Trigrammaton and Magic Squares

 

            A magic square is created when a series of numbers are placed on a square grid in such a way that each row and column of the grid sums to the same number, known as the magic sum of the square.  Usually the diagonals also add to the magic sum, and in some cases other symmetrical subsections of the grid will also equal the magic sum.  The classic magic squares are formed from a numerical sequence, such as all the numbers from 1 to 25, but there is no requirement that the numbers used be in sequence; they might be 25 odd numbers or 25 multiples of 3, etc. 

These diagrams have held a powerful fascination for occultists and mathematicians over the centuries.  In the Western occult tradition they are known by the Hebrew name Kamea, and have been used as talismans or in the art of sigil magic relating to the planetary energies.  Each planet presides over a different Kamea; Saturn over the 3 x 3, Juppiter over the 4 x 4, Mars over the 5 x 5, Sol over the 6 x 6, Venus over the 7 x 7, Mercury over the 8 x 8, and Luna over the 9 x 9.  These numbers are derived from the number of the Sephira on the Tree of Life  that each planet is attributed to.

            Trigrammaton provides for 729 possible Hexagrams, which may be arrayed on a 27 x 27 grid with very interesting results.  Numerous subsets of Hexagrams may also be used to create magic squares, and I have provided some examples of these, based on 7 x 7, 8 x 8, and 12 x 12 grids.  What is remarkable about these grids is that they display a new property of magic squares, one impossible to duplicate using Base Ten numerals.  This is the property of reversal; each of these magic squares may be reversed, or rotated 180 degrees, and still remain a magic square.

By this I mean that if the numbers in each cell are expressed in base 3, by reversing the digits of each individual number, the grid remains a magic square.  If this grid were displayed using hexagrams, then the same effect would be demonstrated by turning the page upside down.

 All but 27 of the Hexagrams become a different number when reversed, but the reversed number array, although composed of the same numbers in a completely different arrangement, is still a magic square.  A general rule which explains this can be found by examining the 8 x 8 squares; the rule is that any pair of Hexagrams that sum to 728 will reverse their digits to become another pair of numbers that equal 728.

All of the magic squares are represented in base 10 notation, to make it easier to understand the grid and use the totals.  But it should be kept in mind that these are all meant to be understood in terms of base 3.

            The 7 x 7 magic square illustrated below was created using only the first 7 Trigrams of Trigrammaton, in all 49 of their possible Hexagram arrangements.  These first seven represent the Zero and the 6 Elements; they are 0, 1, 2, 3, 6, 9, & 18.  This kamea was accomplished by placing these 49 Hexagrams in numerical order from lowest to highest, (0-505), then assigning these based on the numbers from 1 - 49 as found on the traditional Kamea of Venus.  Finally, the top row of this grid was removed and placed on the bottom, so that the Zero Hexagram is located in the center of the grid.  The result is that all of the Hexagrams involving the Zero Trigram are found on the diagonals.  These diagonals do not add to the magic sum separately, but when added together they equal the magic sum.

            Following that is an 8 x 8 magic square based on the famous square created by brother Benjamin Franklin.  This squares uses only the last eight Trigrams of Trigrammaton, which are the Planetary, or Classic Trigrams, and thus involves all 64 I Ching Hexagrams.  The array was created by the same method of placing the numbers, (351-728), on the grid in order from lowest to highest based on the position of the numbers 1 - 64.  The Franklin square is mathematically superior  to the kamea of Mercury, insofar as the magic sum can be arrived at in many more ways.

            Also given is a 12 x 12 square, invented by the author, which uses only the 12 Zodiac Trigrams in all 144 of their possible Hexagrams.  This grid has several interesting characteristics, including the fact that each of its diagonals in the smaller 3 x 3 squares are reversals of each other.

            Thus in these examples, I have demonstrated the interactions of the three primary groups of the Trigrams: Elemental, Planetary and Zodiacal, with the other members of their group.

 

 

The Elemental Kamea in Decimal Notation

 

  6        82       495      56       261      30       162

163        9        83       504      57       243      33

 36       164      18       84       486      60       244

245      45       165        0        87       487      63

 72       246      27       168        1        90       488

489      54       249      28       171        2        99

81        492      55       252      29       180        3

 

 

 

Ben Franklin’s Planetary Square in Decimal Notation

 

400      368      724      692      643      611      481      449

 

691      725      367      401      448      482      610      644

 

395      373      719      697      638      616      476      454

 

698      718      374      394      455      475      617      637

 

392      376      716      700      635      619      473      457

 

701      715      377      391      458      472      620      634

 

403      365      727      689      646      608      484      446

 

688      728      364      404      445      485      607      647

 

 

Italics are Hexagrams with doubled Trigrams;

underline shows Hexagrams that remain the same when reversed. 

Magic sum of the 7 x 7 is 1092.  Magic sum of the 8 x 8 is 4368.

 

 

Reversed Elemental Kamea in Decimal Notation

 

81        60       495      28       261      164        3

57        486      33       252      163      18       83

488      30       243      168        9        82       72

 45       245      165        0        87       63       487

244      180        2        84       54       492      36

171        1        99       56       489      27       249

  6        90       55       504      29       246      162

 

                       

Reversed Ben Franklin’s Square in Decimal Notation

 

725      365      727      367      724      364      728      368

 

610      484      608      482      611      485      607      481

 

374      716      376      718      373      715      377      719

 

475      619      473      617      476      620      472      616

 

455      635      457      637      454      634      458      638

 

394      700      392      698      395      701      392      697

 

644      446      646      448      643      445      647      449

 

691      403      689      401      692      404      688      400

 

 

Italics are Hexagrams with doubled Trigrams;

underline shows Hexagrams that are the same when reversed. 

Magic sum of the 7 x 7 is 1092.  Magic sum of the 8 x 8 is 4368.

 

 

 

 

 

12 By 12 Magic Square of the Zodiac in Decimal Notation

 

142      318      424      533      575      213      339      119      275      658      220      552

 

316      412      156      591      209      521      113      285      335      228      550      652

 

426      154      304      197      537      587      281      329      123      544      660      226

 

663      227      545      334      112      282      520      588      208      155      305      429

 

221      555      659      120      280      328      586      196      534      321      425      143

 

551      653      231      274      336      118      210      532      574      413      159      317

 

523      571      201      150      308      410      668      224      564      331      129      289

 

579      199      517      302      420      146      240      560      656      127      277      345

 

193      525      577      416      140      312      548      672      236      291      343      115

 

344      116      294      655      237      559      145      301      417      528      578      194

 

132      290      332      235      547      669      309      415      139      572      204      524

 

278      348      128      561      667      223      409      147      307      200      518      582

 

Italics are Hexagrams with doubled Trigrams. 

Underlines are Hexagrams which are identical when reversed. 

 

The 12 x 12 grid has a substructure of a 4 x 4 grid based on the Elemental nature of the signs.  The doubled Elements occupy the four central squares, and the remaining sub-elements balance each other across the grid.  The entire magic square can be reversed and remain magical.  Each row and column contains 12 different upper and lower Trigrams.  The magic sum of this square is 4368 = 28 x 156. 

            Each of the squares in this 4 x 4 grid is a 3 x 3 magic square, (except for their diagonals).  Each sector has three signs of the same Element for the upper Trigrams and three signs of the same Element for the lower Trigrams, in all nine possible combinations.  The magic sum of these 3 x 3 squares is given under the name of the sector.  Each of the four quarters of the grid also adds to the magic sum, 4368.

 

 

 

Air of Earth                  Water of Air                 Earth of Fire                 Fire of Water

884                       1321                             733                             1430

Earth of Water              Fire of Fire                   Air of Air                      Water of Earth

1435                      728                            1316                               889

Fire of Air                   Earth of Earth             Water of Water               Air of Fire

1295                      868                            1456                               749

Water of Fire                Air of Water                 Fire of Earth                 Earth of Air

754                      1451                              863                             1300

           

 

           

Reversed 12 x 12 Magic Square of the Zodiac in Decimal Notation

 

194      578      528      289      129      331      429      305      155      552      220      658

 

524      204      572      345      277      127      143      425      321      652      550      228

 

582      518      200      115      343      291      317      159      413      226      660      544

 

417      301      145      564      224      668      208      588      520      275      119      339

 

139      415      309      656      560      240      534      196      586      335      285      113

 

307      147      409      236      672      548      574      532      210      123      329      281

 

559      237      655      410      308      150      282      112      334      213      575      533

 

669      547      235      146      420      302      328      280      120      521      209      591

 

223      667      561      312      140      416      118      336      274      587      537      197

 

294      116      344      201      571      523      545      227      663      424      318      142

 

332      290      132      517      199      579      659      555      221      156      412      316

 

128      348      278      577      525      193      231      653      551      304      154      426

 

 

Italics show Hexagrams with doubled Trigrams,

underline shows Hexagrams which are reversible. 

In some cases these sets overlap. 

 

This graph represents reversal of the entire grid at once.  It is also possible to reverse smaller sections of the grid, or to reverse each cell individually.  So long as all cells are reversed, the result will be a magic square.  The nine numbers within each of the sixteen 3 x 3 cells remain together, in a different order, when reversed.  Thus, for example, reversing the Earth of Fire subsection results in the Fire of Air subsection.  Fire signs reverse to Fire signs, being only Tao and Yang lines.  Water signs invert to Water, being only Tao and Yin lines.  Air and Earth signs, having Tao, Yang, and Yin lines, reverse into each other.

 

 

                                   Symbolic Layout of the 12 x 12 Magic Square of the Zodiac Hexagrams

 

 

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