Creating the mass model led to the discovery of some new properties .. such as the moment of inertia  by Peter Loly.  The model also lends itself to degenerating the square to one dimension and looking at its balance in one dimension across the midline

 

 

Figure 1 above … shows the distance from center values for the degenerated order 3 square

 

 

 

 

 

 

 

Figure 2

 

    Shows the 4 possible degenerations for the order 3 square …

 

         Ie read left to right, right to left, top to bottom, and bottom to top

 

 

 

 

 

 

 

 

 

 

 

Figure 3 above …

 

   I wrote a program to search for all order 3 squares where the only criteria was

      That the sum of the  mass x distance totals had to be equal for the square for 

      All four possible degenerations ..        

        Two unique examples other than the standard

        Magic square structure are  found.

 

   The pattern can be extended for higher order of squares but no interesting

   Patterns emerge ( Francis Gaspalou)

 

 

 

 

 

Addendum …

 

 

All magic squares still balance across the midline when degenerated to one dimension.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4  above …

 

 

I have a continued interest in looking at exchanging distance and cell value units for the square.