ATRACTOR

If we start with a polygon with \(n\) sides whose vertices have abscissa \(x_r\), with \(r\in\{0,1,\ldots,n-1\}\), and we link the midpoints of the diagonals defined by points with abscissa \(x_{r-1}\) and \(x_{r+1}\) (where \(x_{-1}=x_{n-1}\) and \(x_n=x_0\), we obtain a new \(n\)-sided polygon whose vertices have abscissa \[x'_r\;=\;\frac{x_{r-1}+x_{r+1}}{2}\] Consider the Fourier representation of abscissa \(x_r\), given by \[x_r\;=\;\sum_{j=0}^{\lfloor n/2\rfloor} \left(P_j\cos\frac{2jr\pi}{n}+Q_j\sin\frac{2jr\pi}{n}\right) \] where \(\lfloor n/2\rfloor=m\) denotes the integral part of \(n/2\). Writing the vector \((P_j,Q_j)\) in polar coordinates \((C_j\cos\theta_j,C_j\sin\theta_j)\), we have \[x_r\;=\;\sum_{j=0}^m C_j\cos\left(\frac{2jr\pi}{n}-\theta_j\right) \] Then \[\begin{array}{ll}x'_r & =\;\frac{x_{r-1}+x_{r+1}}{2}\;=\\ & =\;\frac{1}{2}\sum_{j=0}^m C_j\left(\cos\left(\frac{2jr\pi}{n}-\theta_j-\frac{2j\pi}{n}\right)+ \cos\left(\frac{2jr\pi}{n}-\theta_j+ \frac{2j\pi}{n}\right)\right)\;=\\ & =\;\sum_{j=0}^m C_j \cos\left(\frac{2j\pi}{n}\right) \cos\left(\frac{2j\pi}{n}-\theta_j\right) \end{array}\] and, more generally, \[\begin{array}{ll}x_r^{(k)} & =\;\frac{x_{r-1}^{(k-1)}+x_{r+1}^{(k-1)}}{2}\;=\; \sum_{j=0}^m\cos^k\left(\frac{2j\pi}{n}\right) \cos\left(\frac{2jr\pi}{n}-\theta_j\right)\;=\\ & =\;X+\sum_{j=1}^m C_j \cos^k\left(\frac{2j\pi}{n}\right) \cos\left(\frac{2jr\pi}{n}-\theta_j\right) \end{array}\]

If \(n\) is even, when \(k\) tends to infinite all the summands of the sum above converge to 0 with the exception of the last one, given by \[C_m\cos^k\left(\frac{2m\pi}{n}\right) \cos\left(\frac{2mr\pi}{n}-\theta_m\right)\;=\; C_m\cos^k\pi \cos(r\pi-0)\;=\; (-1)^{k+r} C_m. \] So we may forget them and consider the following approximation for big values of \(k\): \[x_r^{(k)}\;\approx\;X+(-1)^{k+r} C_m. \] We have similarly \[y_r^{(k)}\;\approx\;Y+(-1)^{k+r} D_m \] and, assuming the points in three-dimensional space, \[z_r^{(k)}\;\approx\;Z+(-1)^{k+r} E_m \] Hence the points \(P_r^{(k)}=(x_r^{(k)},y_r^{(k)},z_r^{(k)})\) approach more and more the points \((X-C_m,Y-D_m,Z-E_m)\) e \((X+C_m,Y+D_m,Z+E_m)\) and the polygons in the sequence are more and more closer to the polygons obtained by linking alternately the two given points, starting in point \((X+C_m,Y+D_m,Z+E_m)\) if \(k\) is even and in point \((X-C_m,Y-D_m,Z-E_m)\) otherwise. In this case, the polygon sequence does not converge to a point but rather to a line segment (possibly degenerate). This happens because not all summands in the given sum tend to 0.

If \(n\) is odd, when \(k\) tends to infinite, all the summands of the sum tend faster to 0 than the last one, given by \[C_m\cos^k\left(\frac{2m\pi}{n}\right) \cos\left(\frac{2mr\pi}{n}-\theta_m\right)\;=\; C_m\cos^k\left(\frac{n-1}{n}\pi\right) \cos\left(\frac{n-1}{n}r\pi-\theta_m\right) \] so that we may take the following approximation for big values of \(k\): \[x_r^{(k)}\;\approx\;X+C_m\cos^k\left(\frac{n-1}{n}\pi\right) \cos\left(\frac{n-1}{n}r\pi-\theta_m\right). \] Let \(C=C_m\cos^k\left(\frac{n-1}{n}\pi\right)\), then \[x_r^{(k)}\;\approx\;X+C\cos\left(\frac{n-1}{n}r\pi-\theta_m\right) \] that is, \[x_r^{(k)}\;\approx\;X+P\cos\left(\frac{n-1}{n}r\pi\right)+ Q\sin\left(\frac{n-1}{n}r\pi\right) \] where \(P=C\cos\theta_m\) e \(Q=C\sin\theta_m\). Similarly, we have \[y_r^{(k)}\;\approx\;Y+R\cos\left(\frac{n-1}{n}r\pi\right)+ S\sin\left(\frac{n-1}{n}r\pi\right) \] and, assuming the points in three-dimensional space, \[z_r^{(k)}\;\approx\;Z+T\cos\left(\frac{n-1}{n}r\pi\right)+ U\sin\left(\frac{n-1}{n}r\pi\right) \] Hence, as like in the cases of hexagon and pentagon, the points \(P_r^{(k)}\) approach more and more the vertices of a polygon obtained by applying a linear function to a \(n\)-sided polygon centered on the origin, followed by a translation. The obtained polygon is inscribed inside an ellipse with center \((X,Y,Z)\). For \(n > 3\), the polygon centered on the origin has the shape of a star since its vertices have angles with the positive part of \(xx\)-axis that are multiple of \(\frac{n-1}{n}\pi\) but not of \(\frac{2\pi}{n}\), as in preceding constructions. Therefore, the obtained polygon has also the shape of a star, inscribed in an ellipse.

Note also that \[x_r^{(k+1)}\;\approx\;X+C_m\cos^{k+1}\left(\frac{n-1}{n}\pi\right) \cos\left(\frac{n-1}{n}r\pi-\theta_m\right)\] \[x_r^{(k+1)}-X\;\approx\;\cos\left(\frac{n-1}{n}\pi\right) C_m\cos^k\left(\frac{n-1}{n}\pi\right) \cos\left(\frac{n-1}{n}r\pi-\theta_m\right) \] However, \[x_r^{(k+1)}-X\;\approx\;\cos\left(\frac{n-1}{n}\pi\right) (x_r^{(k)}-X) \] since \[x_r^{(k)}-X\;\approx\;C_m\cos^k\left(\frac{n-1}{n}\pi\right) \cos\left(\frac{n-1}{n}r\pi-\theta_m\right). \] Thus, the new points are approximately the points that are produced by an homothety of center in the centroid of that polygon and ratio \(\cos\left(\frac{n-1}{n}\pi\right)\). The approximation is as better as the value of \(k\) is bigger (note that, in this case, the ratio might be negative).