[Rotating Ring Torus] - A New Kind of Science: The NKS Forum

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Rotating Ring Torus

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Posted by: Lawrence J. Thaden

This cellular automaton ends with a remarkable persistent structure. It is a ring torus that rotates in discrete increments of 60 degrees. The rotation is about the center of the tube in a direction that is clockwise on the left side and counterclockwise on the right side of the cross section through the xz-plane. (See http://mathworld.wolfram.com/Torus.html for the illustration and cross section diagram.)

But in the attached graphic it appears as a tape which after a transient of 314 steps begins six segments of a cycle, each of which has 491 steps. After each segment the pattern shifts right 6 cells and then begins another 491 steps. This repeats until all 2946 steps of the six segments have been traversed. Then the process starts all over again, and it runs for an indefinitely long time.

The transient is shown in a graph along with the six segments of the torus. Then in a second graph the six segments are shown with their cells rotated left by 6, 12, 18, 24, and 30 to depict a reversal in the corresponding rotations of multiples of 60 degrees. Observe that this maneuver makes all of the segments the same.

The rule for this cellular automaton is the algebraic expression: Mod[(2 + q) s - p (-2 + r + s), 3]. Its four variables (p, q, r, and s) are fed information from the two right and left neighbors of a cell. When the algebraic expression is simplified its value becomes the updated cell. Cells have values 0, 1, or 2.

Initial conditions are {1, 2} centered with 17 zeros on both sides.


Posted by: Lawrence J. Thaden

Here is the second graphic. It shows the segments of the cycle adjusted by rotations left in multiples of 6 cells. At the foot of each column in the graphic the first integer represents the last step of the cellular automaton in that column and the second integer indicates the number of cells rotated to the left.




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