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_The Classic of Magick Squares: Research on I Ching and Magick Squares_
By Frater I Nigris
Do what thou wilt shall be the whole of the Law.
The word of Sin is Restriction.
Two questions are followed by some background material for
those who may wish to read it:
A) What is the logic behind King Wen's arrangement of the I Ching?
B) Are there many configurations of an 8 x 8 magick square in which the
central diagonals also add to the same total as the columns and rows??
How many, if so? Is there a method to construct them?
Below is a trail of exploration through the I Ching and a design for a
new arrangement of it based on the Magick Square of Mercury. It presumes
that the reader has a familiarity with the traditional King Wen I Ching.
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Research on the I Ching/I Chou (_Classic of Changes_/_Classic of Chou_)
There are three main arrangement strategies for I Ching including:
1) Sequential; 2) Inversive; and 3) Reflective.
1) Sequential/Binary Arrangement
Whatever its origin (whether the legendary Fu Hsi, Shao Yung or King Wen),
the Sequential (or Binary) arrangement seems to have the most
mathematical support. The hexagrams are assigned numbers within
King Wen's traditional, inversive arrangement.
While it traditionally begins with Hexagram 1, Ch'ien, I side with Leibnitz in
modifying this by reversing the order. This places Hexagram 1 at the end and Hexagram 2, K'un, at the beginning. From a Western point of view this makes
the most sense, since the 'broken' line can be said to 'lack a center' and
therefore best represents the binary '0'. See Illustration 'A' below.
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- - --- - - --- - - --- |
- - - - --- --- --- --- |
- - - - - - - - --- --- |
- - - - - - - - --- --- | 0, 1, 10, 11... 111110, 111111
- - - - - - - - --- --- |
- - - - - - - - ... --- --- |
2 23 8 20 ... 43 1 |
K'un Po Pi Kuan ... Kuai Ch'ien |
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Illustration A. Sequential/Binary Arrangement (Leibnitz)
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2) King Wen's Inversive Arrangement
The traditional arrangement consists of a numbered series of matched pairs,
mostly inversions. There are eight exceptions, the symmetrical
hexagrams, which are assigned numbers 1, 2, 27, 28, 29, 30, 61 and 62.
A logical reason for this arrangement has yet to be discovered. Certainly
the inversions make sense, yet their sequence does not fall into an obvious
pattern. R.L. Wing has this to say about it:
"[This] arrangement of the sixty-four hexagrams is the oldest known and
represents the sequence in which they appear in [the I Ching]. Each odd
numbered hexagram is followed by a hexagram that is either its opposite
or its inverse (that is, stood on its head). There is some mystery
surrounding the sequence of the odd numbered hexagrams. Scholars and
mathematicians are unable to unlock the code which generates the order
of these odd hexagrams. Perhaps the logic of their arrangement is
approachable only on the intuitive level, that is, by considering the
order of the hexagrams in terms of the human affairs they represent."
pg. 163, _The I Ching Workbook_, R.L. Wing. See Appendix III.
Here we encounter the first problem:
Question: What is the logic behind King Wen's arrangement of the I Ching?
Most modern tomes arrange the hexagrams in the traditional fashion out of
respect for the Chinese Classic. Yet when we attempt to integrate them into
occult practice, the traditional arrangement becomes burdensome.
What if they were arranged in this fashion to hide some more profound sequence?
What if the hexagrams were tossed together rather haphazardly so as to allow
the teacher to distinguish the dedicated student, who had reasoned them
into a logical arrangement, from the novice, who had simply followed tradition?
Considering this, and returning to the Sequential/Binary Arrangement,
it is quite natural to place the hexagrams in either a circular or square form.
The circle has the most appeal, since it doubles back upon itself when the hexagrams reach their climax (...111110, 111111, 000000, 1, ....). The circle
closes itself if we then exchange 0's for 1's (...111111/000000, 1, 10, 11...).
The result is a cyclic pictogram which meshes quite well with the
symbol of the Tao (Yin/Yang). The circular sequence may be found in most
translations of the I Ching.
Looking at the linear, binary progression of the 8 x 8 square, there is
still the problem of arriving at a numbering system which is more logical
than King Wen's Inversives. This might be solved by simply numbering
the binary progression from 0 to 63. Yet it may be addressed more subtlely
and syncretically by drawing from the Qabalah's 'magick squares' to create
a reflective arrangement.
3. Reflective Arrangement
Magick Squares
Charms or Talismans have always played a large role in Western Magick.
Magick Squares - grid/matrices into which letters or numbers are fitted
to create a Talisman - are featured prominently in popular systems of
ceremonial magick such as that of Abramelin the Mage, or John Dee.
Those with numbers, while less commonly used, are fairly standardized.
Squares from 3 to 9 numbers in size are traditionally associated with
the seven Planets (Saturn, Jupiter, Mars, Sun, Venus, Mercury and Moon)
and are used as Talismans for invoking the planetary energies in ritual.
Magick squares of numbers are composed of the entirety of a series beginning
at one. They are placed within the matrix such that they add along horizontal
and vertical rows to an identical number. Traditionally, the squares also
total the same number along central diagonals. See Illustration 'B' below.
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Examples of magick squares:
Saturn: 4 9 2
3 5 7
8 1 6
(Refl. (opposing radials): 10; Line (total of each line): 15; Sum: 45)
Jupiter: 4 14 15 1
9 7 6 12
5 11 10 8
16 2 3 13
(Refl.: 17; Line: 34; Sum: 136)
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Illustration B. The Magick Squares of Saturn and Jupiter
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The Magick Square of Mercury is an 8 x 8 matrix, identical in size to the
square arrangement of the I Ching hexagrams. From this we may derive a logical
association of number and a more coherent arrangement of the I Ching.
Illustration 'C' shows Mercury's magick square paired with the Binary I Ching:
---------------------------------------------------------------
The I Ching Magick Square
1 56 48 25 33 24 16 57
000000 000001 000010 000011 000100 000101 000110 000111
63 10 18 39 31 42 50 7
001000 001001 001010 001011 001100 001101 001110 001111
62 11 19 38 30 43 51 6
010000 010001 010010 010011 010100 010101 010110 010111
4 53 45 28 36 21 13 60
011000 011001 011010 011011 011100 011101 011110 011111
5 52 44 29 37 20 12 61
100000 100001 100010 100011 100100 100101 100110 100111
59 14 22 35 27 46 54 3
101000 101001 101010 101011 101100 101101 101110 101111
58 15 23 34 26 47 55 2
110000 110001 110010 110011 110100 110101 110110 110111
8 49 41 32 40 17 9 64
111000 111001 111010 111011 111100 111101 111110 111111
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Illustration C. The numbers of the Magick Square of Mercury paired with
Leibnitz's Sequential/Binary Hexagram Arrangement.
Each row, column and central diagonal total 260.
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Magick squares are diagonally reflective. Individual numbers may be
added to their inverted, symmetrical reflections to achieve identical
totals. In the Jupiter square, for example, we may add the 4 and 13 to
arrive at 17, as well as the 11 and 6, 12 and 5, etc. In the Mercury square
this factors not only into the totals of the numbers assigned to the
hexagrams (consistently 65), but due to the configuration of the
hexagrams themselves, it also highlights the graphic reflection
throughout the I Ching.
Some examples demonstrate this:
1 + 64 = 65
000000 / 111111
20 + 45 = 65
100101 / 011010
27 + 38 = 65
101100 / 010011
Hexagram 1, 000000, is a graphic reflection of hexagram 64, 111111. Hexagram
20, 100101, is a graphic reflection of hexagram 45, 011010.
Rotating the magick square 90, 180 and 270 degrees produces minor variations
on this theme, yet does not allow the aesthetically pleasing placement of
the number 1 upon the 000000 (K'un) hexagram and 64 upon the 111111 (Ch'ien).
It may be possible that other major configurations exist, however.
Here we encounter the second problem:
Question: Are there many major configurations of an 8 x 8 magick square in which
the central diagonals also add to the same total as the columns and rows??
How many, if so? Is there a method to construct them?
If there are other configurations, then we may select among these to arrive
at the most symbolically and aesthetically pleasing. The configuration
above seems very effective in any case. This gives a logical framework
for the sequential/binary arrangement and the assignment of numbers to
the hexagrams of the I Ching. Unless a superior magical square configuration
is found, the above can be said to be the only reasonable solution.
--------------------------------------- END ----- >> Appendices follow:
Appendix I
The Nigris Sequence
Given the Leibnitz/Mercury overlay (Illustration 'C'), the hexagrams
may now be arranged numerically in the following manner:
# Binary Name Base-2 Value King Wen's #
[Notes]
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1 000000 K'un 0 2
[Double 000 Trigram]
2 110111 Lu 55 10
3 101111 T'ung Jen 47 13
4 011000 Sheng 24 46
5 100000 Fu 32 24
6 010111 Sung 23 6
[The only retained #!]
7 001111 Tun 15 33
8 111000 T'ai 56 11
9 111110 Kuai 62 43
10 001001 Ken 9 52
[Double 001 Trigram]
11 010001 Meng 17 4
12 100110 Sui 38 17
13 011110 Ta Kuo 30 28
14 101001 Pi 41 22
15 110001 Sun 49 41
16 000110 Ts'ui 6 45
17 111101 Ta Yu 61 14
18 001010 Chien 10 39
19 010010 K'an 18 29
[Double 010 Trigram]
20 100101 Shih Ho 37 21
21 011101 Ting 29 50
22 101010 Chi Chi 42 63
23 110010 Chieh 50 60
24 000101 Chin 5 35
25 000011 Kuan 3 20
26 110100 Kuei Mei 52 54
27 101100 Feng 44 55
28 011011 Sun 27 57
[Double 011 Trigram]
29 100011 I 35 42
30 010100 Hsieh 20 40
31 001100 Hsiao Kuo 12 62
32 111011 Hsiao Ch'u 59 9
33 000100 Yu 4 16
34 110011 Chung Fu 51 61
35 101011 Chia Jen 43 37
36 011100 Heng 28 32
37 100100 Chen 36 51
[Double 100 Trigram]
38 010011 Huan 19 59
39 001011 Chien 11 53
40 111100 Ta Chuang 60 34
41 111010 Hsu 58 5
42 001010 Lu 13 56
43 010101 Wei Chi 21 64
44 100010 Chun 34 3
45 011010 Ching 26 48
46 101101 Li 45 30
[Double 101 Trigram]
47 110101 K'uei 53 38
48 000010 Pi 2 8
49 111001 Ta Ch'u 57 26
50 001110 Hsien 14 31
51 010110 K'un 22 47
52 100001 I 33 27
53 011001 Ku 25 18
54 101110 Ko 46 49
55 110110 Tui 54 58
[Double 110 Trigram]
56 000001 Po 1 23
57 000111 P'i 7 12
58 110000 Lin 48 19
59 101000 Ming I 40 36
60 011111 Kou 31 44
61 110111 Wu Wang 39 25
62 010000 Shih 16 7
63 001000 Ch'ien 8 15
64 111111 Ch'ien 63 1
[Double 111 Trigram]
NOTE THE PROGRESSIVE CYCLIC NATURE OF THE DOUBLE TRIGRAMS
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Appendix II
Magick Squares for Mars, Sun, Venus and Moon.
The 'Line' value is the total of each column, row or central diagonal. The
'Refl.' value is the consistent total obtained by adding diagonally reflected
numbers. The 'Sum' value is the total of all the numbers in the square.
Magick Square of Mars:
11 24 7 20 3
4 12 25 8 16
17 5 13 21 9
10 18 1 14 22
23 6 19 2 15
(Refl.: 26; Line: 65; Sum: 325)
Magick Square of Sun:
6 32 3 34 35 1
7 11 27 28 8 30
24 14 16 15 23 19
13 20 22 21 17 18
25 29 10 9 26 12
36 5 33 4 2 31
(Refl.: 37; Line: 111; Sum: *666*)
Magick Square of Venus:
22 47 16 41 10 35 4
5 23 48 17 42 11 29
30 6 24 49 18 36 12
13 31 7 25 43 19 37
38 14 32 1 26 44 30
21 39 8 33 2 27 45
46 15 40 9 34 3 28
(Refl.: 50; Line: 175; Sum: 1225)
Magick Square of Moon:
37 78 29 70 21 62 13 54 5
6 38 79 30 71 22 63 14 46
47 7 39 80 31 72 23 55 15
16 48 8 40 81 32 64 24 56
57 17 49 9 41 73 33 65 25
26 58 18 50 1 42 14 34 66
67 27 59 10 51 2 43 75 35
36 68 19 60 11 52 3 44 76
77 28 69 20 61 12 53 4 45
(Refl.: 82; Line: 369; Sum: 3321)
( Refl.: 82; Line: 369; Sum: 3321)
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Appendix III
References (Texts used in this research)
_An Anthology of I Ching_, by W. A. Sherril and W.K. Chu, Arkana Books, 1977.
_I Ching: Book of Changes_, by James Legge, Ed. by Ch'u Chai and Winberg Chai,
Bantam Books, 1969.
_The I Ching or Book of Changes_, by Richard Wilhelm, transl. by C.F. Baynes,
Princeton/Bollingen, 1990.
_Change: Eight Lectures on the I Ching_, by H. Wilhelm, transl. by C.F. Baynes,
Princeton/Bollingen, 1973.
_The I Ching Workbook_, by R.L. Wing, Doubleday, 1979.
_The Secret Lore of Magic_, by Idries Shah, Abacus, 1972.
_The Black Arts_, by Richard Cavendish, Putnam, 1967.
_Godwin's Cabalistic Encyclopedia_, by David Godwin, Llewellyn, 1989.
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Invoke me under my stars. Love is the law, love under will.
I am I!
9208.16 e.v.
Frater (I) Nigris (666) 333
Tyagi@Houseof Kaos.AByss.com
871 Ironwood Dr.
Saint Joseph, Kali Fornica, 95125-2815