Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712 
The first isn't a CA proper, it is the system or process known as "diffusion limited aggregation". You start with any seed of "stuff" which just persists (anything that turns black stays black forever).
Then it works by just having a random walk particle (think of it as gray or a third color) come in from anywhere around the open edge of the area (meaning, for each gray cell it turns white next but one and only one of its neighbors, chosen at random, turns gray on the next step), until it "hits" a spot that contains "stuff" already (a gray next to a black turns black).
In greater detail, the process explained in CA terms  not that you'd implement it this way in programmingspeed terms.
White generally stays white, it can only become gray if it is the randomchoice white cell next to a gray on the previous step. It never directly becomes black.
Black always just stays black.
Gray goes to black if there is any black in the neighborhood.
If there is no black in the neighborhood, gray goes to white *but* one randomly chosen cell in its neighborbood (other than the center cell) becomes gray. (That is the aspect that is different from a regular CA).
Now start with a "seed" of black in the center. Add gray at the edges of the pattern at random step numbers (Poisson e.g.), and random edge locations. Not many of them  in fact the usual method is to have only a single gray and "resolve" its walk (until it becomes black I mean) before the next gray is introduced.
Notice that randomly forming "arms" tend to "catch" more particles in this process, while "hollows" will be "sheltered" from new gray arrivals. It is a well studied system, and it can be implemented as a CA, but it is more straightforward to only worry about updating the gray cell movement, since everything else is "null" transitions. You don't want to "check" all the 0 and 1 regions and waste time computing their dirtsimple behavior.
The second is just ordinary 2D cellular automaton, but what Wolfram has done is *averaged* the site values on successive steps at each location, after the evolution itself. This doesn't change the underlying CA rule. But it gives a "gray level" or hue that is essentially the average sitevalue of that location in that run "to date". You can see the results, quite interesting...
The last, "growth CAs", are a restriction on the full rulespace for ordinary 2D CAs, but otherwise just exactly 2D CAs in the conventional manner. The restriction is, any cell that becomes black has to remain black according to the rule. Outer totalistic CAs in which all the "center cell is black" "digits" are "1" are the typical example. All center cell = 1 digits going to 1, implies any cell that becomes black will remain black. So these are a proper subset of standard outer totalistic 2 color CAs, as we've already discussed.
I hope this helps.
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