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I invent models to explore the physical properties of magic squares.
The illustration below is a 11 x 11 magic square depicted as a water retaining surface. The letter "H" is shown with side specific drainage to the left and to the top. Of the 26 letters, the letter H is the most difficult to make. Harry White and I wrote the American declaration of independence on the surface of a large magic square in 2014.
This illustration shows a efficient way to figure the water retention for one of these squares.
The concept of "most-perfect space" requires all 2x2 planar subsets to have the same sum. This allows most-perfect space to be compared to "reversible space" which operates on the same 2x2 planar subsets.
When a Hilbert space filling curve through "most-perfect space" has positions equidistant along the curve sum to 2x the 2x2 subset sum (260 in this example) the structure is called "magic space".
For cubes the definition of compact is that all 2x2x2 sub cubes add to the same sum. That definition also includes wrap around. Your most perfect space cube is compact. It has the additional constraint that each orthogonal plane is also compact. There are 64 2x2x2 sub cubes that add to 260 and 192 2x2 sub squares that add to 130 in your cube. ( Dwane Campbell)
There are 15 essentially different 4x4x4 most-perfect magic 4x4x4 cubes. Walter Trump and Francis Gaspalou have figured out the operative transformations and cyclic permutations - 15 x 64 ... gives 960 total.
Both the most-perfect magic cubes and the reversible cube are pantriagonal.
Dame Kathleen Ollerenshaw and David Bree showed the 1:1 correspondence of reversible squares to most-perfect magic squares. A most-perfect cube / reversible cube correspondence is considered here. The 108 2x2 planar subset criteria of having diagonals that have the same sum though those sum may differ plus requiring symmetrically opposite pairs on the row/col/pillars to have the same sum (example: 21+55 = 23+53, 21+13 = 5+29 etc) produce the 960 reversible cubes. Or more simply requiring the main diagonals to have this property will suffice (21 + 44 = 3 + 62). Also all possible diagonally opposite pairs in the cube (864) mimic the 2x2 sub square criteria.
Ideally after producing the reversible cube from the simple 2x2 sub square criteria a transformation applied to those cubes would yield the 960 most-perfect cubes.
Percolation constant for a 5 neighbor system
It was on my bucket list to calculate a percolation constant. The square lattice has 4 nearest neighbors with a Pc of 0.59. The cube has 6 nearest neighbors with a Pc of 0.3116. This little bilayer system has 5 nearest neighbors and by my calculation has a Pc of 0.36.