Repeated Addition and Doubling

This Demonstration shows some simple examples of systems with simple rules of a limited size. These systems are displayed in order to illustrate that any system of a limited size that involves discrete elements and follows definite rules must exhibit periodicity.
The first example is repeated addition. A dot moves to the right the number of spaces indicated by the value of "move" and this repeats for a given number of steps. The dot wraps around to the right-hand side after reaching the specified width.
The second example is repeated doubling. The position of the dot doubles on each step for a given number of steps, again wrapping around the right-hand side after reaching the specified width. (The first position is 0, so the dots for 1, 2, 4, 8, … start at the second position, which is 1.)



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See Stephen Wolfram's A New Kind of Science, pp. 255–258.
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