For without it there will tend to be no particular way to match up corresponding parts in successive slices through the causal networks that are produced.

Yet it is rather close to being complete—since as we saw earlier one has to go through at least millions of statements before finding ones that it cannot prove true or false.

So what this means is that if one were just to go through a list of the simplest few thousand axiom systems one would already be quite likely to find one that represents logic.

In general the density for an arrangement of white squares with offsets v is given in s dimensions by (no simple closed formula seems to exist except for the 1 × 1 case) Product[With[{p = Prime[n]}, 1 - Length[Union[Mod[v, p]]]/p s ], {n, ∞ }] White squares correspond to lattice points that are directly visible from the origin at the top left of the picture, so that lines to them do not pass through any other integer points.

But ever since the dawn of civilization humans have increasingly been exposed to things that were explicitly constructed through engineering, architecture, art, mathematics and other human activities.

And beginning in the early 1980s—particularly through work by Alvy Ray Smith and later Przemyslaw Prusinkiewicz —models based on L systems and fractals became routinely used for producing images of plants in practical computer graphics.

But in the late 1980s it became popular to think that in many systems nesting (as well as the largely unrelated phenomenon of 1/f noise) might be the result of fine tuning of parameters achieved through some automatic process of self-regulation. … The picture below shows slices through the evolution at several successive cycles.

Yet being convinced that more complicated behavior must be possible, I decided to persist, and so I wrote a program that would automatically search through large numbers of mobile automata.

Nanotechnology Popular since the late 1980s, especially through the work of Eric Drexler , nanotechnology has mostly involved investigation of several approaches to making essentially mechanical devices out of small numbers of atoms.

The quantity expr //. { ℯ  0, x_[y_]  2 x + y} turns out to remain constant through the evolution, so this gives the final value of m for any initial condition.