Cellular Automaton Rule In Figs example essay topic

This section discusses patterns formed by the evolution of cellular automat a from simple seeds. The seeds consist of single nonzero sites, or small regions containing a few nonzero sites, in a background of zero sites. The growth of cellular automat a from such initial conditions should provide models for a variety of physical and other phenomena. One example is crystal growth. The cellular automaton lattice corresponds to the crystal lattice, with nonzero sites representing the presence of atoms or regions of the crystal.

Different cellular automaton rules are found to yield both faceted (regular) and dendritic (snowflake-like) crystal structures. In other systems the seed may correspond to a small initial disturbance, which grows with time to produce a complicated structure. Such a phenomenon presumably occurs when fluid turbulence develops downstream from an obstruction or orifice. (3) Figure 2 shows some typical examples of patterns generated by the evolution of two-dimensional cellular automat a from initial states containing a single nonzero site.

In each case, the sequence of two-dimensional patterns formed is shown as a succession of 'frames. ' ' A space-time 'section' is also shown, giving the evolution of the center horizontal line in the two-dimensional lattice with time. Figure 3 shows a view of the complete three-dimensional structures generated. Figure 4 gives some examples of space-time sections generated by typical one-dimensional cellular automat a.

Examples of classes of patterns generated by evolution of two-dimensional cellular automat a from a single-site seed. Each part corresponds to a different cellular automaton rule. All the rules shown are both rotation and reflection symmetric. For each rule, a sequence of frames shows the two-dimensional configurations generated by the cellular automaton evolution after the indicated number of time steps. Black squares represent sites with value 1; white squares sites with value 0. On the left is a space-time section showing the time evolution of the center horizontal line of sites in the two-dimensional lattice.

Successive lines correspond to successive time steps. The cellular automaton rules shown are five-neighbor square outer, with codes (a) 1022, (b) 510, (c) 374, (d) 614 (sum modulo 2 rule), (e) 174, (f) 494. With some cellular automaton rules, simple seeds always die out, leaving the null configuration, in which all sites have value zero. With other rules, all or part of the initial seed may remain invariant with time, yielding a fixed pattern, independent of time. With many cellular automaton rules, however, a growing pattern is produced. View of three-dimensional structures formed from the configurations generated in the first 24 time steps of the evolution of the two-dimensional cellular automat a shown in Fig. 2.

Rules (a), (b), and (c) all give rise to configurations with regular, faceted, boundaries. Rules (d), (e), and (f) yield dendritic patterns. In this and other three-dimensional views, the shading ranges periodically from light to dark when the number of time steps increases by a factor of two. The three-dimensional graphics here and in Figs. 10 and 14 is courtesy of M. Pruitt at Los Alamos National Laboratory.

Examples of classes of patterns generated by evolution of one-dimensional cellular automat a from a single-site seed. Successive time steps are shown on successive lines. Nonzero sites are shown black. The cellular automaton rules shown are nearest-neighbor, with possible values at each site: (a), code 14, (b), code 6, (c), code 10, (d), code 21, (e), code 102, (f), code 138. Irregular patterns are also generated by some, rules (such as that with code 10), and by asymmetric, rules (such as that with rule number 30). Rule (a) in Figs.

2 and 3 is an example of the simple case in which the growing pattern is uniform. At each time step, a regular pattern with a fixed density of nonzero sites is produced. The boundary of the pattern consists of flat (linear) 'facets,' ' and traces out a pyramid in space-time, whose edges lie along the directions of maximal growth. Sections through this pyramid are analogous to the space-time pattern generated by the one-dimensional cellular automaton of Fig. 4 (a).

Cellular automaton rule (b) in Figs. 2 and 3 yields a pattern whose boundary again has a simple faceted form, but whose interior is not uniform. Space-time sections through the pattern exhibit an asymptotically self-similar or fractal form: pieces of the pattern, when magnified, are indistinguishable from the whole. Figure 4 (b) shows a one-dimensional cellular automaton that yields sections of the same form.

The density of nonzero sites in these sections tends asymptotically to zero. The pattern of nonzero sites in the sections may be characterized by a Hausdorff or fractal dimension that is found by a simple geometrical construction to have value. Self-similar patterns are generated in cellular automat a that are invariant under scale or blocking transformations. Particular blocks of sites in a cellular automaton often evolve according to a fixed effective cellular automaton rule.

The overall behavior of the cellular automaton is then left invariant by a replacement of each block with a single site and of the original cellular automaton rule by the effective rule. In some cases, the effective rule may be identical to the original rule. Then the patterns generated must be invariant under the blocking transformation, and are therefore self-similar. (All the rules so far found to have this property are additive.) In many cases, the effective rule obtained after several blocking transformations with particular blocks may be invariant under further blocking transformations.

Then if the initial state contains only the appropriate blocks, the patterns generated must be self-similar, at least on sufficiently large length scales. Cellular automaton (c) gives patterns that are not homogeneous, but appear to have a fixed nonzero asymptotic density. The patterns have a complex, and in some respects random, appearance. It is remarkable that simple rules, even starting from the simple initial conditions shown, can generate patterns of such complexity. It seems likely that the iteration of the cellular automaton rule is essentially the simplest procedure by which these patterns may be specified.

The cellular automaton rule is thus 'computationally irreducible' (cf. Ref. 19). Cellular automat a (a), (b), and (c) in Figs. 2 and 3 all yield patterns whose boundaries have a simple faceted form. Cellular automat a (d), (e), and (f) give instead patterns with corrugated, dendritic, boundaries.

Such complicated boundaries can have no analog in one-dimensional cellular automat a: they are a first example of a qualitative phenomenon in cellular automat a that requires two or more dimensions. Cellular automaton (d) follows the simple additive rule that takes the value of each site to be the sum modulo two of the previous values of all sites in its five-site neighborhood. The space-time pattern generated by this rule has a fractal form. The fractal dimension of this pattern, and its analogs on -dimensional lattices, is given by: , or approximately for. The average density of nonzero sites in the pattern tends to zero with time. Rules (e) and (f) give patterns with nonzero asymptotic densities.

The boundaries of the patterns obtained at most time steps are corrugated, and have fractal forms analogous to Koch curves. The patterns grow by producing 'branches' along the four lattice directions. Each of these branches then in turn produces side branches, which themselves produce side branches, and so on. This recursive process yields a highly corrugated boundary.

However, as the process continues, the side branches grow into each other, forming an essentially solid region. In fact, after each time steps the boundary takes on an essentially regular form. It is only between such times that a dendritic boundary is present. Cellular automaton (e) is an example of a 'solidification' rule, in which any site, once it attains value one, never reverts to value zero. Such rules are of significance in studies of processes such as crystal growth. Notice that although the interior of the pattern takes on a fixed form with time, the possibility of a simple one-dimensional cellular automaton model for the boundary alone is precluded by non local effects associated with interactions between different side branches.

The boundaries of the patterns generated by cellular automat a (a), (b), and (c) expand with time, but maintain the same faceted form. So after a rescaling in linear dimensions by a factor of, the boundaries take on a fixed form: the pattern obtained is a fixed point of the product of the cellular automaton mapping and the rescaling transformation (cf. Refs. 20 and 21). The boundaries of Figs. 2 (d, e, f) and 3 (d, e, f) continually change with time; a fixed limiting form after rescaling can be obtained only by considering a particular sequence of time steps, such as those of the form.

The result depends critically on the sequence considered: some sequences yield dendritic limiting forms, while others yield faceted forms. The complete space-time patterns illustrated in Figs. 3 (d, e, f) again approach a fixed limiting form after rescaling only when particular sequences of times are considered. It appears, however, that the forms obtained with different sequences have the same overall properties: they are asymptotically self-similar and have definite fractal dimensions. The limiting structure of patterns generated by the growth of cellular automat a from simple seeds can be characterized by various 'growth dimensions. ' ' Two general types may be defined.

The first, denoted generically, depend on the overall space-time pattern. The second, denoted, depend only on the boundary of the pattern. The boundary may be defined as the set of sites that can be reached by some path on the lattice that begins at infinity and does not cross any nonzero sites. The boundary can thus be found by a simple recursive procedure (cf.

Ref. 22). For rules that depend on more than nearest-neighboring sites, paths that pass within the range of the rule of any nonzero site are also excluded, and so no paths can enter any 'pores' in the surface of the pattern. Growth dimensions in general describe the logarithmic asymptotic scaling of the total sizes of patterns with their linear dimensions. For example, the spatial growth dimension is defined in terms of the total number of sites (interior and boundary) contained in patterns generated by a cellular automaton as a function of time by the limit of as. Figure 5 shows the behavior of as a function of for the cellular automat a of Figs.

2 and 3. For those with faceted boundaries, for all sufficiently large: the total size of the patterns scales as the square of the parameter that determines their linear dimensions. When the boundaries can be dendritic, however, varies irregularly with. In case (d), for example, depends on the number of nonzero digits in the binary decomposition of the integer (cf. Ref. 4): is thus maximal when, and is minimal when.

One may define upper and lower spatial growth dimensions and in terms of the upper and lower limits (sup and inf) of as. For case (d), , while. For cases (e) and (f), oscillates with time, achieving its maximal value at, and its minimal value at or near. However, in these cases numerical results suggest that the upper and lower growth dimensions are in fact equal, and in both cases have a value. Sizes of structures generated by the two-dimensional cellular automat a of Fig. 2 growing from single nonzero initial sites as a function of time. (Although the sizes are defined only at integer times, their successive values are shown joined by straight lines.) gives the number of sites on the boundaries of patterns obtained at time. gives the total number of sites contained within these boundaries. is the number of sites in the boundary (surface) of the complete three-dimensional space-time structures illustrated in Fig. 3 up to time, and is the number of sites in their interior.

The large- limits of and so on give various growth dimensions for the structures. In cases (a), (b), and (c), structures with faceted boundaries are produced, and the growth dimensions have unique values. In cases (d), (e), and (f) the structures have dendritic boundaries, and the slopes of the bounding lines shown give upper and lower limits for the growth dimensions. In many of the cases shown, the number cal values of these upper and lower limits appear to coincide. An alternative definition of the spatial growth dimension includes only nonzero sites in computing the total sizes of patterns generated by cellular automaton evolution. With this definition, the spatial growth dimension has no definite limit even for cellular automat a such as that of case (b) which give patterns with faceted boundaries.

The spatial growth dimensions for the boundaries of patterns generated by cellular automat a are obtained from the limits of at large, where gives the number of sites in the boundary at time (cf. Ref. 23). Figure 5 shows the behavior of with for the cellular automat a of Figs. 2 and 3. For the faceted boundary cases (a), (b), and (c), .

In cases (d), (e), and (f), where dendritic boundaries occur, varies irregularly with. is minimal when and the boundary is faceted, and is maximal when the boundary is maximally dendritic, typically at. No unique limit for exists. In case (d), , while. In case (e), and, while in case (f), and. The limiting forms obtained after rescaling for the spatial patterns generated by the dendritic cellular automat a (d), (e), and (f) depend on the sequences of time steps used in the limiting procedure, so that there are no unique values for their spatial growth dimensions. On the other hand, the overall forms of the complete space-time patterns generated by these cellular automat a do have definite limits, so that the growth dimensions that characterize them have definite values.

The total growth dimensions and (4) may be defined as and, where is the total number of sites contained in the space-time pattern generated up to time step, and is the number of sites in its boundary. [Notice that.] Figure 5 shows the behavior of and as a function of for the cellular automat a of Figs. 2 and 3. Unique values of and are indeed found in all cases. Rules that give patterns with faceted boundaries have, . The additive rule of case (d) gives, .

Cases (e) and (f) both give, . Growth dimensions may be defined in general by considering the intersection of the complete space-time pattern, or its boundary, with various families of hyperplanes. With fixed-time hyperplanes one obtains the spatial growth dimensions and. Temporal growth dimensions and are obtained by considering sections through the space-time pattern in spatial direction.

(The section typically includes the site of the original seed.) The total growth dimension may evidently be obtained as an appropriate average over temporal growth dimensions in different directions. (The average must be taken over pattern sizes, and so requires exponentiation of the growth dimensions.) The values of the temporal growth dimensions for the patterns of Figs. 2 and 3 depend on their internal structure. Cases (a), (c), (e), and (f) have; case (b) has, and case (d) has. The temporal growth dimensions for the boundaries of the patterns are equal to one for the faceted boundary cases. These dimensions vary with direction in cases with dendritic boundaries.

They are equal to one in directions of maximal growth, but are larger in other directions. In general the values of growth dimensions associated with particular hyperplanes are bounded by the topological dimensions of those hyperplanes. Empirical studies indicate that among all (symmetric) two-dimensional cellular automat a, patterns with the form of case (c), characterized by, , are the most commonly generated. Fractal boundaries are comparatively common, but their growth dimensions are usually quite close to the minimal value of two. Fractal sections with are also comparatively common for five-neighbor rules, but become less common for nine-neighbor rules. The rules for the the two-dimensional cellular automat a shown in Figs.

2 and 3 are completely invariant under all the rotation and reflection symmetry transformations on their neighborhoods. Figure 6 shows patterns generated by cellular automaton rules with lower symmetries. These patterns are often complicated both in their boundaries and internal structure. Even though the patterns grow from completely symmetric initial states consisting of single nonzero sites, they exhibit definite directionalities and as a consequence of asymmetries in the rules. Asymmetric patterns may be obtained with symmetrical rules from asymmetric initial states containing several nonzero sites. For example, some rules should support periodic structures that propagate in particular directions with time.

Other rules should yield spiral patterns with definite. Structures of these kinds are expected to be simpler in many rules than for rules (cf. Ref. 24) just as in one-dimensional cellular automat a. Notice that spiral patterns in two-dimensional cellular automat a have total growth dimensions. Examples of patterns generated by growth from single-site seeds for 24 time steps according to general nine-neighbor square rules, with symmetries: (a) all, (b) horizontal and vertical reflection, (c) rotation, (d) vertical reflection, (e) none.

Figure 7 shows the evolution of various two-dimensional cellular automat a from initial states containing both single nonzero sites, and small regions with a few nonzero sites. In most cases, the overall patterns generated after a sufficiently longtime are seen to be largely independent of the particular form of the initial state. In cases such as (c) and (e), features in the initial seed lead to specific dislocations in the final patterns. Nevertheless, deformations in the boundaries of the patterns usually occur only on length scales of order the size of the seed, and presumably become negligible in the infinite time limit. As a consequence, the growth dimensions for the resulting patterns are usually independent of the form of the initial seed (cf. Ref.

20 for additive rules). Examples of patterns generated by evolution of two-dimensional cellular automat a from minimal seeds and small disordered regions. In most cases, growth is initiated by a seed consisting of a single nonzero site; for some of the rules shown, a square of four nonzero sites is required. There are nevertheless some cellular automaton rules for which slightly different seeds can lead to very different patterns. This phenomenon occurs when a cellular automaton whose configurations contain only certain blocks of site values satisfies an effective rule with special properties such as scale invariance. If the initial seed contains only these blocks, then the pattern generated follows the effective rule.

However, if other blocks are present, a pattern of a different form may be generated. An example of this behavior for a one-dimensional cellular automaton is shown in Fig. 8. Patterns produced with one type of seed have temporal growth dimension, while those with another type of seed have dimension. Cellular automaton rules embody a finite maximum information propagation speed.

This implies the existence of a 'bounding surface' expanding at this finite speed. All nonzero sites generated by cellular automaton evolution from a localized seed must lie within this bounding surface. (The cellular automat a considered here leave a background of zero sites invariant; such a background must be mapped to itself after at most time steps with any cellular automaton rule.) Thus the pattern generated after time steps by any cellular automaton is always bounded by the poly tope (planar-faced surface) corresponding to the 'unit cell' formed from the set of vectors specifying the displacements of sites in the neighborhood, magnified by a factor in linear dimensions (cf. Ref. 14). Thus patterns generated by five-neighbor cellular automaton rules always lie within an expanding diamond-shaped region, while those with nine-neighbor rules may fill out a square region.

The actual minimal bounding surface for a particular cellular automaton rule often lies far inside the surface obtained by magnifying the unit cell. A sequence of better approximations to the bounding surface may be found as follows. First consider a set of sites representing the neighborhood for a cellular automaton rule. If the center site has value one at a particular time step, there could exist configurations for which all of the sites in the neighborhood would attain value one on the next time step. However, there may be some sites whose values cannot change from zero to one in a single time step with any configuration. Growth does not occur along directions corresponding to such sites.

The poly tope formed from sites in the neighborhood, excluding such sites, may be magnified by a factor to yield a first approximation to the actual bounding surface for a cellular automaton rule. A better approximation is given by the poly tope obtained after two time steps of cellular automaton evolution, magnified by a factor. Example of a one-dimensional cellular automaton in which space-time patterns with different temporal growth dimensions are obtained with different initial seeds. The cellular automaton has, , and rule number 218. With an initial state containing only the blocks and, it behaves like the additive rule 90, and yields a self-similar space-time pattern with fractal dimension. But when the initial state contains and blocks, it behaves like rule 128, and yields a uniform space-time pattern.

The actual bounding surfaces for five-neighbor two-dimensional cellular automaton rules usually have their maximal diamond-shaped form. However, many nine-neighbor rules have a diamond-shaped form, rather than their maximal square form. Some nine-neighbor rules, such as those of Figs. 7 (g) and 7 (h) have octagonal bounding surfaces, while still others, such as those of Fig. 7 (i) have do decagonal bounding surfaces. The cellular automat a rules with lower symmetries illustrated in Fig. 6 in many cases exhibit more complicated boundaries, with lower symmetries. Patterns that maintain regular boundaries with time typically fill out their bounding surface at all times.

Dendritic patterns, however, usually expand with the bounding surface only along a few axes. In other directions, they meet the bounding surface only at specific times, typically of the form. At other times, they lie within the bounding surface. Dendritic boundaries seem to be associated with cellular automaton rules that exhibit 'growth inhibition' (cf. Ref. 14).

Growth inhibition occurs if there exist some for which, but, or vice versa. Such behavior appears to be common in physical and other systems. Figures 9 and 10 show examples of two-dimensional cellular automat a that exhibit the comparatively rare phenomenon of slow, diffusive, growth from simple seeds. Figure 11 gives a one-dimensional cellular automaton with essentially analogous behavior. The phenomenon is most easily discussed in the one-dimensional case. The pattern shown in Fig. 11 is such that it expands by one site at a particular time step only if the site on the boundary has value one.

If the boundary site has one of its other three possible nonzero values, then on average, no expansion occurs. The cellular automaton rule is such that the boundary sites have values one through four with roughly equal frequencies. Thus the pattern expands on average at a speed of about sites per time step (on each side). Examples of two-dimensional cellular automat a that exhibit slow diffusive growth from small disordered regions. The cellular automaton rules shown are nine-neighbor square outer, with codes (a) 256746, (b) 736, (c) 291552. The origin of diffusive growth is similar in the two-dimensional case.

Growth occurs there only when some particular several-site structure appears on the boundary. For example, in the cellular automaton of Fig. 9 (a), a linear interface propagates at maximal velocity. Deformations of the interface slow its propagation, and a maximally corrugated interface with a 'battlement' form does not propagate at all. Since many boundary structures occur with roughly equal probabilities, the average growth rate is small. In the cases investigated, the growth rate is asymptotically constant, so that the growth dimensions have definite values. A remarkable feature is that the boundaries of the patterns produced do not follow the poly topic form suggested by the underlying lattice construction of the cellular automaton.

Instead, in many cases, asymptotically circular patterns appear to be produced.