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]] .

Nevertheless, if one looks at overall shapes into which these proteins fold, there is some evidence that the same patterns of behavior are often seen.

{w___, x_, y_, ∘ , z___}  {w, y[x], z}] (Pictures of symbolic system evolution made with Polish notation differ in detail but look qualitatively similar to those made as in the main text with functional notation.) The tree representation of an expression can be obtained using expr//.x_[y_]  {x, y} , and when each object has just one argument, the tree is binary, as in Lisp .

[Cellular automata with] two-cell neighborhoods By having cells on successive steps be arranged like hexagons or staggered bricks, as in the pictures below, one can set up cellular automata in which the new color of each cell depends on the previous colors of two rather than three neighboring cells.

Note that after just a few steps, the sequences produced always seem to consist of white elements followed by black, with possibly one block of black in the white region.

Typically the idea of these models is to approximate those elements of a system about which one does not know much by random variables.

In most cases the only credible models seem to be ones based on intrinsic randomness generation.

And following my work on cellular automata in the early 1980s, David Young in 1984 considered a model even more similar to the one I use here.

If one looks at many terms, then their geometric mean is finite, and approaches Khinchin's constant Khinchin ≃ 2.68545 . … The largest individual term is the 432th one, which is equal to 20,776. In the first million terms, there are 414,526 1's, the geometric mean is 2.68447, and the largest term is the 453,294th one, which is 12,996,958.

And so, as one example, it might appear from the pictures on the previous page that (c), (d) and (e) always stay systematically above the axis.