Lawrence J. Thaden
Registered: Jan 2004
Posts: 356 
Rules with expressions  (x  1),  (x), and  (x + 1)
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 swarmlike 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 swarmlike layer and reaching the more structured layers will bring us finally in spiral fashion to a ringlike 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 swarmlike 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 swarmlike layers for both polar opposites.
Lawrence J. Thaden has attached this image:
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L. J. Thaden
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