Does Day and Night on a Klein bottle have a steady state?

Question

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)?

Answer 1

Here's a still-life position on the infinite plane: the marker on $(x,y)$ is oriented one way if $x>0$, the other if $x<0$. If $x=0$, the orientation depends on whether $y$ is even or odd.

Each marker on the $y$-axis $x=0$ has three neighbors of its own orientation. Each marker on $x = \pm 1$ has $6$ or $7$ neighbors if its own orientation, and all markers with $|x| \geq 2$ are entirely surrounded by markers of their own orientation. Hence the position is stable.

Now roll the plane into a Klein cylinder: choose some large negative $x_1$ and large positive $x_2$, remove all markers with $x$ outside $[x_1,x_2]$, and identify $(x_1,y)$ with $(x_2,-y)$. This gives a still-life position on a Klein cylinder of circumference $x_2 - x_1$.

The position is periodic in the $y$ direction with period $2$, so we can get a still-life position on an $(x_2-x_1) \times n$ Klein bottle for any even $n$.

There's likely some variation on this that handles odd $n$ too.