[Revised T5 = ({0, +1, x, y, -1}; +, -; 0, +1, -1) Part 3: Rules Numbered (19) - (27)] - A New Kind of Science: The NKS Forum

A New Kind of Science: The NKS Forum

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Revised T5 = ({0, +1, x, y, -1}; +, -; 0, +1, -1) Part 3: Rules Numbered (19) - (27)

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Posted by: Lawrence J. Thaden

Now to consider rules numbered 19 through 27.

There are two with a single phase, five with two phases, one with three phases, and one with four phases.

The rules, their phase counts and cycle component counts together with notes follows:

(19) 7348211845533 (x + y): 2 phases, cycle: phases 1-2.

Compared to the original phase, phase 2 exchanges triangles composed of all 1s to triangles composed of all 2s.

But this is not the only difference between the two phases. A noticeable difference is the location of an arc of diminishing triangles in both phases.

In the original phase, it is to the left of the collision points. In phase two it is to the right of the collision points.

The original phase is the outside surface of a torus. Phase two is the inside surface of the same torus. Figure 1 shows both surfaces side by side. Figure 2 shows a condensed version of the original surface painted on a torus.

Condensing was warranted due to huge memory and computation requirements for mapping the surface onto the torus. And although only one cell every fifth row and column appears in the condensed version, you can still follow the diagonal of red triangles as they wrap their way around the torus. (Thanks to Wolfram's support personnel for their assistance on code to paint the torus.)

(20) 7348404768497 (-1 - x + y): 3 phases, cycle: phases 1-3.

Original and second phases are arrangements of Sierpenski triangles.

The last phase in the cycle, phase three, is very unusual. It is populated heavily by cell values of 2. (See figure 3.) But is similar in structure to two other rules in their third phase which are heavily populated by 0s and 1s respectively: rules (6) 146430861993: (-x + y) and (16) 3943560596989: (1 - x + y). (See figure 4.)

(21) 7348577761291 (1 + y): 2 phases, cycle: phase 1-2.

Original phase is vertical stripes and phase 2 is a checkerboard pattern of cells with 0s and 1s. (See figure 5.) This rule is similar to rule (4) 146064945221: (-1 + y) in which the original phase is vertical stripes and phase two has a checkerboard pattern of 0s and 2s.

(22) 7479166622993 (-1 + x - y): 2 phases, cycle: phase 1-2.

Both phases are arrangements of Sierpenski triangles with a predominance of cells with 2s. (See figure 6.) This rule is similar to rule (8) 277192716489: (x - y), which has a predominance of cells with 0s and to rule (12) 3682036887997: (1 + x - y), which has a predominance of cells with 1s.

(23) 7479339588409 (1 - x - y): 2 phases, cycle: phase 1-2.

The original phase is the characteristic pattern for this set of 27 rules, but with a predominance of cell values of 2. Phase two is an arrangement of Sierpenski triangles. (See figure 7.)

(24) 7479532539765 (-y): 2 phases, cycle: phase 2.

The original phase has vertical stripes. Phase 2 is all 0s. This rule is similar to rule (17) 3943753520967: (y) in which the original phase has vertical stripes and the second phase is all 0s. (See figure 8.)

(25) 7625210074339 (1 + x): 4 phases, cycle: phases 2-4.

Phase 3 self-replicates original phase in a side-by-side arrangement.

The original has a cycle length of 1458 rows and the second half is equivalent to the first half after applying the rule: {1->0, 0->1}. (See figure 9.)

This rule is similar to rule (15) 3812992433055: (x) in which the third phase replicates the original phase, and which also has a cycle from phase 2 through phase 4. But rule (15) has the property that the second half of the original is equivalent to the first half once this rule is applied: {2->1, 1->2}.

(26) 7625403764901 (-x): 1 phase, cycle: phase 1. (See figure 10.)

Interestingly variable y's original phase has a graph composed of vertical stripes, whereas variable -x has a graph composed of diagonal stripes. One might have thought that variable -x's graph would have been horizontal stripes.

Also of note is that (-x) in T5 for x and z has just the original phase, but its graph is slightly different due to the difference in initial conditions for T5 in x and y compared with those for T5 in x and z.

(27) 7625597484986 (-1): 1 phase, cycle: phase 1.

This is the negative multiplicative identity. It generates a CA with all cell values of 2.


Graph of the torus is attached to this post.
Graphs for figures numbered 1 and 3 through 10 are attached to the following post.


Posted by: Lawrence J. Thaden

Remaining figures attached.

Correction (1/21/09): In "Figure 4: Comparison of Three Similar Structures, Each the Third Phase of Their Respective Cycles" the rightmost figure is mislabeled "Phase 2". It should be labeled "Phase 3".




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