### 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.

**Instructions:**

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?