8
81 72 63 54 45 36 27 18 9
This ternary number (+1) arrangement, according to Tony Smith, is similar to the Fu Xi binary number arrangement of the I Ching (although this is not a magic square arrangement). The positions in Table 3b can now be coloured in Table 3c (below) to highlight these three groups.

37:Encycling: Self-organizing of myriad things 52: Placing: Understanding insignificant beginnings

31: Pretending: Using weapons, when there is no choice, with a calm, still mind

66: Departing: Following rather than leading 81: Nourishing: Doing without outdoing 60: Storing: Allowing potentially disruptive forces to have their place 11: Mistaking: Benefiting from what is not 26: Scrutinizing: Becoming insightful through assiduous handling of obligations 5: Lessening: Engendering through complementarity
65: Questioning: Being in ignorance of appropriate action 80: Attending: Enjoying the freedom of movement in relation to what is to hand 59: Assembling: Gathering insight to ensure staying power 10: Praising: Centering through learning 25: Contending: Following the unnamable 4: Obstructing: A Barrier: Having been there; having done that 39: Dwelling: Enwholing to sustain the integrity of the subtle 54: Ensuring: Ensuring that rules for oneself are consistent with those for the world 33: Internalizing: Applying to oneself the skills developed successfully to deal with others
24: Enjoying: Avoiding disproportion and self-satisfaction 3: Slowing: Cultivating non-engagement 18: Waiting: Failing to exalt merit 50: Doddering: Living in recognition that this implies dying 29: Severing: Doing "nothing" to the world 44: Cooking: Self-constraining fruitfully 76: Distressing: Bending in response to pressure 55:Diminishing: Knowing harmony as knowing the eternal 70: Depriving: Being obscure
49: Escaping: Enminding the world to see the ordinary through the eyes of children 28: Changing: Knowing the other, and retaining one's identity and quality 43: Meeting: Ensubtling to enliven the impenetrable 78: Taking: Recognizing the counter-intuitive, paradoxical nature of appropriate action 57: Protecting: Avoiding instrumental thinking, initiation of innovative change and regulation of action 72: Persevering: Fearing the dangers of acting inappropriately 23: Harming: Being in the moment 2:Surrounding: Engaging without engaging 17: Softening: Acting simply, enabling others to value their own initiatives
77: Domesticating: Redistributing from those who have to those who have not 56: Humming: Knowing that discourages talking 71: Stopping: Knowing without knowing 22: Patterning: Acting contrarily 1:Centering: Journeying through unnaming the myriad patterns of the past 16: Exchanging: Accepting wisely the enduring cycle of beginning and ending 51: Ruling: Nurturing life according to natural processes 30: Enduring: Leading through inspiration that does not seek to win 45: Appreciating: Remaining calm and clear to ensure that the capacity for appropriateness is renewed
9: Penetrating: Avoiding excess 15: Perceiving: Subtly stilling to clarify the troubled 21: Releasing: Knowing the strange uncertainties offered in the moment 35: Accumulating: Holding fast to the eternal process through the very ordinary 41: Befitting: Understanding appropriateness 47: Comprehending: Understanding the truth and opportunity of the moment 61: Adorning: Lying low to ensure integrity and continuity 67: Obscuring: Leading the mightiest by not presuming to do so 73: Completing: Acting silently, non-competitively, and non-directively
34: Relating: Achieving greatness without great doings 40: Regulating: Returning from weakness 46:Extending: Knowing that enough is enough 63: Inspecting: Focusing on the challenge of beginnings 69: Lacking: Yielding to antagonism 75: Erring: Living for more than the pursuit of wealth 8: Opposing: Easing forward, going wherever, without competition 14: Focusing: Living the present 20: Advancing: Living uncertainty, confusion and strangeness
62: Doubting: Honoring the appropriate as a gift 68: Dimming: Avoiding competition 74: Willing: Avoiding the presumptuousness of usurping the judgement on others 7: Ascending: Enduring 13: Increasing: Governing others appropriately 19:Conforming: Being untroubled through needing little and wanting less 36: Enclosing: Prevailing through weakness 42: Greeting: Losing as the key to the cycle of winning and losing 48: Fitting: Unlearning
12: Rejuvenating: Sensing the inner 27:Transacting: Educating the challenged as the inspiration of the wise 6: Suffering: Completing 38: Abounding: Abiding in letting go and doing nothing 53: Perpetuating: Ensuring modesty 32: Multiplying: Knowing when to cease making essential distinctions 64: Sinking: Attending to what may have been neglected in the achievement of undertakings 79: Troubling: Fulfilling obligations 58: Re-uniting: Bumbling on without forcing

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
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.

Bimagic squares

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.

Table 4: Magic squares from which bimagic squares can be generated
43 51 29 66 80 58 14 19 9
26 4 12 46 36 41 78 56 70
63 68 73 2 16 24 31 39 53
76 57 71 27 5 10 47 34 42
32 37 54 61 69 74 3 17 22
15 20 7 44 49 30 64 81 59
1 18 23 33 38 52 62 67 75
65 79 60 13 21 8 45 50 28
48 35 40 77 55 72 25 6 11

22

3

81

42

34

47

17

59

64

37

54

15

71

76

57

32

20

7

33

38

8

55

72

77

52

13

21

68

73

43

12

26

4

63

51

29

2

16

58

46

41

36

24

66

80

53

31

19

78

56

70

39

9

14

61

69

30

5

10

27

74

44

49

75

62

50

25

6

11

67

28

45

18

23

65

35

48

40

1

79

60

28 13 9 59 66 79 51 44 20
50 8 19 81 58 65 43 30 5
11 77 70 42 46 35 4 27 57
75 33 53 22 2 18 68 61 37
6 72 56 34 41 48 26 10 76
45 21 14 64 80 60 29 49 7
25 55 78 47 36 40 12 5 71
67 52 39 17 24 1 63 74 32
62 38 31 3 16 23 73 69 54
G. Pfeffermann: the first 9th-order bimagic square (Les Tablettes du Chercheur, 15 July 1891) J. R. Hendricks (Bimagic Squares: Order 9, Dec. 1999). David M. Collison (1991)

Most-perfect magic squares

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.

Magic cubes

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:

To construct the Tai Hsuan Ching, consider the Magic Square sequence as a line 3 8 4 9 5 1 6 2 7 with central 5 and opposite pairs at equal distances. If you try to make that, or a multiple of it, into a 9x9 Magic Square whose central number is the central number 41 of 9x9 = 81 = 40+1+40, you will fail because 41 is not a multiple of 5.

However, since 365 = 5x73 is the central number of 729 = 364+1+364, you can make a 9x9x9 Magic Cube with 9x9x9 = 729 entries, each 9x9 square of which is a Magic Square. The Magic Cube of the Tai Hsaun Ching gives the same sum for all lines parallel to an edge, and for all diagonals containing the central entry. The central number of the Magic Cube, 365....

The total number for each line is 3,285 = 219 x 15. The total of all numbers is 266,085 = 5,913 x 45.

Since 729 is the smallest odd number greater than 1 that is both a cubic number and a square number, the 729 entries of the 9x9x9 Magic Cube with central entry 365 can be rearranged to form a 27x27 Magic Square with 729 entries and central entry 365. 27 = 3x3x3 = 13+1+13 is a cubic number with central number 14, and there is a 3x3x3 Magic Cube with central entry 14 (14 is the dimension of the exceptional Lie algebra G2) and sum 42...

Magic hypercubes

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!

Conclusion

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):

In the Way, which was also Yang Hsiung's Mystery, science and ethics were one. This Han vision was not a reduction of science to subjectivity. Nature and human nature constitute a single order.... "Hsuan" carries a range of meaning from "black" to "darkness" to "hidden" to "mystery." Its overtones are stillness, solitude, isolation, nondifferentiation, and inaccessibility by purely rational processes. In Chinese thought the ideas at the philosophical end of this range bear no unpleasant connotations. They express that aspect of experience that can be known only by quiet and deep contemplation, or by illumination. Yang Hsiung uses hsuan in his book's title and throughout to mean the profound darkness, silence, ambiguity, and indefiniteness out of which creation comes. In cosmogony it is the undifferentiated state out of which yin-yang and eventually the myriad phenomena separate. In Nature as humans experience it, it is the latency out of which individual things are spontaneously born, and out of which events shape themselves. In the sage--that is, the human being as he should be, as the student of the Mystery is striving to be--it is the spiritual inwardness that precedes conscious decision and action and spontaneously accords them with natural process.

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:

Despite the superficiality of this essay, we have adduced more than enough evidence to illustrate Yang's aim in writing the Mystery: to instigate and guide the personal striving for integrity that is the only possible basis for a sound polity. This virtue is more than a matter of moral and psychic integration; it involves union with the Way of Nature and its Mystery. As the inquirer aligns his own decisions and actions with the cosmic course, he is able to promote harmony in every external sphere, renovating society on the pattern of the Tao.

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
The name that can be named is not the common name
What has no name is the beginning of heaven and earth
What has a name is the mother of the myriad creatures
Those without desires contemplate its secrets
Those who have desires contemplate its periphery
These two emerge together, but differ in name
Being together, they are called Mystery
Mystery upon mystery
Gateway to the myriad secrets.

[Nylan amd Sirvin]

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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