Notes

Section 6: The Phenomenon of Continuity

Generalized aggregation models

One can in general have rules in which new cells can be added only at positions whose neighborhoods match specific templates (compare page 213). There are 32 possible symmetric such rules with just 4 immediate neighbors—of which 16 lead to growth (from any seed), and all seem to yield at least approximately circular clusters (of varying densities). Without symmetry, all sorts of shapes can be obtained, as in the pictures below. (The rule numbers here follow the scheme on page 927 with offsets {{-1,0},{0,-1},{0,1},{1,0}}). Note that even though the underlying rule involves randomness definite geometrical shapes can be produced. An extreme case is rule 2, where only a single neighborhood with a single black cell is allowed, so that growth occurs along a single line.

If one puts conditions on where cells can be added one can in principle get clusters where no further growth is possible. This does not seem to happen for rules that involve 4 neighbors, but with 8 neighbors there are cases in which clusters can get fairly large, but end up having no sites where further cells can be added. The pictures below show examples for a rule that allows growth except when there are exactly 1, 3 or 4 neighbors (totalistic constraint 242).

The question of what ultimate forms of behavior can occur with any sequence of random choices, starting from a given configuration with a given rule, is presumably in general undecidable. (It has some immediate relations to tiling problems and to halting problems for non-deterministic Turing machines.) With the rule illustrated above, however, those clusters that do successfully grow exhibit complicated and irregular shapes, but nevertheless eventually seem to take on a roughly circular shape, as in the pictures below.

At some level the basic aggregation model of page 331 has a deterministic outcome: after sufficiently many steps every cell will be black. But most generalized aggregation models do not have this property: instead, the form of their internal patterns depends on the sequence of random choices made. Particularly with more than two colors it is however possible to arrange that the internal pattern always ends up being the same, or at least has patches that are the same—essentially by using rules with the confluence property discussed on page 1036.

The pictures below show 1D generalized aggregation systems with various templates. The second one is the analog of the system from page 331.

From Stephen Wolfram: A New Kind of Science [citation]