- The most mathematically tractable CA rules are additive (or
linear), meaning that the behavior of the global system can be
reconstructed from that of its local components by an exact superposition
principle. We have briefly discussed such one-dimensional systems before in the
recipes of June 5 - 11, 1995 and
November 27 - December 3, 1995. The prototypical
example, rule 90 (Pascal's Triangle mod 2), generates the Sierpinski lattice -
a digital fractal - in space-time starting from a single occupied cell.
- One common thread of our extensive empirical investigation of
nonlinear CA rules here in the Kitchen has been the emergence of
replicators along phase boundaries. Roughly, a replicator is a
finite configuration which clones itself exactly within an appropriate
periodic slice of space-time according to the linear Sierpinski mechanism.
This week's soup shows one such replicator, consisting of a pair of Larger
than Life 'skeeters' for range 3 Box LtL with birth and survival intervals [6,6] and
[6,6] (the exactly 6 rule). The yellow configuration at the top of the
graphic spreads out within a thin horizontal strip of the two-dimensional
integer lattice until it exactly reproduces itself at times 8, 16, 24, 32,
..., in the manner of rule 90. Our soup shows successive updates of the
skeeter pair along this strip, each below the last, in a period 8 cyclic
palette. Note the complexity of the interactions at intermediate times.
- In her thesis on Larger than Life dynamics, Kellie Evans presents a
formalism which allows one to verify, by computer-aided proof, that a
candidate finite configuration is indeed an exact replicator for a given CA
rule. Moreover, she has collected a large menagerie of examples with
varying architectures. Another extraordinary instance that has been discussed
on the Web is a 'bowtie pasta' for HighLife, the variant of Conway's rule
in which birth also occurs if an empty site has exactly 6 occupied
neighbors. The bowtie dynamics, which have period 12, are featured in our
replica.xpt WinCA demo experiment available from the
Kitchen Sink.
- The real mystery is WHY replicators seem to emerge along phase
boundaries of nonlinear spatial population models. A cynical reponse is to
point out that for such parameter choices the population is delicately perched
between survival and extinction, thereby facilitating the chance creation
of well-separated exact clones of certain small crystals. But the
prevalence of the replicator phenomenon in widely warying CA rules,
often involving configurations of surprisingly large size and exotic desigh that seem
to emerge spontaneously from disordered initial conditions, raises at least the possibility
of a deeper explanation, a kind of cartoon for the emergence of fractals at critical values.
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