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Lagniappe #10

1 or 2 or 6 solidification

One or More Recursive Carpets

In 1985, Packard and Wolfram observed that certain nearest-neighbor totalistic CA rules grow crystals of a fractal nature. They wrote,

The boundaries of the patterns obtained at most time steps are corrugated, and have fractal forms analogous to Koch curves. The patterns grow by producing "branches" along the ... lattice directions. Each of these branches then in turn produces side branches, which themselves produce side branches, and so on. This recursive process yields a highly corrugated boundary. However, as the process continues, the side branches grow into each other, forming an essentially solid region. In fact, after each 2 j time steps the boundary takes on an essentially regular form. It is only between such times that a dendritic boundary is present.

We recently discovered that Exactly 1 solidification on the Moore neighborhood, a rule of this type, fills the lattice with an asymptotic density which can be computed exactly and equals 4/9. The computation appears in our recent survey of Cellular Automaton Growth on Z2. A current research project seeks to extend that analysis to as many as possible of the Moore solidification rules in which an empty cell joins the crystal if exactly one of its 8 neighbors is occupied. Of the 128 such rules, it appears that about 70% admit exact calculation of the density starting from a singleton. Nevertheless, subtle distinctions give rise to crystals of various beautiful forms. Here we show four representative examples in "time-trace" gradient colorings which highlight their complex structure. All images are the result of 220 updates from an initial configuration consisting of a single occupied cell. For instance, the thumbnail image above shows 1 or 2 or 6 solidification; click on it to view an enlargement of the elegant local structure. One can prove that rules of the form 1 or 2 or ... have the same characteristic fractal recursion starting from any finite seed.


Change to 1 or 5 or 7 solidification, and note the different fractal boundary structure (with a different fractional Hausdorff dimension along suitable dyadic subsequences).

For reasons we now partially understand, the 1 or 3 or 5 or 7 rule does not have a recursively computable density, and does not grow like a Koch snowflake, but instead spreads steadily with chaotic boundary dynamics.


Our final example, 1 or 3 or 6 solidification grows a Koch snowflake with yet another fractal structure. Rather curiously, in some cases a fractal crystal arises from a single cell, but other small seeds give rise to chaotic growth. Mysteries such as this will be illuminated in a forthcoming paper.

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