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Two-State Range 1 Automata

Some of the simplest and most widely studied planar cellular automata have two possible states for each cell: 1 = alive or 0 = dead, say, and an update rule which depends on the states of a cell and its eight nearest ("Moore") neighbors: N, S, E, W, NE, SE, NW, SW. This applet lets you experiment with all 262,144 such 'totalistic' rules by setting the checkboxes of birth and survival for the respective population counts. Configurations are enhanced to display the lifetime of active cells, progressing cyclically through colors 1,2,...,7 of the chosen palette. Alternatively, select the Life palette to observe unfiltered two-state evolutions. We offer 8 demo experiments, focusing on some nice illustrations of shape theory for CA growth. The interface is the same as for our earlier applets: 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 simulator.

1. HighLife
Hit Start to see a variant of the Game of Life in which birth also occurs when a cell has 6 occupied neighbors, starting from symmetric randomness. You'll notice some of the same blinkers and gliders as in Conway's game. Click the title of this demo to dowload David Bell's exhaustive experimental study ( By watching long and hard you will eventually catch a glimpse of the star of the show, a bowtie pasta replicator. For your convenience, we have isolated one in the next experiment.

2. The Bowtie Replicator
Load the initial image Bowtie to see perhaps the most beautiful known example of a CA replicator, as discussed in Recipes 75, 77, and 85 on the Kitchen Shelf.

3. Threshold Growth Shapes
Now change to the Range 1 Box, Threshold 3 Growth by checking birth boxes 3 through 8 and all the survival boxes. Load TinySeed and run the dynamics to observe the rapid emergence of an octagonal asymptotic shape L. The same shape is obtained starting from any sufficiently large initial configuration. See Recipes 21, 35, and 37 for more about Threshold Growth. The limit shape for threshold 1 (the additive case) is clearly a square, but can you guess L for threshold 2? Check '2' in the birth column to discover the answer.

4. A Nonconvex Asymptotic Shape
For monotone growth rules in which 1's always survive, the limit shape L must be convex. Nonmonotone rules can have nonmonotone shapes, however, and the shape can depend on the initial seed. Run the '2 or 5' rule from TinySeed to observe such an example, one of many described in Recipes 47, 48, and 60.

5. Fractal Growth
Some nonmonotone CA rules give rise to recursive self-similar growth. Perhaps the simplest is the 'exactly 1' growth model, which generates perfect von Koch crystals starting from many symmetric seeds. Even our little demo on a 100 x 100 grid should convince you that there is no limit L in this case. Along one subsequence of times the crystal fills a square with density 1/2, but along another it has an intricate fractal boundary.

6. Dendritic Growth
Change to the 'exactly 3' birth rule and Flake palette, then load a 14 x 14 square seed to catch at least a glimpse of the exotic nonlinear growth of one of our favorite CA rules, Life without Death, featured in Recipes 16, 26, and 47 on the Kitchen Shelf.

7. Hickerson's Amoeba
Saving the best for last, here is the most mysterious CA growth model we know, compliments of Dean Hickerson. Load a 2 x 59 box with one corner cell removed to get things rolling. A rough and amorphous diamond shape forms rapidly, but then nothing much happens for 2,000 updates. By about time 5,000, however, the amoeba has grown enough to touch the edge of the array, and at about time 7,200 it wraps around to interact with itself, after which the 'all 1's' state is reached quite rapidly. Given enough room, this same evolution fills an 8K by 8K box after approximately 750,000 updates. Nevertheless, there is no assurance that it will keep growing forever, and little or no indication of an asymptotic shape. Click on this demo's title to view a 'level-set' rendering at time 50,000.

8. Oscillating Collapse
In light of the previous experiment, it is quite remarkable that Hickerson's amoeba dies out completely from any arbitrarily large square seed with sides having an odd number of cells. In order to obtain a multi-colored rendition of this collapse, we have interchanged the role of 0's and 1 in the rule. At first, a 49 x 49 square of 0's surrounded by 1's expands to form a diamond. But then the 1's proceed to fill the 'hole,' which cycles repeatedly through a sequence of characteristic shapes as it shrinks. Click on this demo's title to see a more illustrative 'level-set' rendering after 1776 updates from an n x n seed, where n = 301. You can send a postcard of this image to a friend by clicking the envelope at the top of the page. Even more remarkably, there is an exact formula with 5 terms for the time to fill the square in terms of n.

# Palette Birth Survival Initial Condition
1. Bright 3,6 2-3 Random
2. Bright 3,6 2-3 Bowtie
3. Bright 3-8 0-8 TinySeed
4. Bright 2,5 0-8 TinySeed
5. Bright 1 0-8 TinySeed
6. Flake 3 0-8 14square
7. Flake 3,5-8 5-8 2x59-1
8. Flake 4-8 4,6-8 49square

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