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Cellular Automata rules lexicon |
|
Family: Cyclic CA |
Type: cyclic totalistic
Cyclic Cellular Automata (CCA) exhibit complex self-organization by iteration of an extremely simple update rule. A specified # of Colors are arranged cyclically in a "color wheel." Each color can only advance to the next, the last cycling to 0. Each update a cell's color advances by 1 if there are at least Threshold cells of the next color within its neighbour set of size Range in extended Moore or von Neumann neighbourhood. These simple dynamics exhibit complex self-organization starting from randomness. This class of CAs was discovered and explored by David Griffeath (http://psoup.math.wisc.edu/kitchen.html).
The MCell's implementation of CCA features also a special case of CCA - the Greenberg-Hastings (GH) Model, perhaps the simplest CA prototype for an excitable medium. A prescribed number of colors N are arranged cyclically in a "color wheel." Each color can only advance to the next, the last cycling to 0. Every update, cells change from color 0 (resting) to 1 (excited) if they have at least Threshold 1's in their neighbor set. All other colors (refractory) advance automatically. Starting from a uniform random soup of the available colors, the excitation dies out if the threshold is too large compared to the size of the neighbor set, while a disordered soup virtually indistinguishable from noise results if the threshold is too low. For intermediate thresholds, however, waves of excitation self-organize into large-scale spiral pairs that stabilize in a locally periodic state.
The notation of Cyclic Cellular Automata has the "R/T/C/N" form, where:
R - specifies the neighbourhood range (1..10).
T - specifies the threshold - minimal count of cells in the neighbourhood having the next
color, necessary for the cell to advance to the next state.
C - specifies the count of states in the rule (0..C-1).
N - specifies the neighbourhood type: NM stands for extended Moore, NN for extended von
Neumann.
Range 'R' von Neumann neighborhood includes all sites which can be reached from
the origin in at most R steps by N, S, E and W moves, whereas range 'R' Moore
neighbourhood also allows NE, SE, NW and SW moves at each step.
In general Cyclic CA rules should be started from uniformly randomized boards.
MJCell Java applet is able to run all rules from this group.
| Name | Rule (R/T/C/N) | Character | Description |
| 313 | R1/T3/C3/NM | Cyclic | "This is the only three-color rule in the range-1 CCA rulespace that exhibits significant self-organization. And it's no
slouch in that regard, either. It sports a number of distinct spiral types, its demographics are eminently watchable, and in general it
has a flavor all its own." - J.Elliott A rule by David Griffeath. |
| 3-color bootstrap | R2/T11/C3/NM | Cyclic | The term bootstrap refers to systems with local clusters which can only
propagate by means of external support from random noise or other clusters. Such critical
dynamics, and related near-critical rules, often fixate in complex 'fossilized' final
configurations. This rule shows a 3-color CCA with competing bootstrap growth. A rule by David Griffeath. |
| Amoeba | R3/T10/C2/NN | Stable | Similar to Vote 4/5, this rule smoothes patterns and leaves irregular
islands with characteristic oscillating edges. A rule by Jason Rampe. |
| Black vs White | R5/T23/C2/NN | Chaotic | Well-balanced rule where both death (black) and life (white) have equal
chances. Seed patterns with 50% density. A rule by Jason Rampe. |
| CCA | R1/T1/C14/NN | Cyclic | A basic CCA. Starting from a uniform random distribution over 14 colors,
droplets of color waves nucleate fairly quickly. Soon virtually all of the initial
"debris" are overrun by the droplets. As the last vestiges of debris are
eliminated, vortices emerge from the disordered wave fronts, creating diamond-shaped
spirals. By about time 300 the array is completely covered with periodic spirals, out of
phase with one another and not all of minimal period 14. Typically it takes much longer
for the period 14 spiral cores to displace their feebler competitors. A rule by David Griffeath. |
| Cubism | R2/T5/C3/NN | Stable | This interesting rule creates rectangular regions. A rule by Jason Rampe. |
| Cyclic spirals | R3/T5/C8/NM | Cyclic | A rule by David Griffeath. |
| Fossil debris | R2/T9/C4/NM | Cyclic | A rule by David Griffeath. |
| GH Macaroni | R2/T4/C5/NM/GH | Cyclic | The threshold is sufficiently high that wave fragments have a hard time
bending inwards to make spiral cores. Consequently, cyclic bands self-organize over time
into weakly aligned parades of wave fragments. The pattern is not unlike that of the
monolayer of macaroni that adheres to a colander during the preparation of a $.059 box of
macaroni and cheese. CAM6 experiments have revealed that if these dynamics are allowed to
run for thousands of updates on a larger array, say 2048 by 2048, then the wave fragments
continue to align into ever longer parades. But more than this, the pasta also merge end
to end until they transform into a new variety: spaghetti. We expect that on an infinite
array the process would continue to enormous length scales, giving rise to linguine, then
angel hair, then ... A rule by David Griffeath. |
| GH Multistrands | R5/T15/C6/NM/GH | Cyclic | A rule by David Griffeath. |
| GH Percolation mix | R5/T10/C8/NM/GH | Cyclic | With larger neighbor sets and low thresholds, stable periodic cycles occur
by chance in the initial random soup. Thus one encounters a mixture of partial
self-organization and percolation effects. A rule by David Griffeath. |
| GH Weak spirals | R4/T9/C7/NM/GH | Cyclic | A rule by David Griffeath. |
| GH | R3/T5/C8/NM/GH | Cyclic | A basic Greenberg-Hastings model. A rule by David Griffeath. |
| Imperfect | R1/T2/C4/NM | Cyclic | From Elliott's
description: "A little magic carpet generator of cellsize 4. One thing
that sets this rule apart from other CCAs I know of is a phenomenon that occurs in the occasional orbit, in which a garish living stain
forms and grows till it blots out the entire carpet. This cancerous growth (to mix my metaphors) starts from a small anomalous patch of
cells - though not inevitably, for some such tumors are benign. So, in a way this rule reverses the situation we encounter in its
higher threshold cousin, Perfect. In the latter rule, regular
spirals invade a "broken" but seemingly stable dynamic, whereas here the tables are turned and it is an orderly spiral regime that finds
itself overthrown." A rule by John Elliott, May 2000. |
| LavaLamp | R2/T10/C3/NM | Cyclic | The rule produces blobs that seperate and combine resembling a lava lamp. A rule by Jason Rampe. |
| Maps | R2/T3/C5/NN | Cyclic | A rule by Mirek Wojtowicz. |
| Perfect | R1/T3/C4/NM | Cyclic | A particularly interesting excitable system. From a uniform random
configuration it quickly self-organizes into a chaotic soup with large length scale. But
later on, often after more than one hundred updates, perfect, widely separated stable
spiral cores emerge and slowly take over the lattice. A rule by David Griffeath. |
| Squarish Spirals | R2/T2/C6/NN | Chaotic | A rule by Jason Rampe. |
| Stripes | R3/T4/C5/NN | Cyclic | A rule by Mirek Wojtowicz. |
| Turbulent phase | R2/T5/C8/NM | Cyclic | When the threshold is high enough that wave fronts cannot wind, but still
low enough that they can advance, CCA rules exhibit a chaotic equilibrium phase which
combines the small-scale structure of failed cores with large-scale disordered fronts. Our
animations give only a glimpse of this behavior, since the typical final length scale of
several hundred cells means that large arrays are needed to support a viable steady state.
Small systems inevitably fixate. Note how impossible it is to predict, until the very end,
which color will predominate. A rule by David Griffeath. |
| Webmaster: Mirek Wojtowicz http://www.mirekw.com |
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Last update: 15 Sep 2001