ATRACTOR

In general, given an initial \(n\)-sided polygon with vertices with abscissa \(x_r\), \(r\in\{0,1,\ldots,n-1\}\), we have \[x_r\;=\;\sum_{j=0}^{\lfloor n/2\rfloor} \left(P_j\cos\frac{2jr\pi}{n}+Q_j\sin\frac{2jr\pi}{n}\right) \] where \(\lfloor n/2\rfloor\) denotes the floor of \(n/2\), that is, its integer part. We have two distinct cases:

• If \(n\) is even, then \(n=2m\) for some \(m\) and \(\lfloor n/2\rfloor=m\). Thus \[\begin{array}{ll}x_r & =\;\sum_{j=0}^m \left(P_j\cos\frac{2jr\pi}{n}+Q_j\sin\frac{2jr\pi}{n}\right)\;=\\ & X + P_1\cos\frac{2r\pi}{n}+Q_1\sin\frac{2r\pi}{n}+\ldots +P_m\cos r\pi + Q_m\sin r\pi \end{array}\] with \(X=P_0\).
• If \(n\) is odd, that is, \(n=2m+1\) for some \(m\), then \(\lfloor n/2\rfloor=m\) and \[\begin{array}{ll}x_r & =\;\sum_{j=0}^m \left(P_j\cos\frac{2jr\pi}{n}+Q_j\sin\frac{2jr\pi}{n}\right)\;=\\ & X + P_1\cos\frac{2r\pi}{n}+Q_1\sin\frac{2r\pi}{n}+\ldots +P_m\cos\frac{(n-1)r\pi}{n} + Q_m\sin\frac{(n-1)r\pi}{n} \end{array}\] with \(X=P_0\).

In both cases we have a system of \(n\) equations in \(n\) unknowns.

Writing the vector \((P_j,Q_j)\) in polar coordinates \((C_j\cos\theta_j,C_j\sin\theta_j)\), we get \[P_j\cos\frac{2jr\pi}{n}+Q_j\sin\frac{2jr\pi}{n}\;=\; C_j\cos\left(\frac{2jr\pi}{n}-\theta_j\right) \] so that \[x_r\;=\;\sum_{j=0}^m C_j\cos\left(\frac{2jr\pi}{n}-\theta_j\right) \] Then \[\begin{array}{ll}x'_r & =\;\frac{x_r+x_{r+1}}{2}\;=\\ & =\;\frac{1}{2}\sum_{j=0}^m C_j\left(\cos\left(\frac{2jr\pi}{n}-\theta_j\right)+ \cos\left(\frac{2jr\pi}{n}-\theta_j- \frac{2j\pi}{n}\right)\right)\;=\\ & =\;\sum_{j=0}^m C_j \cos\frac{j\pi}{n} \cos\left(\frac{2jr\pi}{n}-\theta_j+\frac{j\pi}{n}\right) \end{array}\] and, more generally, \[\begin{array}{ll}x_r^{(k)} & =\;\frac{x_r^{(k-1)}+x_{r+1}^{(k-1)}}{2}\;= \;\sum_{j=0}^m C_j \cos^k\left(\frac{j\pi}{n}\right) \cos\left(\frac{2jr\pi}{n}-\theta_j+\frac{kj\pi}{n}\right)\;=\\ & =\;X+\sum_{j=1}^m C_j \cos^k\left(\frac{j\pi}{n}\right) \cos\left(\frac{2jr\pi}{n}-\theta_j+\frac{kj\pi}{n}\right) \end{array}\] When \(k\) tends to infinite, the summands tend to zero quicker than the first one, and so we may forget them and consider the following approximation to the sum for big values of \(k\): \[x_r^{(k)}\;\approx\;X+ C_1 \cos^k\left(\frac{\pi}{n}\right) \cos\left(\frac{2r\pi}{n}-\theta_1+\frac{k\pi}{n}\right). \] Taking \(C=C_1\cos^k(\frac{\pi}{n})\) and \(\theta=\theta_1=\frac{k\pi}{n}\), 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 \(P=C\cos \theta\) and \(Q=C\sin \theta\). Similarly, \[y_r^{(k)}\;\approx\;Y+R\cos\frac{2r\pi}{n}+S\sin\frac{2r\pi}{n} \] and, assuming the points in 3-dimensional space, \[z_r^{(k)}\;\approx\;Z+T\cos\frac{2r\pi}{n}+U\sin\frac{2r\pi}{n} \] Hence, similarly as in the cases of hexagon and pentagon, the points \(P_r^{(k)}=(x_r^{(k)},y_r^{(k)},z_r^{(k)})\) 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.

Likewise, also the points that are obtained by applying the bisection process twice to a polygon with \(n\) sides are, approximately, the points that are obtained by a homothety with center in the centroid of that polygon and ratio \(cos^2(\frac{\pi}{n})\) (the greater the value of \(k\), the better the approximation is). They appear alternately in the same position, in the sequence of constructed polygons, only with a smaller size.

Note that we may apply the method described above for representing the coordinates of a polygon, referred to as Fourier representation, to other polygon construction processes. For example, what happens if, instead of joining the midpoints of each side of the initial polygon, we join the points that trisect each side, that is, that divide each side into two segments, the first having twice the length of the second one? (see"other processes")