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The patterns are arranged on the page so that the pattern shown at a particular position corresponds to what is obtained with a rule in which the tip of the right-hand stem goes to that position (corrected for the aspect ratio of the array) relative to the original stem shown as a vertical line on the left-hand side of the page.

The system generates a dark gray stripe on the left at all positions that correspond to any product of numbers other than 1. White gaps then remain at positions that correspond to the prime numbers 2, 3, 5, 7, 11, 13, 17, etc.

Indeed in many ways the only real difference is that instead of
The digit sequences of positions of points on successive steps in the two examples of kneading processes at the bottom of the previous page . … In general, a point at position x on a particular step will move to position FractionalPart[2x] on the next step.

But with almost any scheme it will eventually be possible to determine the colors of cells at each of the positions across any register of limited width. … The pictures on the right show which cells in the top row and which cells in the right-hand column determine the cells at successive positions in the right-hand column and in the top row respectively. These pictures can be thought of as matrices with 1's at the position of each black dot, and 0's elsewhere.

For all one ever need do is to work out the remainder from dividing the position of a particular square by the size of the basic repeating block, and this then immediately tells one how to look up the color one wants.
… What one does is to look at the digit sequences for the numbers that give the vertical and horizontal positions of a certain square. … So as the picture illustrates this means that new squares always have positions that involve numbers containing one extra digit.

Cyclic addition
After t steps, the dot will be at position Mod[m t, n] where n is the total number of positions, and m is the number of positions moved at each step. … An alternative interpretation of the system discussed here involves arranging the possible positions in a circle, so that at each step the dot goes a fraction m/n of the way around the circle.

Implementation [of branching model]
It is convenient to represent the positions of all tips by complex numbers. One can take the original stem to extend from the point -1 to 0; the rule is then specified by the list b of complex numbers corresponding to the positions of the new tip obtained after one step. And after n steps the positions of all tips generated are given simply by
Nest[Flatten[Outer[Times, 1 + #, b]] &, {0}, n]

Each random walk is made by taking a discrete particle, and then at each step randomly moving the particle one position to the left or right. If one starts off with several particles, then at any particular time, each particle will be at a definite discrete position. But what happens if one looks not at the position of each individual particle, but rather at the overall distribution of all particles?

for the substitution system that generates a particular nested pattern, and from these construct a procedure for finding the color of a square in the pattern given its position. … Procedures for determining the color of a square at a given position in various nested patterns.

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A mobile automaton in which the position of the active cell moves in a seemingly random way.