Width 3 presents certain peculiar features due to a number theoretic
property of Life
's rule of evolution. On a 
torus, the still
lifes are those configurations which contain exactly four live cells.
If the central cell is live, it has three live neighbors and thus
survives; otherwise the four live neighbors prevent a live cell from
forming and again the arrangement is stable.
For a 
torus, the same principle applies, but the
arrangement of the live cells in a cross section can vary as one moves
along the long dimension of the torus. A constant sum of 4 can be
realized in the forms 3 + 1 + 0, 2 + 2 + 0, and 2 + 1 + 1, and
their permutations.
There is no way of breaking out of the sequence 2 + 1 + 1, but the
presence of zeroes in the other two allows them to terminate in
quiescent regions. Moreover, the sequence 
can
be judiciously interspersed in the latter two sequences to produce
still further variation.
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