Lawrence J. Thaden
Registered: Jan 2004
Posts: 350 |
Predecessor state to simple conditions
Here is a CA that can be started with the simple conditions having the middle two cells {2, 2} while the rest of the cells are all zeros. It has a complex symmetric behavior and concludes after 127 steps with a persistent structure 356 steps in length.
But is it possible to have this set of simple initial conditions be the result of updating a previous set of conditions that is not simple? Yes, when the predecessor state is: {0,1,0,2,0,1,0,2,0,1,0,2,0,1,0,2,0,2,2,0,2,0,1,0,2,0,1,0,2,0,1,0,2,0,1,0}.
In the attached image I actually start the CA with initial conditions: {2,1,1,1,2,1,2,2,2,2,2,1,0,1,0,0,2,1,1,2,0,0,1,0,1,2,2,2,2,2,1,2,1,1,1,2}. It then evolves 532 steps before reaching the predecessor state to the simple conditions.
I have marked the spot on the graph with a horizontal red line.
The rule used in this CA is a four variable, three color rule: Mod[-2 (-1 + p) r + q (2 + s), 3]. The rule is an algebraic expression which has variables p, q, r, and s. When the rule is evaluated for a cell, the value of the next nearest neighbor on the left is assigned to variable p; the value of the nearest neighbor on the left is assigned to variable q; the value of the nearest neighbor on the right is assigned to variable r; and the value of the next nearest neighbor on the right is assigned to variable s. The algebraic expression is then simplified modulo 3, and the result is stored as the updated cell.
Lawrence J. Thaden has attached this image:
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L. J. Thaden
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10-14-2004 07:31 PM
