From: Harold V. McIntosh (MCINTOSH@unamvm1.dgsca.unam.mx)
Date: Thu May 13 1993 - 00:43:20 UTC
John Walsh <zed(at)MATHS.TCD.IE> asks: > > Does anyone know why the the rule 2233 in 3d-life should yield > (appearently) population expansion from a sufficiently randomly > populated space (In the tests that I have run the probability of > a cell being initially alive is circa 1/100). Has anybody else tried > this rule and gotten different results? > Having tried this with von Neumann neighborhoods and seen that it doesn't quite work that way, the conclusion is that one should keep with Moore neighborhoods, as Bays specified originally (at least, to answer Walsh's question). - Neither our equipment nor programs handle Moore (2,3) (evidently I ought to write one - it's not hard) so the following comments are of a general nature. - First comment: at density 1/100 it takes a 5x5x5 cube in which to expect a live cell, but the Poisson distribution still gives a significant chance of finding 2 or 3, so it is not entirely out of the question that small clusters exist, keep themselves alive, and even start growing. You just need a big enough volume that some clusters are likely. - Second comment: Mean Field Theory often gives an indication of how a given rule is going to behave. Here we have a totalistic rule based on 3 live cells, therefore the probability is (27,3)p*3q*24. This seems to have a maximum of about 0.3 at p=3/27, or at 0.11. This is well above the diagonal, so the rule should be expansive with a stable density around 1/4 or 1/3. - Applying the foregoing to the von Neumann neighborhood, the maximum lies at 3/7, comfortably below the diagonal, as is the whole curve. Thus one expects rather different results. - My arithmetic is lousy and mean field theory is only an indicator, but off hand, the observation Walsh reports seems reasonable. Anyone really interested should graph this polynomial, and then try varying the rule until the curve rises only slightly over the diagonal at some point. - The degree of tangency at the origin is also important: non-tangency means isolated cells can grow, which should always be expansive. Quadratic behavior means pairs could proliferate, also undesirable. So cubic is the place to begin. - Harold V. McIntosh |Depto. de Aplicaci'on de Microcomputadoras MCINTOSH@UNAMVM1.BITNET |Instituto de Ciencias/UAP mcintosh@unamvm1.dgsca.unam.mx |Apdo. Postal 461 (+52+22)43-6330 |72000 Puebla, Pue., MEXICO
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