generating isotropic discrete waves on cellular automata · 2. cellular automata let us now...

October 25, 2001 9:23 WSPC/115-IJPRAI 00130

International Journal of Pattern Recognition and Artificial IntelligenceVol. 15, No. 7 (2001) 1007–1021c© World Scientific Publishing Company

GENERATING ISOTROPIC DISCRETE WAVES

ON CELLULAR AUTOMATA

FABIEN FESCHET

LASS Laboratory, UMR 5823 – MA2D – Universite Lyon 1, Bat 101,43 Bd du 11 Novembre 1918 – 69622 Villeurbanne Cedex, France

[email protected]

LAURE TOUGNE

E.R.I.C. Laboratory, Bat L – Universite Lyon 2, 5 av. Pierre Mendes-France,69676 Bron Cedex, France

[email protected]

Cellular automata are a massively parallel computation model with discrete time andlocal rules. They are well adapted to biological or physical simulations. However, theyare intrinsically anisotropic. The possibility of computing isotropic figures on cellularautomata such as circles has already been proved.4 Moreover, the previous constructionenables to compute all the major discretizations known in the literature. We present inthis article an extension of this work to the construction of spheres in three dimensions.A local characterization of a sphere is presented based upon the relationship betweenspheres and circles. This leads to the possibility of constructing a family of concen-tric discrete spheres in real time. Moreover, the approach can use many discretizationschemes leading to the construction of various discrete spheres as done for circles.

Keywords: Discrete sphere; 3D cellular automata; discrete circle; discrete waves.

1. Introduction

A cellular automaton is a regular homogeneous network of identical simple machines

which locally communicate with each other. The subjacent network is the graph

Zn so that cellular automata are often viewed as a discrete massively parallel com-

putation model. All machines evolve from one state to another at the same top

following a discrete time. The dynamic of cellular automata is very sensitive to the

initial conditions. This last property leads physicians or biologists to use cellular

automata to model real phenomena with the goal of predicting evolution. However,

the physical universe is usually seen as a three-dimensional continuous and isotropic

space through the classical Euclidean metric. By using cellular automata, a discrete

universe is considered. This is able to have several behaviors but the isotropy is not

natural since directions are privileged. Many physical phenomena are anisotropic,

but some of them, such as waves propagation, are isotropic which means that their

space extension is not direction-depending.

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1008 F. Feschet & L. Tougne

By constructing isotropic figures with a cellular automaton, we refer to the

construction of an approximation of an isotropic object. In particular in Refs. 4

and 7, it has been proved that we can construct discrete circles in real time by

cellular automata. Starting from an initial configuration in which all cells of the

discrete plane are in a same state except one (the center of the concentric circles),

the cellular automaton highlights at time t the cells that belong to a digitization

of the circle of radius t. In fact, it is possible to construct the families of all the

well-known good digitizations of circles found in the literature.

The goal of this article is to extend the previous works and more precisely to

present how we can generate discrete isotropic waves in dimension two (circles) and

in dimension three (spheres) with the help of cellular automata. In fact, whatever

the dimension is, the principle remains the same: we have to find a local charac-

terization of the cells that belong to a wave at a given time. Then each cell, in the

neighborhood of a cell belonging to the discrete wave at time r, enters the state

wave at time r+ 1 according to the states of all its neighbors at time r and the set

of transition rules.

The following section deals with definitions concerning cellular automata. In

Sec. 3, we explain the strategy used for the construction. This leads us to the

construction of figures, called paraboloids, in Sec. 4, from which we can construct

waves in Sec. 5. Finally we conclude and present some future works.

2. Cellular Automata

Let us now remember some classical definitions concerning the cellular automata.

We first recall the definition itself of a cellular automaton and the notion of con-

figuration. Then we introduce the notion of signal used to describe the behavior of

a cellular automaton. Finally, we define the notions of construction of figures and

construction in real time.

2.1. Standard definitions

Definition 1. An n-dimensional cellular automaton (or n-CA), A, is a 4-uplet

(n, S,B, δ) such that:

• n is the dimension of the subjacent grid,

• S is a set of elements of which are the states of A,

• B is the neighborhood of A and is a finite subset of Zn, with cardinal |B|,• δ is a function from S|B| to S, called the local transition function of A.

At each point of the grid Zn is attached the same finite automaton, and such

a decorated point is called a cell. Each cell locally communicates with a finite

number of neighbors. The neighborhood is fixed and geometrically uniform. In this

paper we use the 8-neighborhood in dimension two and the 26-neighborhood in

dimension three. The neighbors of a point x = (x1, . . . , xn) ∈ Zn are the points

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Generating Isotropic Discrete Waves on Cellular Automata 1009

(x1 +x1,1, . . . , xn+xn,1), (x2 +x1,2, . . . , xn+xn,2), . . . , (x1 +x1,|B|, . . . , xn+xn,|B|)

where B = {(x1,1, . . . , xn,1), (x1,2, . . . , xn,2), . . . , (x1,|B|, . . . , xn,|B|)}.The local communications, which are deterministic and uniform, take place

synchronously according to discrete times.

Definition 2 (Configuration). A configuration CA of the cellular automaton Ais an application from Zn to S.

Definition 3 (Evolution). For all time t, the configuration CtA evolves at time

t+ 1 into the configuration Ct+1A defined by: x = (x1, . . . , xn) ∈ Zn and Ct+1

A (x) =

δ(CtA(x1 + x1,1, . . . , xn + xn,1), . . . , CtA(x1 + x1,|B|, . . . , xn + xn,|B|)).

The function which associates the configuration Ct+1A to the configuration CtA

is called the global function of A.

A state q such that δ(q, . . . , q) = q is called a quiescent state. In the following,

we denote by I0 the initial configuration such that all the cells of the grid are in a

quiescent state except one cell, the center of the waves.

2.2. Signals

The natural way to describe the behavior of a cellular automaton is to exhibit all

the possible configurations of the cells during the evolution. When there are too

many possible configurations, it is better to conceive the succession of configurations

of this automaton, introducing its dynamic nature. To do this, we introduce the

notion of a signal.

Definition 4. At a given time t, a signal St on an n-dimensional cellular automa-

ton is a connected subset of cells {(x1(t), . . . , xn(t))/t ∈ N, t ≥ t0} that are in a

given subset of states of S.

In fact, the notion we use to describe the behavior of the cellular automata is not

exactly the signal but the “trace” of the signal along time. This leads to consider

surfaces in the space.

2.3. Construction of figures

We now clarify what building figures means in this paper. A figure is a finite subset

of Zn, and a family of figures is an application from N into the set of finite subsets

of Zn, that we denote F = (Fi)i∈N.

Constructing a figure F by a cellular automaton A is to select a subset SF of

A-states such that the automaton starting from a convenient initial configuration

reaches a configuration where a cell (x1, . . . , xn) is in a state belonging to SF if and

only if (x1, . . . , xn) ∈ F .

Definition 5 (Figures construction). Let A be an n-CA. We say that A con-

structs the family of figures F = (Fi)i∈N of Zn, according to the times (ti)i∈N if

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and only if there exists a subset SF and a sequence (ti)i∈N of times such that the

automaton starting from an initial configuration C0 at time t = 0 enters, at time

t = ti, a configuration where all the cells belonging to Fi are in a state of SF and

are the only ones to be in such states.

We are interested in constructing families of figures “as soon as possible”, that

means in real time.

Definition 6 (Real-time construction). Let A be an n-CA. We say that Aconstructs the family F in real time if and only if A constructs F according to the

times (i+ c)i∈N with c an arbitrary natural constant.

3. Strategy of Construction

3.1. From isotropy to anisotropy

The evolution of a cellular automaton is intrinsically local such that the construc-

tion of a discrete isotropic figure must be founded on local characterizations. The

fundamental idea is to proceed by duality, replacing the construction of a family of

isotropic figures by the construction of a family of anisotropic figures. For instance,

if we consider in dimension two the intersection between all the real concentric

circles and the two-dimensional grid, it is possible to prove that there are only

eleven different kinds of intersections (see Fig. 1).

In this figure, we see that starting from a cell of coordinates (x, 0) and continuing

on increasing y, the intersection remains constant and we reach something that

seems to be a “parabola” and, then it is always the same until we reach another

Fig. 1. We associate a color to each kind of intersection between the family of concentric circlesand the 2D-grid.

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one and so on. This construction might be easily extended to three dimensions

leading to a more complex system of intersections. Such regularities lead us to

study more precisely the intersections between waves and grids.

3.2. Discrete paraboloids for discrete waves

In dimension two, we have the following duality: ∀R, x, y ∈ N, ∀ k ∈ N,{x2 + y2 = R2

x = R− k⇔{y2 = 2kx+ k2

x = R− k .The intersections between the family of concentric circles and the lines x = R−k

belong to the bundle of parabolas of equation y =√

2kx+ k2 that we denote by hk.

Consequently, the proposed definition of discrete two-dimensional isotropic waves

is based on a digitization of this bundle of parabolas which constitutes a family

of anisotropic figures. According to the digitization procedure of the bundle we

consider, we obtain a digitization of the circle. For example, the “floor circle” is

defined as follows.

Definition 7. For every positive integer R, the floor circle Cfloor(0, R) centered

on (0, 0) and of radius R is defined in the first octant by:

Cfloor(0, R) =R⋃k=0

({(x, y) ∈ N2/x = R− k, b√

2kx+ k2c ≤ y

< b√

2(k + 1)x+ (k + 1)2c and x ≥ y ≥ 0}) .

We can find again the “well-known” discrete circles of the literature using this

process and various discretization schemes. Figure 2 gives examples of floor circles

in the first octant.

Fig. 2. Example of circles (light gray).

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A sphere is the natural extension of the notion of circle in dimensions three.

A hypersphere is the generalization of a sphere to arbitrary dimensions. However,

real geometry is based upon the Euclidean metric making the extension easy in

the way that the space is isotropic. This property disappears in a discrete space

such as Zn on which the cellular automata are based. Thus there have been several

definitions of spheres in the literature depending on the type of construction. One

way to draw a sphere consists in applying the discretization process to the points

of the sphere6 as done with the Bresenham-like approximation of circle.5 The goal

is to provide a good approximation of the real sphere but no attention is paid to

the geometrical properties of the resulting figure. Another definition of a discrete

sphere can be obtained with algebraic definition. For instance, Andres1 defined the

sphere from a Diophantine equation which generalizes the Diophantine equation of

a circle (called an arithmetical circle in his work). Again, no geometric property is

imposed leading to a good approximation of a real sphere but without significant

geometrical properties. This work has been extended by Andres and Jacob3 to an

arbitrary dimension and arbitrary thickness. However, only the drawing algorithm

is currently known and for instance the recognition problem of an arithmetical

hypersphere remains open.

The intersections between an arithmetical hypersphere with thickness one and a

plane parallel to the axis are not circles but rings.2 From this remark, it is possible to

reverse the problem of the definition of the sphere: instead of extending a definition

of a circle, we impose a relationship between spheres and circles. This leads to a

simple but efficient definition of a sphere.

The simplest way to construct a sphere having the geometrical property that

the intersection with a plane is a circle is to define the discrete sphere as a stack

of discrete circles. The question to solve is to determine the radius of the circles

in order to construct a discrete sphere which is a “good” approximation of the

real sphere.

The discrete sphere is thus obtained from the discretizations of real circles which

are the intersections of the real sphere with the set of planes z = i. From this

definition, as we can construct the discretizations of real circles, a discrete sphere

can be constructed as the stack of such circles. In two dimensions, we need to

construct the discretization of two bundles of parabolas. To build a discrete sphere,

we thus have to construct their equivalents in three dimensions. Let us remark that

since we want to impose the intersections between a sphere and a plane to be a

circle, we simply compute figures by rotating the discrete 2D parabolas about the

axes z and y (see Fig. 3).

The figures ck are obtained by the rotation of the parabola hk (with equation

z =√

2ky + k2 in the y − z plane) about the y-axis and the figures pk by the

rotation of the same parabola about the z-axis.

The following step consists in digitizing the figures ck and pk. For example,

we can take the floor of√

2ky + k2 and then obtain the “floor sphere” as defined

as follows.

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2k’z+k’ 2

2ky+k2

R

x

y

z

R

x

y

z

R-k’

R-k

Fig. 3. The extensions of the 2D parabolas into C and P by rotations about the y- and z-axes.

Definition 8. For every positive integer R, the floor sphere denoted by Sfloor(0, R)

is the set of points (x, y, z) such that (x, y) belongs to the floor circle of center

(0, 0, z) and of radius b√

2kz + k2c with the constraint z = R− k.

As the discrete isotropic waves (circles and spheres) are locally defined from the

paraboloids (of dimension two or three), the problem of isotropic waves construction

is then reduced to the problem of constructing the discrete bundles of paraboloids by

cellular automata. As a matter of fact, a given cell, with some neighbors belonging

to the wave at time r, can decide whether it belongs to the wave at time r + 1

according to its position compared to the bundle (see Sec. 5). With this approach,

we also replace by duality the construction of isotropic objects by anisotropic ones.

4. Paraboloids Generation

The goal of this section is to construct the floor paraboloids. Such paraboloids, in

dimension two, are defined by:

Hk = {(x, y) ∈ [(i, j), (i + 1, j′)]; i, j, j′ ∈ Z, j = b√

2ki+ k2c and j′ =

b√

2k(i+ 1) + k2c} with k a positive integer.

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Fig. 4. Parabolas Hk.

Fig. 5. Parabolas Hk and Vk.

Figure 4 shows them. In fact, we do not construct only these paraboloids but

also their symmetric (denoted by Vk′ with k′ a positive integer) according to the

first diagonal (see Fig. 5).

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Fig. 6. Examples of figures Ck.

The three-dimensional paraboloids Ck are obtained by the rotation of the two

dimensional ones (Hk) about the y-axis and Pk by the rotation about the z-axis

(see Fig. 6).

4.1. 2D construction

In this part, we summarized the work already done in Refs. 4 and 7. Following

Fig. 5, we can remark that the bundles of parabolas Hk and Vk separates the plane

into three parts as shown in Fig. 7.

The construction is done by mainly three automata. The first step consists in

generating Hk from one of its parts.

Property 1. There exists a CA which builds Hk from Hk ∩ (DV0 ∪K0).

The principle of this automaton is to use a diagonal signal which propagates

the diagonal movement state on a parabola (see Fig. 8).

As a corollary, we can construct all Hk from the intersection of the bundles Hand DV0 ∪ K0. The question to construct this intersection remains. Let us define

K0 = K01 ∪K02 by splitting K0 using the first diagonal and setting K01 the lower

part. Then,

DV0

DH0

K 0

K 02

K 01

Fig. 7. Decomposition of the plane: inDH0 the parabolas are horizontal, inDV0 they are vertical(symmetry of DH0) and in K0 this is the heart of the construction separated by symmetry intoK01 and K02.

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Fig. 8. Diagonal signals for the first automaton.

k’=70

0x

y

t=1t=2t=3t=4t=5t=6t=7t=8t=9t=10t=11t=12t=13t=14t=15t=16t=17t=18t=19

Fig. 9. Symmetrization for the second automaton.

Property 2. There exists a CA which builds,

• Vt+1 ∩ (DH0 ∪K01) from Ht ∩ (DV0 ∪K02)

• Ht+1 ∩ (DV0 ∪K02) from Vt ∩ (DH0 ∪K01).

Figure 9 shows such a construction. The construction of Hk in the heart K01

is still not done. It is based on the detection of specific patterns called “A” (see

Fig. 10, left). It has been proved that these patterns are generated by diagonal

signals thrown from Hk (see Fig. 10, right). Then,

Property 3. There exists a CA which builds Hk+1 in K01 from Hk in K01 and the

origins of the patterns “A”.

From theses constructions, it is possible to build discrete circles in real time

neglecting the initialization time of the construction. Let us remark that specific

rules are needed to locally decide if one cell belongs to a discrete circle or not. Such

rules in three dimensions are given in Sec. 5.

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0(x , y )1 10

1 1(x , y )1 1

1 1(x , y )2 2

1 1(x , y )3 3

1 1(x , y )4 4

y

x

Fig. 10. Patterns “A” and signals thrown from H1 to mark some origins of some patterns “A”.

4.2. 3D construction

In this part, we propose an extension of the previous construction to dimension

three. As we have defined, the sphere as a stack of circles, one might think of

constructing the circles and then of propagating a signal between the stack to decide

whether or not a cell belongs to the sphere or not. However, such an approach leads

to a construction of the sphere which is not done in real time. To achieve such a

result, we must extend the automata of the 2D construction to the case of three

dimensions. To do this, as Ck and Pk are obtained by rotation from Hk and Vk, it is

sufficient to rotate the signals from Hk and Vk with the same rotation parameters.

It is clear then that one circle will be it instead of one pixel. This allows us to easily

extend the first automata by generating circular diagonal propagation, Fig. 11.

This extension is possible by generating the discrete circles plane by plane with

an extension from one plane to the next one instead of a simple bi-dimensional

Ck

Pk

Fig. 11. Diagonal propagation in 3D.

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Fig. 12. Symetrization in 3D.

Fig. 13. Link between the radii and the parabola Hk.

one. This allows us also to extend the second automaton through a simple rotation,

Fig. 12.

Let us remark that the radii of the circles are determined by the parabola Hk

(Fig. 13).

For similar reasons, the links we have in the heart between a parabola Hk and

its successor Hk+1 are also available. As a matter of fact, it has been proven that

the parabolas in the heart were only composed of landing of lengths one and two

and of stairs of length one, so that we could obtain the parabola Hk+1 from the

parabola Hk using small signals that compute one or two knowing the position of

specific patterns “A”. In our case, the signals become circular propagation with

radii varying by one or two.

In fact, we must know the position of the patterns “A” which is the place where

the parabola Hk+1 is just one cell away from the parabola Hk in the heart even

if the parabola Hk contains an “hollow”. In dimension three, the equivalent of the

diagonal signals is, the same as previously, a diagonal wave propagation.

Moreover, the symmetry of revolution of Ck and Pk′ induces that the relations

between the parabolas Hk and Vk′ are preserved. In particular, it was proved that

when two parabolas meet, if there is a “slope” (respectively a “landing”) on one of

them, then there is a “slope” (respectively a “landing”) on the other. The equivalent

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of such a property is the fact that if the radius of the circle grows on Ck, then it

grows on Pk′ , and if it is constant on Ck, it is constant on Pk′ .

Thus, it is possible to construct Pk′ from the family of Ck and by symmetry Ckfrom the family of Pk′ . We can remark that when k equals k′, the parabolas Hk

and Vk′ meet on the first diagonal and thus the radius of the circle which is the

intersection of P ′k and Ck is the same as the one of the circle at the intersection of

C′k and Pk.

To summarize, all the properties proven in dimension two are preserved in

dimension three and thus also is the existence of the cellular automaton used for

the construction.

5. Isotropic Waves

5.1. Construction of spheres from figures Ck and Pk

Let us suppose that we have already constructed the sphere of radius R and the

families Ck and Pk. The question that remains is to locally decide which cells belong

to the sphere of radius (R+ 1).

Let (x, y, z) be a cell such that there exists at least one cell in its 26-neighborhood

that belongs to the sphere of radius R. Then there are two cases that depend on

the position of the cell compared to the figures Ck:

• if the cell is on a figure Ck, then it belongs to the sphere of radius (R+1), because

the intersection between the sphere and a plane y = constant is the intersection

between a figure Ck and the same plane;

• if the cell is “inside” a figure Ck, then it belongs to the sphere of radius (R+ 1)

if and only if it has a 6-neighbor that belongs to the sphere of radius R.

Such rules are available for x ≥ y ≥ z ≥ 0, and symmetrical ones allow us to

obtain the spheres in all space. Figure 14 shows what we obtain for radii equal to

3, 5, 10 and 20.

Fig. 14. Examples of spheres (R = 3, 5, 10, 20).

5.2. What about time?

As the 2D-cellular automaton used to construct the circles generates the point of

coordinates (n, b√

2kn+ k2c) for every integer n and k at time (n+ k), we obtain

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the intersection between the sphere and the plane y = n at time t = n+k: this is the

circle of center (0, n, 0) and of radius b√

2kn+ k2c. Consequently, if we consider the

sphere of radius R, its intersection by the plane y = R−k is obtained at time t = R.

So, the spheres are constructed in real time.

6. Conclusion and Future Works

In this article, we have presented a procedure to construct a family of concentric

spheres in real time on cellular automata. This extends a previous work.4 To do so,

we have defined a discrete sphere by its intersections with planes having one axis

x, y or z as normal vector. From this point of view, the work done for constructing

discrete circles can be used as for the generation of the discrete spheres. We have

presented a definition of the discrete sphere for the floor discretization, but any

well known discretization schemes in the literature can be used. For instance, it has

been proven that arithmetical circles can be constructed in real time, leading to a

new arithmetical sphere as compared to the one defined in Ref. 1. The presented

constructions remain essentially the same and we hope this can serve as a basis for

the generation of isotropic computations with cellular automata by using the duality

principle we propose between circles and parabolas, and spheres and paraboloids.

In future works, we plan to use this algorithm to the study of the intersections

between spheres in order to simulate wave interferences. This has applications in

physics and volume rendering. We also plan to study the consequences of a non-

integer center and/or noninteger radius for the presented work. The extension to

higher dimension is also of interest even if the principle of imposing the intersection

between an hypersphere of dimension n + 1 with an hyperplane to be an hyper-

sphere of dimension n will probably lead to a natural and recursive extension of

the proposed method.

References

1. E. Andres, Cercles Discrets et Rotations Discretes, Ph.D. Thesis, Universite LouisPasteur, Strasbourg, 1994.

2. E. Andres, “Discrete circles, rings and spheres,” Comput. Graph. 18, 5 (1994) 695–706.3. E. Andres and M.-A. Jacob, “The discrete analytical hyperspheres,” Trans. Visual.

Comput. Graph. 3, 1 (1997) 75–86.4. M. Delorme, J. Mazoyer and L. Tougne, “Discrete parabolas and circles on 2D cellular

automata,” Theoret. Comput. Sci. 218 (1999) 347–417.5. A. Kauffman, “Efficient algorithms for 3D scan conversion of parametric curves,

surfaces, volumes,” Comput. Graph. 4 (1987) 171–179.6. A. Kauffman and E. Shimony, “3D scan-conversion algorithms for voxel-based

graphics,” Proc. 1986 Workshop on Interactive 3D Graphics, eds. F. Crow andS. M. Pizer, October 1986, pp. 45–75.

7. L. Tougne, “Circle digitization and cellular automata,” Discrete Geometry forComputer Imagery (DGCI’96), Lecture Notes in Computer Science 1176, SpringerVerlag, pp. 283–294.

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Fabien Feschet re-ceived his Ph.D. in com-puter science from theInstitut National desSciences Appliquees inLyon in 1999. His re-search has been in medi-cal image analysis, med-ical image registrationprocedures for patient

positioning in cancer therapy and paral-lel processing. He is currently an AssistantProfessor in computer science in the LASSLaboratory of the University Lyon 1.

His main scientific interests lie in dis-crete geometry, general topology, operationresearch and image processing.

Laure Tougne receivedher Ph.D. degree incomputer science fromthe Ecole Normale Sup-erieure in 1997. Her re-search has been in thealgorithmic on 2D cel-lular automata. She is

currently an AssistantProfessor in computer

science in the ERIC Laboratory of the Uni-versity Lyon 2.

Her main scientific interests lie in discretegeometry.

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generating isotropic discrete waves on cellular automata · 2. cellular automata let us now...

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