[(-x +z) Self Organization Example] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
Pages:1
(-x +z) Self Organization Example
(Click here to view the original thread with full colors/images)
Posted by: Lawrence J. Thaden
Figure 1 lists a set of 27 three color rules that correspond to algebraic expressions for a ternary logic in two variables:
T5 = ({0, +1, x, z, -1}; +, -; 0, +1, -1)
where the operations, + and -, are modulo 3 sum and difference and
where the three identities are,
0: additive identity
+1: positive multiplicative identity and
-1: negative multiplicative identity.
T5 is a closed set. It is an example of a nondegenerate, complemented, distributive lattice.
Notice that this logic is very similar to that in another thread on the forum. The difference is that this logic has variables x and z, whereas the other logic had variables x and y. This might not seem like much of a difference until you consider that the variables represent three color rules, and the three color rule for y is 3812992433055, but the three color rule for z is 4399383164415. Actually, there is a very large difference in the behavior of these two algebras, even though at times there are instances where their behavior matches up.
However, this post addresses only one of these algebras as mentioned at the outset, and to begin with it also emphasizes just one of the twenty-seven rules, 1466669662809: (-x + z).
This rule can be characterized as self organizing.
Yet this property is not immediately evident from the first output of the cellular automaton as illustrated in Figure 2. There we see 253 time steps before a one row cycle begins and continues on as long as the cellular automaton runs. Figure 2 cuts it off at 729 time steps.
But if we use a phased approach by taking the last column of Figure 2 as initial conditions and continue with the phase 2 cellular automaton for 729 steps, we get Figure 3. And this one does manifest self organization. For the first two thirds of the rows have random cell content. Then the last one third of the rows are a set of six layers, each one more organized than the previous, with the last layer being 729 cells with the same content.
And it appears that each successive phase has a similar structure, but with different content, at least out to the 40,000th phase. (Beyond this I have yet to compute it.)
But let’s start at the beginning, the choice of initial conditions for the original phase. On page 881 of the NKS book Wolfram mentions that in 1983 his study of cellular automata "concentrated on characterizing behavior obtained from all possible initial conditions."
No doubt he means that he ran many cellular automata with the same rule but all possible initial conditions and then analyzed the bulk of output in an attempt to categorize cellular automata types.
But I asked myself: Might it not be possible to represent all possibilities in one set of initial conditions? With that in mind I came up with the idea of studying simple closed sets of three color rules, such as this one with variables x and z. And as a starting point, I decided to use the twenty-seven members of the closed set as initial conditions, representing each in its digit expansion form and maintaining a sequential order of the positions of these rules from left to right in the line of initial condition digits. So I arrived at a set of initial conditions 729 digits in length. And these are just the concatenation of entries under the digit expansions column in Figure 1.
I reasoned: These initial conditions should reflect the fact that every possible rule within the closed set is having an initial effect in the subsequent operation of the cellular automaton.
Moreover, I decided to use the phased approach, taking the last column from the first cellular automaton output as initial conditions to a subsequent cellular automaton. Again, I reasoned: The operation of the rule on the digit expansion of the closed set of three color rules was the causal precedent to this last column. So continued operation on the column will produce a chain of causal results derived from the original precedent.
This choice to use the phased approach was also influenced by my previous discovery with rule 3681843964019 (-1 - y), where subsequent phases were the previous phase with rows and columns switched and then the rows shifted left one cell. The effect was the same as rotating the cellular automaton output 180 degrees about its primary diagonal and then shifting it all over to the left one cell. But since the cellular automata that we are exploring take place on the surface of a tube or torus, this amounts to looking at the inside surface of the cellular automaton after generating the outside surface. It is as though we are following a causal chain of events where the outside precedes the inside and then the inside precedes the outside surface. Oscillating, if you will. So even though I may not be able to express as clearly as in the case of rule 3681843964019 (-1 - y) how the oscillation steps occur, I am lead to believe that the phased approach will lead to interesting behavior. For instance, it supports the concept of persistent structures that are tubes rather than tori.
Certainly, with the rule being discussed now, there is most interesting behavior. For rule 1466669662809: (-x + z) manifests a pattern of self organization consistently predictable from the second phase onward. And this happens from a set of fixed, well defined initial conditions.
I am aware that Wolfram discusses self organization in chapter 6 of the NKS book. But that discussion always proceeds from random initial conditions.
Here we have the reverse. The initial conditions are not random. They are causally determined from the digit expansions of the closed set of three color rules. But right out of the starting gate, with phase two onward, the first two thirds of the rows of cellular automaton output are always random. And thereafter the cellular automaton organizes until it reaches lowest entropy, a row of cells that are all the same.
And phase after phase this behavior is repeated with different content. In fact, as the self organization occurs it always proceeds through six steps, each with its unique number of rows, varying only slightly from phase to phase.
Step 0: 1 object horizontally arranged 729 columns by 486 rows.
Step 1: 3 equivalent objects horizontally arranged 243 columns by nearly162 rows.
Step 2: 9 equivalent objects horizontally arranged 81 columns by 54 rows.
Step 3: 27 equivalent objects horizontally arranged 27 columns by 18 rows.
Step 4: 81 equivalent objects horizontally arranged 9 columns by 6 rows.
Step 5: 243 equivalent objects horizontally arranged 3 columns by 2 rows.
Step 6: 729 equivalent objects horizontally arranged 1 column by 1 or more rows.
The “nearly” and “one or more” phrases highlight that sometimes step 1 drops a few rows and step 6 picks up these rows. But step 0, the random part is consistently the same number of rows.
Notice that the number of objects (or partitions) is a series of powers of three. The step zero is the random section. It has 3^0 objects. Then subsequent steps have these numbers of objects: 3^1, 3^2, 3^3, 3^4, 3^5, and 3^6.
From observing this regular behavior one is tempted to say that random and non-random are on a scale of self organization. However, it is customary to make the clear distinction between random and non-random. So I will remain silent. Although it is tempting to ask: What would happen if there were a step (-1)? Is there such a thing as negative organization?
One final note: the steps are layered. Step 6 fits under step 5 fits under step 4, etcetera through step 1. From this viewpoint step 1 only gets completely organized by the time it reaches step 6 and it includes the layers below it. Self organization is seen as discretely progressive. (See Figure 4 which is the leftmost instance of three completed boxes for step 1. It is broken out into layers.)
This behavior evokes associations from cosmology and subatomic particle collisions. Both with cosmology that is based on general relativity and the collisions that are based on quantum mechanics there are well defined initial conditions, and then there are instantaneous random manifestations of energy that quickly cool down to organized objects. Compare this with the fixed initial conditions that immediately produce 486 rows of random results before self organizing through six distinct steps into the observed local structures of the cellular automaton.
I hope that some on the forum will want to play with this. In the following post I have attached a simple notebook to get you started.
Posted by: Lawrence J. Thaden
Attached is the notebook that goes with the previous post.
Posted by: Lawrence J. Thaden
I have been uncomfortable about rule 1466669662809: (-x + z) not having a cycle of phases. Since the original post in this thread, I have run the phases 5 million times without detecting a cycle.
Then it hit me. The order of the initial conditions begins on the left with all 0s and ends up on the right with all 2s. This is inconsistent with place value mechanisms. What would happen if the order were reversed?
So I reversed the order, and the effect was to generate left-right mirror images of the phases from the original run, again with no cycle, at least out to 4 million phases.
But what I had overlooked is that the last columns were still being fed, as initial conditions, into successive phases in non-reversed order. That is, the least organized part was on the left and the most organized part was on the right of the row of initial conditions for successive phases. That is, high to low entropy, left to right.
When I reversed this order as well, the cycle appeared after 1943 phases. Phase 1944 is identical to the original phase.
Moreover, the last phase has a slight indication that something is about to happen. There appears to be a hint of structure in its random section. (See Figure 1.) But the last one third of the rows still carry out a self organizing process through the powers of 3^1 to 3^6, as do the other phases.
That the phases cycled to the original, led me to ask: how many phases are like the original in that they have many more than one row with uniform cell content?
It turns out that every 243 phases (seven times, not counting the original) there appears a phase that has approximately two thirds of the rows with cells of uniform content (all 0s). And it takes another phase to get the process back into the pattern.
Remarkably, not one phase immediately previous to each of these 243 phases shows any hint about what is going to occur on the next phase with the exception of phase 971. This phase is at the half way point and is followed by a phase that looks almost identical to the first phase.
Figure 2 shows four phases for each of these seven cases. The ones on the left appear right before the 243rd and its multiples. Next is the 243rd phase or one of its multiples. And then one immediately following, showing that things are not yet back to normal. Finally, on the right is the next phase, where the pattern resumes.
For those who downloaded the notebook, here are the two changes to reverse the order of the initials.
Initial conditions were specified as:
inits = Flatten[Table[IntegerDigits[seed[[i]], 3, 27]
Should be specified as:
inits = Flatten[Table[IntegerDigits[Reverse[seed][[i]], 3, 27]]
Last column was picked out with:
inits = Table[lastCA[[i, Length[inits]]], {i, Length[inits]}];
Should be:
inits = Reverse[Table[lastCA[[i, Length[inits]]], {i, Length[inits]}]];
This change affects all of the rules and results previously posted with respect to these simple algebras in two variables. For example, rule 3681843964019: (-1 - y), where successive phases after the original had the rows and columns switched and then all the rows were shifted left, turns out to be the same for successive phases without transposing and shifting. So the inside of the torus for rule 3681843964019: (-1 - y) is identical to the outside. Also, the orientation of the rainbows is perpendicular to the orientation observed when initials were not reversed. That is, the rainbows are now perpendicular to the secondary axis, not the primary axis. (See Figure 3, attached to the next posting.)
Posted by: Lawrence J. Thaden
Attached is figure 3.
Posted by: Lawrence J. Thaden
In the previous post on rule 1466669662809: (-x + z) I mistakenly forgot to count the original phase as phase 1. So instead of 1943 phases there are 1944 phases in the cycle of phases for this rule.
This also means that the figures are incorrectly labeled. The label for Figure 1 should be: 1466669662809: (-x + z) Reversed Phase 1944.
In Figure 2, all of the phase numbers should be 1 greater than as represented in their labels.
Now that that admission is behind me, let’s consider the relation between four pairs of phases in the second column of figure 2.
Using the corrected phase numbers, the pairs are:
1 and 973
244 and 1216
487 and 1459
730 and 1702
Notice that every other pair is made up of odd or even numbered phases.
And if even numbered phases are on one side of space time and odd numbered are on the flip side, then there is an equal representation from both sides of space time.
But here is the relation that adheres for each pair: The red is the same in both phases of each pair while the green and blue switch places. That is, zero is always zero while 1 and 2 cell values switch. But 2 in this algebra is (-1). So the paired members can be considered as having opposite polarity.
Also, the absolute difference between the phase number pair members is the same, 972. So not only are the phases evenly spaced 243 apart, the pairs also manifest a regularity of frequency.
It will be interesting to watch for the other rules in this closed set (T5 = ({0, +1, x, z, -1}; +, -; 0, +1, -1) to see if they have phases that intersect these pairs. Or, if not intersect, are related in some other way.
A word about the geometry: It has been noted that each phase except the paired phases begins with two thirds of the row having no pattern. Then the last one third of the rows begins to self organize through a power series from 3^1 through 3^6. That is, same cell content emerges over this span of powers. So first to emerge within each phase is three sets of 243 cells and the sets are the same. And the last to emerge is a single set of 729 cells all with the same content.
But if these phases are allowed to continue past row 729, there always occurs a cycle of a single row of 0s. So the geometry as we are letting it run is that of a tube. If we let it run a single row more, the geometry would be a tube with a lip on it.
How though do we picture the geometry as it advances to the second phase and beyond?
I do not know. But one image occurred to me comparable to a slice through a pile of soap suds. As one phase completes, the next phase begins on the flip side of space time. So a first tube is covered on the inside and outside surfaces. Next, the third phase begins to make a tube in the same space time as the first, but adjacent to it, generating space time for itself. And this goes on until 972 tubes are made. At which point it is back to the original, beginning a new cycle.
And those irregular pairs of phases from Figure 2 must somehow be prominent points in the geometry, something like twists or pass-thrus.
But as I said: I don’t know. In fact, it is evident I have trouble keeping track of something as simple as phase numbers!
Posted by: Lawrence J. Thaden
Column 2 of Figure 2 in the previous post had phases that could be paired, the absolute difference in phase number being 972. And these pairs were seen to be polar opposites. That is, while both members of each pair maintained an unchanged position for each 0, the 1s and 2s switched cell values. And since 2s behave as -1s in this logic, the switch in cell values from 1 to -1 and from -1 to 1 amount to phases that are polar opposites of each other.
But going further, each of the 1944 phases pair up with another phase in this same manner. And since the phase numbers of the pairs are all the absolute difference 972, we can look at the geometry of the phases as unrolling in space time from the original up to phase 972. At which point they continue unrolling over the same space time (or under it) but as polar opposites. This goes on until phase 1944. At which time the original CA repeats and there is a cycle of phases.
Here is the code to check this out.
First, code to check that 0s do not change in either member of the pairs.
Union[Table[(collectedCAs[[i]] /. {0 -> 1, 1 -> 0, 2 -> 0}) == (collectedCAs[[i + 972]] /. {0 -> 1, 1 -> 0, 2 -> 0}), {i, 972}]]
{True}
Next, code to check that cell values 1 and 2 get switched to 2 and 1.
Union[Table[(collectedCAs[[i]] /. {, 1 -> 2, 2 -> 1}) == collectedCAs[[i + 972]], {i, 972}]]
{True}
You would think that any geometric arrangement of polar opposites of this kind would collapse to all zeros. But remember that there is another activity that might prevent this. Every other phase is in space time “anti” to that previous to it. This might act as a preventive buffer from the collapse of the polar opposites.
Posted by: Lawrence J. Thaden
It is remarkable that the second half the phases are polar opposites of the first half.
But each phase begins with 486 rows that appear to have random cell values and positions. In the original post of this thread, this is referred to as the 3^0 section since it has no structure greater than the 1 object 486 rows by 729 columns. It is the first step in self organization which proceeds through levels 3^0 thru 3^6. Each successive level having more distinct equivalent objects.
However, since each phase has its polar opposite, what does random really mean? For clearly we see in one member of a pair of phases a random pattern which in the other member has opposite polarity but in all other aspects is the same. How can we say that the 486 rows of one member of a pair have random cell content when the other member of the pair has the exact same pattern in these 486 rows, only with opposite polarity?
Maybe there is one kind of random and then there is random of another definition. Maybe random that is subject to self organization is different from random that no rule can self organize.
It would be interesting to hear if Wolfram or others that have access to supercomputers have ever run ECA 30 out until it cycles. I am curious to know if its behavior shows a polarity difference at midpoint. But then, again, perhaps it takes more than two states to set up polarity.
Posted by: Lawrence J. Thaden
Rule (8) in the list for T5 = ({0, +1, x, z, -1}; +, -; 0, +1, -1) is 2053105087449: (x - z).
This is very similar to rule (6) 1466669662809: (-x + z) which we have been discussing.
It is another example of self organization with the same structure as (-x + z). That is, most all its phases start out with 486 rows of random cell content which self organize through levels 3^0 through 3^6 so that the last level has uniform cell content over its rows.
And it also has sets of phases that are polar opposites each other.
However, there are differences. Whereas (-x + z) has 1944 phases in its cycle, (x - z) has 953 phases. And instead of eight phases outstanding for the number of rows of zero cell content, there are only two, the original and one at phase 468 which is its polar opposite.
And this is interesting because the original, phase 1, is on one side of space time while phase 486, being an even integer, is on the other side of space time.
Along with this is the observation that the total number of phases in the cycle is an odd number. This means that there will not be a match one to one of polar opposites.
In fact, it is even stranger than this. Polar opposites for phases 2 thru 467 are found at phases 488 thru 953. This is 466 pairs of phases. Or 932 phases in all.
Phase 1 does not have a pair because it is on the wrong side of space time.
That leaves 20 phases, 468 through 487, none of which pair with themselves.
So where do we find the polar opposites for these 20 phases? Amazingly, there they are at the first 20 phases in the next cycle of phases that starts off on the other side of space time.
It is not until the third cycle of phases begins that we are back on the same side of space time that we started with.
And the original phase is still hanging there unpaired, while the 20 unmatched phases 468 through 487 from the last cycle are always “looking” for a next cycle for their polar opposites.
If, by the third cycle of phases, it somehow gets back to where it started maybe it will overlay what is there, pairing up with the first 20 phases leaving a net polarity in the cycle. The alternative is that it just keeps cycling anew creating space time as it goes.
I have tried to imagine geometry for this. But it is hard. With (-x + z) it was easy. Everything just looped back to the halfway point and formed a second loop with opposite polarity.
But for (x - z) there is an interruption with 20 phases being paired in the next cycle. How do you represent a loop that is interrupted?
Perhaps a seamstress is more qualified to address the problem. When tailors sew by hand they work on opposite sides of the cloth. They can interrupt an interval by going completely through the cloth to the other side, do something there and come back to the exact same spot they left off and then continue on.
If some geometry based on this idea could be formulated, you could have a loop with a broken interval, and the part taken from the break in the interval could be paired up with its polar opposite on the other side of space time in the next cycle.
Perhaps it would look like a series of figure eights, but up close it would be seen that there was no intersection at the meet of the top and bottom halves of the figures. The top half would represent the next cycle. The bottom would represent the previous cycle.
Also, the figure would be made of a double line of little rolls which are the phases and their polar opposites. And each little roll starts off with random circumference content and ends up uniform.
Posted by: Lawrence J. Thaden
Post in haste. Eat crow at your leisure.
I was working in the dark so as not to wake anyone when I read the number of phases as 953. It should have been 972. So this rule does not have a broken interval and the nonsense about the seamstress and figure eights is best forgotten. With that misinformation I succeeded in spinning quite a convoluted error.
In fact, rule 2053105087449: (x - z) has half the cycles that rule 1466669662809: (-x + z) has. And the second half the cycles are polar opposites of the first half. The structure of both is the same. It just gets (x - z) done in half the time.
Posted by: Lawrence J. Thaden
Whereas rules 1466669662809: (-x + z) and 2053105087449: (x - z) are examples of mechanisms illustrating self organization, this rule is an example of a mechanism of self destruction.
And where the previous two rules exemplified self organization again and again phase after phase, this rule demonstrates the systematic breakdown of structure through a process that takes just eight phases. (See attached figure.)
A highly structured and complex CA goes through these phases, each becoming more uniform, until the last phase is a CA with every cell content zero.
Phase 1: Overall random appearance of localized objects. 486 rows without a cycle. This is not the same random appearance seen with the first 486 rows of the phases for rules 1466669662809: (-x + z) and 2053105087449: (x - z). There the random appearance had much less structure. And the localized objects were more uniform. Here the 486 rows form a cycle. For row 487, if it is generated, is the same as the initial conditions.
Phase 2: Local concentration of individual cell values in a variable pattern gives way to a uniform pattern of vertical concentrations of cell values.
Phase 3: Another uniform pattern but horizontal and less complex.
Phase 4: Another vertical pattern, still simpler. Appears that only the red cells retain some semblance of the original pattern.
Phase 5: Diffusion of cell values to the point that color is less observable. Horizontal or vertical orientation no longer tied to odd and even phases. Rather, more associated with colors.
Phase 6: Marked differentiation between arrangements of cell values that are 0 and those that are either 1 or 2, together with a total loss of orientation as to horizontal or vertical.
Phase 7: A return of horizontal and vertical properties, but assigned to cell value rather than associated with odd or even phase number. This is like a last gasp before the loss of all structure.
Phase 8: Finally, complete uniformity with all cell values zero.
All phases 2 through 8 have the last column of the previous phase as initial conditions.
Here is a more detailed description of the structure of phases 2 through 8. In each phase there various numbers of cycles. These can be considered as forming tori. Also, in each phase the number of cells that have 1s is the same as the number of cells that have 2s. What changes from phase to phase is the number of cells that have zeros. This number always increases or decreases by a factor of two.
Phase 2: Vertical appearance. Six cycles of 81 rows, but no cycles among columns. Seems to have attempted a second cycle of columns but failed. Overall pattern is dissipating and replication is taking its place. There is a fracture running down just left of center that is not exactly vertical. Equal number of cells with 1 and with 2. Slightly fewer cells with 0 than the original phase. This array of tori is a stack 6 high.
Phase 3: Horizontal appearance. Three cycles of 162 rows, and six cycles of 81 columns. Further dissipation of original pattern and overall increase in replications is taking its place. There is a fracture just below the center running across that is not exactly horizontal. Equal number of cells with 1 and with 2. Slightly fewer cells with 0, but an increase over the previous cycle. This array of tori is 6 wide by 3 long.
Phase 4: Vertical appearance. Eighteen cycles of 27 rows, and three cycles of 162 columns. Further dissipation of original pattern and threefold increase in row cycles is taking its place. Colors are beginning to aggregate along vertical lines. Equal number of cells with 1 and with 2. Significantly lower number with 0, showing a decrease over previous cycle. This array of tori is 3 wide by 18 long.
Phase 5: Loss of orientation, so strictly neither a horizontal nor a vertical appearance. Eighty one cycles of 6 rows, and fifty four cycles of 9 columns. No trace of original pattern, and four and a half fold increase in row cycles is taking its place over the previous cycle, while column cycles are increasing eighteen fold. Colors are regularly beginning to fade because of similar local distributions of cell values. For the first time equal number of cells with 0, 1, and 2. This array of tori is 54 wide by 81 long.
Phase 6: Loss of orientation, so strictly neither a horizontal nor a vertical appearance. One hundred sixty two cycles of 3 rows, and the same number of cycles of alternating 4 and 2 columns. This is the only phase that has alternating numbers of columns in its cycles. Original pattern has completely given way to uniformity. A two fold increase in row cycles is taking its place over the previous cycle, while column cycles are increasing three fold. Colors are regularly continuing to fade because of similar local distributions of cell values. Again, equal number of cells with 0, 1, and 2. This array of tori is 162 wide by 162 long.
Phase 7: Return of vertical orientation due to concentration of cells with zero (red stripes) and even distribution of cells with 0, 1, and 2 (gray spots). Two hundred forty three cycles of 2 rows, and one hundred sixty two cycles of 3 columns. That is an increase of one and a half fold for the rows and retention of the same amount for the columns over the previous cycle. Uniformity has weakened because of the segregation of most of the cells with 0 (red stripes). It appears like a last gasp that might turn things around. Again, equal number of cells with 0, 1, and 2. This array of tori is 162 wide by 243 long.
Phase 8 has cell content all 0s. It is finished.
At first it appears that all of the even numbered phases will be vertical. But this is no longer the case by phase 6. By then vertical and horizontal seem to have no relation to even and odd.
Similarly, the odd numbered phases began to appear horizontal. But by phase 5 this is no longer the case.
By phase 5 most color is absorbed. Phases 5 and 6 have large gray areas. But some color returns on phase 7, and on phase 8 every cell has zero as content and so it is all red.
If this mechanism of self destruction is applicable to any physical process, I should think that phase 6 will be most telling. For this is the only phase that has alternating column cycle widths, a distinct feature that should be detected with unequivocal physical results.
If the arrays of tori themselves curl up, then these arrays of tori are the surface of the curled up tori.
Posted by: Lawrence J. Thaden
Here is another example of self destruction.
As with rule 3226154728017: (-x - z), a highly structured and complex CA goes through eight phases, each becoming more uniform, until the last phase is a CA with every cell content zero.
In contrast to that rule, this rule has phases with half the number of rows, 486 for the previous rule compared to 243 for the present rule.
Moreover, in that rule every phase has an equal number of cells with 1 and 2 as content. But in this rule the first three phases have a net polarity.
That is, the number of cells with 1 as content is not equal to the number of cells with 2 as content.
Here is a summary description of the phases.
The original phase completes a cycle within 243 time steps. But it itself does not have any cycles either within the rows or columns.
Using the last column of this original CA we generate phase 2.
Phase 2 contains 3 cycles each 81 rows long, but no cycles among the columns. It can be considered a stack of 3 tori.
Phase 3 contains 3 cycles each 81 rows long, and 3 cycles each 81 columns wide. This array of tori is 3 wide by 3 long.
Phase 4 contains 9 cycles each 27 rows long, and 3 cycles each 81 columns wide. This array of tori is 3 wide by 9 long.
Phase 5 contains 81 cycles each 3 rows long, and 27 cycles each 9 columns wide. This array of tori is 27 wide by 81 long.
Phase 6 contains 81 cycles each 3 rows long, and 162 cycles alternating 1 and 2 columns wide. This is the only phase that alternates column cycle widths. This array of tori is 162 wide by 81 long.
Phase 7 contains 243 cycles each 1 row long, and 81 cycles each 3 columns wide. This array of tori is 81 wide by 243 long.
Phase 8 has cell content all 0s.
In the first half odd phases appear horizontal and even phases appear vertical.
Then beginning with phase 5 this pattern switches. Phase 5 and 7 appear vertical and phase 6 appears horizontal. The last phase has no orientation since it is uniformly 0s.
By phase 5 most of the pattern has disappeared and given way to uniform row and column structure.
As stated above, phases 1 through 3 do not have equal numbers of cells with 1 and 2. That is, their phases have a net polarity.
I have not delved into whether this polarity is evenly dispersed over the arrays of tori or whether it has concentrations of density.
Phases 4 through 7 have equal numbers of cells with 1 and 2. In fact, phases 5 through 7 have equal numbers of cells with 0, 1, and 2.
Here are the cell content type summaries for each of the phases:
Phase.......0s........1s........2s...Polarity
1.........59220, 58788, 59139 Negative
2.........20124, 19422, 19503 Negative
3.........19143, 20439, 19467 Positive
4.........21627, 18711, 18711 None
5.........19683, 19683, 19683 None
6.........19683, 19683, 19683 None
7.........19683, 19683, 19683 None
8.........59049, ........0, ........0 None
The reason for the high counts in the first phase is that it is 729 cells wide by 243 cells long while the other phases are 243 by 243.
Polarity is measured as negative when the predominance of cell content is 2. This is because 2 in T5 algebra acts as a (-1).
Notice that this rule 5572343404797: (x + z) is a modulo 3 sum expression, whereas the other example of self destruction, rule 3226154728017: (-x - z), is a modulo 3 difference expression. So the type of operation is not a source of explanation for why the phases manifest self organization or self destruction.
As with rule 3226154728017: (-x - z), so also here, if this mechanism is applicable to any physical process, I should think that phase 6 will be most telling. For this is the only phase that has alternating column cycle widths.
If the array of tori within a phase curl up to form a single torus, then these tori within the array are the surface of the single torus. This holds true for each phase.
Attached is a graphic depicting the eight phases.
Posted by: Lawrence J. Thaden
In the previous post I stated that modulo 3 sum could not be the source for an explanation of why expressions self destruct.
However, if (x + z) is looked upon as the positive of - (x + z), then the two rules that self destruct both are expressions of modulo 3 sum.
This can be seen from the following Mathematica code:
Let the rule for (-1) be 7625597484986 and the rule for (x + z) be 5572343404797.
Then
FromDigits[Mod[IntegerDigits[7625597484986, 3, 27] IntegerDigits[5572343404797, 3, 27], 3], 3]
3226154728017
But rule 3226154728017 corresponds to the expression (-x - z).
So (-1) (x + z) == (-x - z), and both expressions (x + z) and its negative - (x + z) are modulo 3 sum expressions.
However, whether modulo 3 sum is the source of an explanation of why these two expressions self destruct awaits further evidence.
Posted by: Lawrence J. Thaden
Not all modulo 3 sum expressions terminate in self destruction. That is, the final phase of the rules may not always be with every cell content zero.
For example, this is the case with the “rainbow” rules, where the final phase is a set of repeating diagonal rows of some arrangement of uniform 0s, 1s, and 2s.
This post considers another modulo 3 sum example that does not self destruct: rule 1466610065713: (1 + x + z). The original phase has 243 rows per cycle.
Attached is a graphic that shows the eight phases for this rule. It ends with phase eight repeating endlessly.
In the next post is another rendering of phase eight, rescaled to manifest its “rainbow” properties. Notice that the orientation is perpendicular to the secondary diagonal.
Here are the orientations of each of the phases. Orientation here means physical appearance and is based upon the frequency of sub-cycles, either row or column, as well as the density alignment of same cell values with respect to row and column.
Phase Orientation
1.........None
2.........Vertical
3.........Horizontal
4.........Vertical
5.........Vertical and horizontal
6.........Vertical
7.........Horizontal
8.........Diagonal
Here are the cell content type summaries for each of the phases:
Phase.......0s........1s........2s...Polarity
1.........59058, 59031, 59058 Negative
2.........19503, 19800, 19746 Positive
3.........19305, 19872, 19872 None
4.........18711, 20169, 20169 None
5.........26244, 19683, 13122 Positive
6.........39366, 19683, ........0 Positive
7.........19683, 19683, 19683 None
8.........19683, 19683, 19683 None
Here is a summary of the sub-cycles within rows and columns:
Phase...Row Sub-cycles......Column Sub-cycles
1...........No sub-cycles..........No sub-cycles
2...........3 of 81 rows each...No sub-cycles
3...........3 of 81 rows each...3 of 81 columns each
4...........9 of 27 rows each...3 of 81 columns each
5...........81 of 3 rows each...27 of 9 columns each
6...........243 of 1 row each...81 of 3 columns each
7...........81 of 3 rows each...243 of 1 column each
8...........81 of 3 rows each...81 of 3 columns each
Notice that there is no phase with alternating column widths as there is with the self destructive expressions considered previously.
If this is a source of explanation of some physical process, the telling aspect should be that an original negative polarity changes from positive to non-polar twice before settling on non-polar.
In contrast, the self destructing rule 5572343404797: (x + z) begins with negative polarity and only after another cycle changes to positive polarity for one cycle before going non-polar and staying that way.
And in contrast with the other self destructing rule 3226154728017: - (x + z), previously considered, none of that rule's phases had a polarity.
Posted by: Lawrence J. Thaden
Here is phase 8 rescaled to bring out the "rainbow" aspect of the repetitative diagonal cell values.
Posted by: Lawrence J. Thaden
Rule 2053254080189: (-1 - x - z) = - (1 + x + z) is the negative of rule 1466610065713: (1 + x + z).
Both are modulo 3 sum expressions. We just saw that rule 1466610065713: (1 + x + z) did not self destruct, but ended with an endlessly repeating phase that had the appearance of rainbows perpendicular to the secondary diagonal.
Now the question is: what will its negative version end up as?
So here is rule 2053254080189: - (1 + x + z). The original phase has 486 rows per cycle, twice as many as the positive version of the expression.
But there is another difference. It has nine phases instead of eight. And the last two phases form an endlessly repeating cycle, each switching columns for rows and rows for columns. That is, each of the last two phases is equivalent to the other by a rotation of 180 degrees about the secondary diagonal.
Attached is a graphic that shows the nine phases for this rule.
Here are the orientations of each of the phases. Once again, orientation means physical appearance and is based upon the frequency of sub-cycles, either row or column, as well as the density alignment of same cell values with respect to row and column.
Phase Orientation
1.........None
2.........Vertical
3.........Horizontal
4.........Vertical
5.........Vertical
6.........Horizontal
7.........Vertical
8.........Vertical
9.........Horizontal
Here are the cell content type summaries for each of the phases.
Note that the last two, which form the endless cycle, are not only net negative, but absolutely negative. They do not have any positive cell content.
Phase.......0s.........1s...........2s.....Polarity
1.........118359, 117576, 118359 Negative
2...........78984, ..78228, ..78984 Negative
3...........79002, ..78192, ..79002 Negative
4...........80676, ..74844, ..80676 Negative
5...........78732, ..78732, ..78732 None
6...........78732, ..78732, ..78732 None
7...........78732, ..78732, ..78732 None
8.........118098, ..........0, 118098 Negative
9.........118098, ..........0, 118098 Negative
Here is a summary of the sub-cycles within rows and columns:
Phase...Row Sub-cycles.............Column Sub-cycles
1.............No sub-cycles..............No sub-cycles
2...............6 of ..81 rows each....No sub-cycles
3...............3 of 162 rows each.......6 of ..81 columns each
4.............18 of ..27 rows each.......3 of 162 columns each
5.............81 of ....6 rows each.....54 of ....9 columns each
6...........162 of ....3 rows each...162 alternating 4 and 2 columns each
7...........243 of ....2 rows each...162 of ....3 columns each
8...........486 of ....1 row each.....243 of ....2 columns each
9...........243 of ....2 rows each...486 of ....1 column each
Notice that phase 6 has alternating column widths whereas the positive version, rule 1466610065713: (1 + x + z) had none.
If these sub-cycles are curled up both horizontally and vertically, they form tori. Then we can regard each phase as being an array of tori with dimensions the same as the number of sub-cycles.
If we further consider the entire array as being curled up into a torus, then the surface of the torus is composed of these tori.
Finally, if we consider each of the odd numbered tori arrays being the outside surface of the torus and each of the even numbered arrays being the inside surface of the torus, then we have an object with a dynamically changing shape.
It begins with a flat outside surface. Next the outside surface curls up into 3 by 6 tori. Then 81 by 54. Then 243 by 162. Finally, 243 by 486 which remains the same indefinitely.
On the inside it proceeds from flat to 6 by flat to 18 by 3 to 162 by 162 to 486 by 243 which remains the same indefinitely.
It is easy to imagine why some astrophysicists refer to space as being like suds.
If this is a source of explanation for some physical process, the telling aspects should be:
(1) the alternating column widths in phase 6,
(2) no phase with positive polarity,
(3) and an absolute negative polarity in the endlessly repeating two phases.
So modulo 3 sum expressions are certainly not exclusively related to self destruction, for now we have two examples of modulo 3 sum expressions that do not end up in all zero content. Perhaps a more generally applicable description of modulo 3 sum is self simplification.
But there are still more modulo 3 sum expressions in T5 = ({0, +1, x, z, -1}; +, -; 0, +1, -1) to be examined. We will have to see how they behave before making any final statements as regards this property of modulo 3 sum.
Posted by: Lawrence J. Thaden
Recall that this thread started with a rule that self organizes. Namely, 1466669662809: (-x + z). But this is an instance of a modulo 3 difference expression.
We observed:
(1) The rule is self organizing.
(2) It has 1944 phases.
(3) The second half of the phases are polar opposites of the first half.
As regards the third point, recall that for (-x + z) phase 973 was almost like phase 1, but then it was observed to complete another cycle up to phase 1944. And although the rule cycles back to the original, the second time around the phases are polar opposites of the first time around. Only when it reaches phase 1945 is it back to the original state.
This cycling twice calls to mind another object with a mechanism that does a similar thing. It is the spinor, the object that, for fermions, goes through a first rotation in complex space through 2 Pi to become its negative and through a second rotation through 2 Pi to become itself again. (The Road to Reality, Roger Penrose, Alfred A. Knopf, (2005), p.100)
Now we also observed that for this rule the cycles alternate phases on the inside and outside of a tube. The last column of the outside becomes the initial conditions for the next phase which is on the inside. And the last column of the inside becomes the initial conditions of the next phase which is on the outside of the tube.
In fact, from this view, we can consider the outside phases during the first cycle being all odd numbered phases from 1 through 971. And then with changed polarity a second cycle of odd numbers from 973 through 1943.
Similarly, the inside phases cycle through even numbered phases from 2 through 972 and then with a change in polarity from 974 through 1944.
What happens is that the outside and inside go through 486 alternating phases with positive polarity and then the same number of alternating phases with negative polarity.
It is one tube dynamically changing states, first with a positive polarity and then with a negative polarity.
And while the phases are alternating, each phase goes through a unique self organization quantized in seven distinct steps. Each step manifests a number of horizontal repetitions that follow the power series segment: 3^0 through 3^6.
But the 1st step is flat. After that the next steps have an additional dimension that allows them to form tori of diminishing size but greater number. These tori lie on the surface of the tube.
However, one may wonder if there is a net polarity to the tube as it goes through these changing states. The answer is that 6 phases have zero net polarity. And 969 phases each have positive and negative net polarity. So, overall, the tube has zero net polarity.
Another interesting property is found if we decompose each phase into its three constituents. You might think of this as comparable to color separation in the printing industry. We assign the color red to 0s, the color green to 1s, and the color blue to 2s. Then separating or decomposing the image into these colors, the first constituent has only 0s displayed, the second, 1s and the third 2s.
What does this show us? The 0s are the constituent that carry the triangular shaped objects. The other two do not.
The attached figure shows six instances of phases color separated for each of the three types of polarity. In every instance the 1s carry the triangles. (Only layers 3^1 through 3^6 are shown.)
This is in contrast to the behavior of another rule, which we have yet to consider, 4399234167133: (1 - x + z) in which the 1s carry the triangular shaped objects and the others do not.
Interestingly this rule also has 1944 phases but it does not cycle twice in the sense that one cycle is polar opposite the other.
It remains to be seen if the 0s carrying the triangles correlates with a double cycle while 1s carrying the triangles correlates with not having a double cycle.
Also, it remains to be seen what correlates to 2s carrying the triangles, if we find a rule for which 2s carry the triangles.
Posted by: Lawrence J. Thaden
Here is the first of three parts of the figure that goes with the previous post.
No polarity samples.
Posted by: Lawrence J. Thaden
Here is the secondof three parts of the figure that goes with the previous post.
Positive polarity samples.
Posted by: Lawrence J. Thaden
Here is the third of three parts of the figure that goes with the previous post.
Negative polarity samples.
Posted by: Lawrence J. Thaden
Attached is a set of graphs depicting the net polarities for rule 1466669662809: (-x + z).
Shown are two graphs for the outside surface and two for the inside surface of the tube. (Or, if you prefer, the odd number of phases and the even number of phases.)
For each of these, outside and inside, there are two graphs. The first represents the first cycle, the one comparable to the fermion rotating through a 2 Pi angle in complex space.
The second represents the second cycle in which each phase is the polar opposite of the corresponding phase in the first cycle.
It is no surprise to see that the green line in each of the first cycles, both inside and outside, is colored blue in the corresponding second cycle.
That is because the cycles are polar opposites of one another. Their lines switch places.
But what is interesting is that, while the lines for the outside (the odd numbered phases) never cross one another, this is not true for the inside (the even numbered phases).
The x axis runs out to 247, the greatest number of phases of a particular polarity.
The y axis runs out to 485, the total number of phases for the inside surface.
Since the y axis represents the position of the phase within the 485 phases, when the lines representing net polarity cross each other, it indicates that the positions of phases of net polarity up to that point suddenly switch places. The polarity that was always in a phase prior becomes the polarity that now comes later and visa versa.
The actual count of phases having which net polarity for each cycle is shown in the following table. Note that all of the phases do not add up to 1944. The sum is 6 phases short. These are the phases that have no net polarity.
....................................Cycle 1..........Cycle 2
Outside Net Positive.........246.................238
Outside Net Negative........238.................246
Inside Net Positive............247.................238
Inside Net Negative...........238.................247
Posted by: Lawrence J. Thaden
Now let us consider the modulo 3 difference rule 4399234167133: (1 - x + z).
We referred to this rule earlier in this thread. As mentioned, it has a cycle of 1944 phases, the same as rule 1466669662809: (-x + z). Also, as with rule 1466669662809: (-x + z), each of the phases is self organizing with the same quantized structure that is manifest in the phases of rule 1466669662809: (-x + z).
However, there is no comparison to a spinor here, since no phase is polar opposite to any of the other 1944 phases.
Nonetheless, these two rules can be contrasted as to polarity. Rule 1466669662809: (-x + z) has 969 phases with positive and 969 with negative net polarity and 6 phases with no polarity.
Rule 4399234167133: (1 - x + z) has 967 phases with positive and 975 phases with negative net polarity, and 2 phases with no polarity.
So rule 1466669662809: (-x + z) is without a net polarity, while rule 4399234167133: (1 - x + z) has a slightly negative net polarity.
There is another point of contrast between the two rules. Whereas the original phase of rule 1466669662809: (-x + z) ends with cell content all 0s, rule 4399234167133: (1 - x + z) ends with cell content all 1s.
Also rule 1466669662809: (-x + z) has a phase every 243 phases that is all 0s content in rows 254 through 729. But only the very first phase for rule 4399234167133: (1 - x + z) has this property; and, as said, it is all 1s instead of all 0s.
Attached is a set of graphs depicting the net polarities for rule 4399234167133: (1 - x + z). Shown are two graphs, one for the outside surface and one for the inside surface of the tube. (Or, if you prefer, the odd number of phases and the even number of phases.)
In addition at the end there are two extra graphs that magnify certain ranges of the inside surface.
Whereas rule 1466669662809: (-x + z) has two graphs for the outside and two for the inside, that is because the first graph represents the first cycle, the one comparable to the fermion rotating through a 2 Pi angle in complex space. It has 486 phases for its cycle.
And as seen previously, the second graph represents the second cycle in which each phase is the polar opposite of the corresponding phase in the first cycle.
However, for this rule, 4399234167133: (1 - x + z), there is no comparison to the fermion rotating through a 2 Pi angle in complex space. It has 972 phases for the outside cycle and 972 phases for the inside cycle with no polar opposites.
So, where the graphs for rule 1466669662809: (-x + z) showed identical lines in green and blue switching places, for this rule, 4399234167133: (1 - x + z), no lines are identical.
However, the graph for the outside of rule 4399234167133: (1 - x + z) does share a property with that of rule 1466669662809: (-x + z). Namely, after the origin both green and blue lines separate and never cross.
This is not the case for the inside for either rule. But with rule 4399234167133: (1 - x + z) there are more intersections. The green and blue lines cross five times within the first hundred phases and then two more times closer to the center of their cycles. This is illustrated in the last two close up graphs of sections taken from the inside cycle.
The actual count of phases having which net polarity is shown in the following table. Note that all of the phases do not add up to 1944. The sum is 2 phases short. These are the phases that have no net polarity. Both are found on the inside surface of the tube.
..................Positive..........Negative
Outside.........479.................493
Inside............488.................482
Posted by: Lawrence J. Thaden
Now for rule 6158927822177: - (1 - x + z), the additive inverse of rule 4399234167133: (1 - x + z), covered in the last post.
This is similar to rule 4399234167133: (1 - x + z) in that it has no polar opposites and each of the phases is self organizing with the same quantized structure that is manifest in the phases of rule 4399234167133: (1 - x + z).
However, it has only half the cycles. And the original phase ends in rows of cells with 2s as content in contrast to rule 4399234167133: (1 - x + z) which ended in 1s. (See Figure 1.)
Also, the color separation shows that it is the blue cells that carry the triangles. That is, the 2s carry the triangles for this rule where it was the 1s carrying the triangles for rule 4399234167133: (1 - x + z). (See Figure 2.)
Figure 3 illustrates the net polarities of the phases both for the outside surface of the tube and the inside surface. For this rule there is only one place where the lines cross over. It is on the inside surface at the beginning. This is in contrast to the inside surface for rule 4399234167133: (1 - x + z) which has the lines crossing seven times.
It is interesting that no phase is without a net polarity. This is in contrast to rule 4399234167133: (1 - x + z) which has two inside phases with no net polarity.
The following table summarizes the net polarities of phases for outside and inside surfaces of the tube.
..................Positive..........Negative
Outside.........224.................262
Inside............223.................263
A glance at the table reveals that both outside and inside surfaces have a net negative polarity. For rule 4399234167133: (1 - x + z) the outside has a net negative polarity and the inside a net positive polarity. But, as we have seen in the previous post, if both outside and inside are considered as having one polarity, rule 4399234167133: (1 - x + z) has a negative net polarity.
Posted by: Lawrence J. Thaden
Rule 3226363317853: (1 + x - z) is another example of a modulo 3 difference.
The original CA has 253 steps before it begins a cycle of one row all 1s. (See Figure 1.)
There is a cycle of 972 phases which have no polar opposites.
Nonetheless, the phases are self organizing with the same quantized structure that is manifest in the phases of rule 4399234167133: (1 - x + z), discussed previously.
That is, each phase after the original begins with a random appearing set of 253 rows by 728 columns which then stratify into successively smaller sets of rows that are divided into greater sets of columns. The pattern follows descending powers of three for the number of rows and ascending powers of three for the columns.
Color separation shows that the 1s carry the triangles. (See Figure 2.)
Figure 3 is a series of plots that shows the net polarities for the phases. For the outside surface the plot lines for positive and negative cross twice within the first twenty phases. But after that they remain apart.
For the inside surface the plot lines for positive and negative only cross once within the first ten phases. After that they remain separate.
Overall, there are 494 phases with positive net polarity and 478 phases with negative net polarity. So overall this rule produces positive net polarity.
However, when outside and inside surfaces are considered separately, the outside has positive net polarity while the inside surface has negative net polarity.
Here is a table summary of the net polarities:
..................Positive..........Negative
Outside.........253.................233
Inside............241.................245
Not one phase is without a net polarity.
Posted by: Lawrence J. Thaden
Rule 5572492397537: - (1 + x - z) is the additive inverse of the rule considered in the previous post, rule 3226363317853: (1 + x - z). It is also an example of modulo 3 difference.
The original CA has 253 steps before it begins a cycle of one row all 2s. (See Figure 1.)
There is a cycle of 1944 phases which have no polar opposites. Nonetheless, the phases are self organizing with the same quantized structure that is manifest in the phases of rule 4399234167133: (1 - x + z), discussed previously. That is, each phase after the original begins with a random appearing set of 486 rows by 729 columns which then stratify into successively smaller sets of rows that are divided into greater sets of columns. The pattern follows descending powers of three for the number of rows and ascending powers of three for the columns.
Color separation shows that the 2s carry the triangles. (See Figure 2.)
Figure 3 is a series of plots that shows the net polarities for the phases. For the outside surface the plot lines for positive and negative cross four times within the first fourty-five phases. But after that they remain apart. For the inside surface the plot lines for positive and negative cross each other twice somewhere between phases 50 and 100. After that they remain separate.
Overall there are 984 phases with positive net polarity and 960 phases with negative net polarity. So overall this rule produces positive net polarity.
And taken separately, the outside has positive and the inside has negative net polarity.
Here is a table summary of the net polarities:
..................Positive..........Negative
Outside.........501.................471
Inside............483.................489
Not one phase is without a net polarity.
In contrast to its additive inverse, this rule has twice as many phases and has the 2s carrying the triangles as opposed to the 1s.
In common with its additive inverse, the overall net polarity is positive while the outside net polarity is positive and the inside is negative. Also, both rules have no polar opposites.
There is also a point of comparison with rule 6158927822177: - (1 - x + z), for which the 2s carry the triangles. However, that rule has half the number of phases and has net negative polarity both overall and separately on the inside and outside surfaces.
Another rule that has an interesting set of comparisons and contrasts with this is rule 4399234167133: (1 - x + z). While it has the same number of phases in its cycle and has no polar opposites, the 1s are observed to carry the triangles under color separation. It has an overall negative net polarity, but taken separately the outside surface is negative and the inside surface is positive. Also it has two phases on the inside with no net polarity.
However, a most curious property in both is at phase 4. They each have a large triangle split on the right and left hand sides of the phase. And this triangle is made up of 1s for rule 4399234167133: (1 - x + z), while it is made up of 2s for rule 5572492397537: - (1 + x - z). But both are in the exact same location.
In another post that discussed T5 expressions in x and y, it was observed that colors could be “flipped” by applying one or more of the rules on the output of a phase. Indeed the colors of the large triangle do flip by applying rules that were referred to as gamma flips. These are rules 3681670999617: (-x - y), 3812992433055: (x), and 3943753520967: (y).
However, it must be emphasized that, while “flipping” in the T5 that has expressions in x and y includes every cell, in T5 with expressions in x and z, and specifically regarding rules 5572492397537: - (1 + x - z) and 4399234167133: (1 - x + z), not all of the cells are flipped. It can only be affirmed that the cells that make up the large split triangle in phase 4 have their colors flipped from blue to green.
Certainly, this area deserves more in depth study.
But in the next post a more interesting point of comparison will be considered. That is, what happens when the signs of the variables are switched? This thread began with rule 1466669662809: (-x + z).
In contrast rule 2053105087449: (x - z) has the signs of the variables switched. We will see in what ways is this rule it similar and different due to the change of signs in the modulo 3 expression.
Edited 4-10-08 to correct the size of the random section of each phase.
Posted by: Lawrence J. Thaden
Rule 2053105087449: (x - z) is another example of a modulo 3 difference expression that generates a self organizing structure.
It is similar to the rule that was posted at the beginning of this thread. Namely, rule 1466669662809: (-x + z).
The difference between the two expressions is a change of signs of the variables x and z. Rule 1466669662809 expresses the modulo 3 difference (z - x) and rule 2053105087449 expresses the modulo 3 difference (x - z).
Each phase after the original for both rules follows the same self organizing structure. First there is a section of 486 rows by 729 columns that appears to be random. This is followed by stratification into successively smaller sets of rows that are divided into greater sets of columns. The pattern follows descending powers of three for the number of rows and ascending powers of three for the columns.
Recall that rule 1466669662809: (-x + z) has a cycle of phases 1944 in length and that the second half of the phases is the opposite polarity of the first. That is, each phase in the second half has a 1 where the corresponding phase in the first half has a 2, and each phase in the second half has a 2 where the corresponding phase in the first half has a 1.
In this respect its cycle resembles a spinor which rotates in complex space by an angle of 2 Pi to become the negative of itself, and rotates a second time to again become positive.
Also recall that the original CA, phase 1, is 253 rows before it begins to loop with a row of 0s. So it has the shape of a tube with a closed loop at one end.
Now what about rule 2053105087449: (x - z)?
It has an original CA, phase 1, which is also 253 rows before it begins to loop with a row of 0s. So it also has the shape of a tube with a closed loop at one end. (See Figure 1.)
And the cycle of phases for this rule 2053105087449: (x - z) is 972 in length, half of the size for the rule with the (z - x) expression.
But it also has the second half of the phases as polar opposites of the first. So it also resembles the spinor.
Moreover, in color separation it is observed that the 0s carry the triangles just as with the rule with the expression (z - x). (See Figure 2.)
Even when it comes to net polarities, the two rules are similar in that overall they both have no net polarity both on the outside and the inside surfaces.
But while there are six phases with no net polarity for rule 1466669662809: (-x + z), two on the inside and four on the outside, rule 2053105087449: (x - z) only has two phases with no net polarity, phases 73 and 559. They are both on the outside.
Also, when graphs of the net polarity are compared, there is much more crossing with rule 2053105087449: (x - z) than with rule 1466669662809: (-x + z). The latter does not cross on the outside at all and only appears to cross once on the inside surface.
However, with rule 2053105087449: (x - z) the positive and negative net polarity lines cross each other on the outside surface seven times and on the inside surface at least six times. (See Figure 3.)
Here is a table summary of the net polarities:
..................Positive..........Negative
Outside.........242.................242
Inside............243.................243
Notice that the outside surface has phases with net polarity that do not sum to 486. It is short two phases. These are the ones with no net polarity.
Edited on 4-10-08 for spelling error.
Posted by: Lawrence J. Thaden
If ever I though that self organization was linked to modulo 3 difference, this rule dispels it.
This rule has a modulo 3 difference expression and the phases manifest breakdown of complexity until they reach a two step cycle of rainbows. So it is an example of so called self simplification.
There is no self organization in the phases as with the other modulo 3 difference rules that have been presented earlier in this thread.
The original CA, phase 1, has 243 rows, and it cycles back to the first row. (See Figure 1.) So it is a torus rather than the tubes we have been seeing in the other modulo 3 difference rules. And the object is one third the size of those generated by the other modulo 3 difference rules.
For presentation purposes we still run the phases out 729 steps. It makes the graphics pattern stand out more.
Now for some statistics:
The outside surface of the original torus, the first phase, is flat. After that, the surfaces, both inside and outside curl up into tori. So the shape keeps changing through the pairs of odd and even phases. The number of curled up objects gets greater as the phases progress.
Think of each phase as the inside or outside of a torus covered with beads that keep getting smaller and more plentiful through the four pairs of phases only to finally keep repeating the last phase indefinely.
Phase 1: Repeats every 243 rows, but no repetition across columns.
Phase 2: 9 sets of 81 rows and 3 sets of 81 columns.
Phase 3: 9 sets of 81 rows and 9 sets of 81 columns.
Phase 4: 27 sets of 27 rows and 9 sets of 81 columns.
Phase 5: 243 sets of 3 rows and 81 sets of 9 columns.
Phase 6: 243 sets of 3 rows and 486 sets of columns alternating 2 and 1 column(s) in width.
Phase 7: 243 sets of 3 rows and 243 sets of 3 columns.
Phase 8: 243 sets of 3 rows and 243 sets of 3 columns.
Phase 8 keeps repeating indefinitely.
See Figure 2 for a depiction of these phases.
Comments on the phase figures:
Phases 1 through 4 are not remarkable in any obvious way.
Phase 5 is interesting because it appears as a set of twisted columns.
Phase 6 is amazing. It presents the rainbows but distorted into an array of tight, inverted arcs. It reminds one of something like the lensing that is observed in astrophysics when the light from an obscured galaxy is seen repeated around the star that is obstructing its view. And this phase is on the outside of the torus.
Also of interest is the alternating width of the beads from two cells to one.
In figure 2 color separation is used to show the concentrations of color in local regions and the contrast of neighboring cross-hatched patterns of more diffused color. This helps to explain the distribution of inverted tight arc shaped rainbows.
Phase 7 presents the rainbows parallel to the secondary diagonal.
Phase 8 presents the rainbows parallel to the primary diagonal.
Note: the images were edited in Adobe Photoshop for contrast. This is because of the prevalence of even distribution of colors, cell values, making a gray and thus a dark image.
Edited 4-11-08: Attachment corrected for label on last graph.
Posted by: Lawrence J. Thaden
Rule 6158927822177: - (1 - x + z) is the additive inverse of rule 4399442756969: (-1 + x + z) which was discussed in the previous post.
It is an example of a self simplifying rule with 10 phases and a cycle of phases between phases 9 and 10 that clearly illustrates the rotation by 180 degrees on the secondary diagonal.
The original phase is a torus with 486 rows. Thus it cycles back on the initial conditions.
And we run it that way. Not as having 729 rows. There is no repetition across the columns of the original phase.
Here are the orientations of each of the phases:
1. No orientation.
2. Vertical.
3. Horizontal.
4 - 8. Vertical.
9. Horizontal.
10. Vertical.
Here are the statistics for subsets of phases 2 through 10:
Phase 2: 6 sets each with 81 rows and
3 sets each with 162 columns.
Phase3: 3 sets each with 162 rows and
6 sets each with 81 columns.
Phase 4: 18 sets each with 27 rows and
3 sets each with 162 columns.
Phase 5: 27 sets each with 18 rows and
18 sets each with 27 columns.
Phase 6: 54 sets each with 9 rows and
27 sets each with 18 columns.
Phase 7: 81 sets each with 6 rows and
54 sets each with 9 columns.
Phase 8: 486 sets with one row each and
162 sets with 3 columns each.
Phase 9: 243 sets with 2 rows each and
486 sets with 1 column each.
Phase 10: 486 sets with 1 row each and
243 sets with 2 columns each.
Notice that the two phases that cycle (9 and 10) in one phase have the same number of sets of columns as the sets of rows in the next phase. The same relation holds for the number of rows and corresponding columns.
Also the following is found to be true:
Flatten[phase10For23] == RotateLeft[Flatten[Transpose[phase9For23]]]
So, the rotation about the secondary diagonal needs to be understood in the context of the shift of a single cell that ripples through the entire phase.
The attached graphic presents these 10 phases.
Posted by: Lawrence J. Thaden
The next six rules have expressions with one or two terms, 1 and z, with various combinations of signs for the terms.
One property they have in common is that the cycle of phases for each is two phases in length, and those two phases are diagonal reflections of each other.
By diagonal reflections is meant that when the one phase is placed above and to the left of the other phase, the stripes in the one mirror the stripes in the other.
Less there be some ambiguity, here is code which establishes diagonal reflection:
p1 = firstphase;
p2 = secondphase;
r = Table[Reverse[p2[[i]]], {i, Length[p2]}]; (*Reverse order of contents in each row*)
rr = Reverse[r]; (*Reverse the order of the rows*)
t = Transpose[rr]; (* Exchange rows and columns of 2nd phase of cycle*)
t == phase1 (*Compare result with 1st phase of cycle*)
True
Another property they have in common is that every phase is a torus.
Finally, the count of different types of cell contents is the same for each phase. This means that every phase for all six rules has no polarity.
To begin we consider rule 1466461054805: (-1 + z).
The original phase cycles back to the initial conditions every 1458 steps.
While every phase is a torus, the outside phases are 1458 rows and the inside phases are 729 rows.
It has four phases, but it is the last two phases that form a cycle of phases.
The appearance of the phases is of stripes parallel to the secondary diagonal. These stripes have varying thickness and appear solid purple and green. The purple indicates that the stripe has both 0 and 1 cell content distributed evenly within it.
See attached figure.
Posted by: Lawrence J. Thaden
The additive inverse for the rule in the previous post is rule 2053045476727: - (-1 + z).
This is a simple two phase cycle, each phase having 729 rows. The surfaces of the two phases are flat. That is, there is no subset in either the rows or the columns.
It has the appearance of stripes parallel to the secondary diagonal. And these stripes have a pattern of horizontal and vertical colored lines which repeat the rainbow’s colors within each stripe. Some of the groups of stripes are darker than others. Where the darker groups lie makes the difference between the two phases.
See attached figure.
Posted by: Lawrence J. Thaden
Now for rule 3226214320571: (-1 - z). This has similarities with the two previously posted rules.
Each phase is 729 rows in length. The surfaces are flat. It has light and dark groups of colored stripes parallel to the secondary diagonal. And the difference between the phases that cycle, is the placement of the light and dark groups of stripes.
But there are dissimilarities. This rule has three phases, the last two of which form the cycle of phases. Also, the groups of stripes, while horizontal and vertical, are of fewer colors rather than repeating the entire rainbow of colors within a stripe.
See attached figure.
Posted by: Lawrence J. Thaden
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.
Posted by: Lawrence J. Thaden
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.
Posted by: Lawrence J. Thaden
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.
Posted by: Lawrence J. Thaden
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.
Posted by: Lawrence J. Thaden
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.
Posted by: Lawrence J. Thaden
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).
Posted by: Lawrence J. Thaden
Here is the 2nd figure.
Posted by: Lawrence J. Thaden
Here is the 3rd figure.
Posted by: Lawrence J. Thaden
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.
Posted by: Lawrence J. Thaden
Here are more of the figures.
Posted by: Lawrence J. Thaden
Here is the last of the figures.
Posted by: Lawrence J. Thaden
Here is part 1 of the notebook.
Posted by: Lawrence J. Thaden
Here is part 2 of the notebook.
Posted by: Lawrence J. Thaden
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.
Forum Sponsored by Wolfram Research
© 2004-2008 Wolfram Research, Inc. | Powered by vBulletin 2.3.0 © 2000-2002 Jelsoft Enterprises, Ltd. |
Disclaimer
vB Easy Archive Final - Created by Xenon and modified/released by SkuZZy from the Job Openings