Although the fraction of squares that violate the constraints is less than 20% after 100,000 steps, the overall patterns still do not look much like the exact results.

The evolution of the system for t steps can be obtained from SSEvolve[rule_, init_, t_, d_Integer] := Nest[FlattenArray[# /. rule, d] &, init, t] FlattenArray[list_, d_] := Fold[Function[{a, n}, Map[MapThread[Join, #, n] &, a, -{d + 2}]], list, Reverse[Range[d] - 1]] The analog in 3D of the 2D rule on page 187 is {1  Array[If[LessEqual[##], 0, 1] &, {2, 2, 2}], 0  Array[0 &, {2, 2, 2}]} Note that in d dimensions, each black cell must be replaced by at least d + 1 black cells at each step in order to obtain an object that is not restricted to a dimension d - 1 hyperplane.

Such discrete transitions are somewhat less common in one dimension than elsewhere.

For even though the detailed placement of black and white cells in the first two pictures does not seem simple to describe, at an overall level these pictures still admit a quite simple description: in essence they just involve a kind of uniform randomness in which every region looks more or less the same as every other.

For even in the very best case any block of cells in the input can never be compressed to less than one cell in the output.

And in each case higher mathematical constructs that seem in some sense no less implementable immediately allow the problems to be solved.

What rules for natural objects might in effect have been tried in the Judeo-Christian tradition is less clear—though for example the Book of Job does comment on the difficulty of "numbering the clouds by wisdom".

Attitudes of mathematicians Mathematicians often seem to feel that computer experimentation is somehow less precise than their standard mathematical methods.

With typical random initial conditions the most common structures to occur are: The next most common moving structure is the so-called "spaceship": The complete set of structures with less than 8 black cells that remain unchanged at every step in the evolution are: More complicated repetitive and moving structures are shown in the pictures below. … It is also known that from less than 10 initial black cells no unbounded growth is ever possible.

But as the pictures below demonstrate, it is also possible to find less trivial initial conditions that still make rule 30 behave in a simple way.