[Cells, precursors, and the many to one mapping relation] - A New Kind of Science: The NKS Forum

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Cells, precursors, and the many to one mapping relation

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Posted by: Lawrence J. Thaden

Here is an interesting three color cellular automaton rule that illustrates in a striking way the many to one mapping relation that holds between cells and their precursors.

In figure 1 of the attached graphics the cellular automaton is shown as it evolved for 1218 steps from the simple initial condition of {2, 1} centered with 17 zeros on either side.

In figure 2 the same rule is shown as it evolved 1620 steps from initial conditions based on a randomly selected left hand side of 18 cells with values 0, 1, or 2. The right hand side is the reflected complement of the left side.

Both evolutions concluded in the exact same 450 step persistent structure shown in figure 3.

The only way for this to happen is that at some point the two separate evolutions had to have their different sets of cells updated to the exact same next set of cells.

Figures 4 and 5 shows the exact step where the two are different and then where they are the same on the next step. In the figures the fourth row of cells from the bottom is the last time they are different. That is, row 486 of figure 4 and row 889 of figure 5 are different, but the remaining rows are the same.

The rule used is the algebraic expression: Mod[-(-2 + p) r + q (2 + s), 3]. Its four variables, {p, q, r, s}, are fed information from two cells to the right and left of the cell being updated. The algebraic expression is then evaluated and assigned as the updated value of the cell.


Posted by: Lawrence J. Thaden

Figures continued


Posted by: Lawrence J. Thaden

Figures concluded




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