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However, with the rule
n If[EvenQ[n], 3n/2, Round[3n/4]]
it is always possible to go backwards by the rule
n If[Mod[n,3] 0, 2n/3, Round[4n/3]]
The picture shows the number of base 10 digits in numbers obtained by backward and forward evolution from n = 8 .
In base 6, (3/2) n is a cellular automaton with rule
{a_, b_, c_} 3 Mod[a + Quotient[b, 2], 2] + Quotient[3 Mod[b, 2] + Quotient[c, 2], 2]
(Note that this rule is invertible.)
In rule 126, the lengths of the shortest newly excluded blocks on successive steps are 0, 3, 12, 13, 14, 14, 17, 15. … (As an example, in rule 54 they decrease from 9 to 7 between steps 4 and 5.)
Each node is then connected to the nodes associated with the strings reached by one application of the multiway rule, as on page 209 .
… For example, starting with n A 's the rule {"A" "AB","AB" "A"} yields a regular n -dimensional grid, as shown below.
In the case of Cos[x] – Cos[α x] each step in the generalized substitution system has a rule determined as shown on the left from a term in the continued fraction representation of (α–1)/(α+1) .
In rule 126, the only effect at step 2 is that black cells can no longer appear on their own: they must always be in groups of two or more.
This rule captures the basic growth inhibition effect that occurs in snowflakes.
At each step the configuration of particles is updated according to the simple collision rules shown above.
For even though this pattern is produced by a simple one-dimensional cellular automaton rule, and even though one can see by eye that it contains at least some small-scale regularities, none of the schemes we have discussed up till now have succeeded in compressing it at all.
Note that for both systems the majority of cases in their rules are not used in the specific computations for which they were constructed.