From: Harold V. McIntosh (MCINTOSH@unamvm1.dgsca.unam.mx)
Date: Mon May 17 1993 - 23:36:24 UTC
Last month, when inquiries about 3-D Life were just beginning, two messages appeared, reading (in part): > > Wed, 14 Apr 1993 13:16:11 GMT > Richard J. Gaylord <gaylord(at)UX1CSO.UIUC.EDU> > i think the latest issue of Complex Systems has a new article by Bays > on 3D Life. > > Tue, 13 Apr 1993 18:01:53 GMT > From: John Pedersen <jfp(at)GOEDEL.MATH.USF.EDU.> > Carter Bays gives a new rule for 3D Life in the latest issue of > Complex Systems (volume 6, number 5, pages 433-442). > The rule in question is one which Bays calls 6855; in his last paragraph he asserts: > > In conclusion, it is apparent that Life 6855 holds at least as much > promise as the original games Life 4555 and Life 5766. Life 6855 is > the first rule where three gliders have been discovered; and although > there is no two-dimensional analog, the unusual 'barrier life' > construction certainly deserves further investigation. > It also looks like this rule is a classical textbook example of mean field theory. There is a fifth degree horizontal tangency at the origin (65780 p*5q*22+...), a slightly unstable fixed point at 0.13, and a superstable fixed point at 0.25 (superstable means fixed point with slope zero) (and, of course, an 18th degree horizontal tangency at 1.0). - It also, considering many case studies which we have seen, meets the criterion of 'a diagonal tangency;' these are never exact, and give their best results when the curve skims just above the diagonal for a ways. - But what is significant is that the first few generations ITERATE (the others haven't been tested yet). The curve falls to the value of the first fixed point at around 0.37, which means that eventually any density within the interval 0.13-0.37 will rise to 0.24 and stay there. Outside this range density will inexorably drop to zero (not implying that there won't be ANY live cells, but few and far between). - You can't always be sure what goes on at those low densities. In 2-D 3-4 Life, there is a somewhat similar curve (not superstable, however) but there are some very small discrete objects which can devour the universe if they are either present initially or form sometime later. There don't seem to be any of those around in 6855 Life, and the unstable fixed point splits the densities very cleanly into prospering or languishing populations. - Bays observes this (I won't copy three or four paragraphs): when he seeds the universe with well defined blobs of suitable density, they stabilize, ~ can remain in constant turmoil forever.' - At issue here is an important point concerning automata - what happens to grossly nonuniform populations. Some have 'positive surface tension' wherein globules tend to form and retain their identity. Others have 'negative surface tension,' tending to spread out and occupy more and more territory. Life 6855 seems to lie in the positive category, 3-4 Life apparently in the negative. - Globules in the first category are really finite state machines, the more so the smaller they are; thus should settle down into static or periodic structures. What about gliders? You could work this into mean field theory, but Chris Langton's L parameter already tells you a great deal. How many, amongst all rules for a given automaton, lead to shifted cells? This is a good way to hack up automata with lots of gliders in 1-D, at least. Has it been tried or evaluated in 2-D or 3-D? (Another statistic to incorporate in my new program) - 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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