Rigid spaceships in Conway's Game of Life

Question

(1) Is it true that there are no rigid spaceships in Conway's Game of Life, i.e. spaceships with period 1 i.e. spaceships of constant shape (only allowed to rotate) of non-zero translational velocity?

If so: why do such spaceships have to change their shape while moving?

How can this be seen most easily?

Are there Life-like cellular automata with rigid spaceships?

(2) Can it be shown (and how) if a) almost all spaceships have a fixed number of living cells, b) almost all spaceships have a varying number of living cells, c) neither of both?

Answer 1

Let us assume that the spaceship is bounded or maybe that it has only finitely many live cells. Notice that if we put live cells on every even numbered row we get a fixed configuration.

$$\begin{matrix}...&x&x&x&...\\...&o&o&o&...\\...&x&x&x&...\\...&o&o&o&...\end{matrix}$$

Now, since it is bounded we can find a live cell that has live neighbors only on a semi-plane of its neighbors. Graphically this is that at most the neighbors can be as in

$$\begin{matrix}o&x&x\\o&x&x\\o&x&x\end{matrix}$$

or $$\begin{matrix}x&x&x\\x&x&x\\o&o&o\end{matrix}$$

You get the idea. Now, if the center living cell must be preserved we can have at most $3$ living neighbors as in

$$\begin{matrix}o&x&o\\o&x&x\\o&x&o\end{matrix}$$

But the neighboring dead cells must also be preserved. So, only two living neighbors and in a row like

$$\begin{matrix}o&x&o\\o&x&o\\o&x&o\end{matrix}$$

With this same argument the row of living cells must continue indefinitely. But this can't be if the spaceship must be unbounded.

Answer 2

In response to the first question: It can be shown that any such spaceship in the Game of Life would be of one of the two speeds c orthogonal or c/2 diagonal. Both of these speeds have been proven to be impossible, so none can exist. However, there are, in fact, life-like cellular automata that contain such spaceships. Specifically, B2 rules (i.e. ones with birth from two neighbors) often have this property.