ATRACTOR

For an animated version of this applet, with the parameter \(p\) changing continuously between \(0\) and \(1\), click here.

Instructions

Click on the vertices of the initial hexagon (red dots) and drag them, observing the new hexagons constructed by dividing each side in two segments of lengths, respectively, \(p\) and \(1-p\) times the length of the side, with \(0 < p < 1 \). To change parameter \(p\), move the black dot.

Note that for values of \(p\) close to \(\frac{1}{2}\), the new hexagons appear to have a more "regular" shape than for values far from \(\frac{1}{2}\). However, regardless of the value of \(p\), the new hexagons always tend to "regular" hexagons, that is, with opposite sides parallel to each other and parallel to one of the diagonals. The convergence rate increases with the value of \(p\) approaching \(\frac{1}{2}\). Why?