But inevitably we tend to notice only those features that somehow fit into the whole conceptual framework we use. … So in the decades to come, when the science in this book has been absorbed, it is my expectation that it will not only suggest many new ways to achieve existing technological purposes but will also suggest many new purposes that technology can address.

Turing machine enumeration Of the 4096 s = 2 , k = 2 Turing machines (see page 888 ) 560 are distinct after taking account of obvious symmetries and equivalences. Ignoring machines which cannot escape from one of their possible states or which yield motion in only one direction or cells of only one color leaves a total of 237 cases. … Of the 2,985,984 s = 3 , k = 2 machines, 125,294 survive after taking account of obvious symmetries and equivalences, while imposing analogs of the other conditions above yields in the end 16,400 distinct cases.

It is however also possible to consider systems in which there is only a single active node, and operations are performed only on that node at any particular step. … The rule for this system is {{1, 1}  {{{{}, {1, 1}}, {2}}, 2}, {1, 2}  {{{2, 2}, {{}, {2, 2}}}, 2}, {2, 1}  {{{}, {2, 2}}, 2}, {2, 2}  {{{1, 2} ,{{1}, {2}}}, 1}, {2, 3}  {{{{1, 2}, {1}}, {{2}, {2, 1}}}, 2}, {2, 4}  {{{2, 2}, {{2, 1}, {}}}, 1}}

The third element of the rule is at first used only on some steps—but after step 50 it appears to be used somewhere in every step. … Without this additional block of black, only the first case in the rule can ever apply.

Number of [Turing machine] rules With k possible colors for each cell and s possible states, there are a total of (2 s k) s k possible Turing machine rules. Often many of these rules are immediately equivalent, or can show only very simple behavior (see page 1120 ).

[Biological] self-assembly Some growth—particularly at a microscopic level—seems to be based on objects with particular shapes or affinities sticking together only in specific ways—much as in the systems based on constraints discussed on page 210 (and especially the network constraint systems of page 483 ).

Specially constructed transcendental numbers Numbers known to be transcendental include ones whose digit sequences contain 1's only at positions n! , 2 n or Fibonacci[n] .

But only some schools of Greek philosophy ever supported atomism, and it soon fell out of favor. … But it was normally assumed that one should think only about explicit particles with realistic mechanical properties—so that abstract idealizations like cellular automata did not arise.

In rule 126, the only effect at step 2 is that black cells can no longer appear on their own: they must always be in groups of two or more. … Networks representing possible sequences of black and white cells that can occur at successive steps in the evolution of several class 1 and 2 cellular automata. These networks never have more than about t 2 nodes after t steps.

Among elementary cellular automata with just two colors and nearest-neighbor rules, the only types of examples are the fairly trivial ones shown in the pictures below.