In all these examples, however, the randomness that is involved comes from the same basic mechanism: it is explicitly inserted from outside at each step in the evolution of the system.

And so to cause any substantial change one would presumably have to insert a comparable amount of data with the special properties one wants.

And the point is that these maps actually do intrinsically generate complexity and randomness; they do not just transcribe it when it is inserted in their initial conditions.

Examples of patterns produced by inserting a single extra black cell into repetitive backgrounds for rule 45.

The idea is to set up a configuration in rule 30 so that if one inserts input at particular positions the output from the underlying rule 30 evolution corresponds exactly to what one would get from a single step of rule 90 evolution.

To access location n the left-hand initial conditions must contain Flatten[{0, i, IntegerDigits[n, 2] /. {1  {0, 11}, 0  {0, 2}}}] inserted in a repetitive {0, 1} background.

And so one decides to look at a more complicated system—usually with features specifically inserted to produce some specific form of behavior.

Instead, it is just that randomness that was inserted in the digit sequence of the original number shows up in the results one gets.

And for a while what one sees is that the randomness that has been inserted persists.

Structures [in rule 110] The persistent structures shown can be obtained from the following {n, w} by inserting the sequences IntegerDigits[n, 2, w] between repetitions of the background block b : {{152, 8}, {183, 8}, {18472955, 25}, {732, 10}, {129643, 18}, {0, 5}, {152, 13}, {39672, 21}, {619, 15}, {44, 7}, {334900605644, 39}, {8440, 15}, {248, 9}, {760, 11}, {38, 6}} The repetition periods and distances moved in each period for the structures are respectively {{4, -2}, {12, -6}, {12, -6}, {42, -14}, {42, -14}, {15, -4}, {15, -4}, {15, -4}, {15, -4}, {30, -8}, {92, -18}, {36, -4}, {7, 0}, {10, 2}, {3, 2}} Note that the periodicity of the background forces all rule 110 structures to have periods and distances given by {4, -2} r + {3, 2} s where r and s are non-negative integers.