In each case there is a dot that can be in one of six possible positions. … A simple system that contains a single dot which can be in one of six possible positions. At each step, the dot moves some number of positions to the right, wrapping around as soon as it reaches the right-hand end.

The positions of leaves or other elements are indicated by black dots. … The rule for the system places a new black dot at whatever position this concentration is largest. … It turns out that successive black dots rapidly become spaced at almost exactly 137.5°.

And the basic reason for the repetitive behavior is that whenever the dot ends up in a particular position, it must always repeat whatever it did when it was last in that position. But since there are only six possible positions in all, it is inevitable that after at most six steps the dot will always get to a position where it has been before. … More examples of the type of system shown on the previous page , but now with 10 and 11 possible positions for the dot.

The particular rule is at each step to double the number that represents the position of the dot, wrapping around as soon as this goes past the right-hand end. A system where the number that represents the position of the dot doubles at each step, wrapping around whenever it reaches the right-hand end. (After t steps the dot is thus at position Mod[2 t , n] in a size n system.)

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.

The black dots indicate the elements that are replaced at each step.

In each case specifying the colors of the cells that are marked with dots immediately determines the colors of the cells that are marked with diamonds. The final diamond cell is black if an odd number of the dotted cells are black, and is white otherwise. … These pictures can be thought of as matrices with 1's at the position of each black dot, and 0's elsewhere.

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. But if n  k s , the dot reaches 0 after s steps, and then stays there.

Note the presence of triangles and other small structures dotted throughout all of the pictures.

(The three dots in the representation of each rule stand for the rest of the elements in the sequence.)