And what this means is that instead of depending only on quantities like f[n – 1] and f[n – 2] , the rule for f[n] can also for example depend on a quantity like f[n – f[n – 1]] . There is some subtlety here because in the abstract nothing guarantees that n – f[n – 1] will necessarily be a positive number. And if it is not, then results obtained by applying the rule can involve meaningless quantities such as f[0] , f[–1] and f[–2] .

Evolution from a succession of initial conditions is shown corresponding to inputs of numbers from 1 to 20. … The maximum for input of length n is (a) 5 , (b) 6n+3 , (c) 4n+3 , (d) 2n+3 , (e) 2n 2 + 8n + 7 , (f) 2 n+1 -1 (though the average is n+2 ), (g) 2n+1 , (h) 3 , (i) 2n+1 , (j) 4n-1 , (k) roughly 2.5 n 2 .

And starting with 1, the sequence of numbers one gets is 1, 3, 6, 9, 15, 24, 36, 54, 81, 123, 186, 279, 420, 630, 945, 1419, 2130, 3195, 4794, ... … As an example, consider the following procedure: if the number obtained at a particular step is even, then multiply this number by 5/2 ; otherwise, add 1 and then multiply the result by 1/2 . If one starts with 1, then this procedure simply gives 1 at every step.

With the "above" connection labelled as 1 and the "below" connection as 2, these rules correspond to replacing connections {{1}, {2}} at each node by (a) {{2, 1}, {2}} , (b) {{1, 1}, {2}} , (c) {{}, {2}} , and (d) {{}, {1}} .

But if the number is odd, then first add 1—so as to get an even number—and only then multiply by 3/2 . … Results of starting with the number 1, then applying the following rule: if the number at a particular step is even, multiply by 3/2; otherwise, add 1, then multiply by 3/2. … The system here can be represented by the rule n  If[EvenQ[n], 3n/2, 3(n + 1)/2] , while the one on page 100 follows the rule n  If[EvenQ[n], 3n/2, (3n + 1)/2] .

In the early 1900s—notably with the work of Ferdinand de Saussure —there began to be more emphasis on the general question of how languages really operate, and the point was made that the verbal elements or signs in a language should be viewed as somehow intermediate between tangible entities like sounds and abstract thoughts and concepts.

Rule 22—like rule 90 from page 26 —gives a pattern with fractal dimension Log[2,3] ≃ 1.58 ; rule 150 gives one with fractal dimension Log[2, 1+Sqrt[5]] ≃ 1.69 .

All the rules shown start with f[1]=f[2]=1 .

Rule 110 Turing machines Given an initial condition for rule 110, the initial condition for the Turing machine shown here is obtained as Prepend[4 list, 0] with 1 's on the left and 0 's on the right. The Turing machine {{1, 2}  {2, 2, -1}, {1, 1}  {1, 1, -1}, {1, 0}  {3, 1, 1}, {2, 2}  {4, 0, -1}, {2, 1}  {1, 2, -1}, {2, 0}  {2, 1, -1}, {3, 2}  {3, 2, 1}, {3, 1}  {3, 1, 1}, {3, 0}  {1, 0, -1}, {4, 2}  {2, 2, 1}, {4, 1}  {4, 1, 1}, {4, 0}  {2, 2, -1}} with s = 4 states and k = 3 possible colors also emulates rule 110 when started from Prepend[list + 1, 1] surrounded by 0 's. The s = 3 , k = 4 Turing machine {{1 , 0}  {1, 2, 1}, {1, 1}  {2, 3, 1}, {1, 2}  {1, 0, -1}, {1, 3}  {1, 1, -1}, {2, 0}  {1, 3, 1}, {2, 1}  {3, 3, 1}, {3, 0}  {1, 3, 1}, {3, 1}  {3, 2, 1}} started from Append[list, 0] with 0 's on the left and 2 's on the right generates a shifted version of rule 110.

Rule 60 Turing machines One can emulate rule 60 using the 8-case s = 3 , k = 3 Turing machine (with initial condition Append[list + 1, 1] surrounded by 0 's) {{1, 2}  {2, 2, 1}, {1, 1}  {1, 1, 1}, {1, 0}  {3, 1, -1}, {2, 2}  {2,1, 1}, {2, 1}  {1, 2, 1}, {3, 2}  {3, 2, -1}, {3, 1}  {3, 1, -1}, {3, 0}  {1, 0, 1}} or by using the 6-case s = 2 , k = 4 Turing machine (with initial condition Append[3list, 0] with 0 's on the left and 1 's on the right) {{1, 3}  {2, 2, 1}, {1, 2}  {1, 3, -1}, {1, 1}  {1, 0, -1}, {1, 0}  {1, 1, 1}, {2, 3}  {2, 1, 1}, {2, 0}  {1, 2, 1}} This second Turing machine is directly analogous to the one for rule 110 on page 707 .