Cellular Automata in the Triangular Tessellation

For t he discussion below t he following definit ions are helpful. A semito-talistic CA rule is a rul e for a cellular aut omaton (CA) where (a) we tally t he living neighbors t o a cell without regard to their orient atio n wit h respect t o th at cell, and (b) t he ru le applied t o a cell may depend upon its current status. A lifelike rule (LFR rul e) is a semit otalist ic CA rul e where (1) cells have exac t ly two states (alive or dead); (2) t he rule giving t he state of a cell for t he next generation depends exact ly upon (a) it s state this generation and (b) t he tot al count of t he number of live neighbor cells; and (3) when tallying neighbors of a cell, we consider exactly t hose neighbor ing cells t hat t ouch t he cell in quest ion. An LFR rule is written EIEhFIF h , where EIE h (t he "environment" ru le) give th e lower and upp er bo un ds for t he tally of live neighbors of a cur rent ly live cell C so t hat C remains alive, and FlF h (the "fert ility " rul e) give t he lower and upp er bounds for t he tally of live neighbors required for a cur rent ly dead cell t o come t o life. For an LFR rul e t o specify a game of Life we impose two fur t her condit ions: (A) t here must exist at least one glider (t ra nslat ing oscillator) t hat is discoverab le wit h probability one by st arting with finite random initi al configurat ions (somet imes called "random primordial soup") , and (B) th e probability is zero t hat a finite rando m initi al configuration leads t o unbound ed growth. Note t ha t this second condition does not elimina te th e possibility t hat some unusual highly organized configuration can be const ruct ed where t he growt h is unbounded. Note also t hat we may be able to const ruct some ext remely complex config-ur ati on t hat translates; however, if t he possib ility of discovering t his wit h a random experiment is …