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Rule 5572552008259: - (-1 - z) is the additive inverse of the rule in the previous post. However, in one respect it is more like rule 1466461054805: (-1 + z) the original of which has 1458 rows that cycle back to the initial conditions.

There are only three phases to this rule, whereas rule 1466461054805: (-1 + z) has four phases.

But just as with rule 1466461054805: (-1 + z), so also here, the outside phases are 1458 rows and the inside phase is 729 rows.

The last two phases form the cycle of phases.

The appearance of the phases is of stripes parallel to the secondary diagonal. These stripes have varying thickness and appear solid gold and blue. The gold is due to a blend of cell values, whereas the blue is due to cell values always being 2.

See attached figure.

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Rule 4399383164415: (z) is similar to the rule in the previous post. The outside phases have 1458 rows and the inside phase has 729. The last two phases form a cycle of phases.

The difference is in the appearance. Both have solid groups of stripes running parallel to the secondary axis. It appears as red and turquoise stripes. The turquoise is due to a blend of cell contents within the stripe, whereas the red indicates the cell content is always 0.

See attached figure.

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Rule 6159136430181: - (z) is the additive inverse of rule 4399383164415: (z). This rule only has two phases, both with 729 rows.

Its appearance is as of groups of stripes parallel to the secondary axis. They are solid colored. No pattern. Also the colors are simply red, green, and blue. No blends.

See attached figure.

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Now we consider three rules:

193720085: (x - 1)
3812992433055: (x)
7625210074339: (x + 1)

Taken by themselves, there is nothing extraordinary about these rules, but when compared with the three corresponding rules for z, things get interesting.

We will first consider the rules by themselves. Then in the next post we will compare them with the corresponding rules having expressions (z -1), (z), and (z + 1).

First of all, note that these rules belong to the set of T5 with variables x and z. That is, the initial conditions for these rules are the digit expansions of the 27 rule numbers for T5 with variables x and z.

This is different from the same set of rules taken from T5 with variables x and y which have initial conditions that are the digit expansions of a slightly different set of rules. The difference in initial conditions determines the unique behavior of the cycle of phases for the same rule numbers.

All three are tori in each of their phases and each has a two phase cycle. Rule 193720085: (x - 1) has four phases, while the other two rules have three phases. All phases of each rule are 1458 rows in length. Each phase has an equal amount of cell content types, thus making them all polar neutral.

After the original phase, even phases repeat the pattern once down the rows, but there is no repetition over the columns. Odd phases repeat the pattern once across the columns, but there is no repetition down the rows. Original phases have no repetitions of the pattern either down the rows or over the columns.

All of the phases have the same chirality. The pattern is a solenoid about the surface of the torus. It follows the left hand rule of thumb. That is, it is made of stripes parallel to the primary diagonal.

There is a common mechanism for advancing from the first phase in the cycle to the second for each of the three rules:

(1) Reverse the order of the cells in each row of the first phase.
(2) Then reverse the order of the rows of the first phase.
(3) Finally, exchange the rows and columns of the first phase (Transpose).

The mechanism for advancing from the second phase in the cycle of phases back again to the first phase is the same but actions are applied to the second phase.


Edited: 4/29/08
The three steps in this mechanism are 180 degree rotations about the vertical axis, the horizontal axis, and the primary diagonal axis. If you are viewing this from the opposite side, it simply appears as a counter-clockwise rotation of 90 degrees.

Correction: 4/29/08
Not counter-clockwise. Clockwise.
Also, when the three steps are performed a second time, it takes the figure back to the original. So, if you are still viewing it from the opposite side, then it is a 90 degree rotation counter-clockwise. It rocks.


Figures 1 through 3 in the attached graphic display the phases for each of the rules.

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Last edited by Lawrence J. Thaden on 04-29-2008 at 08:26 PM

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Now let’s compare these rules with three corresponding rules for z; namely:

1466461054805: (z - 1)
4399383164415: (z)
5572552008259: (z + 1)

Figure 4 presents the two cycling phases of each of these three rules.

The first thing to note is the reverse chirality. This is an affirmation of what was presented in the thread: Reflections. Namely, that it is sometimes possible to predict the chirality of a CA from knowledge of the algebraic expression of its rule.

The second observation is that certain graphs on the left hand side of Figure 4 appear to be mirror images of those on the right hand side. But this is an illusion.

However, they are related. Let’s take each pair of cycles at a time.


Left hand side: rule 193720085: (x - 1) and
.....Right hand side: rule 1466461054805: (z - 1)

Observe that phase 3 for rule 1466461054805: (z - 1) needs to be rotated 180 degrees counter-clockwise in order to even have the appearance of being the mirror image of its counterpart on the left, rule 193720085: (x - 1). But even then, the appearance is still an illusion.

However, if for each row of rule 193720085: (x - 1), we exchange every cell content of 0 for a 2 and every cell content of 2 for a 0, and then shift left one cell, we will end up with the mirror image of rule 1466461054805: (z - 1) after its rotation of 180 degrees.

For phase 4 of rule 193720085: (x - 1), it is only necessary to shift left one cell in each row in order to have the mirror image for phase 4 of rule 1466461054805: (z - 1).


Left hand side: rule 3812992433055: (x) and
.....Right hand side: rule 4399383164415: (z)

The graphs for phase 2 appear to be mirror images. But, as before, this is an illusion.

If for each row for phase 2 of rule 3812992433055: (x), we shift left one cell, then (x) will be the mirror image for phase 2 of rule 4399383164415: (z).

For phase 3, we first have to rotate the graph of rule 4399383164415: (z) 180 degrees counter-clockwise. Then for each row for phase 3 of rule 3812992433055: (x) exchange cells with a 1 for a 2 and cells with a 2 for a 1. Finally, shift left one cell in these same rows and the result will be the mirror image of rule 4399383164415: (z) after rotation of 180 degrees.


Left hand side: rule 7625210074339: (x + 1) and
.....Right hand side: rule 5572552008259: (z + 1)

Again, the graphs for phase 2 appear to be mirror images. But, as before, this is an illusion.

If for each row for phase 2 of rule 7625210074339: (x + 1), we shift left one cell, then (x + 1) will be the mirror image for phase 2 of rule 5572552008259: (z + 1).

For phase 3, we first have to rotate the graph of rule 5572552008259: (z + 1) 180 degrees counter-clockwise. Then for each row for phase 3 of rule 7625210074339: (x + 1) exchange cells with a 0 for a 1 and cells with a 1 for a 0. Finally, shift left one cell in these same rows and the result will be the mirror image of rule 5572552008259: (z + 1) after rotation of 180 degrees.

It is curious that (x) and (x + 1) as well as (z) and (z + 1) have the same requirements. I would have though that (x - 1) and (x + 1) as well as (z - 1) and (z + 1) would have had the same requirements in order to make mirror images.

But the difference in requirements follows the type of operation (modulo 3 sum and modulo 3 difference) rather than the stand alone variable versus the variable together with an identity.

Another observation: The modulo 3 difference rule 193720085: (x - 1) begins its cycle of phases with an odd numbered phase, phase 3. The modulo 3 sum operations both begin their cycle of phases with even numbered phase 2.

If you are inclined to accept that odd numbered phases unfold on the outside surface of the torus while even numbered phases are on the inside surface, then in this case cycles for a modulo 3 difference rule start from the outside and modulo 3 sum rules start from the inside surface of the torus.

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Rule 1466669662809: (-x + z) is the rule that has a cycle of phases 1944 in length, the first and second half of which are polar opposites of each other, phase for phase.

I compared it to a spinor since both cycle once to become their polar opposite and a second time to return to the original polarity. So phases 1 through 972 make one revolution and phases 973 through 1944 make the other revolution. After that, it just keeps repeating without end.

But there is more. Four pairs of phases are different from all the rest in that the bottom half or more of the rows are all with cells having 0s. And the first of each pair of these phases has polarity opposite the second of each pair. Moreover each one of them occurs every 243 phases from 1 through 1944 at phases: 1, 244, 487, 730, 973, 1216, 1459, and 1702.

We might compare these phase pairs to Riemann surface branch points of order 2 at 1, omega, omega^2, and infinity, where omega is e^(2 Pi I / 3). (See Penrose, Roger (2006). The Road to Reality: A Complete Guide to the Laws of the Universe. Alfred A. Knopf, New York, pp. 136-137.)

Notice that this list of phases varies from odd to even, and the second half of the list is polar opposites of the first list, phase for phase.

If we cut the cycles into four pairs of groups, each with odd and even phase numbers, we find that one “branch point” occurs at the beginning of each odd set of 243 phases and one “branch point” occurs at the beginning of each even set of 243 phases.

Moreover, according to Penrose’s analysis, each of the two Riemann surfaces forms a tube and the tubes are joined at omega through omega^2 branch point and at 1 through infinity branch point to form a torus. But he does not account for an inside difference on the Riemann surfaces.

Now comparing this to rule 1466669662809: (-x + z), we find that the odd phases describe the outside surfaces of two tubes and the even phases the inside surfaces. And these join at the “branch point” phases to make a torus. Going around the torus once keeps all the polarity positive and going around a second time, we see that each of the phases has opposite or negative polarity.

One point that Penrose explains is the branch point for infinity. He describes it as just going around a very large circle (cross section of the torus).

But here is where the comparison breaks down, for there are two properties that carry the notion of infinity for this rule. One is the periodic boundaries of each phase and the other is the endless cycling of the 1944 phases. The fact is that Penrose is dealing with surface branch points of order 2, but we have many more surfaces to account for.

So it’s time to back on out of this cave of comparisons.

But there is an interesting property that the 1944 phases manifest that needs no point of comparison. So let’s consider it.

If you select the first 486 rows of each phase for all 1944 phases but drop out the “branch points”, all except 72 of these subsets appear to be random. When animated, they swarm.

But they swarm with some order. For example since the second half of the 1944 phases has opposite polarity compared to the first half, the second half of these subsets, with the “branch points” excluded, have a format that is in all respects except polarity identical to the first half.

There is more.

Within each fourth of the 1944 phases, less two of the “branch points”, there are 18 anomalous phases. They are still random in appearance, at least to begin with. But they have definite features that set them apart from the rest of the swarming phases. For some of them start to self organize before the 487th row. (See Figures 1 and 2.)

Half of the 18 anomalous phases are on the outside and half are on the inside of each fourth of the full set of 1944 phases.

But what is interesting is that, among all 72 phases, there are only 9 different kinds, but each is unique. So, “different kinds” means resembling each other.

Moreover, the 9 on the inside are not in the same relative positions as the 9 on the outside. And the relative positions are more disordered in the second half of the first and second revolutions than they are in the first half of the first and second revolutions. That is, the second fourth and last fourth of the entire set of phases is less ordered on the inside with respect to the outside than the first and third fourths. (See Figure 3.)

There is one more difference. Outside phase number 649 in the second half of the first revolution has many little red triangles in the bottom section and an emerging set of three evenly spaces triangles made of little red triangles in the top section. And this phase is 4th in the list of anomalous phases for the outside second half of the first revolution. By contrast, its match (outside phase 325) in the first half is found at position 7.

Then for outside phase 1621 in the second half of the second revolution it is in position 4 with its match in the first half of the second revolution outside phase 1297 in position 7.

It might be that all of the outside and inside phases are related in some way, just as these anomalous phases are. I do not have the analytical tools to find out. However, I think that knowing how they are all related would contribute greatly to the understanding of the overall behavior of rule 1466669662809: (-x + z).

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Here is the 2nd figure.

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Here is the 3rd figure.

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The last three rules in T5 for (x and z) to be discussed are the additive inverses of the rules that have expressions (x - 1), (x), and (x + 1):

387410647: - (x - 1)
7625403764901: - (x)
3812605051931: - (x + 1)

We begin with rule 7625403764901: - (x) because it stands alone, unrelated to the other rules and because the other two rules have some things in common.

The most singular thing about rule 7625403764901: - (x) is that it only has one phase, the original. When the last column of cells is used in reverse order as initial conditions, the result is the same as the original CA.

It has 729 rows and columns and an equal number of cell types. So it has no polarity.

And it is a torus.

Its CA has the appearance of groups of solid stripes, each of one cell type (color) running parallel to the primary axis. (See Figure 1.)

We can compare it to rule 6159136430181: - (z), which has two phases that cycle, but whose solid stripes are parallel to the secondary axis.

The original phase for rule 7625403764901: - (x) is the mirror image of the original rule for rule 6159136430181: - (z), once - (z) has been rotated 180 degrees and each of its rows shifted left one cell.

The original phase for rule 7625403764901: - (x) is the mirror image of phase two for rule 6159136430181: - (z) so long as - (z) has each of its rows shifted left one cell. (See Figure 2.)


The next rule is 387410647: - (x - 1).

It has three phases and these constitute a three phase cycle. There are 729 rows and columns to each phase and each phase is a torus.

Each phase has the same number of cell types. So it has no polarity.

The appearance of each phase is of groups of stripes parallel to the primary axis. But the content of the stripes is not solid. Rather it has repetitions of vertical or horizontal patterns that range over a portion of the rainbow’s colors. And the groups are offset from one another in the following sense: some of the groups of stripes have cell values more evenly distributed, and these appear darker than the others. (See Figure 3.)

What is the mechanism for advancing from one phase to the next in the cycle of phases? I do not know, but it certainly is not obvious.

We can compare this rule: 387410647: - (x - 1) with rule: 2053045476727: - (z - 1), which has only two phases in its cycle and opposite chirality. The last phases for both rules are mirror images, provided that - (z - 1) is shifted left one cell on each row. The original phases appear at first to be mirror images, but on closer examination it is seen that they are not. (See Figure 4.)


The last rule to be considered in this thread is 3812605051931: - (x + 1).

This rule has three phases, but the cycle is phases 2 and 3. All three phases are tori with 729 rows and columns.

Each phase has the same number of cell types. So it has no polarity.

The appearance of each phase is similar to the phases of the previous rule. There are individual stripes parallel to the primary axis. But the content of the stripes is not solid. Rather it has repetitions of vertical or horizontal patterns that range over a portion of the rainbow’s colors. And the groups are offset from one another in the following sense: some of the groups of stripes have cell values more evenly distributed, and these appear darker than the others. (See Figure 5.)

I do not know what the mechanism is to pass from the first phase in the cycle to the second. In the case for - (z +1) it was that the two phases were diagonal mirrors of each other.

We can compare the phases of this rule to those of - (z + 1), which also has a cycle of two phases, but with opposite chirality. Interestingly, phase 3 of rule 3812605051931 - (x + 1) appears to be the mirror image of phase 2 of rule 3226214320571 - (z + 1), but this is an illusion.

By way of contrast, phase 2 of rule 3812605051931: - (x + 1) does not even appear to be a mirror image of phase 3 of rule 3226214320571 - (z + 1). (See Figure 6.)

Observe that with rule 387410647: - (x - 1) the comparison with rule 2053045476727: - (z - 1) there are not an equal number of phases. So the second phase for rule 387410647: - (x - 1) has no match among the phases for rule 2053045476727: - (z - 1).

Also observe that with rule 3812605051931: - (x + 1) the comparison of mirror image appearance with rule 3226214320571 - (z + 1) is between phases 3 and 2 respectively, while phases 2 and 3 respectively have no mirror image appearance. This cross over of phases was least expected.

I have not taken the time to analyze these apparent mirror images to see what adjustments are needed to make them actual mirror images.

I am going to leave this for now and go back to correct the thread for T5 for x and y, in which I failed to reverse the initial conditions. I am anticipating finding some very simple cycles of phases that are analogous to the spinor. The ones found so far have been with very many phases. But T5 for x and y might have some that have just a few phases.

Note: Because of the size of the graphics, I have had to put some of the figures in successive posts. The restriction on upload size is my internet provider’s.

At the end of the figures there is a notebook which was used to develop all of these posts on T5 in x and z. The notebook has some things not included in the posts, but for the most part it follows what has been presented here. However, the notebook presents the material in rule number sequence. (again it had to be divided into three parts because of my internet provider's restrictions on size.)

It also shows some of the evolution of my thoughts as I was learning more and more about T5 in x and z. For example, with respect to rule 1466669662809: (-x + z), I am tending to accept that the layers within the phases are not on the surface of a tube, but are like revolutions of a spiral.

Think of it in terms of a baker who has a sheet of dough. It has two sides. I have been calling these the outside and the inside. Now if the baker rolls the dough up from the bottom, he will end up with what in the cross section is viewed as a spiral. And the bottom of the sheet completes revolutions of the spiral that are very tightly wound, while the last revolution is loosely wound. These revolutions correspond to the layers down the rows of the phases, with the last revolution, the one farthest from the center of the spiral, being the first layer, the one having a swarm-like random appearance. But there is another thing the baker can do once the sheet is rolled. He can join the right and left ends together. This corresponds to the phases having periodic boundary conditions.

This describes any phase except the eight phases that have a predominance of cells with 0 contents. And there are 243 of these phases for the outside and 243 for the inside within what was compared to the spinor’s Pi radians rotation in complex space. After the 243rd there is the phase with the predominance of cells with 0 content. Then another 243 which brings us full circle where polarity switches and the process begins again.

It as though the baker bent two of his spiral creations into a circle (cross section of the torus), with the spaces between corresponding to those phases with a predominance of cells having 0 content. And then ramped up a second round of his creations on top, corresponding to the spinor’s second 2 Pi radians rotation in complex space. But this baker’s two creations have the strange property of changing appearance 242 times on each level of the two ramps before repeating the change process.

Those pairs of spaces, four in all, might be compared to the branch points of order 2 for Reimann surfaces.

But this comparison to a baker’s creations leaves out the detail of what is happening on the rotations of the spiral. In the model each phase is observed to self organize into greater numbers of tori on each rotation in its journey to the center of the spiral. It does this by powers of three.

The first power of 3 surface is flat. The rest of the powers of 3 surfaces are increasing numbers of tori arranged around the object: 3^1 tori, 3^2 tori, 3^3 tori, 3^4 tori, 3^5 tori, and 3^6 tori. This last, 3^6, is always a row of cells with each cell content 0, corresponding to the tightest revolution of the baker’s spiral as he begins to role the sheet of dough.

If this has any relevance to reality, it might be that traveling through the swarm-like layer and reaching the more structured layers will bring us finally in spiral fashion to a ring-like singularity. That is, to physical existence that corresponds, in our model, to rows of discrete cells of the same stuff. But that stuff for this rule is cell content 0.

However, if we just remained looking at the model without journeying into the spirals we would only observe the 243 phases of the swarm-like layers, repeating over and over again.

But, amazingly, while there is no net polarity for any of the polar opposite phases, there is an overall positive net polarity for these swarm-like layers for both polar opposites.

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Here are more of the figures.

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Here is the last of the figures.

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Here is part 1 of the notebook.

Attachment: t5 = ( {0, +1, x, z, -1}; +, -; 0, +1, -1) part 1.nb
This has been downloaded 1081 time(s).

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Here is part 2 of the notebook.

Attachment: t5 = ( {0, +1, x, z, -1}; +, -; 0, +1, -1) part 2.nb
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Here is part 3 of the notebook.

This notbook and even its parts create very large output, so the warning is do not expect to be able to execute all of the cells at once, unless you have enormous resource capacitiy on your system.

Attachment: t5 = ( {0, +1, x, z, -1}; +, -; 0, +1, -1) part 3.nb
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