Very nice!<p>You might be interested in a simple proof I found of why c/2 and c/3 are speed limits for orthogonal and diagonal spaceships respectively.<p>Definition: In a gameplay of life, an "infinite lifeline" is a sequence of pairs (c_i,n_i) such that each c_i is alive in generation n_i and either c_(i+1)=c_i or c_(i+1) is adjacent to c_i.<p>Lemma ("Two Forbidden Directions"): Let x,y be any two 'forbidden' directions from among N,S,E,W,NE,NW,SE,SW. In any gameplay of life that starts finite and doesn't die out, there is an infinite lifeline that never goes in either direction x or y.<p>The lemma's proof uses biology. Say that (c,n) is a "father" of (c',n+1) if c' is the cell adjacent to c in direction x or y. Otherwise, (c,d) is a "mother" of (c',n+1). By the rules of the game of life it's easy to show every living (c,n+1) has at least one living father and at least one living mother. It follows (modulo some more details) that since the gameplay doesn't die out, there must be an infinite lifeline where each cell is a mother of the next, i.e., an infinite lifeline that never goes direction x or y.<p>Proof of c/2 orthogonal speed limit: If a spaceship went faster than c/2, say, northward, by the lemma, it would have an infinite lifeline that never goes N or NE. The only way it could ever go northward would be to go NW. Every NW step would have to be balanced out by an eastward step (of which NE is forbidden) or the spaceship would drift west. So every northward step requires a non-northward step, QED.<p>Proof of c/3 speed limit for diagonal: A diagonal spaceship faster than c/3, say, northeastward, would have an infinite lifeline that never goes N or NE. The only way for it to go northward would be to go NW. Each NW step would need at least two eastward steps in order for the ship to go eastward, QED.