Properties of the 2 x 2 Toroidal Polyhex Matrix

It was explained above that the three-element code used to characterize a toroidal polyhex is derived from a convenient special case of the matrix 

A useful general property is that the number of hexagons is equal to the absolute value of ad-be. 

Rules for manipulation of the matrix elements have been given which enable isomorphic equivalences of matrices to be detected, as an alternative to the graphical approach of the preceding section. 

Here h is the highest common factor of a and b, and x is the multiplicative inverse of bih modulo a/h (or, put another way, x is an integer such that xb -ya = h for some integer y). Rules 2 and 3 are self-inverse, i.e., a second application restores the original matrix. 

Finally, a fifth, and obvious, mle needs to be stated in order to construct a rule set that is useful for comprehensive enumeration.  

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