And what this means is that in a sense there can be no abstract basic science of the human condition—only something that involves all sorts of specific details of humans and their history. So while we might have imagined that science would eventually show us how to rise above all our human details what we now see is that in fact these details are in effect the only important thing about us. … Looking at the progress of science over the course of history one might assume that it would only be a matter of time before everything would somehow be predicted by science.

The only difference in initial conditions from the picture on the previous page is that each block now encodes rule 90 instead of rule 254.

But a crucial point noted by Carl-Friedrich Gauss in the 1820s is that the product of such curvatures—the so-called Gaussian curvature—is always independent of how the surface is laid out, and can thus be viewed as intrinsic to the surface itself, and for example determined purely from the metric for the 2D space corresponding to the surface. … One can then compute the Ricci tensor (R ik = R ijk j ) using RicciTensor = Map[Tr, Transpose[Riemann, {1, 3, 2, 4}], {2}] and this has 1/2 d(d + 1) independent components in d > 2 dimensions. (The parts of the Riemann tensor not captured by the Ricci tensor correspond to the so-called Weyl tensor; for d = 2 the Ricci tensor has only one independent component, equal to the negative of the Gaussian curvature.)

Once again, the only difference in initial conditions from the facing page is that each block now encodes rule 30 instead of rule 90.

In the simplest cases, f[n] depends only on the number immediately before it in the sequence, denoted f[n – 1] . But it is also possible to set up rules in which f[n] depends not only on f[n – 1] , but also on f[n – 2] , as well as on numbers still earlier in the sequence. … Sequence (c) is the powers of two; (d) is the so-called Fibonacci sequence, related to powers of the golden ratio (1 + √ 5)/2 ≃ 1.618 .

As a rough guide, it seems that continuous patterns of growth are possible only when the rate at which small-scale random changes occur is substantially greater than the overall rate of growth. For in a sense it is only then that there is enough time for randomness to average out the effects of the underlying discrete structure. … In general the point is that continuous behavior can arise in systems with discrete components only when there are features that evolve slowly relative to the rate of small-scale random changes.

In both the rules shown on the facing page , the only replacement specified is for the block . … And any substitution system whose rules specify replacements only for blocks such as these is guaranteed to yield the same causal network regardless of the order in which replacements are performed. … For it is not necessary that no overlaps exist at all in the replacements—only that no overlaps occur in whatever sequences of elements can actually be generated by the evolution of the substitution systems.

In a sense the fundamental reason for this—as we discussed on page 252 —is that such class 1 and class 2 cellular automata never allow any transmission of information except over limited distances. And the result of this is that they can only support processes that involve the correlated action of a limited number of cells. … These cellular automata are necessarily all class 1 or class 2 systems.

The rules shown are of the same kind as on the facing page , and include most of the 64 possibilities that leave a state that contains only white cells unchanged.

In the lower block of pictures, only the center cell is taken to be black on these steps.