Log24

(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—