ATRACTOR

Given a \(n\)-sided polygon with vertices with abscissa \(x_r\), \(r\in\{0,1,\ldots,n-1\}\), if we link the points that divide each side in two line segments of length \(p\) and \(1-p\) times the length of the side, with \(0 < p < 1\), we get a new \(n\)-sided polygon whose vertices have abscissa \[x'_r-x_r\;=\;p(x_{r+1}-x_r) \] that is, \[x'_r\;=\;(1-p)x_r+p x_{r+1} \] with \(x_n=x_0\). 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 floor of \(n/2\), that is, its integer part. Writing 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 \[\begin{array}{ll}x'_r & =\;(1-p)x_r+p x_{r+1}\;=\\ & =\;\sum_{j=0}^m C_j\left( (1-p)\cos\left(\frac{2jr\pi}{n}-\theta_j\right)+ p\cos\left(\frac{2jr\pi}{n}-\theta_j+ \frac{2j\pi}{n}\right)\right)\;=\\ & =\;\sum_{j=0}^m C_j d_j \cos\left(\frac{2jr\pi}{n}-\theta_j+\psi_j\right) \end{array}\] where \[d_j\;=\;\sqrt{\left((1-p)+p\cos\frac{2j\pi}{n}\right)^2+ \left(p\sin\frac{2j\pi}{n}\right)^2}\;=\; \sqrt{1-2p(1-p)(1-\cos\frac{2j\pi}{n})} \] and \[\psi_j\;=\;\arctan\frac{p\sin\frac{2j\pi}{n}}{(1-p)+ p\cos\frac{2j\pi}{n}} \] More generally, \[x_r^{(k)}\;=\;(1-p)x_r^{(k-1)}+p x_{r+1}^{(k-1)}\;=\; X+\sum_{j=1}^m C_j d_j^k \cos\left(\frac{2jr\pi}{n}-\theta_j+k\psi_j\right) \] When \(k\) tends to infinite, the summands above converge faster than the first one to 0. Therefore, we approximate the sum by \[x_r^{(k)}\;\approx\;X+C_1 d_1^k \cos\left(\frac{2r\pi}{n}-\theta_1+k\psi_1\right) \] for big values of \(k\). Taking \(C=C_1 d_1^k\) and \(\theta=\theta_1-k\psi_1\), we have \[x_r^{(k)}\;\approx\;X+C \cos\left(\frac{2r\pi}{n}-\theta\right) \] that is, \[x_r^{(k)}\;\approx\;X+P\cos\frac{2r\pi}{n}+Q\sin\frac{2r\pi}{n} \] where \(P=C\cos\theta\) e \(Q=C\sin\theta\). In a similar way, \[y_r^{(k)}\;\approx\;Y+R\cos\frac{2r\pi}{n}+S\sin\frac{2r\pi}{n} \] and, assuming the points in 3-dimensional space, \[z_r^{(k)}\;\approx\;Z+T\cos\frac{2r\pi}{n}+U\sin\frac{2r\pi}{n} \] Thus, like in the preceding processes, points \(P_{r}^{(k)}\) get closer and closer to the vertices of a polygon obtained by applying a linear function to a regular \(n\)-sided polygon centered at the origin followed by a translation. This polygon is also inscribed in a ellipse centered at \((X,Y,Z)\) and the relations of parallelism of the sides of the regular polygon are kept in the corresponding sides of the new polygon.

Note also that, given a pair of integers \(a\) and \(b\) such that \(p\approx\frac{a}{b}\), \[x_r^{(k+b)}\;\approx\;X+C_1 d_1^{k+b} \cos\left(\frac{2r\pi}{n}-\theta_1+(k+b)\psi_1\right) \] \[x_r^{(k+b)}-X\;\approx\;C_1 d_1^k d_1^b \cos\left(\frac{2r\pi}{n}-\theta_1+k\psi_1+b\psi_1\right) \] \[x_r^{(k+b)}-X\;\approx\;d_1^b C_1 d_1^k \cos\left(\frac{2(r+a)\pi}{n}-\theta_1+k\psi_1+b\psi_1-\frac{2a\pi}{n}\right). \] Since \[x_{r+a}^{(k)}-X\;\approx\;C_1 d_1^k \cos\left(\frac{2(r+a)\pi}{n}-\theta_1+k\psi_1\right), \] we have \[x_r^{(k+b)}-X\;\approx\;d_1^b (x_{r+a}^{(k)}-X) \] provided that the approximation \(b\psi_1\approx\frac{2a\pi}{n}\) is valid, that is, \(\psi_1\approx p\frac{2\pi}{n}\). In fact, \[\psi_1\;=\;\arctan\frac{p\sin\frac{2\pi}{n}}{(1-p)+ p\cos\frac{2\pi}{n}}\;=\; p\frac{2\pi}{n}+\frac{p(1-p)(1-2p)}{6}\left(\frac{2\pi}{n}\right)^3+\ldots \] so that, for big values of \(n\), we take the approximation \(\psi_1\approx p\frac{2\pi}{n}\). Thus, the points we get after aplying the described process \(b\) times are, approximately, the points obtained by an homothety centered in the polygon centroid and ratio \(d_{1}^{b}\). The approximation is the better the greater the values of \(k\) and \(n\) are. However, we need to realize that, by the assumed approximation, there might be some difference between the shapes of the initial polygon and of the polygon obtained by \(b\) iterations of the described process, and that this difference may not converge to 0 when \(k\) tends to infinite (that is, after a large number of iterations). In fact, that always happens, except for \(p=1/2\), which corresponds to the bisection method. If, for instance, \(p=2/3\), which corresponds to the trisection method, there is always such a difference, as seen.