Moore neighborhood

Fig. 3.53 A few Snapshots of two-dimci sioiial Moore neighborhood totalistic rule 1,2,3,4 -> 1.

Of particular interest is the long-term behavior of voting-rule systems, which turns out to very strongly depend on the initial density of sites with value cr = 1 (= p). While all such systems eventually become either stable or oscillate with period-two, they approach this final state via one of two different mechanisms either through a percolation or nucleation process. Figure 3.60 shows a few snapshots of a Moore-neighborhood voting rule > 4 for p = 0.1, 0.15, 0.25 and 0.3. 

A similar behavior is obtained in the full 9-neighbor Moore neighborhood rule... 

Table 3.7 list,s the critical density and type of process for several von Neumann and Moore neighborhood rules. In the first and fourth columns, the rules are defined by the fractions (m/n), which specify a threshold of rn cr = 1 sites out of a total of n possible votes. The table entries for are taken from published results [vich84j using the CAM-6 hardware simulator [marg87] whether some or all of these values can be determined analytically remains an open problem. 

Notice that p,. = 1/2 exactly for both the von Neiimaim neighborhood 3/5 rule and Moore neighborhood 5/9 rule . This simple result follows immediately from the 0 e-r 1 symmetry of these simple majority rules. A simple majority rule is one in which the total possible number of votes is evenly split between yes (cr = 1) and no (cr = 0) votes, and can obviously exist only if the neighborhood size is odd. 

Perhaps the single most studied (and joyfully played with) rule - certainly the most famous is the two-dimensional Moore neighborhood binary-valued CA invented by John H. Conway, and popularized extensively by Martin Gardner in his Mathematical Gaines column in Scientific. American in the early 1970 s ([gardnerTO], [gardnerTl], [gardner78]). 

We begin by recalling that Conway s original deterministic rule is an outer-totalistic (code OT224) k = 2 rule defined on the two-dimensional Moore neighborhood ... 

As an example, consider a system of size 20 x 20, and take N = 100, px = 2 and Pj = 3. What happens if we vary the parameter u between the value 1 and, say, 20 Gerhardt and Schuster found that when this system evolves from a random initial state (using the Moore neighborhood for updates), some combination of four basic behavioral types emerges [gerh89]. Behavioral types - which appear to depend most strongly on the value of the parameter u - are characterized by both the manner in which the fraction of sites that are infected (= / ) varies as a function of time and the kind of transition-wave spatial patterns that develop ... 

Figure 2.5. Cell neighborhoods (a) the von Neumann neighborhood, (b) the Moore neighborhood, and (c) the extended von Neumann neighborhood of cell A...

Figure 5 The cellular automata neighborhoods associated with cell i. (a) The von Neumann neighborhood, (b) The Moore neighborhood, (c) The extended von Neumann neighborhood.

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