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Symmetry

 

 

The examples use integers to illustrate magic squares and hyper cubes.  It is easy to visualize the symmetry (more properly "anti-symmetry") because each integer is matched by its negative in the opposite position of a square or hyper cube.

This arrangement also helps to visualize that the sum of each row, column, and principal diagonal is equal to zero.

It is instructive to view magic squares and hyper cubes  that are illustrated with tuples instead of integers.  

The number of digits in each tuple corresponds to the Dimensions for the square or hyper cube.  The range for the digits is from

(-Order - 1)/2  to (+Order - 1)/2 for odd squares and hyper cubes; and (-Order )/2  to (+Order )/2  for even.

For example,

notice the right-most digit in column "L" in the magic square above.  These digits form an ascending series from +1 through 0.  Operations on these digits are similar to a digital "half-adder" where the most negative value that can be represented follows the most positive value that can be represented.

As a further example,

notice the pattern of digits around the center cell.  Each series of digits follows the pattern established by the Direction_Controller as explained in Methods.