Counterimage matrices

Since the evolved cell has been inserted into the symbolic matrix, it is hardly surprising that strings of evolved cells appear as elements of the product matrices. But it is decidedly inconvenient to have to look up all their indices to recover the original string, especially when the same information could have been encoded in the symbolic matrix from the beginning.

Therefore let a subscript such as denote the letter of the word w, beginning with the first; would be the state of the central cell of its neighborhood for integral r.

This time vary Eq. 8 by defining


which is likewise decomposable into a sum

by defining


Continuing to use Rule 22 as an example, 0.30em

There are still seven nonempty matrix elements. Expanded by adjoining the implicit boundary cells to obtain all seven precursors of the sequence 00, the evolution shown below results.

There is no reason to insist that the central cell be used in defining the matrices ; frequently an ancestor at one side is chosen, particularly if the radius of the neighborhood is half integral.

Harold V. McIntosh