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. They first appeared in Heinrich Cornelius Agrippa’s 16^th century work, Three Books of Occult Philosophy.

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 Sefira each planet is attributed to on the Hebrew Kabbalistic Tree of Life.

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, especially 7 x 7, 8 x 8, 9 x 9 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 10 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. 

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 to become another pair of numbers that equal 728.

A version of the 8 x 8 magic square is given below, based on the Kamea of Mercury.  These squares use only the last eight trigrams of Liber Trigrammaton. Although these look identical to the 64 I Ching hexagrams, their numerical value is derived from Base 3, not binary. The numerical values of the Planetary trigrams are 13, 14, 16, 17, 22, 23, 25, 26. These eight numbers align as four pairs of antigrams, from highest to lowest value: 13 & 26, 14 & 25, 16 & 23, 17 & 22. Because of this, the patterns created by this magic square are full of antigrams.

The array was created by placing the numbers on the grid in order from lowest to highest, based on the position of the numbers 1 – 64 in the original Kamea of Mercury.

The Kamea of Mercury in Planetary Hexagrams

The lower left to upper right diagonal contains the 8 doubled-Trigrams, while the other diagonal contains hexagrams that have antigrams in upper and lower position. Any pair of hexagrams that are equidistant from the center, i.e., a line joining them goes through the center of the magic square, are a pair of antigrams. Thus the hexagram directly below the upper left corner is the antigram of the one directly above the lower right corner. This kamea is reversible; it can be turned upside-down and still remain magic.

Magic sum of the 8 x 8 is 4368.