Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^{l}, l=1,2,3,..., with Information Entropy

Abstract: Frierson used a powerful parameterization of the pattern of the order 3 associative magic square to construct a family of six related order 9 compound (or composite) magic squares, several of them ancient. Stimulated by Bellew's 1997 extension to order 27, we extend these ideas to all orders that are powers of 3, and in addition find simple formulae for the matrix spectra and entropic measures for all those orders. This construction is fractal and we give numerical results to order 243 which show an information entropy measure converging to a constant value of about 1.168.. for the lowest entropy members. We also briefly consider compounding of an order 4 magic square with the lowest entropy, for which we find a similar trend to constant entropy.