Jafet

(* Conway's life rules *)
rule = {224, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}};

(* hashlife algorithm *)
lastId = 1;
h[x_] := (lastId = lastId + 1; h[x] = lastId; q[lastId] = x; lastId);
q2[i_] := ArrayFlatten[Map[q, q[i], {2}]];
step[i_, d_] := step[i, d] = Block[{
    i2 = q2[i],
    half = Map[step[h[#], d - 1] &, Partition[#, {2, 2}, 1], {2}] &},