ATRACTOR

To consider other methods for building new polygons, try out the following applet:

Instructions

Click on the vertices of the initial polygon \(P\) (in blue) and drag them to obtain a new polygon \(Q\) (in red). You may choose the number of vertices in the initial polygon (up to a maximum of six) and the values of \(\alpha_0,\alpha_1,\alpha_2,\ldots,\alpha_{n-1}\in\mathbb{R}\) that appear in the matrix \[A=(\alpha_{n-i+j})_{0\leq i,j \leq n-1}= \left(\begin{array}{cccc}\alpha_0 & \alpha_1 & \ldots & \alpha_{n-1}\\ \alpha_{n-1} & \alpha_0 & \ldots & \alpha_{n-2}\\ \ldots & \ldots & \ldots & \ldots\\ \alpha_1 & \alpha_2 & \ldots & \alpha_0 \end{array}\right)\] which allow to obtain the coordinates of \(Q\) from the coordinates of \(P\).

You may also get the value of the determinant of this matrix, which will be zero if the initial polygon uniquely determines the new polygon and different from zero otherwise. Note that the centers of gravity (also known as "centers of mass" or "centroids") of the two polygons (points with the shape of squares) do not always coincide, contrary to what happened with the methods in the previous applets. In fact, this happens only when the sum of \(\alpha_0,\alpha_1,\alpha_2,\ldots,\alpha_{n-1}\in\mathbb{R}\) is equal to \(1\).

You may consider only non-negative values for \(\alpha_0,\alpha_1,\alpha_2,\ldots,\alpha_{n-1}\in\mathbb{R}\) whose sum equals \(1\) (designated as usual values, since the methods in the previous apps correspond to this kind of values. Notice that:

• if \(\alpha_0=\alpha_1=\frac{1}{2}\) and \(\alpha_j=0\) for \(j > 1\), we have the bisection method;
• if \(\alpha_0=\frac{1}{3}\), \(\alpha_1=\frac{2}{3}\) and \(\alpha_j=0\) for \(j > 1\), we have the trisection method;
• if, for some \(p\) between \(0\) and \(1\), \(\alpha_0=1-p\), \(\alpha_1=p\) and \(\alpha_j=0\) for \(j > 1\), we have the method of general division of the sides of the polygon;
• if \(\alpha_1=\alpha_{n-1}=\frac{1}{2}\) and \(\alpha_j=0\) for \(j\neq 1,n-1\), we have the method of bisection of diagonals that link alternating vertices;

and so on, we may get any of the methods considered in the previous applets (note that, for any of these values, the centroids of the two polygons coincide).

If the number of vertices is odd, it is also possible to obtain the inverse process of bisection (that is, to find the polygon that, by bisection, gives rise to the initial polygon), although not with the usual values of \(\alpha_0,\alpha_1,\alpha_2,\ldots,\alpha_{n-1}\in\mathbb{R}\) since some of them are negative. Can you find out those values?