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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. … The pictures below show more examples of the same setup, where now the number of possible positions is 10 and 11.

In the type of system shown on the facing page , it turns out that the repetition period is maximal whenever the number of positions moved at each step shares no common factor with the total number of possible positions—and this is achieved for example whenever either of these quantities is a prime number.
… 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.)

The positions of new plant organs or other elements around a stem are presumably determined by what happens in a small ring of material near the tip of the growing stem. And what I suspect is that a new element will typically form at a particular position around the ring if at that position the concentration of some chemical has reached a certain critical level.
… Nevertheless, general processes in the growing stem will presumably make the concentration steadily rise throughout the ring of active material, and eventually this concentration will again get high enough at some position that it will cause another element to be formed.

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.

Indeed, the simple pegboard shown below exhibits the same phenomenon, with balls dropped at even infinitesimally different initial positions eventually following very different trajectories.
The details of these trajectories cannot be deduced quite as directly as before from the digit sequences of initial positions, but
Paths followed by four idealized balls dropped from initial positions differing by one part in a thousand into an array of identical circular pegs. … If balls are assumed to fall randomly on each side of each peg then with a large number of balls the final positions will approximate a binomial distribution.

Just like in other pictures in this book, position goes across the page, and time down the page. … Similarly, D[u,x] is the rate of change with position in space, and D[u,x,x] is the rate of change of that rate of change.

At each step a new element, indicated by a black dot, is taken to be generated at whatever position the concentration is maximal. … 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.

has the property that its vertical position ends with a 0, and its horizontal position ends with a 1. So if the numbers that correspond to the position of a particular square contain this combination of digits at any point, it follows that the square must be white.

The distribution of positions by reached particles that follow random walks. The top left shows three individual examples of random walks, in which each particle randomly moves one position to the left or right.

The vertical black line on the left-hand side of the page represents in effect the original stem at each step, and the pictures are arranged so that the one which appears at a given position on the page shows the pattern that is generated when the tip of the right-hand new stem goes to that position relative to the original stem shown on the left.
… Page 407 gives results for four choices of the position of this region relative to the original stem. … The areas of solid black thus correspond to ranges of parameters in the underlying rule for which the patterns obtained always reach a particular position.