2D class 4 cellular automata No 5- or 9-neighbor totalistic rules nor 5-neighbor outer totalistic ones appear to yield class 4 behavior with a white background. But among 9-neighbor outer totalistic rules there are examples with codes 224 (Game of Life), 226, 4320 (sometimes called HighLife), 5344, 6248, 6752, 6754 and 8416, etc.

The pattern after n steps is then given by Nest[Flatten[f[#]] &, {0}, n] , where for the rule on page 189 f[z_] = 1/2 (1 -  ) {z + 1/2, z - 1/2} ( f[z_] = (1 -  ){z + 1, z} gives a transformed version). For the rule on page 190 , f[z_] = 1/2 (1 -  ) {  z + 1/2, z - 1/2} . For rules (a), (b) and (c) (Koch curve) on page 191 the forms of f[z_] are respectively: (0.296 - 0.57  ) z - 0.067  - {1.04, 0.237} N[1/40 {17 ( √ 3 -  ) z, -24 + 14 z}] N[(1/2 (1/ √ 3 - 1)(  + {1, -1}) -  - (1 + {  , -  }/ √ 3 ) z)/2]

"Firing squad" synchronization By choosing appropriate rules it is possible to achieve many forms of synchronization directly within cellular automata. One version posed as a problem by John Myhill in 1957 consists in setting up a rule in which all cells in a region go into a special state after exactly the same number of steps. … If one drops the requirement of cells going into a special state, then even the 2-color elementary rule 60 shown on the left can be viewed as solving the problem—but only for widths that are powers of 2.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

The patterns on each row are obtained from rules that are set up to give branches with particular relative lengths.

The particular rules shown are ones that lead to slow growth in the total number of elements.