Larger than Life: threshold-range scaling of Life’s coherent structures

Physica D 183 (2003) 45–67

Larger than Life: threshold-range scaling of Life’s coherent structures Kellie Michele Evans∗ Department of Mathematics, California State University, Northridge, 18111 Nordhoff Street, Northridge, CA 91330, USA Received 12 October 2002; received in revised form 7 April 2003; accepted 13 May 2003 Communicated by C.K.R.T. Jones

Abstract The Game of Life has many coherent structures known as still lifes, oscillators, and spaceships. The most intriguing of these structures are the spaceships due to their ability to carry information across long spatial distances. Similar structures are supported by Larger than Life (LtL), which is a four-parameter family of two-dimensional cellular automata that generalizes the Game of Life to large neighborhoods and general birth and survival thresholds. Numerous examples of large range versions of Life’s spaceships are provided along with descriptions of the experimental methods used to find these objects. The empirical work illustrates that these structures are quite common, scale in a fairly coherent manner, and have a distinct geometry. A mix of rigorous results, questions, and conjectures are made about the existence of the generalized spaceships and other coherent structures for LtL rules with arbitrarily large neighborhoods as well as the convergence of such rules to “Euclidean automata”. Euclidean automata are deterministic rules that have Euclidean, rather than discrete universes. © 2003 Elsevier B.V. All rights reserved. PACS: 02.90; 07.05; 05.70; 46.10 Keywords: Cellular automata; Game of Life; Larger than Life; Threshold-range scaling; Spaceships; Gliders; Bugs

1. Introduction A two-dimensional cellular automaton (CA) consists of a regular spatial lattice of cells (this can be finite or infinite) each of which is in one of a finite number of states. This is the initial configuration of the system. Each cell is affected by the states of the cells in its neighborhood, which consists of a finite number of other cells. Every time step, each site checks the states of its neighbors. It then makes a computation, which is based on the status of its neighbors, to decide ∗

Tel./fax: +1-818-677-2705. E-mail address: [email protected] (K.M. Evans).

which state it will be in next time. All the lattice cells do the same thing, and after making their computations, they update to their new states simultaneously. CAs were originally imagined by the mathematicians von Neumann [1] and Ulam [2] in the late 1940s as possible models for biological self-reproduction [1,2]. In the late 1960s John Horton Conway discovered the Game of Life (Life), which has since become the most famous example of a CA. Its fame is due in part to the fact that its update rule is very simple, yet it generates extremely complicated dynamics [3–5]. For Life, as is the case with most CA rules, the initial state has an enormous impact on the resulting dynamics. When Life is started from a random initial state

0167-2789/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-2789(03)00155-6

46

K.M. Evans / Physica D 183 (2003) 45–67

Fig. 1. The glider’s trajectory starting in the northwest and moving along the diagonal to the southeast. The phase depicted at time 17 also appears in Fig. 3.

with an appropriate density of occupied cells, complex structures emerge. For example, gliders appear and their trajectories take them across the infinite lattice (Fig. 1). If not stopped by some other Life pattern, they will walk on forever. The glider is Life’s most commonly occurring spaceship, which is a finite pattern that reappears (without additions or losses) after a number of generations and is displaced by a nonzero distance [6]. Spaceships are able to carry information across long spatial distances. This ability leads to interesting reactions when spaceships collide with other coherent structures. Some of these reactions are crucial ingredients in the proof that Life is computation universal [4]. New reactions and their outcomes continue to be designed and tested by a group of “Life enthusiasts” who are fascinated by the many challenging open problems posed by Life. Some of these results are posted regularly to the websites [7,8]. The complex interactions among Life’s coherent structures make it difficult to classify the rule’s global dynamics, meaning the limiting dynamics of the rule on an infinite system starting from a random initial state. Various researchers have done empirical studies that resulted in different answers [5]; nevertheless, Life seems to be “between” rules whose global dynamics are periodic, meaning every site on the lattice is eventually periodic with probability 1 or aperiodic, meaning the sequence of 0’s and 1’s that occurs at each site of the lattice never cycles with probability 1. Its apparent location on a phase boundary led David Griffeath in the early 1990s to wonder whether Life might be a clue to a critical phase point in the threshold-range scaling limit. In order to explore this question, he proposed Larger than Life (LtL), which is a four-parameter family of two-dimensional cellular

automata that generalize Life to large neighborhoods and general birth and survival thresholds [9]. The LtL family of rules was also motivated by two mathematical prototypes for continuous time models of population growth: the Lotka–Volterra model, which is the simplest model of predator–prey interactions and the logistic map, which is a simple population growth model. LtL is the simplest CA analogue of these famous nonlinear models. That is, LtL is a spatial version of nonlinear population dynamics in which both space and time are discrete. The LtL family of rules includes Life as well as a rich set of two-dimensional rules, some of which exhibit dynamics vastly different from Life [10]. For example, the Threshold Growth models in which the update rules depend on a fixed threshold and occupied sets can only grow [5], comprise a subset of LtL parameter space. A family of “Life-like” rules comprises a much larger subset of LtL parameter space [10,11], than initially imagined by Griffeath. Like Life, these rules appear to “self-organize” over time and coherent structures emerge. Gravner and Griffeath [5] describe the behavior of this type of CA, which includes Life, as being, “so exotic that, at least for now, empirical study seems the only avenue to understanding”. In this paper, our focus is on the complex viable structures supported by these “exotic” CAs. The most intriguing such structures are called bugs and are generalizations of Life’s spaceships. Part of the fascination with LtL’s bugs comes from the fact that they are very common, they scale in a fairly coherent manner, and their geometry is distinct. Additionally, bugs are reminiscent of solitons, which are found in nonlinear partial differential equations that have dispersive and nonlinear terms which neutralize each other [12]. In the sections that follow, we present a mix of empirical and rigorous results about LtL’s coherent structures. We provide numerous examples of LtL’s bugs for rules with large neighborhoods as well as descriptions of the experimental methods used to find these objects. We describe how this empirical evidence leads to a section of conjectures that include: a family of LtL rules converge to “Euclidean automata”, which are deterministic rules whose universe is Euclidean space and that support complex viable

K.M. Evans / Physica D 183 (2003) 45–67

Euclidean structures. These Euclidean automata and their coherent structures are the continuum limits to the respective LtL examples described in this paper. Various things, including the nonlinearity of this set of rules, make these conjectures difficult to prove. However, we prove less complicated things, which include: a class of LtL rules with arbitrarily large ranges supports bugs and a one-dimensional “Euclidean automaton” supports bugs. We also present speculative results which suggest further experiments.

2. LtL: definition and notation The LtL rules are defined as follows. Each site of the two-dimensional lattice Z2 is in one of two states, live (1) or dead (0). This is the initial configuration of the system. The neighborhood Nρ of a site consists of the (2ρ+1)×(2ρ+1) sites in the box surrounding and including it. That is, the neighborhood of the origin is Nρ = {y ∈ Z2 : y∞ ≤ ρ} (ρ a natural number), so that its translate Nρx = x + Nρ is the neighborhood of site x ∈ Z2 . Nρ is called the generalized Moore or “range ρ” box neighborhood. Each time step, all the sites update (meaning change states or not) simultaneously according to the deterministic LtL rule, which in words is • Birth. A site that is dead at time t will become live at time t + 1 if and only if the number of live sites in its neighborhood at time t is in the closed interval [β1 , β2 ], 0 ≤ β1 ≤ β2 . • Survival. A site that is live at time t will remain live at time t + 1 if and only if the number of live sites in its neighborhood (itself included) at time t is in the closed interval [δ1 , δ2 ], 1 ≤ δ1 ≤ δ2 . • Death. A site that is dead at time t and does not become live at time t + 1 will remain dead at time t +1. A site that is live at time t and does not remain live at time t + 1 will become dead at time t + 1. Let us introduce the notation needed for the precise definition of the LtL update rule and the remainder of the paper. 2 Let T denote the CA rule. That is, T : {0, 1}Z → 2 {0, 1}Z .

47

Let ξt (x) ∈ {0, 1} denote the state of the site x ∈ Z2 at time t and let ξ t represent the state of all sites in Z2 at time t. As is customary we will often think of the CA as a set-valued process, confounding ξ t with {x : ξt (x) = 1}. For example, this allows us to use the notation ξtΛ = T t (Λ) = Λt to mean that starting from configuration ξ0 = Λ we arrive at the set Λt of occupied sites after t iterations of rule T. With this notation, a range ρ LtL update rule is  1 if ξt (x) = 0 and |Nρx ∩ ξt | ∈ [β1 , β2 ]      or ξt+1 (x) =  if ξt (x) = 1 and |Nρx ∩ ξt | ∈ [δ1 , δ2 ],     0 otherwise. For each fixed range ρ the LtL CA rules form a four-parameter family indexed by the endpoints β1 and β2 of the birth intervals and the endpoints δ1 and δ2 of the survival intervals. We denote each rule by the 5-tuple (ρ, β1 , β2 , δ1 , δ2 ). In this framework Life has LtL parameters (1, 3, 3, 3, 4). (Note that a live site counts itself and so the survival interval is [3,4] rather than the more standard [2,3].) The LtL CAs are totalistic because their update rules depend only on a site’s state and the number of its occupied neighbors, but not on the arrangement of those neighbors [13]. 2.1. Limiting neighborhoods: remembering Riemann The range ρ LtL neighborhood can be thought of as a grid that covers a region of Euclidean space. As the range size ρ increases, the grid’s mesh becomes a better approximation of a Euclidean neighborhood. For a clearer picture, consider the map f : Z2 → R2 , defined by f(x, y) = (x/ρ, y/ρ). This map scales Z2 by 1/ρ and takes the range ρ neighborhood Nρ into the 2×2 Euclidean square. As the range increases, the neighborhood scaling becomes an increasingly better approximation to the Euclidean square. For example, range 1 provides a glimpse of only 9 points of each Euclidean square while range 2 gives 25 points (Fig. 2). In general, range ρ gives (2ρ + 1)2 points. In Section 4.1.1 we will describe experimental results which suggest that as the range increases, the

48

K.M. Evans / Physica D 183 (2003) 45–67

Fig. 2. From left to right are f(N1 ) and f(N2 ) which are scalings of the range 1 and 2 neighborhoods, respectively, to the 2 × 2 Euclidean square.

boundaries of such coherent structures as LtL’s bugs smooth out.

3. Coherent structures: definitions and examples Life’s still lifes, oscillators, and spaceships are sets of cells that are periodic under the rule but that may be unstable when put on backgrounds other than all 0’s. Part of Life’s intrigue is due to the numerous coherent structures that emerge when the rule updates starting from a random initial configuration (Fig. 3). We will present a mix of experimental and rigorous results to show that these local structures generalize to LtL rules with arbitrarily large ranges. Let us define the specific configurations that we will generalize. These definitions arose through the study of the LtL family of rules, however, they apply to any two-state CA rules and for the most part conform to the Life terminology described in [4,6]. For the following, let T be a CA

Fig. 4. From left to right are still lifes supported by the LtL rules (8, 79, 106, 79, 137) and (10, 120, 160, 120, 209), respectively.

rule from the LtL family and let Λ ⊂ Z2 be a set of sites in state 1. The first two definitions are for still lifes and oscillators, which are configurations that are fixed and periodic, respectively, under the CA rule. These are less interesting than bugs, which are configurations that are periodic mod translation. They are also easier to generalize and rigorously analyze, which we will do in Section 4.3.3: • A still life is a configuration Λ which is a fixed point for T. That is, T (Λ) = Λ. • An oscillator or periodic object is a finite configuration Λ for which there exists a positive, finite integer n so that T n (Λ) = Λ. The smallest such n is called the period of Λ. Example 1. Fig. 4 depicts a still life supported by a range 8 LtL rule on the left and a range 10 still life on the right. Fig. 5 depicts the phases of a period 8 oscillator supported by a range 8 LtL rule. A bug is a finite configuration Λ for which there exists a finite time, τ, and a nonzero displacement

Fig. 3. From left to right are times 400 and 1200 of Life run on a 100 × 100 torus starting from a random initial configuration with density 1/10 of live sites. At time 400 there is at least one glider (Fig. 1) heading southeast. By time 1200 the configuration is periodic and only still lifes and oscillators remain.

K.M. Evans / Physica D 183 (2003) 45–67

49

Fig. 5. From left to right are the distinct phases of a period 8 oscillator supported by LtL rule (8, 79, 107, 79, 135).

Fig. 6. Orthogonal bug supported by LtL rule (8, 77, 123, 77, 129), τ = 11, d = (4, 0).

 The vector, d = (d1 , d2 ), such that T τ (Λ) = Λ + d. smallest such τ is a bug’s period, mod translation, in  the direction of d. LtL’s bugs are generalizations of Life’s famous spaceships. We consider bugs to be spaceships supported by rules whose neighborhoods are in ranges 2 and higher. We make this distinction because the geometry of the large range bugs is reminiscent of various insects. In some cases, the bugs are composed of line segments and appear to have legs. We will show in Section 4.3.1 that these bugs generalize to arbitrarily large ranges. Other bugs, like those in the following examples and in Section 4.1.1 are composed of connected regions of 1’s with holes in them that look like stomachs. Let us define such bugs more formally: • A bug with a stomach is a bug such that all phases satisfy: the live sites divide the dead sites into exactly two connected regions. (A region is connected if any two points can be joined by a nearest-neighbor path.) In the next section we will present empirical results which suggest that bugs with stomachs also generalize to arbitrarily large ranges. Bugs are characterized according to their trajectories and their periods. An orthogonal bug is a bug whose displacement vector d has exactly one component equal to 0. A diagonal bug is a bug whose displacement vector satisfies d1 = d2 or d1 = −d2 . A disoriented bug is a bug that is neither orthogonal nor diagonal. Most orthogonal, diagonal, and disoriented bugs have trajectories along a straight line. Some bugs

have trajectories that are more sinusoidal. Such a bug, Λ, is called a “jitter bug” if there are integers k and l such that T k (Λ) lies on one side of a line parallel to the bug’s displacement vector and T t (Λ) lies on the other side and neither set intersects the line. Example 2. The trajectories of a range 8 orthogonal bug, a range 15 diagonal bug, a range 10 disoriented bug, and a range 4 orthogonal jitter bug are depicted in Figs. 6–9, respectively. In each of the figures, the numbers below the phases of the bugs are the times at which they were “captured”. These capture times were chosen in order to present as many distinct and nonoverlapping phases of each bug as possible. We are interested in the limiting shape of bugs as the neighborhood size approaches infinity. However, as illustrated in the previous example, many of the bugs are very complicated in that they have large periods and in some cases move along a sine wave. A translation invariant bug is less complicated since it is a bug for which τ = 1, and thus invariant mod  In Section 4.1.1 we translation in the direction of d. will focus on such bugs, which it turns out are quite common. 4. Threshold-range scaling In [11] we described strategies for finding LtL rules which support bugs in two distinct regions of arbitrary range ρ LtL parameter space. In one of the regions the rule parameters are linear functions of the range

50

K.M. Evans / Physica D 183 (2003) 45–67

4.1. Experimental results

Fig. 7. Diagonal bug supported by LtL rule (15, 268, 429, 268, 467), τ = 2, d = (1, 1).

and in the other region the parameters are quadratic functions of the range. We gave empirical evidence which suggests that the distinct regions are actually connected. That is, the rules that lie between the linear and quadratic regimes also support bugs. Here we focus on the geometries of these bugs and how they scale as the range and rule parameters vary. In Section 4.3.1 we will show that the bugs supported by rules in the linear regime comprise sets which are discrete approximations to Euclidean line segments or curves and are relatively straightforward to generalize to arbitrarily large ranges. They scale linearly and their Euclidean limits are thus measured using arc length. In Section 4.1.1 we will give numerous examples to show that the more complex bugs supported by rules in the quadratic regime comprise sets which are discrete approximations to nonconvex Euclidean regions. These bugs scale quadratically and their Euclidean limits are thus measured using area.

4.1.1. Large range bugs with stomachs Experimental work suggests that there exist LtL rules with arbitrarily large ranges that support bugs. The general problem is quite complicated due to various factors which include four parameters for each rule, the nonlinearity of this class of rules, and the many phases of a bug. In this section we simplify the problem by restricting our attention to translation invariant bugs supported by LtL rules with β1 = δ1 . Let us begin with a sample of the varied translation invariant bug geometries supported by range 5 LtL rules in the quadratic regime of LtL parameter space. Example 3. Fig. 10 depicts a collection of translation invariant bugs supported by range 5 LtL rules. Each bug has τ = 1 and d = (0, 1) and each supporting rule satisfies β2 < δ2 . One translation invariant bug supported by an LtL rule with β1 = δ1 is presented for each β1 = δ1 from 24 to 42. In the examples that follow several of the range 5 translation invariant bugs depicted in Fig. 10 are scaled up to ranges 25, 50 and 100. Let us describe the algorithm used to do this threshold-range scaling and then we will present the examples. The algorithm assumes the existence of a bug supported by an LtL rule from a different range and has three main steps. The first is to determine the LtL

Fig. 8. Disoriented bug supported by LtL rule (10, 123, 170, 123, 212), τ = 26, d = (26, 3).

Fig. 9. Orthogonal jitter bug supported by LtL rule (4, 22, 31, 24, 38), τ = 40, d = (4, 0).

K.M. Evans / Physica D 183 (2003) 45–67

51

Fig. 10. Collection of range 5 translation invariant bugs. Below each is its supporting rule, (β1 = δ1 , β2 , δ2 ).

parameters for the desired bug’s supporting rule. The second is to construct an appropriate initial condition that the rule will “sculpt” into a bug and the third is to let the rule update. If a “good” initial configuration has been constructed, then after a relatively small number of time steps a bug of some period, which most likely is not 1, will emerge. Let us discuss how to determine the rule parameters and initial condition. In order to determine the rule parameters we use the following mapping that takes a given range ρ¯ rule to the desired range ρ LtL rule: (ρ, ¯ β1 , β2 , δ1 , δ2 )      

1 1 → ρ, β1 − γ , β2 + γ , 2 2   

 1 1 δ1 − γ , δ2 + γ , 2 2   2ρ + 1 2 . γ= 2ρ¯ + 1 The mapping scales the range ρ¯ LtL rule parameters by γ, which is quadratic in ρ. In order to construct the initial configuration, use a paint program to scale the given range ρ¯ bug into an initial configuration for a range ρ LtL rule by “resizing” the given bug by ρ/ρ. ¯ Our goal in this threshold-range scaling is to show the existence of bugs that are invariant mod translation. Attaining the translation invariance usually requires a fair amount of experimentation. This includes modify-

ing rule parameters and letting various rules “sculpt” the bug until one leaves it translation invariant. It is much easier, however, to construct bugs without the translation invariant constraint. In fact, in certain regions of parameter space the algorithm we have just described will immediately yield a bug with period larger than 1. For each of the following examples various characteristics for the bugs are included. These are |Λ|, which is the number of live sites the bug comprises; W × H, which is the size of the smallest rectangle that contains the bug; and w × h, which is the size of the smallest rectangle that contains the bug’s interior region of dead sites. Example 4. The bugs depicted in Fig. 11 were found by scaling up the bug that is supported by the LtL rule with β1 = δ1 = 28 and depicted in Fig. 10. The bugs’ characteristics are given in the following table: Fig. 11

|Λ| W ×H w×h

Range 25

50

100

1058 46 × 46 22 × 23

4019 87 × 88 41 × 43

15862 172 × 175 80 × 86

Example 5. The bugs depicted in Fig. 12 were also found by scaling up the bug that is supported by the

52

K.M. Evans / Physica D 183 (2003) 45–67

Fig. 11. Translation invariant bugs for increasing ranges. Left to right, their supporting LtL rules and displacement vectors are as follows: (25, 591, 805, 591, 915), d = (0, 3); (50, 2319, 3161, 2319, 3596), d = (0, 5); (100, 9185, 12 513, 9185, 14 239), d = (0, 10).

Fig. 12. Translation invariant bugs for increasing ranges. Left to right, their supporting LtL rules and displacement vectors are as follows: (25, 593, 790, 593, 938), d = (0, 4); (50, 2323, 3095, 2323, 3708), d = (0, 8); (100, 9198, 12 260, 9198, 14 704), d = (0, 16).

LtL rule with β1 = δ1 = 28 that is depicted in Fig. 10. However, in this example, the supporting rules’ δ2 parameters are larger than those from the previous example and the β2 parameters are smaller. The resulting effect on the bug’s geometry is that the smallest rectangle that contains it is larger, the number of live sites is also larger, and the flat pieces of the boundaries surrounding the dead sites on the bug’s interior are smaller. The bugs’ characteristics are given in the following table:

Fig. 12

|Λ| W ×H w×h

Range 25

50

100

1131 49 × 49 23 × 28

4621 101 × 100 47 × 59

18703 205 × 203 97 × 122

Example 6. The bugs depicted in Fig. 13 were found by scaling up the bug supported by the LtL rule with β1

K.M. Evans / Physica D 183 (2003) 45–67

53

Fig. 13. Translation invariant bugs for increasing ranges. Left to right, their supporting LtL rules and displacement vectors are as follows: (25, 697, 865, 697, 1243), d = (0, 4); (50, 2733, 3408, 2733, 4875), d = (0, 8); (100, 10 821, 13 500, 10 821, 19 295), d = (0, 16).

Fig. 14. Translation invariant bugs for increasing ranges. Left to right, their supporting LtL rules and displacement vectors are as follows: (25, 706, 958, 706, 1216), d = (0, 3); (50, 2773, 3755, 2773, 4738), d = (0, 5); (100, 10 983, 14 849, 10 983, 18 764), d = (0, 10).

= δ1 = 33 that is depicted in Fig. 10. These bugs are larger than those in the previous examples; they have more live sites and larger bounding rectangles. The bugs’ characteristics are given in the following table: Fig. 13

|Λ| W ×H w×h

Range 25

50

100

1428 52 × 52 18 × 30

5790 108 × 104 38 × 62

22696 212 × 207 74 × 123

Example 7. The bugs depicted in Fig. 14 were found by scaling up the bug supported by the LtL rule with β1 = δ1 = 34 that is depicted in Fig. 10. In this example, the supporting rules’ δ2 parameters are smaller than those from the previous example, while the β2 parameters are larger. The resulting effect on the bug’s geometry is that the smallest rectangle that contains it is smaller, the number of live sites is also smaller, and the flat pieces of the boundaries surrounding the dead sites on the bug’s interior are larger. The bugs’ characteristics are given in the

54

K.M. Evans / Physica D 183 (2003) 45–67

following table: Fig. 14

|Λ| W ×H w×h

Range 25

50

100

1317 47 × 48 17 × 21

5016 91 × 93 31 × 41

19854 180 × 185 62 × 80

The translation vectors for the bugs in Examples 4 and 7 are given by the formula d = (0, ρ/10 + 1/2(ρ bmod 2)). We conjecture that if these bugs are scaled up to even larger ranges the formula will still hold. That is, it will hold for any range ρ = 25 × 2k , where k is a whole number. Similarly, the translation vectors for the bugs in Examples 5 and 6 are given by the formula d = (0, 4ρ/25), which we conjecture holds for any range ρ = 25 × 2k , k is a whole number. The following table compares the bugs’ characteristics from Examples 4–7. That is, the relative sizes of the various characteristics with respect to the neighborhood sizes are given so that the values for different ranges may be compared. The following notation is used to express these values: (β1 , β2 , δ2 )ρ = (β1 /|Nρ |, β2 /|Nρ |, δ2 /|Nρ |); |β|ρ = |β2 − β1 |/|Nρ |; |δ|ρ = |δ2 − δ1 |/|Nρ |; |Λ|ρ = |Λ|/|Nρ |; Wρ = W/(2ρ+1); Hρ = H/(2ρ+1); wρ = w/(2ρ + 1); hρ = h/(2ρ + 1). The table values are rounded to the nearest hundred thousandth:

As shown in the table, as the range increases from 25 to 50 to 100, the relative sizes of the bugs in Figs. 11 and 14 decreases, while that of the bug in Fig. 12 increases and the relative size of the bug in Fig. 13 increases from range 25 to 50 and decreases from range 50 to 100. In each of the four examples, the relative bug size appears to be converging to a limit as do their bounding rectangles. We also see in the table that the rules supporting the bugs in Figs. 11 and 12 have nearly identical β1 = δ1 values and the relative sizes of the β2 values are slightly larger in Fig. 11 while the δ2 values are slightly smaller. The ratio wρ /Wρ , which represents the width of the smallest rectangle that contains the bug’s interior region of dead sites compared to the width of the smallest rectangle that contains the bug is larger for all the bugs in these figures as compared to those in Figs. 13 and 14. All the parameters for the supporting rules for Fig. 14 are larger than those for Fig. 11. However, the bugs in both figures have various similarities which  the boundinclude the same displacement vectors, d; aries of their stomachs contain large flat pieces; and the size of the smallest rectangle that contains each bug relative to the neighborhood size is smaller for all the bugs in these figures as compared to those in Figs. 12 and 13. All the parameters for the supporting rules for Fig. 13 are larger than those for Fig. 12. However, the bugs in both figures have various similarities

Range 25

50

100

Fig. 11 (β1 , β2 , δ2 )ρ |β|ρ ; |δ|ρ |Λ|ρ Wρ × Hρ w ρ × hρ

(0.22722, 0.30950, 0.35179) 0.08228, 0.12457 0.40677 0.90196 × 0.90196 0.43137 × 0.45098

(0.22733, 0.30987, 0.35251) 0.08254, 0.12518 0.39398 0.86139 × 0.87129 0.40594 × 0.42574

(0.22735, 0.30972, 0.35244) 0.08237, 0.12510 0.39261 0.85572 × 0.87065 0.39801 × 0.42786

Fig. 12 (β1 , β2 , δ2 )ρ |β|ρ ; |δ|ρ

(0.22799, 0.30373, 0.36063) 0.07574, 0.13264

(0.22772, 0.30340, 0.36349) 0.07568, 0.13577

(0.22767, 0.30346, 0.36395) 0.07579, 0.13628

K.M. Evans / Physica D 183 (2003) 45–67

55

(Continued ) Range 25

50

100

|Λ|ρ Wρ × H ρ w ρ × hρ

0.43483 0.96078 × 0.96078 0.45098 × 0.54902

0.45299 1 × 0.99010 0.46535 × 0.58416

0.46293 1.01990 × 1.00995 0.48259 × 0.60697

Fig. 13 (β1 , β2 , δ2 )ρ |β|ρ ; |δ|ρ |Λ|ρ Wρ × H ρ wρ × h ρ

(0.26797, 0.33256, 0.47789) 0.06459, 0.20992 0.54902 1.01961 × 1.01961 0.35294 × 0.58824

(0.26791, 0.33408, 0.47789) 0.06617, 0.20998 0.56759 1.06931 × 1.02970 0.37624 × 0.61386

(0.26784, 0.33415, 0.47759) 0.06631, 0.20975 0.56177 1.05473 × 1.02985 0.36816 × 0.61194

Fig. 14 (β1 , β2 , δ2 )ρ |β|ρ ; |δ|ρ |Λ|ρ Wρ × H ρ w ρ × hρ

(0.27143, 0.36832, 0.46751) 0.09689, 0.19608 0.50634 0.92157 × 0.94117 0.33333 × 0.41176

(0.27184, 0.36810, 0.46446) 0.09627, 0.19263 0.49172 0.90099 × 0.92079 0.30693 × 0.40594

(0.27185, 0.36754, 0.46444) 0.09569, 0.19259 0.49142 0.89552 × 0.92040 0.30846 × 0.39801

 their which include the same displacement vectors, d; stomachs are not convex, rather pieces of their boundaries are “ribbed” and become more intricate as the range increases; and the size of the smallest rectangle that contains each bug relative to the neighborhood size is larger than 1 for both of the range 100 examples in these figures. The next example illustrates how the geometry of the bugs in Fig. 14 changes as the range increases from 25 to 50. Example 8. The relative decrease in bug size as the range increases from 25 to 50 in Fig. 14 is illustrated in Fig. 15. The range 25 and 50 bugs from the figure are mapped to the 2 × 2 Euclidean square by f, which was defined in Section 2.1. The illustration shows that the outer boundary of the range 50 bug has the same general shape as that of the range 25 bug, but covers less area. This suggests that the bug geometry might be converging to a smaller shape and thus the area of the range 50 bug is an upper bound approximation for the area of the Euclidean bug to which it is converg-

ing. If the convergence is monotone then this is reminiscent to the convergence of the sum of the areas of circumscribed rectangles converging to the area under the curve as the width of the rectangles approaches zero. Here we imagine the grid mesh becoming finer as the range size increases. Along the same line of thinking, we might think of the bugs from Fig. 12 expanding to their limiting shape, just like the sum of the areas of a curve’s inscribed rectangles increases to the area under the curve as the width of the rectangles shrinks to zero. If the convergence is not monotone the story is more complex. The threshold-range scaling bug construction algorithm given earlier required the existence of a bug supported by a rule from a different range. There are other strategies for constructing initial configurations. One such strategy is to use a more general initial configuration. A configuration in which both the outer and inner boundaries enclose convex regions is a rough starting point. For example, a region of live sites bounded by a “circle” containing a region of dead sites that is also

56

K.M. Evans / Physica D 183 (2003) 45–67

first appears at time 24. The bug’s translation vector is d = (0, 16).

Fig. 15. Ranges 25 (gray) and 50 (black) scaled by 1/ρ and superimposed on the 2 × 2 Euclidean square.

bounded by a “circle” may be used as an initial configuration that the rule will “sculpt” into a bug. Appropriate radii and centers for the outer circle that bounds the live sites and the inner circle that bounds the dead sites must be chosen. Let us illustrate this strategy, assuming the LtL rule is given, in the next example. Example 9. In this example, the LtL rule (100, 9198, 12 260, 9198, 14 704) is given. The radius of the outer circle is 80 and its center is (0, 0), while the inner circle has radius 35 and center (0, −25). In Fig. 16 the sculpting is illustrated. By time 6 the shape is nearly identical to the eventual translation invariant bug which

In the previous example, the rule sculpted the initial configuration into a translation invariant bug, which in a sense was lucky since it seems to be more common for a bug with period larger than 1 to emerge. The strategy used in that example may be used to construct a bug with a stomach for a given rule as follows. The initial configuration is a circle of live sites centered at (0, 0) with radius Ro (3ρ/4 ≤ Ro ≤ 5ρ/4) and a circle of dead sites centered at (0, −a) (0 < a < Ro /2) with radius RI (ρ/4 ≤ RI ≤ 3ρ/4) removed from the larger circle. Let the given rule update on this initial configuration. If a bug is not “sculpted” by the rule, modify the values of the radii and center of the smaller circle of dead sites. 4.1.2. Rule parameters and bug geometry The experimental results presented in the previous section lead to various questions about a bug’s geometry. For example, why do some bugs have large flat boundary pieces while others are more curved or jagged? Which of the rule parameters determine these characteristics? In this section we begin to explore the

Fig. 16. From left to right, top row to bottom are the configurations that appear as the LtL rule (100, 9198, 12 260, 9198, 14 704) updates. The rule also supports the range 100 bug depicted in Fig. 12. The rule “sculpts” the initial configuration into a translation invariant bug. The number below each phase represents the time. The sculpting is complete at time 24.

K.M. Evans / Physica D 183 (2003) 45–67

effect that each of the four LtL rule parameters has on a bug’s geometry. To do this, we need the following definitions. Let Λ be a set of live sites. Let T1 (Λ) = {x : |Nρx ∩Λ| ≥ β1 }. In other words, site x will be live next time iff there are at least β1 live sites, including x, in its neighborhood this time. Thus, T1 is determined by β1 . Similarly, let T2 (Λ) = {x : |Nρx ∩ Λ| ≤ β2 }, so that site x will be live next time iff there are at most β2 live sites, including x, in its neighborhood this time. T2 is thus determined by β2 . The next two maps are just like the previous two, but determined instead by δ1 and δ2 , respectively. Let T3 (Λ) = {x : |Nρx ∩ Λ| ≥ δ1 } and T4 (Λ) = {x : |Nρx ∩ Λ| ≤ δ2 }. The rules T1 and T2 are monotone nondecreasing in live sites, while the rules T2 and T4 are monotone nonincreasing in live sites. That is, T1 and T3 map larger sets of live sites to larger sets of live sites. In other words, if A and B are sets of live sites and A ⊂ B then T1 (A) ⊆ T1 (B). Similarly for T3 . On the other hand, T2 and T4 map larger sets of live sites to smaller sets of live sites. In other words, if C and D are sets of live sites and C ⊂ D then T2 (C) ⊇ T2 (D). Similarly for T4 . Monotonicity makes these rules easier to rigorously analyze than the LtL rules of interest in this paper since the latter rules are not monotone. An LtL rule may, however, be defined in terms of the monotone rules as follows. Let T be an LtL rule and Λ a set of live sites. Then T(Λ) = (T1 (Λ) ∩ T2 (Λ) ∩ Λc ) ∪ (T3 (Λ) ∩ T4 (Λ) ∩ Λ). Now T2 (Λ) ∩ Λc is nonincreasing and T3 (Λ)∩Λ is nondecreasing so every LtL rule is formed by four monotone rules. To see how each of the four LtL rule parameters affects a bug’s geometry, we update a given bug under each of the four rules separately. The results are described in the following example. Example 10. Fig. 17 depicts a translation invariant bug along with its outline after one update of the rule. The numbers in the figure indicate the various regions determined by the boundaries of the bug and its outline. Each of the monotone rules was run on the bug, which we denote by Λ, and yielded the following:

57

Fig. 17. Range 100 translation invariant bug from Fig. 14 in black and the outline of its translate after one iteration of the rule. The numbers 1–7 represent the connected regions formed by the bug and its outline.

Birth: T1 (Λ) ∩ Λc = 2 ∪ 5 ∪ 6 and T2 (Λ) ∩ Λc = 1 ∪ 2 ∪ 6. Survival: T3 (Λ)∩Λ = 3∪4 and T4 (Λ)∩Λ = 3∪7. The above shows that under the LtL rule, the dead sites in region 1 do not become live next time because they see fewer than β1 live sites in their neighborhoods; however, the dead sites in region 2 see between β1 and β2 live sites so they do become live next time. Thus the boundary of the front end of the bug is determined by the parameter β1 . Similarly, the dead sites in region 5 do not become live next time because they see more than β2 live sites; however, the dead sites in region 6 do become live next time. Thus, the boundary of the back end of the bug’s stomach is determined by the parameter β2 . The live sites in region 4 do not remain live next time because they see more than δ2 live sites; however, the live sites in region 3 do remain live next time. Thus, δ2 determines the boundary of the front end of the bug’s stomach. Similarly the live sites in region 7 do not remain live next time because they see fewer than δ1 live sites and as we have already said the live sites in region 3 do remain live next time. Thus, the boundary of the back end of the bug is determined by the parameter δ1 . The previous example is representative of the other translation invariant bug examples described in this paper.

58

K.M. Evans / Physica D 183 (2003) 45–67

Fig. 18. Range 50 translation invariant bugs. Left to right, their supporting LtL rules and displacement vectors are as follows: (50, 2773, 3745, 2773, 4740), d = (0, 5); (50, 2773, 4621, 2773, 4740), d = (0, 1).

Let us give two more examples which illustrate how the geometry of a translation invariant bug is affected by a change in rule parameters. Example 11. The bugs depicted in Fig. 18 are supported by nearly identical range 50 LtL rules. The only difference is that it is easier to be born for the bug on the right. In other words, the β2 parameter on the right is larger than that on the left. The bug on the right is very slow, moving forward only one site every time step (compared to 5 for the bug on the left). The front and back ends of the bug on the right are nearly the same size, making it almost symmetrical. The previous example suggests that this is caused by the fact that β2 and δ2 are closer together for the bug on the right. Example 12. In Fig. 19 the rule supporting the bug on the right is different from that on the left in that it is easier to be born and more difficult to survive. In other words, β2 is larger and δ2 is smaller. The

Fig. 20. Bug supported by the LtL rule (100, 9350, 12 360, 10 350, 15 740). The reader is encouraged to determine the period of this bug which appears to have wings.

other parameters are identical to those for the bug on the left. The bug on the right is very slow, moving forward only one site every time step, compared to the seven sites moved by the bug on the left. As in the previous example, the bug on the right is almost symmetrical. As stated at the beginning of this section, we have simplified the bug generalization problem by focusing on bugs supported by LtL rules that satisfy β1 = δ1 . To emphasize that this does indeed simplify matters, let us end this section with Fig. 20, which is an example of a bug supported by an LtL rule with β1 = δ1 . 4.2. Questions and conjectures

Fig. 19. Range 50 translation invariant bugs. Left to right, their supporting LtL rules and displacement vectors are as follows: (50, 2773, 3755, 2773, 4859), d = (0, 7); (50, 2773, 4549, 2773, 4700), d = (0, 1).

The experimental results presented in Section 4.1.1 and in [11] suggest more formal results which we present here in the form of questions and conjectures. Where appropriate we elaborate on the intuition or empirical results that suggest them. In [10,11] we gave parameters for families of LtL rules that support bugs in ranges 2 and 5, respectively. In the previous section we described a strategy for scaling these bugs to larger ranges along with examples of the scaling to ranges 25, 50, and

K.M. Evans / Physica D 183 (2003) 45–67

100. This experimental work motivates the next two conjectures. Conjecture 1. There exists a constant C > 0 such that for any range ρ ≥ 2 at least Cρ8 LtL rules support “bugs with stomachs”. Conjecture 2. There exists a constant C > 0 such that for any range ρ ≥ 2 at least Cρ8 LtL rules support translation invariant bugs with stomachs. Rough estimates of the rule parameters in the previous two conjectures are as follows: 0.19 ≤ β1 /|Nρ | ≤ 0.34, 0.25 ≤ β2 /|Nρ | ≤ 0.57, 0.19 ≤ δ1 /|Nρ | ≤ 0.34, 0.27 ≤ δ2 /|Nρ | ≤ 0.63, |β2 − β1 | ≈ l1 |Nρ |, |δ2 − δ1 | ≈ l2 |Nρ |, where 0.05 < l1 < 0.21 and 0.07 < l2 < 0.28. Example 10, which is representative of the other translation invariant bug examples described in this paper, motivates the following conjecture. Conjecture 3. Suppose an LtL rule with β1 = δ1 supports a translation invariant bug with a stomach. Then the rule’s parameters must also satisfy β2 < δ2 . The intuition behind the previous conjecture is that the dead sites in region 5 of Fig. 17 see more than β2 live sites and thus do not become live next time. The live sites in region 4 see more than δ2 live sites and thus become dead next time. Since a bug’s front end has more live sites than its back end, the sites on the boundary between regions 4 and 5 see more live sites than the sites on the boundary between regions 5 and 6. Thus, in order for a bug to translate invariantly, its supporting rule must satisfy β2 < δ2 . The previous discussion along with the experimental results in Section 4.1.1 suggest that the dimensions, W × H, of the smallest rectangle that contains a range ρ translation invariant bug with a stomach are contained in a small interval surrounding 2ρ and the dimensions, w × h, of the smallest rectangle that contains the bug’s stomach are contained in a small interval surrounding ρ. If the outer and inner rectangles are not just the right sizes, the bug’s ability to be translation invariant will be destroyed.

59

That is, rather than moving forward invariantly, the configuration of live sites will either implode or explode. These observations suggest the following conjecture. Conjecture 4. The dimensions, W ×H, of the smallest rectangle that contains a range ρ translation invariant bug with a stomach satisfy (8/5)ρ ≤ W ≤ (12/5)ρ and (8/5)ρ ≤ H ≤ (12/5)ρ. The dimensions, w × h, of the smallest rectangle that contains the stomach of a translation invariant bug satisfy (13/20)ρ ≤ w ≤ (7/5)ρ and (13/20)ρ ≤ h ≤ (7/5)ρ. The next conjecture is about a “Euclidean automaton”, which has as its universe R2 ; the “number” of live sites in the neighborhood of x ∈ R2 is replaced by the area of the region of live sites contained in the neighborhood of x. More concretely, let TE denote 2 the Euclidean automaton rule. That is, TE : {0, 1}R → 2 {0, 1}R . The neighborhood of a site x ∈ R2 consists of all sites in the 2 × 2 Euclidean square which has x as its center. That is, the neighborhood of the origin is NE = {y ∈ R : y∞ ≤ 1} so that its translate NEx = x + NE is the neighborhood of site x ∈ R2 . Let A be a set of live sites. Then TE (A) = {x ∈ A : δ1 ≤ Area(NEx ∩A) ≤ δ2 }∪{x ∈ / A : β1 ≤ Area(NEx ∩A) ≤ β2 }. Conjecture 5. There exists a family of Euclidean automata that support Euclidean bugs. The Euclidean bugs are threshold-range scaling limits of the bugs in the previous conjectures as the range approaches infinity. 4.3. Rigorous results Proving the conjectures about the limiting shapes of bugs and the Euclidean automata that support them remains elusive. In this section we prove the following easier things: there exist bug geometries supported by rules in the linear regime of LtL parameter space that rescale to arbitrarily large ranges; the one-dimensional version of a bug with a stomach has a Euclidean limit and easily stated Euclidean rule; and there exist still lifes and oscillators supported by rules in the quadratic

60

K.M. Evans / Physica D 183 (2003) 45–67

regime of LtL parameter space that rescale to arbitrarily large ranges and that have Euclidean limits supported by Euclidean rules. 4.3.1. Bugs to infinity: exact scaling The LtL rules in the following propositions have β1 = β2 = δ1 = δ2 = θ, for some integer θ. In other words, site x will be live next time iff there are exactly θ live sites, including x, in its neighborhood this time. Such LtL rules are called exactly θ rules and denoted by (ρ, θ). In this section we focus on the subset of exactly θ rules known to support bugs. For other exactly θ results, see [9] (note, however, that in our definition a live site includes itself when counting while Griffeath’s definition excludes it) [10, Chapter 4.4], and [14]. In [11] we proved that LtL rules in ranges 2 and higher with exactly θ parameters (ρ, 2ρ) support period 2 diagonal bugs. We also stated the following proposition without proof.

that the live site between the two segments of 1’s sits at the origin. Then the set that Λ comprises is Λ = {(−1, 1), (−2, 1), . . . , (−ρ, 1), (0, 0), (1, −1), (1, −2), . . . , (1, −ρ)}. All the sites of Λ are in the neighborhood of the site (0, 0), which lies between the two segments. Since Λ consists of 2ρ+1 live sites (0, 0) dies of overcrowding. The two sites (−ρ, 1) and (1, −ρ) each have ρ + 1 live sites in their respective neighborhoods so they both die of loneliness. All the other live sites have 2ρ live sites in their respective neighborhoods and so die of overcrowding. Thus, after one time step, every site in Λ dies. All the sites in Λ + (1, 1) have ρ − 1 live sites in their neighborhoods and thus become live next time. No other site of the lattice has exactly ρ − 1 live sites in its neighborhood so we see that after one update the image of Λ under the CA rule T will comprise the set that is the translation of the above set by the vector (1, 1). That is, T(Λ) = Λ + (1, 1) as claimed. 䊐

Proposition 1. For any range ρ ≥ 3, the exactly θ LtL rule (ρ, 2ρ − 1) supports at least one bug.

Corollary 1. For any range ρ ≥ 3 there exists at least one bug for at least one LtL rule.

Let us now prove that such rules support translation invariant diagonal bugs.

The exactly θ rules support many bug varieties that scale in a straightforward way. Let us present just two more of these.

Proof. By definition of a translation invariant diagonal bug we must show that there exists a finite configuration, Λ of 1’s and a displacement vector, d =  Let Λ be the configu(n, n) such that T(Λ) = Λ + d. ration consisting of perpendicular segments of length ρ separated by one live site which lies on a diagonal between them (Fig. 21). The segments are mirror images of each other through the diagonal line that passes through the live site that lies between them. Then T(Λ) = Λ + (1, 1). To see this, orient Λ so

Fig. 21. Invariant bug supported by exactly θ LtL rule (ρ, 2ρ − 1).

Proposition 2. For any range ρ ≥ 2, the exactly θ LtL rule (ρ, 2ρ − 1) supports period 4 diagonal bugs. Proof. By definition of a bug we need to show that there exists a finite configuration, Λ of 1’s and a dis Let λ1 placement vector, d such that T 4 (Λ) = Λ + d. and λ2 be positive integers such that λ1 + λ2 = 2ρ + 1 and 4 ≤ λ1 ≤ λ2 . Let Λ be the configuration consisting of perpendicular segments, one of length λ1 and the other of length λ2 , separated by one site (Fig. 22). We leave it as an exercise for the reader to show that T 4 (Λ) = Λ + (4, 4). The checking can be done by hand or with the help of MCell or WinCA which can be downloaded from [15]. 䊐 Proposition 3. For range ρ = 2k, k = 1, 2, 3, . . . the exactly θ LtL rule (ρ, 2ρ) supports period 2 orthogonal bugs.

K.M. Evans / Physica D 183 (2003) 45–67

61

odic mod translation. The reader is encouraged to see for herself by trying out examples in various ranges. Range 7 would be a good starting point since it is especially intriguing, leading to infinite populations [10,14].

Fig. 22. Diagonal bug supported by the exactly θ LtL rule (ρ, 2ρ − 1), with τ = 4 and d = (4, 4).

Fig. 23. Period 2 orthogonal bug supported by the exactly θ LtL rule (ρ, 2ρ), d = (ρ + 1, 0).

Proof. In this case, Λ is the configuration depicted in Fig. 23 and d = (ρ + 1, 0). It is easy to check by hand (or using the computer) that, after one update T(Λ) is the configuration depicted in Fig. 24, which is displaced by (1, 0) from the rightmost live site of Λ. After one more update, the configuration will displace by the vector (ρ, 0) and will comprise a translate of Λ. That is, T 2 (Λ) = Λ + (ρ + 1, 0), as desired. 䊐 The configuration depicted in Fig. 23 is also supported by the exactly θ LtL rule (ρ, 2ρ), for ρ > 1 odd. However, in those cases the evolution is more complex, generating a variety of objects that are peri-

4.3.2. One-dimensional bugs with stomachs In this section we show that there exist one-dimensional translation invariant “bugs with stomachs” that converge to Euclidean bugs with stomachs, which are composed of disconnected line segments. Each rule described in this section is one-dimensional, meaning the universe is the set of integers, and the neighborhood is an interval. Since the rules are one-dimensional versions of LtL rules, the range ρ neighborhood of a site consists of all sites in an interval of length 2ρ + 1 centered on the given site. That is, the neighborhood of the origin is Nρ = {y ∈ Z : |y| ≤ ρ} (ρ a natural number), so that its translate Nρx = x + Nρ is the neighborhood of site x ∈ Z. Such rules will be referred to as range ρ one-dimensional LtL rules and denoted as usual by the range, birth, and survival parameters (ρ, β1 , β2 , δ1 , δ2 ). The first question is: what is the shape of a one-dimensional bug? It must consist of at least two segments of live sites separated by a segment of dead sites. This is due to the symmetry of a single segment and the rule. For the same reason, if there are exactly two segments, then their lengths must not be equal. In fact, the front segment must be longer than the back segment and the segment of dead sites between them must be long enough, but not too long. Let us give an example of a translation invariant bug supported by a range 6 one-dimensional LtL rule. Example 13. Let Λ be a set of live sites comprising the set {x ∈ Z − {−1, −2, −3} : |x| ≤ 6} (Fig. 25). Then Λ is a translation invariant bug with d = 1 for the one-dimensional LtL rule with parameters (6, 6, 7, 5, 9).

Fig. 24. Intermediate phase of orthogonal bug depicted in Fig. 23.

Example 14. The set of live sites depicted in Fig. 25 also represents a translation invariant bug with d = 2 for the one-dimensional LtL rule with parameters (6, 5, 8, 6, 8), and a translation invariant bug with d =

62

K.M. Evans / Physica D 183 (2003) 45–67

Fig. 25. One-dimensional bug. The number inside each box represents the state of the site. That is, 1 denotes a live site and 0 a dead site. The number below each box represents the number of 1’s in the site’s range 6 neighborhood.

3 for the one-dimensional LtL rule with parameters (6, 4, 9, 7, 7). The next proposition shows that Example 13 generalizes to any range ρ ≥ 2 one-dimensional LtL rule. Proposition 4. For any range ρ ≥ 2, the one-dimensional LtL rule with parameters (ρ, ρ, ρ + 1, ρ/2 + 2, ρ + ρ/2) supports at least one translation invariant bug. Proof. Let Λ be a set of live sites comprising the set {x ∈ Z − {−1, −2, . . . , −ρ/2} : |x| ≤ ρ} if ρ is even and the set {x ∈ Z−{0, −1, −2, . . . , −ρ/2} : |x| ≤ ρ} ∪ {ρ + 1} if ρ is odd. That is, from left to right, Λ consists of a segment of ρ/2 live sites, a “hole” of ρ/2 dead sites, and another segment of ρ + 1 live sites. Then, T (Λ) = Λ+1. This is clear from Fig. 26, which depicts Λ along with the number of live sites in the neighborhood of each x in an interval surrounding Λ. 䊐 The next proposition shows that the range 6 bug with d = 3 given in Example 14 also generalizes to any range ρ ≥ 2 one-dimensional LtL rule. Proposition 5. For any range ρ ≥ 2, the one-dimensional LtL rule with parameters (ρ, ρ −ρ/2+1, ρ + ρ/2−ρ(mod 2), ρ+(ρ+1)(mod 2), ρ+1) supports at least one translation invariant bug.

Proof. Let Λ be the set of live sites given in the proof of Proposition 4. That is, from left to right, Λ consists of a segment of ρ/2 live sites, a “hole” of ρ/2 dead sites, and another segment of ρ + 1 live sites. To see what happens after one time step, consider Fig. 26, which depicts Λ along with the number of live sites in the neighborhood of each x in an interval surrounding Λ. The rightmost ρ/2 + 1 live sites all have ρ + 1 live sites in their neighborhoods and thus will remain live next time. The neighborhoods of the remaining ρ/2 live sites in that segment contain ≥ρ + 2 live sites; thus they will all become dead next time. Exactly ρ/2 of the dead sites just to the right of the rightmost segment of live sites will become live next time since the number of live sites in their neighborhoods is in [ρ − ρ/2 + 1, ρ + ρ/2 − ρ(mod 2)]. The remaining changes depend on whether ρ is even or odd. If ρ is even, then all the ρ/2 dead sites between the two segments of live sites will become live next time since the number of live sites in their respective neighborhoods is in [ρ − ρ/2 + 1, ρ + ρ/2]. All live sites in the leftmost segment will become dead next time since all have <ρ + 1 live sites in their neighborhoods. No other changes will occur. Thus, T (Λ) = Λ + ρ/2. On the other hand, if ρ is odd, then all but the rightmost site of the ρ/2 dead sites between the two segments of live sites will become live next time since they all have at least ρ + 1 live sites and at most ρ + ρ/2 − 1 live sites in their neighborhoods. All but the rightmost live site in the

Fig. 26. Range ρ one-dimensional bug. The number inside each box represents the state of the site. That is, 1 denotes a live site and 0 a dead site. The value below each box represents the number of 1’s in the site’s range ρ neighborhood, where r = ρ, a = ρ/2 and b = ρ/2.

K.M. Evans / Physica D 183 (2003) 45–67

leftmost segment of live sites will become dead next time since they all have <ρ−1 live sites in their neighborhoods. No other changes will occur. Thus, T (Λ) = Λ + ρ/2. 䊐 Propositions 4 and 5 give the extreme cases: when the translation vector d is as small as possible (d = 1) and as large as possible (d = ρ/2). For ranges ρ ≤ 5 these are the only possibilities. However, as the range increases there are certainly cases in between for which the configuration Λ (described in the proofs of the propositions) is also a translation invariant bug. One such range 6 example was presented in Example 14. The reader is encouraged to generalize that one and find others. A slight variation of the one-dimensional bug in Proposition 4 is given in the next proposition. Proposition 6. For any range ρ ≥ 3, the one-dimensional LtL rule with parameters (ρ, ρ, ρ + 1, ρ/2 + 2, ρ + ρ/2 − 1) supports at least one translation invariant bug. Proof. Let Λ be a set of live sites comprising the set {x ∈ Z−({−1, −2, . . . , −ρ/2}∪{ρ}) : |x| ≤ ρ} if ρ is even and the set {x ∈ Z − {0, −1, −2, . . . , −ρ/2} : |x| ≤ ρ} if ρ is odd. That is, from left to right, Λ consists of a segment of ρ/2 live sites, a “hole” of ρ/2 dead sites, and another segment of ρ live sites. Looking at Fig. 27 makes it clear that T (Λ) = Λ + 1. 䊐 Proposition 7. For any range ρ ≥ 3, the one-dimensional LtL rule with parameters (ρ, ρ −ρ/2+2, ρ + ρ/2 − 1, ρ, ρ + 1 − ρ(mod 2)) supports at least one translation invariant bug.

63

Proof. Let Λ be the set of live sites given in the proof of Proposition 6 and depicted in Fig. 27. That is, from left to right, Λ consists of a segment of ρ/2 live sites, a “hole” of ρ/2 dead sites, and another segment of ρ live sites. We leave it as an exercise for the reader to show that T (Λ) = Λ + ρ/2 − 1. 䊐 As before, the previous two propositions give the extreme cases: when the translation vector dis as small as possible (d = 1) and as large as possible (d = ρ/2 − 1). For ranges ρ ≤ 6 these are the only possibilities. However, as the range increases there are certainly cases in between for which the configuration Λ (described in the proofs of the propositions) is also a translation invariant bug. The reader is encouraged to find them. If the previous range ρ one-dimensional LtL rules and their respective Λ configurations are scaled by 1/ρ and the limits as ρ → ∞ are taken, the result is a “Euclidean automaton”, which has as its universe the real line and the “number” of live sites in the neighborhood of x ∈ R; is the sum of the lengths of the intervals of live sites contained in the neighborhood of x. Example 15. Let Λ be a set of live sites comprising the set {x ∈ R −(3/2, 2) : |x−2| ≤ 1} (Fig. 28). Then Λ is a translation invariant bug with d = ε, 0 < ε < 1/2 for the Euclidean automaton with birth interval [1 − ε, 1 + ε], and survival interval [1/2 + ε, 3/2 − ε]. For this automaton, the neighborhood of the origin is N = {x ∈ R : |x| ≤ 1} and its translate N x = x + N is the neighborhood of site x ∈ R. Observe that the above gives the extreme cases: when the translation vector is as small and as large as possible, as well as all the intermediate cases (d = ε,

Fig. 27. Range ρ one-dimensional bug. The number inside each box represents the state of the site. That is, 1 denotes a live site and 0 a dead site. The value below each box represents the number of 1’s in the site’s range ρ neighborhood, where r = ρ, a = ρ/2 and b = ρ/2.

64

K.M. Evans / Physica D 183 (2003) 45–67

Fig. 28. One-dimensional Euclidean bug. The line segments in the upper portion of the figure represent live sites. The graph of the function depicted below represents the length of 1’s in the neighborhood of x for each x ∈ [0, 4].

0 < ε < 1/2). Thus, the previous example is the threshold-range scaling limit of the bugs and supporting rules given in the propositions as well as the intermediate cases that the reader may have discovered. The strategy used above to determine the limiting shape of a translation invariant bug and its supporting rule as the range goes to infinity is very straightforward for the one-dimensional case. However, the two-dimensional case is more complicated for a variety of reasons. For example, the geometry of two-dimensional bugs is vastly more complicated and the graph that represents the “number” of live sites in each site’s Euclidean neighborhood requires three dimensions. Additionally, this “number” is actually an area, which requires the computation of various integrals. Nevertheless, creating the three-dimensional graphs analogous to Fig. 25 for some of the known LtL bug examples may shed light on the more complex Euclidean case. 4.3.3. Blocks and blinkers to infinity Unlike the bugs generalized in Section 4.3.1 and supported by LtL rules in the linear regime of LtL parameter space, the large range bug examples presented in Section 4.1.1 suggest that the local structures supported by rules in the quadratic regime of LtL parameter space scale quadratically. That is, the length, width, and number of live sites that such an object comprises scale quadratically as the range size increases. Let us illustrate this scaling by generalizing a very simple still life and a very simple oscillator to arbitrarily large ranges. First recall that Life’s block is a 2 × 2 square of live sites, which remains fixed under the rule (i.e. it is

a still life or a fixed point). Life’s blinker is a 1 × 3 rectangle of live sites, which oscillates between this and its other phase, a 3 × 1 rectangle of live sites (i.e. it is a period 2 oscillator). Before stating the next proposition we need the following Block Property. Block Property. δ1 ≤ (ρ + 1)2 ≤ δ2 and either β1 > ρ(ρ + 1) or n ∈ [β1 , β2 ] only if n < ρ(ρ + 1) and n is not an integer of the form ab where a and b are integers in the interval [0, ρ + 1]. Proposition 8. For any range ρ ≥ 1 every LtL rule that satisfies the Block Property supports a still life that generalizes Life’s famous block. We call this still life a range ρ block. Proof. By definition of a still life, we must show that there exists a finite configuration, Λ of 1’s such that T (Λ) = Λ. Let Λ be a set of live sites comprising a (ρ + 1) × (ρ + 1) box and suppose it is placed on a background of all 0’s at time 0. That is, ξ0 = Λ. y For all y ∈ / Λ, ξ0 (y) = 0 and |Nρ ∩ ξ0 | ≤ ρ(ρ + y 1). Also the intersection of Nρ with Λ comprises a y rectangle. Thus, |Nρ ∩ ξ0 | = ab where a and b are natural numbers ≤ρ + 1. If the LtL rule satisfies the block property it follows that such sites y will remain dead next time. That is, ξ1 (y) = 0. Therefore no sites that are dead at time 0 will become live at time 1. For all x ∈ Λ, ξ0 (x) = 1 and |Nρx ∩ ξ0 | = (ρ + 1)2 . Since δ1 ≤ (ρ + 1)2 ≤ δ2 it follows that ξ1 (x) = 1. Thus, all sites that are live at time 0 will remain live at time 1. Therefore T (Λ) = Λ so the range ρ block is a still life for the rule as desired. 䊐 Let us illustrate the range 5 block property. A range 5 block will be supported by an LtL rule for which δ1 ≤ 36 ≤ δ2 and either β1 > 30 or n ∈ [β1 , β2 ] only if n ∈ {7, 11, 13, 14, 17, 19, 21, 22, 23, 26, 27, 28, 29}. The next example is a generalization of the above to a “Euclidean automaton”, which has as its universe R2 and the “number” of live sites in the neighborhood of x ∈ R2 is the area of the region of live sites contained in the neighborhood of x.

K.M. Evans / Physica D 183 (2003) 45–67

Example 16. Let Λ be a set of live sites comprising the set {x ∈ R2 : ||x||∞ ≤ 1/2}. Then Λ is a still life supported by a Euclidean automaton with the following birth and survival conditions: β1 > 1 and 1 ∈ [δ1 , δ2 ]. For this automaton, the neighborhood of the origin is N = {x ∈ R2 : ||x||∞ ≤ 1} and its translate N x = x + N is the neighborhood of site x ∈ R2 . Proposition 9. For any range ρ ≥ 1 every LtL rule with parameters that satisfy ρ(ρ + 1) < β1 ≤ ρ(ρ + 2) ≤ β2 and ρ(ρ + 1) < δ1 ≤ ρ(ρ + 2) ≤ δ2 supports a period 2 oscillator that generalizes Life’s famous blinker. We call this still life a range ρ blinker since, in general, a blinker is an oscillator with period 2. Proof. By definition of a period 2 oscillator, we must show that there exists a finite configuration, Λ of 1’s such that T 2 (Λ) = Λ. Let Λ be a set of live sites comprising a ρ × (ρ + 2) rectangle (Fig. 29) and let ξ0 = Λ. y y For all y∈Λ, / either |Nρ ∩ ξ0 | ≤ ρ(ρ + 1) or |Nρ ∩ ξ0 | = ρ(ρ + 2). In the former case, ξ1 (y) = 0 since β1 > ρ(ρ + 1). The only sites in the latter case are the ρ sites adjacent to the left side of the rectangle and the ρ sites adjacent to the right side of the rectangle, not including the top and bottom site on each side, all of which will become 1’s next time since ρ(ρ + 2) ∈ [β1 , β2 ]. For all x∈), either |Nρx ∩ ξ0 | = ρ(ρ + 2) or |Nρx ∩ ξ0 | = ρ(ρ + 1). In the former case, ξ1 (x) = 1 since ρ(ρ + 2) ∈ [δ1 , δ2 ]. The only sites in the latter case are the ρ sites in the lower row of the rectangle and the ρ sites in the upper row of the rectangle, all of which will become 0’s next time since δ1 > ρ(ρ + 1). Therefore at time 1 the configuration of live sites will

Fig. 29. Range ρ blinker.

65

be the initial configuration rotated 90◦ as illustrated in Fig. 29. This dynamics will repeat so that T 2 (Λ) = Λ. Thus the range ρ blinker is a period 2 oscillator for the rule as desired. 䊐

5. Further speculation: questions and experimental results 5.1. Limiting shapes and neighborhood geometry In this paper we have restricted our attention to box neighborhoods. The question thus arises: are box neighborhoods the only shapes that give rise to bugs or more specifically to bugs with stomachs?. As in Section 4.1.1 we have found bugs with stomachs that are supported by range 5 “ball” neighborhoods and scaled these up to ranges 25, 50 and 100. The next example illustrates one such scaling. Example 17. The bugs depicted in Fig. 30 were found by scaling up a translation invariant bug supported by a range 5 LtL rule with a ball neighborhood. The scaling was done using the algorithm described in Section 4.1.1. The previous example is representative of other such scalings. These preliminary results lead to the following natural conjecture.

Fig. 30. From left to right, bugs supported by LtL rules with “ball” neighborhoods and respective parameters (25, 571, 790, 571, 1018), (50, 2228, 3088, 2229, 3970), and (100, 8826, 12 220, 8826, 15 740). The range 25 and 100 bugs have period 2 and the range 50 example has period 4.

66

K.M. Evans / Physica D 183 (2003) 45–67

Conjecture 6. There exists a constant C > 0 and a polynomial f(ρ) with positive coefficients and degree larger than 2 such that for any range ρ ≥ 2 at least Cf(ρ) LtL rules with range ρ ball neighborhoods support bugs with stomachs. In fact, such rules support bugs that move in every rational direction. We suspect that bugs with stomachs are not unique to box and ball neighborhoods. In fact, we believe that bugs with stomachs are supported by numerous rules with convex neighborhoods. This leads to the following conjecture. Conjecture 7. A nonempty subset of LtL rules with convex neighborhoods support bugs with stomachs.

6. Conclusion and future work We have shown that LtL’s coherent structures generalize to large ranges and suggest a “natural” limiting automaton which has as its universe Euclidean space and update rule defined in terms of area. We showed that the one-dimensional version of this problem, where the update rule is defined in terms of length, has such a limit. Though the two-dimensional problem is vastly more complex, the one-dimensional techniques may be extended to two dimensions and perhaps shed light on the Euclidean shapes we seek. Various questions remain about the threshold-range scaling limits of LtL’s coherent structures. As discussed earlier, LtL was motivated, in part, by systems of differential equations that are continuous time nonlinear population models. The question remains whether the Euclidean automata we seek—the threshold-range scaling limits of LtL rules—can be represented by a system of, say, integral differential equations or delay difference equations that approximate LtL rules. Perhaps the coherent structures—the Euclidean limits of LtL’s bugs—are solutions to such a system of equations. That they are exact solutions is perhaps wishful thinking, but they may well be numerical solutions. If this is the case, it is also possible

that studying such a system of equations will shed light on questions such as exactly which Euclidean rules support, for instance, bugs with stomachs. That LtL’s coherent structures appear to have continuum limits poses the question of whether coherent structures of other families of CAs or similar systems have scaling limits. As suggested by our work, for a family with a birth and survival rule much like LtL’s, but with a different convex neighborhood, the answer most likely is yes. We speculate that other kinds of rules and neighborhoods may also yield coherent structures with continuum limits. However, more work on large range CAs is required. Other known CAs that generalize to Euclidean automata include the special case of Majority Vote (which is an LtL rule in a different region of parameter space than the rules discussed in this paper) [9]. In Wolfram’s recent book, A New Kind of Science, he argues that studying simple programs such as cellular automata will shed light on the basic mechanisms that drive such natural phenomena as apparent randomness in a system, the growth of crystals, the breaking of materials, fluid flow, and various problems from biology and fundamental physics [16]. In Gray’s review of A New Kind of Science, he writes: “. . . most mathematical models in science are based on the assumption that time and space are continuous, whereas Wolfram says that time and space are discrete; Wolfram would have us abandon models based on calculus and Euclidean geometry in favor of discrete systems like CAs (ANKS, p. 8). Indeed Wolfram sees the entire universe as a CA-like system that likely follows a simple dynamical rule, and the better part of Chapter 9 consists of some clever speculation on the exact nature of such a rule” [17]. While such questions regarding the nature of the universe remain open, perhaps thinking that there is a simple answer to whether time and space are continuous or discrete is too limiting. Perhaps there is an interplay between continuity and discreteness. Our work explores a bit this interplay in terms of space: perhaps such an interplay also occurs in nature. That is, some natural phenomena may well be limiting “rules”, or Euclidean versions of processes that update much like CAs.

K.M. Evans / Physica D 183 (2003) 45–67

7. Technology All the experimental work described in this paper was done using Bob Fisch and David Griffeath’s WinCA, which is a CA modeling environment for the PC and available as freeware from [15]. WinCA’s editing capabilities are modest (interaction with a paint program is necessary for creating detailed initial states), however, it is excellent for running LtL rules with large ranges on large lattices.

Acknowledgements Thanks to Jarrko Kari for taking the Euclidean bug problem down a dimension, Janko Gravner for helpful (and hilarious) suggestions, Dean Hickerson for carefully editing a draft, Cris Moore for suggestions, and Matt Evans (owner of streetink.com) for assistance with various graphics programs. In memory of Christian Patrick Elie, 4 August 1963–11 July 2002, whose existence, uniqueness, and many years of encouragement will never be forgotten. References [1] J. von Neumann, in: A. Burks (Ed.), Theory of Selfreproducing Automata, University of Illinois Press, Champaign, 1966. [2] S. Ulam, Statistical mechanics of cellular automata, 1952, in: Proceedings of the International Congress on Mathematics, vol. 2, 1950, pp. 264–275.

67

[3] M. Gardner, Mathematical games—the fantastic combinations of John Conway’s new solitaire game, Life, Sci. Am. 223 (1970) 120–123. [4] E. Berlekamp, J. Conway, R. Guy. What is Life? in: Winning Ways for Your Mathematical Plays, vol. 2, Academic Press, New York, 1982, Chapter 25. [5] J. Gravner, D. Griffeath, Cellular automaton growth on Z2 : theorems, examples, and problems, Adv. Appl. Math. 21 (1998) 241–304. http://psoup.math.wisc.edu/extras/ r1shapes/r1shapes.html. [6] S. Silver, Life Lexicon Home Page. http://www.argentum. freeserve.co.uk/lex home.htm. [7] J. Summers, Game of Life status page, 2000. http://home. mieweb.com/jason/life/status.html. [8] D. Eppstein, Which Life-like systems have gliders? 2002. http://www.ics.uci.edu/∼eppstein/ca/. [9] D. Griffeath, Self-organization of random cellular automata: four snapshots, in: G. Grimmett (Ed.), Probability and Phase Transitions, Kluwer Academic Publishers, Dordrecht, 1994. [10] K. Evans, Larger than life: it’s so nonlinear, Ph.D. Dissertation, University of Wisconsin-Madison, 1996. http://www. csun.edu/∼kme52026/thesis.html. [11] K. Evans, Larger than Life: digital creatures in a family of two-dimensional cellular automata, in: Discrete Mathematics and Theoretical Computer Science, vol. AA, 2001, pp. 177– 192. http://dmtcs.loria.fr/proceedings/html/dmAA0113.abs. html. [12] S. Vongehr, Solitons. http://physics.usc.edu/∼vongehr/solitons html/solitons.html. [13] N. Packard, S. Wolfram, Two-dimensional cellular automata, J. Statist. Phys. 38 (1985) 901–946. [14] K. Evans, Replicators and Larger than Life examples, in: D. Griffeath, C. Moore (Eds.), New Constructions in Cellular Automata, Oxford University Press, Oxford, 2003. [15] D. Griffeath, Primordial Soup Kitchen. http://psoup.math. wisc.edu/kitchen.html. [16] S. Wolfram, A New Kind of Science, Wolfram Media, Inc., 2002. [17] L. Gray, A mathematician looks at Wolfram’s new kind of science, Notices Am. Math. Soc. 50 (2) (2003) 200–211.