## Sequence of polygons

### Generalization

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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