Lawrence J. Thaden
Registered: Jan 2004
Posts: 357 
T5 = ({0, +1, y, z, 1}; +, ; 0, +1, 1) Rules (1) – (9)
Introduction
Figure 1 lists a set of 27 threecolor rules that correspond to algebraic expressions for a ternary logic in two variables:
T5 = ({0, +1, y, 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.
These algebraic expressions can be entered in Mathematica and will perform T5 ternary logic operations provided they are specified along with a set of rules to ensure absorptive and modulo properties are preserved.
For example, if the list of algebraic expressions are assigned to the symbol t5, a list of the results for modulo 3 plus: y[[i]] + z[[i]] for all i in t5 can be generated with rules:
{2 > 1, 2 > 1, 2 y > y, 2 z > z}.
Moreover, the results correspond to those obtained by use of the map function described in these posts.
Of course, the map function results in rule numbers, not algebraic expressions.
However, there is a correspondence. It is found between the mapped rule number result and the position it holds in the list of algebraic expressions.
So, if the set of rule numbers are assigned to the symbol t5Subset, and the expressions are assigned to t5, executing the following code in Mathematica gives {True}.
Union[(Table[(t5[[i]] + t5[[#]]&/@Range[Length[t5]]/.{2>1, 2>1, 2 y>y, 2 z>z}) == (t5[[#]]&/@(Flatten[Position[t5Subset, #]&/@(map[t5Subset[[10]], dontcare, t5Subset[[i]], t5Subset[[#]]]&/@Range[Length[t5Subset]])])), {i, Length[t5]}])]
Where t5Subset[[10]] identifies the rule corresponding to the algebraic expression (y) + (z).
Similar tests can be constructed for correspondence between threevalued rule numbers and other simple algebraic expressions that are modulo 3 sums and differences.
Expectations Finally Fulfilled
Towards the end of the thread titled “(x +z) Self Organization Example”, which discussed (at length) the rules and CAs from T5 for x and z, I said:
“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.”
Well I have made those corrections and posted them in the threads: “Revised T5 = ({0, +1, x, y, 1}; +, ; 0, +1, 1)” Parts 1 – 4.
However, there were no rules with “spinorlike” behavior.
But now with the last set of T5 rules for y and z, we hit pay dirt. There are two rules with cycles of phases that have “spinorlike” behavior within the first nine of twentyseven rules. And as anticipated, they have short cycles of phases. One has six phases and the other eight phases. Both begin geometrically as tubular objects with a lip at the bottom.
But where the earlier ones in T5 for x and z were selforganizing, these for the most part are arrangements of Sierpinski triangles. So they will not be able to shed any light on how the selforganizing ones come about, as I had hoped they would.
Rules and Their Behavior
So to begin this discussion lets consider these two spinorlike rules.
Rule (6) 1616787841929: (y + z) has six phases which make up its cycle. The first three have positive polarity and the last three have negative polarity. One to one the first three are identical in all respects to the last three, save for polarity. Wherever there is a cell with a 1 in the first three, the corresponding cell in the last three is a 2, which algebraically behaves as a –1.
This is easily shown by Mathematica code that sums corresponding cellular automaton outputs modulo 3. Whenever the pair consists of polar opposites, the result is all zeros. Here is an example of the code that might be used:
Union[Flatten[Mod[polarPositive[[n]] + polarNegative[[n]], 3]]]
{0}
The cycle is spinorlike in that its phase changes can be compared to a spinor that takes 4 Pi radians to complete a cycle and which after 2 Pi radians changes polarity. Following this comparison, each phase corresponds to a discrete advance of (2/3) Pi radians.
The structure of the first and fourth phases is arrangements of Sierpinski triangles. The structure of the second and fifth phases is an additive version of three sets of Sierpinski triangles. The structure of the third and sixth phases shifts two of those sets down and slightly to the right, while shifting the third fully 50 percent to the left and slightly down. At the same time there appears an additional set of Sierpinski triangles along the left edge. Whether these come from the previous phase or are something knew is hard to determine. If they originate from the previous phase, it might have something to do with modulo 3 sum and difference canceling things out, and with periodic boundaries.
If you accept the concept that the phases alternate as outside and inside surfaces of the tube, then all you could observe is phases 1, 3, and 5. (See Figure 7.)
Rule (8) 2100313177833: (y  z) has eight phases which make up its cycle. The first four have positive polarity and the last four have negative polarity. One to one the first four are identical in all respects to the last four, save for polarity. Wherever there is a cell with a 1 in the first four, the corresponding cell in the last four is a 2, which algebraically behaves as a –1.
The cycle is spinorlike in that its phase changes can be compared to a spinor that takes 4 Pi radians to complete a cycle and which after 2 Pi radians changes polarity. Following this comparison, each phase corresponds to a discrete advance of (1/2) Pi radians.
The structure of the first and fifth phases is arrangements of Sierpinski triangles. The structure of the third and seventh phases is an additive version of three sets of Sierpinski triangles, very similar in appearance to the third and sixth phases for rule (6) 1616787841929: (y + z), except that the bottom edge does not touch the frame. Rather there are several rows with 0 content before the bottom frame.
The structure of the second and sixth phases is similar to that of phases two and five for rule (6) 1616787841929: (y + z), with two exceptions noted:
(1) the additive triangles on the right touch the bottom frame, and
(2) Sierpinski “cut outs” for the additive triangles on the right are not 0s. Rather they are a combination of 1s and 2s.
But for phases four and eight Sierpinski is left behind. These phases resemble phase three of rules (6) 146430861993: (x + y), (16) 3943560596989: (1 – x + y), and (20) 7348404768497: (1 – x + y), all belonging to T5 in x and y.
But these phases for T5 in y and z have a different orientation than the three just referred to. The center diagonal in these two begins in the upper left corner and descends halfway down the figures. In the three others, it begins in the upper right corner. (See Figure 9 for these phases, but click here to see the comparable phases in figure 4 of the presentation for T5 in x and y. Also note that the figure for rule (20) 7348404768497: (1 – x + y) is mislabeled “Phase 2”. It should be labeled “Phase 3”.)
If you accept the concept that the phases alternate as outside and inside surfaces of the tube, then all you could observe is phases 1, 3, 5, and 7.
I should think that it would be quite an achievement for someone to be able to explain the derivation of phases four and eight of rule (8) 2100313177833: (y  z). It might be done in the same terms that are used to explain how the Sierpinski triangle is constructed, including Hausdorff dimensions of the various regions. This assumes that the complex pattern can be broken down into regions that at sometimes intersect and at other times cancel out modulo 3.
Besides these two spinorlike rules, there are two other remarkable rules in the first nine rules for T5 in y and z. These are the selfreplicating rules (4) 1466461054805: (1 + z) and (9) 2201047622813: (1 – y – z). Both begin as tori with 1458 rows and 729 columns.
Rule (4) 1466461054805: (1 + z) has three phases, the second and third of which make up the cycle. All three phases have stripes running from the upper right corner to the lower left corner. The original phase is replicated in a sidebyside double copy in phase three. The second and third phases mirror each other diagonally. That is, if you place one to the right and above of the other, they reflect each other. (See Figure 5a and 5b.)
Also, since the cycle begins with phase 2, this is the outside surface of the torus. So the selfreplicating phase occurs on the inside surface of the torus.
This selfreplicating is dependent upon letting phase 2 run for 1458 rows. Actually, phase 2 will cycle after 729 rows. So you really have odd phases with dimensions: 1458 rows and 729 columns and even phases with dimensions of 729 rows and 1458 columns. And there is no selfreplication. The length of the rows and columns just keep alternating. And it cycles after the second phase. In this scenario the original phase is the outside surface of the torus.
This dependence upon some external control to let phase 2 cycle twice in order to meet the requirements for selfreplicating in phase 3 is just one of the two dependencies on external controls to achieve selfreplication. The other is a mechanism for separating the right and left halves of phase 3 so that they stand on their own.
Rule (9) 2201047622813: (1 – y – z) has thirteen phases and the last phase is a sidebyside double copy of the first phase. However, its cycle runs from phase two through phase thirteen. Instead of stripes running diagonally from upper right to lower left, there are rows of green triangles, and these appear only in the original phase, phase five, phase nine, and phase thirteen.
Moreover, in phases 5 and 9 the image appears to have been rotated clockwise 90 degrees. And each of the phases also has intersecting rows of purple triangles that impart to the images a latticelike structure.
The intervening phases are characterized by Sierpinski triangle structures, some with green “cut outs” and some with purple. And, as was said, the thirteenth phase replicates the original in a sidebyside double copy. (See Figures 10a and 10b.)
Again, achieving replication is by externally controlling phase 2 so that it runs out to 1458 rows. If no control were exercised, the second phase would have stopped at row 248, a row with every cell a 1. And then continuing on to the subsequent phases would present a bewildering variety of dimensions that would not readily result in the selfreplication. For example, by phase three the dimensions are 248 columns by something in excess of 50000 rows.
At first glance phases (3 and 11), (5 and 9), and (8 and 12) appear to be the same. But this is an illusion. Moreover, phase 5 is not the original replicated sidebyside in a 90 degree rotation of itself. If you do take half of phase 5 and rotate it counter clockwise 90 degrees and then animate it with the original, you will notice that the small green triangles in the vicinity of intersections of the larger green and purple triangles scintillate.
This rule is interesting because its cycle starts with phase 2. If you accept the concept that the phases alternate on the outside and inside of the surface of the torus, this means that for this rule only even phases are observable. And the selfreplication would take place on the inside surface of the torus.
Remaining Rules
Now lets sequentially consider each of the remaining first nine rules for T5 in y and z.
The first one is the additive identity 0 and needs no explanation.
Next is rule (2) 146064945221: (1 + y) which has two phases, the second of which cycles on itself. The color is purple because it is made up of evenly distributed cells with 0s (red) and 2s (blue). These cells form a checkerboard pattern, beginning with a red cell in row one and repeating every odd integer row number. The original phase has an alternating pattern of vertical bars, which repeats with each odd integer row number. At first glance the left and right halves appear to mirror each other, but this is an illusion. So both phases are very small tori, but we run them out to 729 rows. (See Figure 3.)
Rule (3) 277019723695: (1  y) also has two phases, the second of which cycles on itself. Both phases appear gray because of the even distribution of all three types of their cell values. The original and phase 2 both repeat every three rows. The original has a vertical bar appearance. At first glance it looks like the left and right sides reflect each other, but they do not. The second phase has repeating diagonal rows of blue, red, and green cells. So both phases are very small tori, but we run them out to 729 rows. (See Figure 4.)
Rule (5) 1576494063937: (1 + y + z) has three phases and it cycles through all three phases. Each phase has a rich texture but at low resolution appears grayish due to the even distribution of its three colors. The phases all present a diagonal of triangles running from lower right to upper left, with each phase a variation on the filler for the triangular shapes. The original phase has horizontal stripes for filler. Phase three has vertical stripes. And phase two has a combination or blend of the horizontal and vertical that gives it a checkered appearance. In addition there are olive green and purple triangles interspersed about the central diagonal triangles. And this feature becomes more pronounced in each phase until phase three has the hint of a latticelike appearance. Each of the phases is a torus with 729 rows. (See Figure 6a for the three phases and Figure 6b for closeup views of the diagonal pattern.)
If you accept the concept that these phases alternate between the outside and inside surface of a torus, then this rule is indeed interesting, since its cycle ends up with the last phase an outside surface.
This means that the next iteration of the cycle of phases begins on the inside surface of the torus. So its cycle starting points alternate in a trinary fashion. Viewed from the outside, the surface of the torus would appear to change its phases from 1 to 3 to 2 to 1 to 3 to 2 ad infinitum.
If this alternating cycle of phases is found to manifest itself in any physical process, I should think detecting it would be very obvious.
In the process of preparing graphics for the three phases I considered making them as “painted” tori, the paint being the 729 rows by 729 columns of cell values converted to RGBColor specifications. However, my computer was not able to process all the graphic data.
So, undaunted, I attempted to extract cell values from every third row and column, thus yielding a subset of cells from rows and columns 1, 4, 7, 10, and so on. The result was three tori with dimensions of 243 rows and columns.
This, my graphics card was able to process. But the image for each phase was very different. And the way in which the differences appeared was unexpected.
The purple and olive colors are gone and only the pure red, green, and blue colors are to be seen. Moreover, each phase has a similar structure, although the phases differ in color arrangements.
There is a center diagonal of modified triangular shapes made of triangles that changes color with each phase. In the original phase it is blue; in phase 2 it is green; and in phase 3 it is red.
In addition, there are three rows of three regions of same colored clusters of triangles. (These clusters include the center diagonals referred to above.) The pattern of colors runs over the regions as follows:
..............................Cell Colors…………Cell Values
Original phase: Blue...green...red.…….....2, 1, 0
………………..........Red….blue…green…….....0, 2, 1
…………….........…Green..red…..blue..……....1, 0, 2
Phase 2:………...Red..…blue.…green.….....0, 2, 1
………………........Green…red..…blue….........1, 0, 2
…………….….......Blue....green....red..….…...2, 1, 0
Phase 3:………..Green…red.….blue.…….....1, 0, 2
……………….........Blue….green...red………....2, 1, 0
…………….…........Red..…blue.....green..…...0, 2, 1
Finally, there is a structural symmetry about the diagonals. The same structure, if not color, is found on either side of the diagonals for all three phases. (See Figure 6c.)
It is remarkable how this symmetry is hidden in every 3rd row and column of the original versions for this rule. It is certainly there in the originals. Only it is disguised by the other detail.
Well I was able to paint tori with these extracted versions of rule (5) and also to ListAnimate them. However it took a long time before the animation started. But when it did, it was clear that the three abstracted phases have the exact same structure. Only the colors change during the animation. (I am not including the set of graphic figures for these extracted tori.)
But where did the olive and purple colors go? They are in complements of these extracted phases. And it takes 486 rows and columns to construct tori from them. (See Figure 6d.)
One is tempted to conjecture about subsets of cells that make up a torus, since in this case the two extracted subsets of the original cellular automaton data are also tori, although of different dimensions.
The conjecture would be stated this way: Given cellular automaton output from a rule with periodic boundaries that cycles, subsets of the cycle data also cycle, but not necessarily with the same number of rows and columns.
There would have to be certain constraints such as that the cellular automaton must be 1d with dimensions (n x n) in order to follow from this example.
But I merely throw out the suggestion. I am not making the conjecture. All I can state is that there is one cellular automaton that meets these criteria.
Rule (7) 2053045476727: (1  z) has two phases and it cycles through these two phases. The appearance is grayish because of the even distribution of cell values. Both phases consist of diagonal stripes running from the upper right to lower left. The two phases are diagonally mirrored. That is, if you place one above and to the side of the other, the two reflect each other. Both phases are tori with 729 rows. (See Figure 8a for the two phases and Figure 8b for closeup views of the original phase.)
Attached are figures 1  5 which include the list of rules and figures for the first five rules.
Two subsequent posts will have attachments for the rest of the rules.
Lawrence J. Thaden has attached this image:
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L. J. Thaden
Last edited by Lawrence J. Thaden on 02062009 at 02:50 AM
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