Analogs of Conway Objects in

We have already alluded to patterns that evolve under Conway s original two-dimensional rule as either having or not having analogs in three-dimensions. We now make this notion of analogous more precise. 

Assume we are on a cubic lattice. Define a Conway object as any pattern of live sites that emerges during Conway s Life. Without loss of generality, we may suppose that the two-dimensional pattern lives on the 2 = 0 plane of the cubic lattice. 

introduce two primitive concepts [bayes87a]. (1) An expansion, (C), of a Conway object C = (xi,yi,z = 0), which is a three-dimensional pattern, P, formed by copying all live cells of that object onto an adjacent plane, so that xi,yi,z = 0), xi,yy,z = 1). (2) A projection of a three-dimensional Life object into two dimensions, which exists if and only if (i) all of its live sites lie in two adjacent planes (say 2=0 and 2 = 1), and (ii) all pairs of sites, (xi,yi,0) and xi,yi, 1) are either both live or both dead. 

A three-dimensional analog of a Conway object is then defined to be an expansion sucli that, when subjected to the appropriate three-dimensional Life-rule, yields after each and every generation a projection that is identical to the original Conway object for the same iteration step under Conway s original two-dimensional 

We conclude this chapter with a few illustrative color plates. Simple B W graphics are sufficient for viewing CA only in the special case when sites take on one of two values. In all other cases, color is indispensable for visualizing the behavior of the system. The plates were produced using the freely available CA software packages MCell, Cellab and WinCA. See Appendix A CA Research Tools for a description of these programs and links to web sites from which they can be HnwnlnnHpH 

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