Game of Life in 3D

Question

Is there a three-dimensional cellular automaton that in some sense generalizes Conway's Game of Life? If so, how might we use the CellularAutomaton command to simulate it? How might we visualize it?

Answer 1

Conway's game of life is a 2D, two-state, outer totalistic, cellular automaton. I guess the natural thing is to try such CAs in 3D. Here's the evolution of one such CA:

twos = Array[2 &, {3, 3}];
twosWithOne = twos;
twosWithOne[[2, 2]] = 1;
outerTotalisticCA3D[ruleNumber_Integer, 
  duration_Integer, init_List] := 
 CellularAutomaton[
  {ruleNumber,
   {2, {twos, twosWithOne, twos}},
   {1, 1, 1}
   },
  {init, 0},
  duration,
  {{0, duration}, Automatic}]
SeedRandom[5];
init = RandomInteger[{0, 1}, {20, 20, 20}];
pics = Image3D /@ outerTotalisticCA3D[47104, 42, init];
ListAnimate[pics]

Most of this is fairly straightforward and explained well in the documentation on CellularAutomaton. The tricky part is the rule specification here:

  {ruleNumber,
   {2, {twos, twosWithOne, twos}},
   {1, 1, 1}
  }

In this bit of code, the {1,1,1} specifies that the neighborhood should have radius 1 in each direction; that list has length three, since we're working in 3D. The 2 in {2,{twos, twosWithOne, twos}} specifies that we've got two states, live or dead. The {twos, twosWithOne, twos} is a $3\times 3\times 3$ block of integers, all of which are 2 except the central entry which is a 1. The rule depends on the sum of the entries corresponding to the twos; the central entry is special. Thus, the state of a cell at time $t$ depends on the state of the cell at time $t-1$, together with the sum of the states neighboring cells at time $t-1$. This is exactly what it means to say that we're dealing with an outer totalistic rule. Since there are 26 neighboring cells, there are 27 possible values of the sum, i.e. anything between 0 and 26 inclusive. There are also two possible states for the central cell, so there are up to $2^{2\times27}$ total such rules.

According to a nice analysis in this paper, rule 47104 is the most "life like" of all the potential 3D life rule and, in fact, we see a characteristic glider appearing in this animation. The paper denotes this by rule $(5766)$, meaning that a live cell stays alive if it has precisely 5, 6, or 7 live neighbors and a dead cell springs to life if it as precisely 6 live neighbors. We can generate the rule number in Wolfram's scheme as follows:

live = dead = Table[0, {27}];
live[[6]] = live[[7]] = live[[8]] = dead[[7]] = 1;
rule = FromDigits[Reverse[Riffle[dead, live]], 2]
(* Out: 47104 *)

More generally, live and dead should be $0-1$ lists of length 27; the element in position $j$ of the live list indicates the subsequent state of a live cell, assuming the sum of its neighbors is $j-1$, and similarly for the dead list.