(Color online) Typical complex spatio-temporal patterns produced by rule 20. After a first transient (short-lived), the surviving nonquiescent activity is that of a few localized gliders: three stationary and time-periodic gliders located roughly at the center of the figure, and a traveling glider on the right. The collision between traveling and static gliders defines a second transient (long-lived) of the system, when the dynamics under rule 20 invariably collapses into tame class 2 behavior (Refs. 30 and 31 ).
(Color online) Typical complex spatio-temporal patterns generated by rule 52. Top: Asymptotically, the system settles into arbitrarily large patches of synchronized activity, 0 (white) or 1 [black (purple online)], interconnected by transition interfaces where cyclic activity occurs. Gliders may subsist inside individual patches but remain ephemeral as in rule 20. Bottom: Transition interfaces are static or may travel. Periodic boundary conditions eventually lead to collisions of interfaces and larger regions of synchronized activity.
(Color online) The elementary interfaces needed to produce all asymptotic complexities computed for rule 52. Interfaces A–E bridge patches with backgrounds of different colors. Time always evolves downwards, here and subsequently.
(Color online) Elementary structures F and G which may be regarded as localized structures that may or may not travel, or interfaces bridging two identical but spatially separated phases. The traveling glider is not elementary but a juxtaposition of and , the conjugate of . It travels at the “speed of light” in the system: one cell per time step. Glider travels at cell per step.
(Color online) Gliders arising when interfaces are glued together. Top: the simplest motifs surviving on a white background representing zeros. Middle: conjugates of the gliders in the top row, white structures surviving on a black background of ones. Bottom: “fat” gliders, i.e., the same gliders seen on the top row but now with “inflated” inner cores of ones. Note the different possibilities of combining interfaces.
(Color online) Top: Static hybrid gliders formed by juxtaposing distinct elementary interfaces. The four black sites separating interfaces may be inflated arbitrarily, as indicated in Fig. 5 . Center: static gliders formed by dephasings due to the different ways of juxtaposing identical period-3 interfaces. Note that has “rotational symmetry” with respect to its central axis while and have “helicoidal symmetry,” i.e., involve a reflection plus a one time-step shift as hinted by the additional coloring. The minimum distance between interfaces is larger for the gliders located on the left, to prevent white cells from interacting. Bottom: Hybrid gliders due to the three possible time-dephasings between different period-3 interfaces.
(Color online) The basic construction used for sidewise updating of the automaton, i.e., for spatial updating, and to locate periodic patterns, illustrated here for the simplest scenario possible: fixed point (i.e., period 1). Since , the first site of the interface is ambiguous, i.e., , where or , because both values satisfy rule 52, Eq. (2) . Whatever the value chosen, it must be copied downward, as indicated by the arrow, to ensure period-1. The evolution for the nontrivial choice is shown in Fig. 8 .
(Color online) Proof that rule 52 supports only one elementary interface/glider of period one. (a) The bit highlighted is determined by the left-neighbor of the starting configuration. Yellow (light gray) indicates that the construction may proceed since none of the deadlock configurations discussed in Fig. 10 below was found. (b–e) The bit determined in the previous step is copied and a new bit is determined. (f) A new darker color (shading) is used to indicate that an ambiguous situation is reached, when both 0 and 1 are valid choices. Repeated choices increase the size of the synchronization patch of 1, producing the interface , the conjugate of in Fig. 2 , preserving the possibility of an ambiguous choice indefinitely (see text).
(Color online) Initial developments for proving existence of interfaces/gliders of period 3, when two sites are considered simultaneously. (a) A pair of 1s is required on the right. (b) The choice and violates rule 52. (c) Deadlock generated by the choice . Conclusion: may only be 10 or 01. These two choices are analyzed in Fig. 12 .
(Color online) Key properties of rule 52. Two uppermost panels: Ambiguous configurations, when the binary value of site at time is true independently of the value of at time . Two lowest panels: Deadlock configurations, when the value at time is impossible, independently of . The precise location of and in the 5-cells neighborhood is irrelevant. The colors marking ambiguous and deadlock configurations will be used subsequently.
(Color online) Proof that there is only one possible arrangement leading to an elementary structure of period-2, that is labeled in Fig. 4 . The ambiguity is preserved as the structure grows to the right by selecting . Selecting in the manner shown in Fig. 9(a) will generate another structure . Arbitrary alternations of structures are possible by suitably intercalating choices and .
(Color online) Proof of the existence of just two elementary interface/gliders of period three for the pair of possible initial configurations. Top two rows: Development of the cases and discussed in Fig. 9(b) . Bottom two rows: Development for the remaining possibility of initial configurations, both leading to deadlock.
(Color online) Spatial information flow for period 7. The pattern above may be extended indefinitely to the right. After a spatial transient of 32 sites one finds a spatial period of 49 sites. Note that it is impossible to find this asymmetric pattern using the time-honored expedient of time-evolving initial conditions on finite lattices under periodic boundary conditions. It can only be found using sidewise spatial updating.
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