Dynamic I

In the geometric proof of incommensurability between the diagonal and the side of a square, we saw that, given a square with diagonal \(d\) and side \(l\), it was possible to obtain a sequence of new squares with diagonals \(d_{i}\) and sides \(l_{i}\), using the formulas \(d_{i}=2l_{i-1}-d_{i-1}\) and \(l_{i}=d_{i-1}-l_{i-1}\), with \(d_{0}=d\) and \(l_{0}=l\). Suppose now that we start from an arbitrary point \((x_{0},y_{0})\) and we consider the sequence of points defined recursively by \((x_{n},y_{n})=(2y_{n-1}-x_{n-1},x_{n-1}-y_{n-1})\). Marking these points in an orthogonal coordinate system, we have an infinite sequence of points which changes according to the choice of the initial point \((x_{0},y_{0})\). What happens to this sequence of points? How does it depend on the choice of the initial point \((x_{0},y_{0})\)? Try the applet below, where you can also check what happens in the case of the regular pentagon (considering the first geometric proof given) and what happens in the case of the regular hexagon.

Click on the red dot to change the initial point; you can also choose if you want to see the lines joining the iterates shown in the graphic, highlight the \(n\)-th iterate for \(n\) between \(0\) and \(5\) or change the regular polygon on whose geometric proof of incommensurability this dynamic process is based.

What is the similarity between these three cases?

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