Place a $m \times n$ ($m,n \ge 3$) square grid on a Klein bottle. On each square, we select an arbitrary non-mirror symmetric marker, and arrange them on the Klein bottle in some way. This arrangement is called a "position".
Now, we introduce the following operation on positions. For each square, we do the following:
- If the symbol on the square is the same orientation as 3, 6, 7, 8 symbols of its moore neighbors, do nothing to that symbol.
- Otherwise, replace it by its mirror reflection.
This definition takes advantage of the fact that although the Klein bottle is not globally orientable, it is locally orientable (which is true of every space). What we have defined basically is Day and Night on a Klein bottle, where life is the mirror symmetry of death.
My question is this: is there a still-life (meaning, a position for which the operation does nothing)?