A New Kind of Science: The NKS Forum > Pure NKS > CA for Three Color Rule Number 3671584492797 Records a Geometric Sequence
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Registered: Jan 2004
Posts: 350

CA for Three Color Rule Number 3671584492797 Records a Geometric Sequence

In the attached figure, three color rule number 3671584492797 records the geometric sequence:

a + ar + ar^2 + ar^3 + ar^4, ...
where a = 1 and r = 2.

The solid inverted triangles down the right side of the figure mark off successive powers of 2.

The left hand side has very small triangles marking the units being sequenced.

But each term in the geometric sequence is also enumerated horizontally at the point of the apex of its inverted triangle. Moreover, it is also enumerated by a diagonal running from the previous term’s inverted triangular apex down to the base of its own inverted triangle.

Here is the Mathematica code that generates the figure:

ArrayPlot[
CellularAutomaton[{3671584492797, 3, {{-1}, {0}, {1}}}, {{1}, 0},
3000], ColorRules -> {0 -> Black, 1 -> Yellow, 2 -> Red}, PlotLabel -> {{"(702: 1+AND[p, 1+q] ~XOR~ "}, {"511758: 1+AND[1+p, q]" " [Rule]
512460)"}, {" ~DIFF~ "}, {"(13626: 1+AND[p, -1+r] ~XOR~ "},
{"7343167948506: 1+AND[-1+p, -1+q]" " [Rule] 7343167962132
)"}, {"
[Rule] 3671584492797"}}, ImageSize -> {850, 750}]

A word on the lengthy plot label.

The label shows that there are four basic AND-type rule numbers as progenitors to the rule number that generates the figure. These four rules are paired as input to modulo 3 sum operations. Then modulo 3 difference of these sums is found. That difference is the rule that generates the figure.

In the label ~XOR~ indicates modulo 3 sum of two operands. Similarly, ~DIFF~ indicates modulo 3 difference of just two operands.

I used the map function with rule number 3681670999617 for ~XOR~ and rule number 146430861993 for ~DIFF~.

But “+” and “-“ may be used just as well. Here is how to do it with “+” and “-“.

(1) Find modulo 3 sum of rule numbers 702 and 511758:

FromDigits[Mod[IntegerDigits[702,3,27] + IntegerDigits[511758,3,27],3],3]

512460

(2) Find modulo 3 sum of rule numbers 13626 and 7343167948506:

FromDigits[Mod[IntegerDigits[13626,3,27] + IntegerDigits[7343167948506,3,27],3],3]

7343167962132

(3) Find modulo 3 difference of these two modulo 3 sums. Note that operands input to modulo 3 difference are not commutable.

FromDigits[Mod[IntegerDigits[512460, 3, 27] - IntegerDigits[7343167962132, 3, 27], 3], 3]

3671584492797

So the lengthy plot label establishes that the rule number generating the geometric sequence is quite complex, being the difference of sums built up from four basic AND-type functions.

Lawrence J. Thaden has attached this image:

Last edited by Lawrence J. Thaden on 04-23-2011 at 02:17 AM

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04-22-2011 03:05 PM

Registered: Jan 2004
Posts: 350

Cellular automaton for three color rule number 302188237325 records a geometric sequence in the length of the base of obtuse triangles. This sequence is useful for locating a subset of the prime numbers. The sequence is:

2 + a + a r + a r^2 + a r^3 + a r^4 , …
Where a = 15 and r = 6.

The lengths of the bases of the first four obtuse triangles are: {2, 17, 107, 647}. All of these are prime numbers.

But the next term has an obtuse triangle with a length that is not a prime: It is a r^3 = 3887 which is divisible by 13.

Mathematica code that generates this sequence is:

n=4; FoldList[Plus, 2, Table[(15 6^i), {i, 0, n}]]

From the first thousand terms there are only 17 prime numbers. Here are their locations in the geometric sequence. Of course these are not the locations of where they are to be found in the list of all primes.

n = 998; subscriptsTrue =
Flatten[Position[
PrimeQ[FoldList[Plus, 2, Table[(15 6^i), {i, 0, n}]]], True]]

{1, 2, 3, 4, 6, 7, 10, 13, 14, 25, 42, 72, 141, 190, 191, 631, 714}

Here are the first ten of these seventeen prime numbers:

Style[TableForm[
FoldList[Plus, 2, Table[(15 6^i), {i, 0, n}]][[#]] & /@
Take[subscriptsTrue, 10]], Medium]

2
17
107
647
23327
139967
30233087
6530347007
39182082047
14215144014964850687

The last seven of the seventeen prime numbers have many digits. Too many to conveniently display. But here is a list of their lengths.

Prime IntegerLength
(11) 33
(12) 56
(13) 110
(14) 148
(15) 149
(16) 491
(17) 556

In the attached figure you can see how the cellular automaton assists in calculating the base of the obtuse triangles.

The triangles appear down the right edge of the foreground of the plot, with the side opposite the obtuse angle being aligned on this right edge. These triangles are dark red.

Along the base of each triangle is a sequence of yellow-green dashes. These mark off the units of length of the base of the obtuse triangle.

Tally these dashes and you have the length of the base, which is also one of the terms in the geometric sequence.

The leadoff term in the geometric sequence is the constant 2. In the plot this term is found represented by the first two yellow-green dashes at the top right edge of the plot.

Note that there is no triangle there. You could say that the base of the triangle that is not there is 2 units in length but it has no height.

Incidentally, the side adjacent to the base of each triangle has units of measure that are indicated by pink dotted lines coming in from the left edge of the plot. However, these units are not on the same scale as those for the base.

I find it fascinating that the cellular automaton has the capability to work with these two disparate scales simultaneously. If we were to work it out on a single scale it would require real number approximations instead of discrete numbers.

The background of the plot is alternating rows of black and red. These indicate time steps in the evolution of the cellular automaton.

Along the left edge of the foreground there is a fine pink line bordering a diagonal row of yellow-green hash marks. I do not know what purpose they serve or what information they convey. If you do, please post your comments.

Here is the Mathematica code used to generate the label and the plot. The cellular automaton does not have periodic boundary conditions. It keeps expanding to the left and right with each time step.

The cellular automaton is only evolved through 1000 time steps. This is sufficient to illustrate the first three terms in the geometric sequence and part of the fourth term. But to see all of the fourth term the number of time steps must be greatly increased.

CAlabel = {{"(26: 1+AND[p, q] ~XOR~ "}, {"10072932714: 1+AND[-1+p,
q]" " [Rule] 10072932740)"}, {" ~DIFF~ "}, {"(586554620346:
1+AND[-1+p, r] ~XOR~ "}, {"511758: 1+AND[1+p, q]" " [Rule]
586555132104)"}, {"  [Rule] 302188237325"}};

ArrayPlot[
CellularAutomaton[{302188237325, 3, {{-1}, {0}, {1}}}, {{1}, 0},
1000], ColorRules -> {0 -> Black, 1 -> Yellow, 2 -> Red},
PlotLabel -> CAlabel, ImageSize -> {950, 950}]

Note that the label indicates rule number 302188237325 is the set difference of two modulo 3 sums, and that those sums are of pairs of basic AND-type statements.

The cellular automata for these basic AND-type statements are all Class 1. So are cellular automata for modulo 3 sums. It is only when we get to modulo 3 difference of sums that the cellular automaton becomes Class 2.

This is an example of a more complex behavior emerging from simpler mechanisms.

Also, the fact that the mechanism of searching for a subset of primes by developing the equivalent of a geometric sequence is achieved by the cellular automaton underscores that cellular automaton behavior is causally prior to traditional mathematical formulations of geometric series.

This should be emphasized, if for no other reason than to point out that the mechanisms of the cellular automaton are all logical: (1) basic AND-type operations with two variables, (2) modulo 3 sum, and (3) modulo 3 difference.

So logic is causally prior to traditional mathematical formulations.

However, all the cellular automaton does is present the sequence of triangles. It takes a person to see that the triangles illustrate the geometric sequence. And the person has to be familiar with proof by mathematical induction in order to know that the next terms in the geometric sequence will be present as the length of the base of triangles when the cellular automaton is evolved a sufficient number of time steps into the future.

In this sense the cellular automaton has predictability about it. Some say that NKS has no capability to make predictions. But this cellular automaton is clearly capable of predicting the length of the base of any future obtuse triangle that may be coming up as it evolves.

Lawrence J. Thaden has attached this image:

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04-29-2011 10:01 PM

Registered: Jan 2004
Posts: 350

A Third Example Combining Class 2 and 4 Behavior

The two previous examples in this thread both used rule numbers whose progenitors were simple AND-type expressions in the form AND-type[operand1, operand2]. This third example likewise uses a rule number whose progenitors are simple AND-type expressions.

The AND-type expression used is (1+AND). There are other types available. Namely, AND, -AND, 1-AND, -1+AND, and -1-AND. But these come in three pairs:

AND and –AND are additive inverses.
1+AND and 1-AND are negative multiplicative inverses.
-1+AND and -1-AND are positive multiplicative inverses.

These two operand AND-type operations ignore signs. So the results are always the absolute value. For this reason, we only use AND, 1+AND, and -1+AND.

Each AND-type operation has operands that also come in six forms, but we only use the three positive versions for each of the three operands: {p, 1+p, -1+p}, {q, 1+q, -1+q}, and {r, 1+r, -1+r}. The reason is the same: negative versions of these algebraic expressions are meaningless, since they point to the same rule numbers as the equivalent positive expressions. In this sense these binary operators and their respective rules are comparable to absolute valued functions in traditional mathematics.

Before presenting this third example I want to review some comments from the Forum that have to do with definitions of the four Classes of cellular automata.

My motive in this is to put this third example in context, because, as will be seen, it illustrates behavior from both Class 2 and 4.

In the thread: “Defining the Classes of Elementary Cellular Automata” I wrote:

There are descriptions of the four classes in the NKS book on pages 231 and 235. …

“In class 1, the behavior is very simple, and almost all initial conditions lead to exactly the same uniform final state.” …

“In class 2, there are many different possible final states, but all of them consist just of a certain set of simple structures that either remain the same forever or repeat every few steps.” …

“In class 3, the behavior is more complicated, and seems in many respects random, although triangles and other small-scale structures are essentially always at some level seen.” …

“… class 4 involves a mixture of order and randomness: localized structures are produced which on their own are fairly simple, but these structures move around and interact with each other in very complicated ways.”

Jason Cawley replied with informative comments on the definition of classes 3 and 4.

Among other things he wrote:

“…universality is not part of the definition of class 4. That class 4s are in general probably capable of universal computation in the infinite size limit, is a result, not something that proceeds from a definition. Class 4 is not meant to designate "such algorithms as are capable of universal computation".

“Class 4 means - from typical (not special or measure zero) random initials (not whole ensembles of initials taken as a programmer's degree of freedom, and not simple initials – italics added - ), typically resolves into a number of localized structures that themselves move about and interact, in often complicated ways.

“Class 3 means - from typical random initials, produces random chaotic patterns.”

Unbounded with Infinite Time

Because we are starting from simple initials, we do not intend to use the class number as defining the cellular automaton in the sense that Jason mentions. Rather it is meant to indicate that the cellular automaton manifests behavioral characteristics in common with the classes Jason comments on.

In addition, we are discussing unbounded cellular automata. The foreground of these cellular automata grow horizontally with each time step to infinity, and they do not cycle.

The Third Example

Now I want to introduce this curious cellular automaton that keeps spawning what appear, at the appropriate resolution, to be diagonally parallel lines.

But, at a finer resolution these lines are seen to be made up of identical triangular particles separated by regular intervals of time steps.

The three color rule number that generates this cellular automaton is 3671598304234. This is another rule number that is the set difference of two modulo 3 sums, and those sums are of pairs of basic AND-type statements.

(13817466: 1+AND[1+p, 1+q] ~XOR~
511758: 1+AND[1+p, q] -> 14329224)
~DIFF~
(1514: 1+AND[p, r] ~XOR~
7343167948506: 1+AND[-1+p, -1+q] -> 7343167950020)
-> 3671598304234

Arithmetic Progression

The time steps at which each parallel line begins is defined by an arithmetic progression, which in traditional form has the expression:

an = a1 + (n - 1) d
where a = 1542, d = 3030, n ranges over the natural numbers, and n and 1 are subscripts.

Of course the traditional form does not yield the desired result in Mathematica, but we can use:

Fold[Plus, a, Table[d, {n}]]
where n = 3

to find the fourth term in the arithmetic progression:

10632

Similarly, to list any number (n + 1) terms in the arithmetic progression, we can use:

FoldList[Plus, a, Table[d, {n}]]

{1542, 4572, 7602, 10632}

Considering just this aspect, the cellular automaton manifests Class 2 behavior, a simple pattern repeating periodically.

The pattern is a stream of triangular shaped particles that appear as a line at an appropriate resolution. The repeating period, after 1542 time steps of start-up, is 3030 time steps in length.

Figure 1 shows a portion of the cellular automaton where there are three parallel lines. The origin of the second and third lines is also shown, but the origin of the first line does not show because the image is just a section taken from the full output.

Here is the Mathematica code that I used to produce Figure 1.

CA3Lines =
Take[CellularAutomaton[{3671598304234, 3, {{-1}, {0}, {1}}}, {{1},
0}, 8500], {4580, 8000}];

Dimensions[CA3Lines]

{3421, 8502}

ArrayPlot[Table[Take[CA3Lines[[i]], {2000, 8502}], {i, 3421}]
ColorRules -> {0 -> Black, 1 -> Darker[Yellow], 2 -> White},
PlotLabel ->
Style[{{"(13817466: 1+AND[1+p, 1+q] ~XOR~ "}, {"511758: 1+AND[1+p,
q]" " [Rule] 14329224)"}, {" ~DIFF~ "}, {"(1514: 1+AND[p, r] ~XOR~
"}, {"7343167948506: 1+AND[-1+p, -1+q]" " [Rule] 7343167950020)"},
{"  [Rule] 3671598304234"}}, Bold, "SubSection"]
ImageSize -> {1100, 748}]

Class 4 Behavior

But now for the curious aspect of this cellular automaton: the patterns at the left foreground before and between the time steps that spawn parallel lines. (We will be using “descriptive definition” in this discussion rather than “explanatory definition”.)

Figure 2 shows the start-up of the cellular automaton and the spawning of the first parallel line. The resolution makes it possible to see that the parallel line is actually made up of triangular shaped particles separated by regular intervals of time steps.

But we want to focus on the local structures in the detail preceding the spawning of the first parallel line. The structures look as though the cellular automaton is repeatedly attempting to spawn a parallel line. But in the failure to do so, it generates a disordered set of triangular particles. Then each next attempt seems to get closer with more triangular particles aligning nearer to what is required to make them appear as a parallel line. Finally, the first parallel line is spawned. In the figure it is at the bottom fourth emanating from the left foreground and appears as a stream of 14 triangular shaped particles.

Figure 3 shows the interval between the beginning of parallel lines two and three. It starts off with a short burst that appears common to all the intervals. This is followed by three small failed attempts. Then there is a large attempt with a succession of five tries before complete failure. This is followed by another large attempt that has a first and second half. The first half appears to be organizing the particles. The second half appears to be trying to spawn. But this also ends in failure. Finally, it spawns the third stream of triangular particles.

This behavior observed close to the left edge of the foreground before the first parallel line and in the interval between parallel lines resembles Class 4 behavior. It is not entirely regular and it is not entirely random. Nor is it simple.

And when the efforts to spawn a parallel line collapse, there appears to be definite lines of communication connecting the collapsing structure to the right edge of the foreground. (See circled areas in Figure 3.)

Identical Class 4 Behavior Repeated Regularly

What about the intervals taken as a whole? Does the interval between the first and second spawning of the parallel lines manifest the same behavior as in the interval between parallel lines two and three?

Figure 4 shows the interval between the spawning of parallel lines one and two. The local structures are the same as for those between lines two and three shown in Figure 3.

So the overall behavior of this cellular automaton is completely predictable.

First, there is the predictability of the spawning of parallel lines explained by the arithmetic progression, expressed in Mathematica by:

FoldList[Plus, a, Table[d, {n}]]

where a = 1542, d = 3030, and n ranges over the natural numbers.

Second, the predictability of the equivalent structure in the intervals between the spawning of parallel lines is defined by comparing pictures of the intervals and using proof by induction.

Periodic Boundary Conditions

Figure 5 shows the same cellular automaton evolved over 8001 steps with periodic boundary conditions. Only one line gets spawned.

When the left edge of the foreground reaches the left edge of the background of the cellular automaton, it wraps around and proceeds across to the right edge of the foreground where it intersects exactly at the left edge of the foreground just before the second line is about to be spawned. The collision prevents the second line from ever being spawned.

Instead there is a burst of triangular shaped particles from which there comes a composite vertical stream of two sizes of triangular shaped particles that continues on without incident. This stream is on the left.

But also, to the right emanates a stream of smaller disintegrating particles flanked by a heavy double line on the right and a single line on the left.

The disintegrating stream bifurcates, being drawn by the lines flanking it. The part that is on the left forms a composite vertical stream that continues on without incident. The part on the right bursts into two clusters of triangular shaped particles.

The cluster on the right breaks up the heavy double line into a triple line which continues on the same track as the heavy double line that is no more. The cluster on the left disintegrates only to burst again into triangular shaped particles from which emanate on the right another composite vertical stream that continues on without incident, and on the left a heavy double line similar to the previous heavy double line to its right.

The three streams of particles are of course parallel since they are vertical, but they are not evenly spaced apart from each other. But just as it seems like they will go on down through the time steps forever unimpeded, along comes the first and only line spawned that has wrapped around due to boundary conditions.

It slams into the leftmost vertical line knocking it to the right, though not changing its angle.

This causes the formation of a cluster of small triangular shaped particles, and the spawned line has its angle changed slightly as it also changes the shape of its constituent particles. There also emanates from it a heavy line that is headed on a collision course with what happens next.

The modified original spawned line now slams into the second vertical line and knocks it to the right without changing its angle. There is an accompanying burst of triangular shaped particles, and it is with these that the heavy line collides.

Well, all pandemonium now breaks loose. Eventually, eight vertical streams emerge on the left while the majority of the momentum of the collision continues on the right following a slightly altered angle of the original spawned line.

What happens next is outside of the frame of the 8001 time steps through which the cellular automaton evolved.

Conclusion

Quite a ride! The descriptive language used was motivated by associations with subatomic particle collisions. Not very objective, I acknowledge.

However, the behavior is definitely Class 4. It is not predictable. And, yes, the cellular automaton will eventually cycle.

Moreover, the initial conditions, as we ran it, were simple, not random (See Jason Cawley's comments above.)

initialConditions =
Flatten[{Table[0, {1000}], {1}, Table[0, {1000}]}];

In conclusion, the cellular automaton for rule 3671598304234, when evolved without boundary conditions, has an infinite number of time steps and entirely predictable behavior that can be explained by an arithmetic progression.

However, when periodic boundary conditions are prescribed, the same rule number evolves a cellular automaton with Class 4 behavior, even with simple initials, which is not predictable, but which at some point cycles.

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05-10-2011 02:37 AM

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Figure 2 shows the first parallel line and the start-up time steps preceding it in a resolution that makes the line appear to be a stream of triangular shaped particles.

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05-10-2011 02:39 AM

Registered: Jan 2004
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Figure 3 shows the Class 4 behavior in the interval between the start of parallel lines two and three. Circles indicate times where Class 4 behavior collapses.

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05-10-2011 02:41 AM

Registered: Jan 2004
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Figure 4 shows the Class 4 behavior in the interval between the start of parallel lines one and two. Circles indicate times where Class 4 behavior collapses. Class 4 behavior in this interval is the same as that in the interval between the start of parallel lines two and three.

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05-10-2011 02:42 AM

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Figure 5 shows the cellular automaton with the same rule number but with periodic boundary conditions.

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05-10-2011 02:43 AM

Registered: Jan 2004
Posts: 350

CA for Three Color Rule Number 10072947729 Illustrating Recursion

In attached figure 1, three color rule number 10072947729 illustrates the recursion:

f[n_] := 2 n - 2
f[0] = 5;

The solid inverted triangles down the center right side of the figure mark off successive terms of the recursion in that the terms indicate the time step for the base of each triangle.

Here are the time steps on which the bases of the first five inverted triangular shaped figures appear.

i = 5; Do[Print[i = f[i]];, {5}]

8
14
26
50
98

The measure in cell units of the base of each triangle in the recursion follows this sequence:

a + (a + r) + (a + r^2) + (a + r^3), ...
where a = 5 and r = 2.

In Mathematica code:

a = 5; r = 2;
ts = FoldList[Plus, a, Table[r^i, {i, 5}]]

{5, 7, 11, 19, 35, 67}

The area of each of the triangles would be 0.5(base^2), but these are not perfect triangles, as figure 2 shows.

So we have to add 0.5(base) to each “triangle’s area to get the true area.

In Mathematica code:

areas = Table[(0.5 (ts[[i]])^2), {i, 5}]

{12.5, 24.5, 60.5, 180.5, 612.5}

additionalPart = Table[0.5 ts[[i]], {i, 5}]

{2.5, 3.5, 5.5, 9.5, 17.5}

{15., 28., 66., 190., 630.}

I post this so that those being introduced to recursion and sequences might have examples to motivate their study.

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05-22-2011 04:25 AM

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In the attached figure 2 the triangle's hypotenuse is seen to be jagged or stair step.

This adds to the area of the triangle one half cell for each time step.

And the base and height of the triangle are the same.

So adding one half cell for each of the time steps of the triangle gives us the additional area required.

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05-22-2011 04:31 AM

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Posts: 350

Here are two cellular automata that on first glance appear to have the same output.

But on closer inspection they are seen to differ starting from a single cell on line 155, counting the initial conditions line as line 1.

Then this difference spreads to cells on the left and (mainly) on the right on subsequent lines.

But the difference area never completely overtakes the CAs. Cells at the right edge remain common.

We can think of this as two different controls, the rules, operating on the same initial cells, the simple initial conditions.

At first the controlled output is the same. The cells manifest equivalent growth patterns.

Then after 151 steps of cell growth, they differentiate and begin to grow apart, but not completely.

The right border remains the same for both.

This mechanism echoes analogously with mechanisms in biochemistry.

The attached figure shows the two similar cellular automata on the left. On the right is the area in which they differ.

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06-15-2011 06:42 PM

Registered: Jan 2004
Posts: 350

The attached figure shows a cellular Automaton whose progenitors have (-1+AND) operators.

It is interesting because it shows what at first appears to be a symmetric structure.

But on closer inspection it can be seen that wherever on the left side a 0 (black) is to be found, on the mirror image right side there will be found a 2 (white).

Also, cells with a 1 (red) first appear on time step three, starting from simple initial condition cells that make up row 1.

If it is agreed that these red cells establish the background, then we can say the black and white foreground space is cleared to let the background appear.

Collectively this background in the cleared space forms the random pattern of inverted triangles.

And although the black and white foreground is asymmetrical, the pattern of red cells is symmetrical.

We might describe this as a random pattern of holes in the asymmtric foreground that lets the symmetric background shine through.

Here is Mathematica code used to generate the image:

ArrayPlot[CellularAutomaton[{2552112607886, 3, {{-1},{0}, {1}}}, {{1}, 0},500],ColorRules->{0->Black,1->Red,2->White},PlotLabel->{{"(3812798742480: -1+AND[p, q] ~XOR~ "},{"3807762276136: -1+AND[-1+p, q]" " -> 7620561018616)"},{" ~DIFF~ "},{"(3812754042400: -1+AND[1+p, 1+r] ~XOR~ "},{"1270803767440: -1+AND[-1+q, -1+r]" " -> 5083557809840)"},{"  -> 2552112607886"}},ImageSize->{1000,600}]

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06-27-2011 03:40 PM

Registered: Jan 2004
Posts: 350

In the previous post, symmetry was absent in the foreground, but present in the background.

Here is a three color rule whose cellular automaton, when run with simple initial conditions, is completely symmetric: 10460885698.

Its digits are all 0s and 1s. And so is the cellular automaton output it generates.

IntegerDigits[10460885698, 3, 27]

{0,0,0, 0,0,1, 0,0,0, 0,0,0, 0,0,1, 0,0,0, 0,0,1, 1,1,0, 0,0,1}

But unlike elementary rule 30, the center digits do not form a random pattern of 0s and 1s.

Rather, they alternate from 0 to 1 with each time step after the initial 1.

Here is the Mathematica code to generate the figure.

ArrayPlot[
CellularAutomaton[{10460885698, 3, {{-1}, {0}, {1}}}, {{1}, 0}, 500],
ColorRules -> {0 -> Black, 1 -> White, 2 -> Red},
PlotLabel -> {{"(3812798742142: -1+AND[p, 1+q] ~XOR~ "},
{"3812798742142: -1+AND[p, 1+q]" " -> 7625597484284)"}, {"
~DIFF~ "}, {"(3812798741736: -1+AND[p, r] ~XOR~ "}, {"3802337857822:
-1+AND[1+q, r]" " -> 7615136599558)"}, {"  ->
10460885698"}}, ImageSize -> {949, 588}]

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06-30-2011 05:42 AM

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