universe, while as an abstract fact it could simply represent an incorrect deduction, and even as a definition it could prove not useful or relevant.

Tag Systems One of the goals of this chapter is to find out just how simple the underlying structure of a system can be while the system as a whole is still capable of producing complex behavior.

One example of a single combinator system can be found using {s  j[j], k  j[j[j]]} , and has combinator rules (whose order matters): {j[j][x_][y_][z_]  x[z][y[z]], j[j[j]][x_][y_]  x} The smallest initial conditions in this case that lead to unbounded growth are of size 14; two are versions of those for s , k combinators above, while the third is j[j][j[j]][j[j]][j[j][j[j][j]]][j[j][j]] .

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.

The number of steps for which a cell at position n will survive can be computed as Module[{q = n + k - 1, s = 1}, While[Mod[q, k] ≠ 0, q = Ceiling[(k - 1)q/k]; s++]; s] If a cell is going to survive for s steps, then it turns out that this can be determined by looking at the last s digits in the base k representation of its position.

Phenomena that involve geometrical shapes, for example, usually require at least two dimensions, while phenomena that rely on the existence of knotted structures require three dimensions.

The basic reason is that intrinsic randomness generation in a sense puts all the components in a system to work in producing new randomness, while getting randomness from the environment does not.

And so for example the sequence of colors of the center cell, while seemingly random, will also be exactly the same.

If one takes some water and continuously increases its temperature, then for a while nothing much happens.

And while this particular example may seem contrived, it turns out that essentially the same mathematical equations also occur in many other situations—such as the evolution of chemical concentrations in various chemical reactions.