Active cell motion [in mobile automata]

The pictures below show the positions of the active cell for 20,000 steps of evolution in various mobile automata. (a), (b) and (c) correspond respectively to the rules on pages 73, 74 and 75. (c) has an outer envelope whose edges grow at rates {-1.5, 0.3} √t. (d) yields logarithmic growth as shown on page 496 (like Turing machine (f) on page 79). In most cases where the behavior is ultimately repetitive, transients and periods seem to follow the same approximate exponential distribution as in the note above. (g) however suddenly yields repetitive behavior with period 4032 after 405,941 steps. (h) does not appear to evolve to strict repetition or nesting, but does show progressively longer patches with fairly orderly behavior. (c) shows no obvious deviation from randomness in at least the first billion steps (after which the pattern it produces is 57,014 cells wide).