ATRACTOR

For an animated version of this applet in space, (with parameter \(p\) changing continuously from \(0\) to \(1\)), click here.

Instructions

Click on the vertices of the initial hexagon (red dots) and drag them, observing the new hexagons formed by dividing the diagonals that link opposite vertices into two segments whose lengths are respectively \(p\) and \(1-p\) times the length of the diagonal, 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 obtained hexagons appear to tend to a triangle (possibly degenerated into a line segment or a point). In fact, no matter the value of \(p\), the new hexagons always tend to a certain triangle. The closer the \(p\) is to \(\frac{1}{2}\), the faster the convergence rate to the triangle is. Why?