Notes

Chapter 5: Two Dimensions and Beyond

Section 6: Multiway Systems


Properties [of multiway system example]

The total number of strings grows approximately quadratically; its differences repeat (offset by 1) with period 1071. The number of new strings generated at successive steps grows approximately linearly; its differences repeat with period 21. The third element of the rule is at first used only on some steps—but after step 50 it appears to be used somewhere in every step.

The pictures below show in stacked form (as on page 208) all sequences generated at various steps of evolution. Note that after just a few steps, the sequences produced always seem to consist of white elements followed by black, with possibly one block of black in the white region. Without this additional block of black, only the first case in the rule can ever apply.

In analogy with page 796 the picture below shows when different strings with lengths up to 10 are reached in the evolution of the system.

Different initial conditions for this multiway system lead to behavior that either dies out (as for "ABA"), or grows exponentially forever (as for "ABAABABA").


From Stephen Wolfram: A New Kind of Science [citation]