ATRACTOR

Given a polygon of \(n\) sides such that the abscissas of the vertices are \(x_i\) with \(i\in\{0,1,\ldots,n-1\}\), we obtain, by bisection, a new polygon with \(n\) sides whose abscissas are given by \[x'_{i'}=\frac{x_i+x_{i+1}}{2}\] below, we may see what happens in the case of the sequences obtained with the simplest polygons (that is, with a small number of sides):

• triangles
• quadrilaterals
• pentagons
• hexagons

After analysing the case of the simplest polygons, we are led to conjecture the following:

• if \(n\) is odd, any polygon with \(n\) sides can be obtained by bisection and the choice of the initial polygon uniquely determines the sequence of polygons.
• if \(n\) is even, not all \(n\)-sided polygons can be obtained by bisection and the choice of the initial polygon does not uniquely determine the sequence of \(n\)-sided polygons. There is an infinite number of different initial polygons that give rise to the same sequence of polygons.

Is this always true? Yes, indeed, this is a fact that may be proved with the help of Linear Algebra (see "generalization").

However, regardless of the number of sides to be considered, the polygons in the sequence have alternately the same shape, although with increasingly smaller size. Why is that so?

What happens if we consider other processes? To find out, see this page.