In all the constructions presented earlier, we started from an initial \(n\)-sided polygon with vertices with abscissa \(x_j\), \(j\in\{0,1,\ldots,n-1\}\), and obtained a new \(n\)-sided polygon with vertices with abscissa given, for certain fixed values \(a_0,a_1,\ldots,a_{n-1} \in \mathbb{R}\), by \[\begin{array}{ll} x'_0= & a_0 x_0+a_1 x_1+\ldots+a_{n-1} x_{n-1}\\ x'_1= & a_0 x_1+a_1 x_2+\ldots+a_{n-1} x_0 = a_{n-1} x_0 + a_0 x_1+\ldots+a_{n-2}x_{n-1}\\ x'_2= & a_0 x_2+a_1 x_3+\ldots+a_{n-1} x_1 = a_{n-2} x_0 + a_{n-1} x_1+\ldots+a_{n-3}x_{n-1}\\ \ldots\\ x'_{n-1}= & a_0 x_{n-1}+a_1 x_0+\ldots+a_{n-1} x_{n-2} = a_1 x_0 + a_2 x_1+\ldots+a_0 x_{n-1}\\ \end{array}\] Let us see some examples:

  1. If \(a_0=a_1=\frac{1}{2}\) and \(a_j=0\) for \(j>1\), then \(x'_i=\frac{x_i+x_{i+1}}{2}\), where \(x_n=x_0\).

  2. If \(a_0=\frac{1}{3}\), \(a_1=\frac{2}{3}\) and \(a_j=0\) for \(j >1 \), then \(x'_i=\frac{x_i+2x_{i+1}}{3}\), where \(x_n=x_0\).

  3. If \(a_0=1-p\) and \(a_1=p\) for some \(p\) between 0 and 1, and \(a_j=0\) for \(j > 1\), then \(x'_i=(1-p)x_i+p x_{i+1}\), where \(x_n=x_0\).

  4. If \(a_1=a_{n-1}=\frac{1}{2}\) and \(a_j=0\) for \(j\neq 1, n-1\), then \(x'_i=\frac{x_{i-1}+x_{i+1}}{2}\), where \(x_n=x_0\) and \(x_{-1}=x_{n-1}\).

Hence, for each \(i\in\{0,1,\ldots,n-1\}\), we have \[x'_i = \sum_{j=0}^{n-1}a_{n-i+j} x_j\] where \(a_{n+l}=a_l\) for every \(l\in\{0,1,\ldots,n-1\}\).
If we take, for each positive integer \(k\), \(x^{(k)}=\left(\begin{array}{c} x^{(k)}_1\\ x^{(k)}_2\\ \ldots\\ x^{(k)}_n \end{array}\right)\), then \[x^{(k)}=A x^{(k)},\;\forall_{k\in \mathbb{N}_0} \] where \(x^{(0)}=\left(\begin{array}{c} x_0\\ x_1\\ \ldots\\ x_{n-1} \end{array}\right)\), and \(A= (a_{n-i+j})_{0\le i,j \le n-1} = \left(\begin{array}{cccc} a_0 & a_1 & \ldots & a_{n-1}\\ a_{n-1} & a_0 & \ldots & a_{n-2}\\ \ldots & \ldots & \ldots & \ldots\\ a_1 & a_2 & \ldots & a_0\\ \end{array}\right)\).
Equivalently, we have \[x^{(k)}=A^{(k)} x^{(0)},\;\forall_{k\in \mathbb{N}_0}\] Then, using some Linear Algebra, we may conclude something about the sequences of constructed polygons. In fact, one may show that matrix \(A\) has \(n\) complex eigenvalues (not all necessarily distinct), given by \[\lambda_t=\sum_{j=0}^{n-1} a_j \omega^{jt}\] for \(t\in\{0,1,\ldots,n-1\}\), with \(\omega\) a primitive nth root of unity. To know more about this, consult this PDF file (only in Portuguese).
How can we apply this to the examples presented earlier?

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