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

TL;DR

This paper extends Frierson's 1907 parameterization of order 3 associative magic squares to all powers of 3, providing formulas for spectra and entropy, and analyzing their fractal and entropic properties up to order 243.

Contribution

It generalizes Frierson's construction to all orders of 3^l, introduces formulas for spectra and entropy, and explores their fractal and entropic characteristics.

Findings

Entropy converges to approximately 1.168 for large orders.

Construction is fractal and extends to order 243.

Similar entropy trend observed in order 4 magic squares.

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.