Had some time recently while flying from one place to another and played with simple Cellular Automata rules in excel. It is very simple to do – you create a rule within the cell as function of other cells and extend it (you can simply copy/paste) to other cells.

Here, if A2 is equal to B1, we define a value within the cell “Set”, otherwise it remains empty. That’s all.

Now, if we will keep the first row and column empty (let’s call them “boundaries”), while extending the rule to bigger area, we are going to get a beautiful pattern:

Here I took the area of 200×200

The size of the “triangles” are recursive 2N+1 (1,3,7,15,31,63…)

So by very simple rule we have created pretty complex pattern

Now what is interesting that by simple change of the reference cell the pattern complexity can dramatically increase.

If we change the “offset” of the reference cell by one i.e. apply this kind of rule:

We will get the next pattern:

If the offset of the reference comparing to original pattern is going to be 2, i.e.:

Then the pattern becomes much less structured and more “pseudo random” (of course everything here is deterministic, so it only “looks” sporadic) :

Offset = 4:

You can think – ok, I’ve got it, it is moving from very structured pattern to a uniformly distributed one or at least say that the entropy of the system is growing.

But then, at certain offsets we start to see suddenly partially of fully returned structures close to boundaries or for all the area.

Offset = 11:

Offset = 12:

Offset = 13:

Offset = 14:

Offset = 29:

In some cases the sensitivity to boundaries or initial condition is small (i.e. changing a value on the boundary is not changing the whole pattern, while in other cases it is big – a single cell change can cause the change of the whole area).

Is not it funny?

P.S. If you would like to play, here is the file that you can easily play with (push arrows or change numbers)

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