15th November 2003 | Draft
9-fold Higher Order Patterning of Tao Te Ching Insights
Possibilities in the mathematics of magic squares, cubes and hypercubes
- / -
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.
| |
a |
b |
c |
d |
e |
f |
g |
h |
i |
| 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 |
Magic squares
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 |
| |
8:204
|
|
|
|
1:15
|
|
|
|
6:150
|
|
| 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 |
| |
3:69
|
|
|
|
5:123
|
|
|
|
7:177
|
|
| 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 |
| |
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 squares
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
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). (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 |
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 ½n 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:
- Simple: Containing no, or less then 3m, orthogonal magic squares.
Only the rows, columns, pillars and triagonals are required to sum correctly
for a simple magic cube.
- Pantriagonal: All pantriagonals must sum correctly. There may be
some simple and/or pandiagonal magic squares, but not enough to satisfy any
other classifications.
- Diagonal: All 3m planar arrays must be ‘simple’ magic squares.
- Pandiagonal: All 3m planar arrays must be ‘pandiagonal’ magic squares.
The 6 oblique squares are always magic. One of them may be pandiagonal magic.
- Perfect: All 3m planar arrays must be ‘pandiagonal’ magic squares.
In addition, all pantriagonals must sum correctly. These two conditions combine
to provide another 6m pandiagonal magic squares.
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).
| 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 arrangement, according to Tony Smith, is similar to the
Fu Xi
binary number arrangement of the I Ching. This is not a magic square
arrangement.
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!
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