A New Kind of Science: The NKS Forum > NKS Way of Thinking > Are elementary CA really simple?
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Registered: Jan 2004
Posts: 357

Ian,

One example of a CA with this simple mechanism uses the elements of the symmetric group S5 as initial conditions along with a single “impurity” cell. Say, zero. Then the update for a site can be had by the code included below.

Sample Mathematica code to implement this simple mechanism:

coefficientIndex[randomlySelectedLeftOrRightNeighbor_, site_]: = FromDigits[IntegerDigits[randomlySelectedLeftOrRightNeighbor, 5, 5] [[IntegerDigits[site, 5, 5] + 1]], 5]

The reason for the “impurity” cell is that the CA remains chaotic without it. But with it, the CA starts to organize the environment.

Attached is a representative result. Representative, because random selection of the neighbor makes for endless variety.

In this representative CA the “impurity” cell is the first cell of the initial conditions. The elements of symmetric group S5 follow in random order.

Notice that organization, the emergence from chaos, starts around the “impurity” cell and proceeds to the right, wrapping as it goes from line to line.

The CA takes place on the surface of a cylinder. And if you use the rule of thumb method to indicate its downward time flow, the flow of organization is characterized by a left chirality.

By the time chaos is “vanquished”, there appear regions of mixed colors. These vary in width as they “contend” with each other for space. The random selection of a neighbor is responsible for this “contending”.

What has happened to the original 120 S5 elements is that they have been reduced during the organization process to just 5 elements. In base 5 they are: 0, 1111, 2222, 3333, and 4444.

Each of the 5 elements has its own color assignment in the graph. The mixed colors are accounted for by juxtaposition of the 5 elements.

Although not in this example, sometimes the regions will be a solid color rather than a mix of colors.

The graph represents only 3000 time steps. Had it run longer eventually a solid color would have won out. Or perhaps alternating lines of solid colors.

Here is a link to a post that has a notebook that does this CA.

The notebook is called: nicecaptionraster samples.nb.

Lawrence J. Thaden has attached this image:

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