Magic Squares have been around a very long time. To some, they are mere mathematical curiousities. To others, they can be gates and keys to powerful magical forces. What they are, essentially, are square arrangements of a set of numbers that has peculiar mathematical properties. The magic squares I have been most interested in are those that use the same numbers as the squares they are based upon.
A magic square has a base square that is a true square, with four sides of equal length. These squares have a Base number that is the length of one side, a Square number that is the base times itself, and a Constant that the square adds up to. There are three basic rules to qualify a Magic Square:
The oldest of these is Square Three. This is the same 3x3 matrix that is the basis for the Tet-Ra Tree of Life. It is said to have come from China in ancient times, appearing on the shell of a turtle. This is the first of what can be considered real magic squares.
We could call the square of One the first. It does obey all three rules above. But, since it has only one cell in the grid (1x1) and only one digit, it can hardly be called magical.
Likewise, the Square of Two is unsolvable according to the three rules. It cannot be made to add to a constant in all directions. Therefore, of nine possible squares using only the base digits, there are really only seven that we can use. It is using these seven (from three to nine) that we find a pattern develop.
Square Three Has a pattern to it in the way the numbers are plotted. The opposite in a line of all seven is Square Nine. Closer examination of Square Nine reveals the numbers to be plotted in the same pattern. If the 9x9 grid is broken into the same nine sections as Square Three, this pattern is apparent.
Each of the nine sections contains numbers that all reduce to the same base number. (The section containing the number 1 has nine numbers that will all add back to 1.) Within this section these nine numbers are also arranged in the same pattern as in Square Three. The result is that the entire square is as an octave higher than the Square Three.
The pattern to arrange Square Four is quite a bit different. The shaded numbers show a pattern of simply leaving the numbers where they would be if we started at the top left and placed them in order, row by row to the bottom. The remaining numbers are then plotted in the exact reverse order from the bottom to the top. On the number line of seven, Square Eight is the opposite of Square Four. As with Squares Three and Nine, Squares Four and Eight each follow the same pattern enlarged to a higher octave.
Squares Five and Seven also follow the same pattern. These have patterns that begin with 1 in the middle of the top row. From here the counting order is followed up and to the right in a diagonal line. When the edge of the square is reached, the count resumes as if another square were placed beside or on top of the original. The strangest part of this pattern is the way the diagonal is shifted down one row when there is already a number in the line ahead of it. Otherwise it continues counting diagonally.
A detailed diagram of this pattern is available here.
So, all seven of these magic squares are perfectly balanced on opposite sides of Square Six which is in the center of the line. Square Six has the most bizarre pattern of design of the seven, and is discussed in detail on this page: The Center Square
