For while there are certainly many examples of elaborate forms and patterns in biological systems, the overall shapes and many of the most obvious features of typical organisms are usually quite simple. … For while The behavior of a sequence of cellular automaton programs obtained by successive random mutations.

For while the individual machine instructions executed by a computer may be quite simple, the sequence of such instructions defined by a program may be long and complicated. … In a system like a cellular automaton the underlying rules can be thought of as rough analogs of the machine instructions for a computer, while the initial conditions can be thought of as rough analogs of the program. … So while practical computing gives a hint of part of what we saw in the previous section , the whole phenomenon is something much larger and stronger.

And while in the past it was often assumed that such complexity must somehow be special to systems in nature, what my discoveries and the Principle of Computational Equivalence now show is that in fact it is vastly more general. … So while science has often made it seem that we as humans are somehow insignificant compared to the universe, the Principle of Computational Equivalence now shows that in a certain sense we are at the same level as it is. … But while science has in the past shown that in many ways there is nothing special about us as humans, the very success of science has tended to give us the idea that with our intelligence we are in some way above the universe.

New structures on both left and right are produced every 77 steps; the central structure moves 20 cells to the left during each cycle so that the structures on the left are separated by 37 steps while those on the right are separated by 107 steps.

Note that the sequences of elements generated by the cellular automata are aligned at the right, while in the pictures of the substitution systems shown they are aligned at the left.

But after a while they reach the edge of the material, and although in the first case they then show quite random behavior, in the second case they instead just show simple repetitive behavior. What differs between the two cases is the detailed digit sequences of the positions of the points: in the first case these digit sequences are quite random, while in the second case they have a simple repetitive form.

In case (c), 3 possible blocks occur; the most common is assigned a codeword consisting of a single white cell, while the others are assigned codewords consisting of two cells. In case (d) 4 out of the 8 possible blocks occur, while in case (e) 6 occur.

With more complicated initial conditions the behavior one sees can sometimes be more complicated, at least for a while—as in the pictures below. … So this means Examples of 3-state 2-color Turing machines which behave for a while in slightly complicated ways.

Black corresponds to complete overlap, while white corresponds to no overlap.

Note that the orderings are shown for 8×8 basic forms, while the actual images are 32×32.