Properties [of operators from axioms] There are k k 2 possible forms for binary operators with k possible values for each argument. … Of the 274,499 axiom systems of the form {…  a} where … involves ∘ up to 6 times, 32,004 allow only operators {6,9} , while 964 allow only {1,7} . The only cases of 2 or less operators that appear with k = 2 are {{}, {10}, {12}, {1, 7}, {3, 12}, {5, 10}, {6, 9}, {10, 12}} .

The rules for the three cellular automata involve only nearest neighbors, and allow 12 possible colors for each cell.

When the initial density of black cells has any value less than 50%, only white stripes ever survive. … If the initial density of black cells is less than 50%, then only white stripes ultimately survive.

Of regular polygons, only squares, triangles and hexagons can be used to do this, and in these cases the tilings are always repetitive. … Repetitive tilings of the plane can only have 3-, 4- or 6-fold symmetry. … In 3D, John Conway has found a single biprism that can fill space only in a sequence of layers with an irrational rotation angle between each layer.

And indeed after just thirty steps, the description of the kneading process given above would imply that two points initially only one atom apart would end up nearly a meter apart. … So the idea that randomness comes purely from initial conditions can be realistic only for a fairly small number of steps; randomness which is seen after that must therefore typically be attributed to other mechanisms. … But once again, in any practical implementation, the light would go around only a few tens of times before being affected by microscopic

And in rules like 0R and 90R shown on page 452 the period of repetition is always very short. … And instead, starting from any particular initial condition, the system will only ever visit a tiny fraction of all possible states. … And by thinking in terms of simple programs we have thus been able in this section not only to understand why the Second Law is often true, but also to see some of its limitations.

Certainly among all possible rules of a particular kind only a limited number can ever be considered simple, and these rules are by definition somehow special. … For if the number of different parts of the rule were, for example, comparable to the number of different situations that have ever arisen in the history of the universe, then we would not expect ever to be able to describe the behavior of the universe using only a limited number of physical laws. And in fact if one looks at present-day physics, there are not only a limited number of physical laws, but also the individual laws often seem to have the simplest forms out of various alternatives.

So what this perhaps suggests is that in the end there might be only certain specific capabilities that can be realized in practical methods of perception and analysis. And certainly it seems not inconceivable that there could be a fundamental result that the only kinds of regularities that both occur frequently in actual systems and can be recognized quickly enough to provide a basis for practical methods of perception and analysis are ones like repetition and nesting. … And what this perhaps suggests is that we choose the methods we use to be essentially those that pick out only regularities with which we are somehow already very familiar from our own built-in powers of perception.

And it does this by successively eliminating cases that do not apply, until eventually only one case remains. … And after all three stripes have passed, only one of the 8 cases ever survives, and this case is then the one that gives the new color for the cell. … But the universal cellular automaton is in no way restricted to emulating only rules that involve nearest neighbors.

But it also means that it is not clear whether the axiom system actually describes only the objects one wants—or whether for example it also describes all sorts of other quite different objects. … One might imagine that if one were to add more axioms to an axiom system one could always in the end force there to be only one kind of object that would satisfy the constraints of the system. … But if an axiom system is close to complete—so that the vast majority of statements can be proved true or false—then it is almost inevitable that the different kinds of objects that satisfy its constraints must differ only in obscure ways.