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Note that in the later cases shown, the head often visits the same position on the grid many times.

numbers that specify the horizontal and vertical positions of the square, the square is white whenever this factor is 1, and is black otherwise.
… Given the horizontal and vertical positions x and y a square is white when GCD[x, y] 1 and is black otherwise.

Cyclic multiplication
With multiplication by k at each step the dot will be at position Mod[k t , n] after t steps. … When GCD[k, n] 1 the dot can never visit position 0. … In general, the dot will visit position m = k^IntegerExponent[n, k] every MultiplicativeOrder[k, n/m] steps.

The idea is to set up a configuration in rule 30 so that if one inserts input at particular positions the output from the underlying rule 30 evolution corresponds exactly to what one would get from a single step of rule 90 evolution. And in the particular case shown, this is achieved by having blocks 3 cells wide between each input position.
… The initial conditions for rule 30 are fixed except at the two positions indicated, where input can effectively be given.

With a concentrations list c , the position p of a new element is given by Position[c, Max[c], 1, 1] 〚 1, 1 〛 , while the new list of concentrations is λ c + RotateRight[f, p] where f is a list of depletions associated with addition of a new element at position 1.

In each picture black cells further back from the position of the slice are shown in progressively lighter shades of gray, as if they were receding into a kind of fog.

The position of this cell is chosen entirely at random, with the only constraint being that it should be adjacent to an existing cell in the cluster.
… The pictures below, for example, show generalizations of the aggregation model in which new cells are added only at positions that have certain numbers of existing neighbors.

Once again, the behavior that results is always repetitive, and the repetition period can never be greater than the total number of possible positions for the dot. … And as it turns out, the repetition period is again related to the factors of the number of possible positions for the dot—and tends to be maximal in those cases where this number is prime.
… For the systems involving a single dot that we discussed above, the possible states correspond just to possible positions for the dot, and the number of states is therefore equal to the size of the system.

And as a result, the notion of being able to make arbitrarily small changes in the position of a point is unrealistic.
… The system is set up so that every time the light goes around, its position is modified in exactly the same way as the position of a point in the kneading process.

The basic idea is to represent the position of each point at each step as a number, say x , which runs from 0 to 1. … What differs between the two cases is the detailed digit sequences of the positions of the points: in the first case these digit sequences are quite random, while in the second case they have a simple repetitive form.