From: Tagi@cup.portal.com (Tagi Mordred Nagashiva)
Newsgroups: alt.magick
Subject: I Ching and Magick Squares (Long)
Date: Sun, 16 Aug 92 19:31:10 PDT
Organization: The Portal System (TM)
Distribution: world

9208.16 e.v.


Do what thou wilt shall be the whole of the Law.
The word of Sin is Restriction.



I Ching and Magick Squares
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Two questions to the newsgroup, 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_)

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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.  

---------------------------------------------------------------------

- -   ---   - -   ---    - -   ---  |
- -   - -   ---   ---    ---   ---  |
- -   - -   - -   - -    ---   ---  |
- -   - -   - -   - -    ---   ---  | 0, 1, 10, 11,..., 111110, 111111
- -   - -   - -   - -    ---   ---  |
- -   - -   - -   - -... ---   ---  |
 2    23     8    20 ...  43    1   |
K'un  Po     Pi  Kuan... Kuai Ch'ien|

----------------------------------------------------------------------
Illustration A.  Sequential/Binary Arrangement (Leibnitz)
----------------------------------------------------------------------


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:

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.

---------------------------------------------------------------------
Examples of magick squares:


Saturn:       4     9     2
              3     5     7
              8     1     6           (line total: 15; grand total 45)


                Jupiter:      4     14     15      1
                              9      7      6     12
                              5     11     10      8
                             16      2      3     13         
                                          
                                       (line tot.: 34; grand tot.: 136)

-----------------------------------------------------------------------
Illustration B.   The Magick Squares of Saturn and Jupiter
-----------------------------------------------------------------------


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        2       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

-------------------------------------------------------------------
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.
-------------------------------------------------------------------


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:

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]
-----------------------------------------------------------------      
1       00000       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

------------------------------------------
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     

                              (Line: 65; Refl.: 26; 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    

                               (LIne: 111; Refl.: 37; Sum: *666*)


Magick Square of Venus:

               22     47     16      41      10     35      4
                5     23     48      17      42     11     29
               30      6      4      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    

                                (Line: 175; Refl.: 50; 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
    
                                                                               
                                (Line: 369; Refl.: 82; 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!


Frater I Nigris (666)