Other procedures

Let us see other methods of construction of new polygons, applied to a given \(n\)-sided polygon with vertices with abscissa \(x_i\), \(i\in \{0,1,\ldots,n-1\}\).

Note: the last case applied to quadrilaterals is very similar to bisection of triangles. If we consider \(x_m=\frac{x_0+x_1+x_2+x_3}{4}\), we have \[\begin{array}{ll} x'_0= & \frac{4}{3} x_m - \frac{1}{3} x_3\\ x'_1= & \frac{4}{3} x_m - \frac{1}{3} x_0\\ x'_2= & \frac{4}{3} x_m - \frac{1}{3} x_1\\ x'_3= & \frac{4}{3} x_m - \frac{1}{3} x_2 \end{array}\] that is, \[\begin{array}{ll} x'_0-x_m = - \frac{1}{3} (x_3-x_m)\\ x'_1-x_m = - \frac{1}{3} (x_0-x_m)\\ x'_2-x_m = - \frac{1}{3} (x_1-x_m)\\ x'_3-x_m = - \frac{1}{3} (x_2-x_m) \end{array}\] This means that the new points are obtained from the previous ones by an homothety of ratio \(-1/3\) with center in a point \(G\) whose coordinates are given by the arithmetic mean of the corresponding coordinates of the four vertices of the given initial polygon. If these vertices are non complanar, they may be regarded as the vertices of a tetrahedron. Thereby, this process produces a sequence of embedded tetrahedrons in space, all similar to each other, with an increasingly smaller size, since the similarity ratio tends to zero.

Observe it in the following applet.