The pattern continues to expand on the left forever, but only the part that fits across each page is shown.

The pattern continues to expand on the left forever, but only the part that fits across each page is shown.

Note that on each line in each picture, only the order of elements is ever significant: as the insets show, a particular element may change its position as a result of the addition or subtraction of elements to its left.

Nearby cellular automaton rules In a range r cellular automaton the new color of a particular cell depends only on cells at most a distance r away. … One can then define nearby cellular automata to be those where the differences in the rule involve only cells close to the edge of the range.

As discussed on page 155 , each cell here can have any gray level between 0 and 1, and at each step the gray level of a given cell is determined by averaging the gray levels of the cell and its two neighbors, adding the specified constant, and then keeping only the fractional part of the result.

Among templates involving nodes out to distance two there are 106 that have no dangling connections, and of these only 8 satisfy the constraints. The main text considers only constraints based on a single template. … For templates involving nodes out to distance one, there are 13 minimal sets in the sense of page 941 , of which only 6 contain just one template, 6 contain two and 1 contains three.

And one way to do this is to require not only that the colors around each cell match a set of templates, but also that a particular template from this set must appear at least somewhere in the array of cells. … In a system like a cellular automaton that is based on explicit rules, it is always straightforward to take the rule and apply it to see Examples of patterns produced by systems in which not only must the arrangement of colors in each neighborhood match one of a fixed set of templates, but also a certain template from this set must occur at least once in the pattern.

And in a 10×10 array the chance of finding a pattern where the fraction of squares that violate the constraints is even less than 50% is only one in a thousand, while the chance of finding a pattern where the fraction is less than 25% is one in four trillion. … Only a handful of patterns satisfy the constraints exactly (so that 0% of the squares are wrong).

On the first step in this particular picture, there happens to be only one replacement that can be performed consistent with the rules, so only a single string is produced.

only at overall features, then typically one would tend to say that the third picture seems more complex than the other two. … Of course, if one goes to an extreme and looks, say, only at how big each picture is, then all three pictures have very short descriptions.