*(Continued from Epiphany and from yesterday.)*

Detail from the current American Mathematical Society homepage—

Further detail, with a comparison to Dürer's magic square—

The three interpenetrating planes in the foreground of Donmoyer's picture

provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer's array into four 4-sets corresponding to the

four rows of Dürer's square, and apply the 4-color decomposition theorem.

Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these

occupy the same positions in the Donmoyer square that 4-sets of

like elements occupy in the diamond-puzzle figure below—

Thus the Donmoyer array also enjoys the *structural * symmetry,

invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem's interpenetrating *lines * explain the structure

of a 4×4 *square** *, the foreground's interpenetrating *planes * explain the structure

of a 2x2x2 *cube** *.

For an application to theology, recall that *interpenetration* is a technical term

in that field, and see the following post from last year—