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Cyclic Cellular Automata

Cyclic Cellular Automata (CCA) exhibit complex self-organization by iteration of an extremely simple update rule. Using a cyclic color wheel, the state of each cell advances to the next in the cycle if and only if it sees a sufficient representation of that next color within its prescribed local neighborhood. Here 'enough' is measured by a threshold parameter. CCA dynamics are featured extensively on the Kitchen Shelf. This Java-based CCA simulator, with 10 demo experiments, provides an overview of the model's behavior at various representative locations in phase space. The interface is essentially the same as for our companion Greenberg-Hastings Page. Many of the capsule descriptions are linked to explanatory recipes from the Shelf. The Table at the bottom prescribes each experiment's parameters. Click on the number index of an experiment to go to its Table entry, and vice versa, or click on the Table's upper left # symbol to jump to the applet. Try other parameter combinations to develop your predictive understanding of these excitable dynamics, and consult our research paper Threshold-Range Scaling of Excitable Cellular Automata for a systematic study.

1. The Basic CCA
Hit Start to see the debris replaced by nucleating droplets and then, unless you're unlucky, by one or more self-organized spirals. If at first you don't succeed, hit Random to try again...

2. Nucleating Droplets
Switch to 16 colors, in which case a spiral is unlikely to form from randomness on such a small (60 by 60) array. Load Egg to improve the chances. Actually, I promise you a 16-color spiral from this particular configuration!

3. Spiral Formation
The formation of basic CCA spirals is instigated by certain stable cyclically colored loops known as defects. Our third experiment places such a defect around the edges of the array, thereby ensuring the dynamic formation of a tight-knit spiral somewhere within. Click on this experiment's title for more information.

4. Percolation Effects
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. Range 2 Box, threshold 1 provides one examples of this effect. For another case with larger parameters, see our recipe of March 6, 1995.

5. Prototypical Cyclic Spirals
Larger neighbor sets and intermediate threshold values give rise to the prototypical spiral-driven periodic states of excitable media. For CCA rules the spiral pairs often emerge with one dominant core and one weak end, in contrast to the more symmetric ram's horns of GH dynamics.

6. Cyclic Turbulence
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 animation gives 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.

7. Majority Vote
Whereas 3 states are needed to create an oriented cycle, CCA parameter space naturally includes 2 state automata such as Voter and Growth models. This experiment implements range 3 Box Majority Vote. The system clusters until it achieves uniformly small curvature. Often this results in complete consensus. But see if you can generate a trajectory which fixates in a final state other than all 0's (purple) or all 1's (red).

8. 3-color Bootstrap
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 experiment shows a 3-color CCA with competing bootstrap growth.

9. Perfect Spirals
Certain 4-color CCA rules support periodic crystals which spread without imperfection over any disordered environment. In the range 1 Box, threshold 3 case these arise spontaneously from noise, often after hundreds of updates. See if you can capture one on our small array, an enterprise which may take patience and/or luck, by hitting Start, then Random repeatedly. Click on this experiment's title for more details and a snapshot of the elusive perfect spiral.

10. Synthetic Spirals
Our final experiment shows how to engineer delicate spirals from an initial configuration special to the 4-color case, in a manner similar to our Ends experiment for GH. Start with threshold 1 and gradually increase the threshold 'on the fly' to generate periodic centers for all values up to 10. Can you predict what happens at threshold 11 ?

[This applet requires a Java-enabled browser...]

# Palette Range Shape Threshold # Colors Initial Condition Boundary
1. CGA 1 Dmd 1 12 Random Wrap
2. CGA 1 Dmd 1 16 Egg Wrap
3. CGA 1 Dmd 1 16 Defect Wrap
4. CCA 2 Dmd 1 16 Random Wrap
5. Bright 2 Box 3 8 Random Wrap
6. Turbulent 2 Dmd 4 5 Random Wrap
7. Turbulent 3 Box 25 2 Random Wrap
8. Haring 1 Dmd 2 3 Random Wrap
9. Haring 1 Box 3 4 Random Wrap
10. Haring 2 Box 1 - 11 4 4 Square Free

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Introduction to the PSK PSK Search Recent Additions CA Archive CA Links Feedback Appreciated !