- / -
This is part of a commentary on the Tao Te Ching Interpreted
order) and (alternative
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.
Also published in modified form in Statistics, Visualizations and Patterns (Vol 5 of the Yearbook of International Organizations, K G Saur Verlag, 6th edition, 2006/2007, as section 10.2.3). Further developed in 9-fold Magic Square Pattern of Tao Te Ching Insights experimentally associated with the 81 insights of the T'ai Hsüan Ching
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.
|I||1: Journeying through unnaming the myriad patterns of the past||2: Engaging without engaging||3: Cultivating non-engagement||4: Having been there; having done that||5: Engendering through complementarity||6: Completing||7: Enduring||8: Easing forward, going wherever, without competition||9: Avoiding excess|
|II||10: Centering through learning||11: Benefiting from what is not||12: Sensing the inner||13: Governing others appropriately||14: Living the present||15: Subtly stilling to clarify the troubled||16: Accepting wisely the enduring cycle of beginning and ending||17: Acting simply, enabling others to value their own initiatives||18: Failing to exalt merit|
|III||19: Being untroubled through needing little and wanting less||20: Living uncertainty, confusion and strangeness||21: Knowing the strange uncertainties offered in the moment||22: Acting contrarily||23: Being in the moment||24: Avoiding disproportion and self-satisfaction||25: Following the unnamable||26: Becoming insightful through assiduous handling of obligations||27: Educating the challenged as the inspiration of the wise|
|IV||28: Knowing the other, and retaining one's identity and quality||29: Doing "nothing" to the world||30: Leading through inspiration that does not seek to win||31: Using weapons, when there is no choice, with a calm, still mind||32: Knowing when to cease making essential distinctions||33: Applying to oneself the skills developed successfully to deal with others||34: Achieving greatness without great doings||35: Holding fast to the eternal process through the very ordinary||36: Prevailing through weakness|
|V||37: Self-organizing of myriad things||38: Abiding in letting go and doing nothing||39: Enwholing to sustain the integrity of the subtle||40: Returning from weakness||41: Understanding appropriateness||42: Losing as the key to the cycle of winning and losing||43: Ensubtling to enliven the impenetrable||44: Self-constraining fruitfully||45: Remaining calm and clear to ensure that the capacity for appropriateness is renewed|
|VI||46: Knowing that enough is enough||47: Understanding the truth and opportunity of the moment||48: Unlearning||49: Enminding the world to see the ordinary through the eyes of children||50: Living in recognition that this implies dying||51: Nurturing life according to natural processes||52: Understanding insignificant beginnings||53: Ensuring modesty||54: Ensuring that rules for oneself are consistent with those for the world|
|VII||55: Knowing harmony as knowing the eternal||56: Knowing that discourages talking||57: Avoiding instrumental thinking, initiation of innovative change and regulation of action||58: Bumbling on without forcing||59: Gathering insight to ensure staying power||60: Allowing potentially disruptive forces to have their place||61: Lying low to ensure integrity and continuity||62: Honoring the appropriate as a gift||63: Focusing on the challenge of beginnings|
|VIII||64: Attending to what may have been neglected in the achievement of undertakings||65: Being in ignorance of appropriate action||66: Following rather than leading||67: Leading the mightiest by not presuming to do so||68: Avoiding competition||69: Yielding to antagonism||70: Being obscure||71: Knowing without knowing||72: Fearing the dangers of acting inappropriately|
|IX||73: Acting silently, non-competitively, and non-directively||74: Avoiding the presumptuousness of usurping the judgement on others||75: Living for more than the pursuit of wealth||76: Bending in response to pressure||77: Redistributing from those who have to those who have not||78: Recognizing the counter-intuitive, paradoxical nature of appropriate action||79: Fulfilling obligations||80: Enjoying the freedom of movement in relation to what is to hand||81: Doing without outdoing|
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: Knowing without knowing||64: Attending to what may have been neglected in the achievement of undertakings||69: Yielding to antagonism||8: Easing forward, going wherever, without competition||1: Journeying through unnaming the myriad patterns of the past||6: Completing||53: Ensuring modesty||46: Knowing that enough is enough||51: Nurturing life according to natural processes|
|66: Following rather than leading||68: Avoiding competition||70: Being obscure||3: Cultivating non-engagement||5: Engendering through complementarity||7: Enduring||48: Unlearning||50: Living in recognition that this implies dying||52: Understanding insignificant beginnings|
|67: Leading the mightiest by not presuming to do so||72: Fearing the dangers of acting inappropriately||65: Being in ignorance of appropriate action||4: Having been there; having done that||9: Avoiding excess||2: Engaging without engaging||49: Enminding the world to see the ordinary through the eyes of children||54: Ensuring that rules for oneself are consistent with those for the world||47: Understanding the truth and opportunity of the moment|
|26: Becoming insightful through assiduous handling of obligations||19: Being untroubled through needing little and wanting less||24: Avoiding disproportion and self-satisfaction||44: Self-constraining fruitfully||37: Self-organizing of myriad things||42: Losing as the key to the cycle of winning and losing||62: Honoring the appropriate as a gift||55: Knowing harmony as knowing the eternal||60: Allowing potentially disruptive forces to have their place|
|21: Knowing the strange uncertainties offered in the moment||23: Being in the moment||25: Following the unnamable||39: Enwholing to sustain the integrity of the subtle||41: Understanding appropriateness||43: Ensubtling to enliven the impenetrable||57: Avoiding instrumental thinking, initiation of innovative change and regulation of action||59: Gathering insight to ensure staying power||61: Lying low to ensure integrity and continuity|
|22: Acting contrarily||27: Educating the challenged as the inspiration of the wise||20: Living uncertainty, confusion and strangeness||40: Returning from weakness||45: Remaining calm and clear to ensure that the capacity for appropriateness is renewed||38: Abiding in letting go and doing nothing||58: Bumbling on without forcing||63: Focusing on the challenge of beginnings||56: Knowing that discourages talking|
|35: Holding fast to the eternal process through the very ordinary||28: Knowing the other, and retaining one's identity and quality||33: Applying to oneself the skills developed successfully to deal with others||80: Enjoying the freedom of movement in relation to what is to hand||73: Acting silently, non-competitively, and non-directively||78: Recognizing the counter-intuitive, paradoxical nature of appropriate action||17: Acting simply, enabling others to value their own initiatives||
10: Centering through learning
|15: Subtly stilling to clarify the troubled|
|30: Lading through inspiration that does not seek to win||32: Knowing when to cease making essential distinctions||34: Achieving greatness without great doings||75: Living for more than the pursuit of wealth||77: Redistributing from those who have to those who have not||79: Fulfilling obligations||12: Sensing the inner||14: Living the present||16: Accepting wisely the enduring cycle of beginning and ending|
|31: Using weapons, when there is no choice, with a calm, still mind||36: Prevailing through weakness||29: Doing "nothing" to the world||76: Bending in response to pressure||81: Doing without outdoing||74: Avoiding the presumptuousness of usurping the judgement on others||13: Governing others appropriately||18: Failing to exalt merit||11: Benefiting from what is not|
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.
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)
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). (NB: Versions in drafts dated prior
to 15 November contained two errors in the following table).
|37: Self-organizing of myriad things||52: Understanding insignificant beginnings||31: Using weapons, when there is no choice, with a calm, still mind||66: Following rather than leading||81: Doing without outdoing||60: Allowing potentially disruptive forces to have their place||11: Benefiting from what is not||26: Becoming insightful through assiduous handling of obligations||5: Engendering through complementarity|
|65: Being in ignorance of appropriate action||80: Enjoying the freedom of movement in relation to what is to hand||59: Gathering insight to ensure staying power||10: Centering through learning||25: Following the unnamable||4: Having been there; having done that||39: Enwholing to sustain the integrity of the subtle||54: Ensuring that rules for oneself are consistent with those for the world||33: Applying to oneself the skills developed successfully to deal with others|
|24: Avoiding disproportion and self-satisfaction||3: Cultivating non-engagement||18: Failing to exalt merit||50: Living in recognition that this implies dying||29: Doing "nothing" to the world||44: Self-constraining fruitfully||76: Bending in response to pressure||55: Knowing harmony as knowing the eternal||70: Being obscure|
|49: Enminding the world to see the ordinary through the eyes of children||28: Knowing the other, and retaining one's identity and quality||43: Ensubtling to enliven the impenetrable||78: Recognizing the counter-intuitive, paradoxical nature of appropriate action||57: Avoiding instrumental thinking, initiation of innovative change and regulation of action||72: Fearing the dangers of acting inappropriately||23: Being in the moment||2: Engaging without engaging||17: Acting simply, enabling others to value their own initiatives|
|77: Redistributing from those who have to those who have not||56: Knowing that discourages talking||71: Knowing without knowing||22: Acting contrarily||1: Journeying through unnaming the myriad patterns of the past||16: Accepting wisely the enduring cycle of beginning and ending||51: Nurturing life according to natural processes||30: Leading through inspiration that does not seek to win||45: Remaining calm and clear to ensure that the capacity for appropriateness is renewed|
|9: Avoiding excess||15: Subtly stilling to clarify the troubled||21: Knowing the strange uncertainties offered in the moment||35: Holding fast to the eternal process through the very ordinary||41: Understanding appropriateness||47: Understanding the truth and opportunity of the moment||61: Lying low to ensure integrity and continuity||67: Leading the mightiest by not presuming to do so||73: Acting silently, non-competitively, and non-directively|
|34: Achieving greatness without great doings||40: Returning from weakness||46: Knowing that enough is enough||63: Focusing on the challenge of beginnings||69: Yielding to antagonism||75: Living for more than the pursuit of wealth||8: Easing forward, going wherever, without competition||14: Living the present||20: Living uncertainty, confusion and strangeness|
|62: Honoring the appropriate as a gift||68: Avoiding competition||74: Avoiding the presumptuousness of usurping the judgement on others||7: Enduring||13: Governing others appropriately||19: Being untroubled through needing little and wanting less||36: Prevailing through weakness||42: Losing as the key to the cycle of winning and losing||48: Unlearning|
|12: Sensing the inner||27: Educating the challenged as the inspiration of the wise||6: Completing||38: Abiding in letting go and doing nothing||53: Ensuring modesty||32: Knowing when to cease making essential distinctions||64: Attending to what may have been neglected in the achievement of undertakings||79: Fulfilling obligations||58: Bumbling on without forcing|
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|
|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)|
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:
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...
The I Ching is based on hexagrams of binary lines. Tony Smith, in his discussion of the Tai Hsuan Ching of ternary line tetragrams "arranged in T'ien" (as in the table below), the ternary numbers are given "plus 1", since the ternary numbers go from 0 to 80 (as indicated by Grogono above) instead of from 1 to 81 (see further discussion in 9-fold Magic Square Pattern of Tao Te Ching Insights experimentally associated with the 81 insights of the T'ai Hsüan Ching).
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!
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