# Why some rules in Elementary Cellular Automata halt and others don't?

I was looking over stephen wolfram's work mainly about Elementary Cellular Automata, i see that for all rules with N steps {i.e 200 steps} they show the same pattern {i,e rule 90 running on 20 steps and 100 steps are the same, like Here}.

Unlike rule 110 which shows different behavior when N = 20, 100 or 200, i know that it is universal but does that mean if a cellular automaton structure does not halt {continue to show different pattern when changing N steps} does that means it is always universal ?

• No, there is absolute no reason that the dichotomy you mention covers all cases. It is probably not difficult to find examples, once you tel us what you mean by "show the same structure". Commented Oct 16, 2015 at 14:22
• i mean for example rule 90 when running on 100 steps it is the same as if you run rule 90 on 200 steps. Commented Oct 16, 2015 at 14:23
• I still don't know what you mean. Try to express yourself formally. According to Wikipedia's account on rule 90, When started from a random initial configuration, its configuration remains random at each time step, whatever that means. This seems to contradict any natural interpretation of your claim. Commented Oct 16, 2015 at 14:25
• We cannot refute your conjectured dichotomy unless you state it formally. Commented Oct 16, 2015 at 14:35
• The example on Wolfram Alpha starts with an empty tape except for a black box in one place. It's true: this leads to a repeating pattern, for some appropriate definition of 'repeating pattern'. However, the universality of rule 110 has to do with initialising the tape so that it codes some problem and having the 110 ECA compute a value. For example, you can paint the tape such that 110 will compute the prime divisors of 15. Many initial conditions of 110 appear chaotic but aren't, and many other rules (e.g. rule 30) appear chaotic (and never repeat) but aren't very meaningful. Commented Oct 16, 2015 at 14:59